Mathematics Challenges
for Classroom Practices
at theLower Primary Level
Based on SEAMEO Basic Education Standards:
Common Core Regional Learning Standards in Mathematics

Mathematics Challenges
for Classroom Practices
at theLower Primary Level
Based on SEAMEO Basic Education Standards:
Common Core Regional Learning Standards in Mathematics

Editors:
TEH Kim HongISODA Masami
GAN Teck HockWith support of:
Maitree Inprasitha (KKU)
Supattra Pativisan (IPST)
Wahyudi & Sumardyono (SEAQIM)

A Collaboration Project:Published by
SEAMEO RECSAM
Jalan Sultan Azlan Shah
11700 Gelugor
Penang, MALAYSIA
Cover design by Nur Syafiyah Binti Mohd Kassim

Mathematics Challenges for Classroom Practices
at the Lower Primary Level
Based on SEAMEO Basic Education Standards:
Common Core Regional Learning Standards in Mathematics

Copyright © 2021 by MASAMI ISODA
Copyright © 2021 by SEAMEO RECSAM
All rights reserved, except for educational purposes with no commercial interests. No part of this publication
may be reproduced, transmitted in any form or by any means, electronic or mechanical, including photocopying,recording or any information storage or retrieval system, without written permission from Masami Isoda or the
Publisher.

For more information of this book contact:
Masami Isoda, PhD.
Arahata 945-8
Tokorozawa, Saitama
Japan 359 -1133
Phone: +81-429228826
Email: Isoda@criced.tsukuba.ac.jp

ISBN: 978-967-930-057-4

or

The Director
SEAMEO RECSAM
Jalan Sultan Azlan Shah, Gelugor
11700 Penang, MALAYSIA
Phone: +604-652 2727 Fax: +604-658 7558
Email: director@recsam.edu.my
URL: http://www.recsam.edu.myForeword |v

FOREWORD
Congratulations to CRICED, University of Tsukuba and SEAMEO RECSAM
for another collaboration project in the publication of three series of guidebook
entitled Mathematics Challenges for Classroom Practices based on the
SEAMEO Basic Education Standards: Common Core Regional LearningStandards (SEA-BES CCRLS) in Mathematics. Two of the series are for the
lower and upper primary level while the third series is for the lower secondary
level. Generally, curriculum standards of subjects are not widely scrutinisedby classroom practitioners and teacher educators compared to the curriculum
specialists. The publication of these three new guidebooks anchor well andconsolidate the role and importance of the SEA-BES Common Core Regional
Standards document which was already introduced to all the 11 SEAMEO member countries andbeyond since published in 2017.
The content of this guidebook series covers across Grade 1 to Grade 9 and consists of tasks written
to understand the learning standards in Mathematics. The transfer of information from the SEA-BES
CCRLS to the newly published book series will create an awareness among classroom teachers
and teacher educators the importance and relevance of curriculum standards in formulating anddesigning learning specifications for students. The presentation of the book series emphasised on
three aspects, namely highlighting the misconceptions, developed new ideas from the previously
learned knowledge and explanation of new concepts in mathematics. The task-based approach will
surely help readers to enhance their own mathematical understanding and ultimately provide bettersupport for classroom teaching and learning.
I sincerely hope that the Minister of Education of SEAMEO members would provide support
and promote the use of this guidebook series among educators and teachers in their respecting
countries. This effort and spirit of cooperation among SEAMEO members and associate members
can be realised to bring benefits for classroom practices, which will eventually benefit children of
our future.
My sincere appreciation and congratulations to CRICED, University of Tsukuba as a project
proponent and provided the financial support, SEAMEO RECSAM as the main collaborator,
other collaborating partner institutions and individual educators and specialists for their expertise,commitment and contributions in this endeavour.

Dr ETHEL AGNES P VALENZUELA
Director, SEAMEO Secretariat, Bangkok, Thailand

vi | Mathematics Challenges for Classroom Practices at the Lower Primary Level

FOREWORD
On behalf of SEAMEO RECSAM, I would like to express my sincerest appreciations
to the Centre for Research on International Cooperation in Educational Development(CRICED), University of Tsukuba for inviting the Centre as the main collaborator
in the publication of the guidebook series, titled “Mathematics Challenges forClassroom Practices” for the i) Lower Primary Level, ii) Upper Primary Leveland iii) Lower Secondary Level. Besides the involvement of SEAMEO RECSAM,
many educators and specialists of other collaborating partners such as Khon
Kaen University, Thailand; the Institute for the Promotion of Teaching Scienceand Technology (IPST), Thailand, SEAMEO QITEP in Mathematics, Indonesia;
mathematics specialists in APEC economies, educators, local teacher educators,
and curriculum specialists involved in the SEAMEO Basic Education Standards:
Common Core Regional Learning Standards (SEA-BES CCRLS) in Mathematics had contributed theirwritings in this guidebook series.
SEA-BES CCRLS in Mathematics and Science was first published in 2017 by SEAMEO RECSAM but
had limited and restricted usage despite being shared with all SEAMEO member countries and beyond.
Today, SEA-BES CCRLS has been given a new life where the learning standards of the Mathematics
component have been adopted and used as the main reference for this guidebook series. Having said
that, I am grateful to the outstanding writing team who made this possible. I shall start by acknowledging
the contribution of Professor Dr Masami Isoda, who initiated the idea and the project and had graciously
invited SEAMEO RECSAM to produce this guidebook series; Ms Teh Kim Hong who coordinated the
project with Mr Pedro Jr. Montecillo; Mr Gan Teck Hock who was later recruited to join the writing teamand other mathematics specialists who were also invited to contribute their writings. Despite facing time
constraints and changes of staff members, the writing team stayed intact with their contributions andcommitment until this guidebook series is published. Besides the writing team, I also like to thank thepanel reviewers of RECSAM who provided their constructive suggestions to improve the content of thisguidebook series.
The guidebook series covers the mathematics content across Grade 1 to Grade 9 with the focus of
utilising written tasks to understand the learning standards of SEA-BES CCRLS in Mathematics. The
transfer of information from the later to the newly published guidebook series will create awarenessamong classroom teachers and teacher educators regarding the importance and relevance of curriculum
standards in the planning of teaching and learning. The presentation of this guidebook series emphasisedon highlighting misconceptions, contradictions, and developed new ideas from the previously learned
knowledge to enhance learning and develop mathematical thinking. Such an approach of contradiction
will foster deeper thinking among readers, thus enhancing mathematical understanding, translating into
better support for classroom teaching and learning.
Without a doubt, much commitment and hard work had been invested to produce these guidebooks. I
hope that this mathematics guidebook series will be used widely by teachers and educators of SEAMEO
member countries for classroom practices. I sincerely hope SEAMEO Secretariat will also provide theirsupport by promoting this guidebook series in the classrooms of educators and teachers in all SEAMEO
member countries.
I am therefore proud to present this guidebook series as the contribution of SEAMEO RECSAM and
CRICED, University of Tsukuba, for the promotion and development of mathematics education in thisregion. This would not have been possible without CRICED, University of Tsukuba’s content expertiseand financial support. I hope this valuable collaboration and cooperation will continue in other future
projects to benefit education development in this region.

Dr SHAH JAHAN BIN ASSANARKUTTY
Centre Director, SEAMEO RECSAM

Foreword | vii

FOREWORD
In addition to the Japanese Ministry of Education, Culture, Sports, Science
Technology (MEXT), the University of Tsukuba has been playing the role of anaffiliate member to collaborate with SEAMEO. As the Director of the Centre
for Research on International Cooperation in Educational Development(CRICED), it is my pleasure to continue working with SEAMEO RECSAM on
the SEAMEO Basic Education Standards for Mathematics (SEA-BES-M). This
project was launched in 2014 as a reference book for curriculum reformers andteachers to develop the 21st century skills and OECD competency (2005) ineducation. For SEA-BES-M, I had collaborated with Pedro Montecillo Jr., Kim
Hong Teh, and the late Mohd Sazali bin Hj. Khalid of RECSAM, along with thecontribution from curriculum developers in SEAMEO countries, specialists in APEC economies andseveral internationally leading researchers. Even though it was published before OECD Education
2030, it is included the same structure as OECD Learning Compass.
Before SEA-BES-M was published, a comparative analysis of the mathematics curriculum documentsof SEAMEO countries revealed that the higher order thinking presented as process standards iscompartmentalised between every content description. Therefore, a major contribution of SEABES-M to the world, particularly the SEAMEO countries is its clear description of the meaning of
higher order thinking for the mathematics curriculum standards to develop mathematical thinking.
After the publication, we recognized that there are difficulties for readers to understand the intended
meaning of every standard. This is mainly due to the fact that readers will interpret what is readbased on their curriculum knowledge and experience teaching mathematics of their own countries.
As such, it is crucial to develop a book series to be used as references for interpreting SEA-BES-M.
This book series is prepared, particularly for teacher educators, textbook authors, and curriculum
developers as well as teachers to understand higher order thinking for developing mathematicalthinking in their classrooms. In this book series, the authors had collaborated with leading researchersand educators in major teacher education institutions, CRME-IRDTP at KKU (Thailand), IPST
(Thailand) and SEAMEO QITEP in Mathematics (Indonesia) with the support of the coordinators.
Furthermore, the authors had made some minor revisions to the SEA-BES-M to align with theneeds of the Era of the 4th Industrial Revolutions to develop stakeholders and users of Artificial
Intelligence and Big Data in business as well as establishing a successful life under the realityof humanity with technology. The minor revision was made based on the curriculum reform
recommendation (2020) by APEC InMside Project with the purpose of promoting mathematical
capitalism under mathematical-statistical-informational sciences on the demands of the Era. It
is hoped that this book series is used in teacher education to develop new curriculum content
knowledge for teaching in this Era. I would like to acknowledge SEAMEO secretariat and centres
for their collaborations, especially Shah Jahan Bin Assanarkutty , the Director of RECSAM, who
made possible this publication. Last but not least, I would like to convey my sincere appreciation
to all contributing writers stated in the contributor list and Gakko Tosho (the Japanese Textbook
Publisher) which provided us innovative ideas.

ISODA MASAMI
Director of CRICED, University of Tsukuba, Japan

Preface | ix

PREFACE
Realising reform in school curriculum beyond the 21st century and revitalising teacher education havebeen set as prioritised agendas in SEAMEO countries. On this demand, SEAMEO Basic Education
Standards: Common Core Regional Learning Standards (SEA-BES:CCRLS) in Mathematic waspublished in 2017 for the main purpose of strengthening collaboration on curriculum standards andlearning assessment across different educational systems in SEAMEO countries. In order for thisdocument to be understood beyond the curriculum developers, supporting materials need to be
developed for helping other users such as classroom teachers and teacher educators to acquirea deeper understanding of the standards. This book is an initiative to provide such support. With
this support, it is anticipated that teachers and teacher educators will be able to innovate theirclassroom practices for developing competency and professional development aligning with thetrends of the 4th Industrial Revolution.
Mathematics Challenges for Classroom Practices at the Lower Secondary Levelconsists of mathematical tasks for the following strands:••••

Numbers and Operations
Quantity and Measurement
Shapes, Figures and Solids
Pattern and Data Representations

These tasks are prepared to be used for pre-service and in-service mathematics teacher
education. Its main purpose is to help readers develop mathematical knowledge for teaching (MKT)
which consists of subject matter knowledge (SMK) and pedagogical content knowledge (PCK)(see, Ball, Thames, and Phelps, 2008)1. In developing the tasks, the English edtion of Japanesemathematics textbooks publised by Gakko Tosho2 had been used as the main reference. These
textbooks provided major guides for learning mathematics through problem solving approach todevelope mathematical thinking. As such, the tasks in this book are focused on mathematical ideas
and ways of thinking. Basically, the tasks are developed in three ways: (a) analysing misconceptionof ideas, (b) developing ideas from previously learnt knowledge, and (c) using inquiry-based
investigation to learn new ideas. In designing the tasks, the importance of local contexts of the
SEAMEO community had been considered. However, some essential elements of Japanese
school mathematics were also incorporated into the tasks. It is hoped that these elements will set
off a new breath of mathematics learning in the SEAMEO community, shaping our students to be
critical and creative thinkers in the era of artificial intelligence and data science.
Each task is written based on a standard in SEA-BES: CCRLS in Mathematics and it servesto clarify (a) the curriculum knowledge of teaching in PCK, and (b) the mathematical ideas and waysof thinking on SMK related to the standard. Apart from that, SEA-BES CCRLS in Mathematics
is used as the basic source for MKT. Since SEA-BES CCRLS in Mathematics was developedwith curriculum specialists of SEAMEO countries, solving the tasks will also provide a bird’s eyeview of their national curriculum to the readers. Furthermore, it will broaden their perspective of
mathematical ideas, ways of thinking and curriculum sequence with respect to their use of local
textbooks.

1. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content Knowledge for Teaching: What makes it special? Journal of
Teacher Education, 49(5), 389-407.
2. Hitotsumatsu, S. et al (2005). Study with your friends: mathematics for elementary school (G. 1-6). Tokyo: Gakko Tosho./
Isoda, M., Murata, A., Yap, A. (2011, 2015, 2020) Study with your friends: mathematics for elementary school (G. 1-6). Tokyo:
Gakko Tosho/ Isoda, M., Tall, D. (2019). Junior High School Mathematics (G. 1-3). Tokyo: Gakko Tosho. For developing tasks
in this book, authors are inspired by the tasks and task sequence of these mathematics textbooks.

x| Mathematics Challenges for Classroom Practices at the Lower Primary Level

Although the targeted readers of this book are teachers and teacher educators, most of thetasks can also be solved by students in classrooms as they are also aligned with school mathematics
curriculum. In addition, teachers and teacher educators are expected to solve them without muchdifficulties. However, studying the tasks carefully will raise the awareness of the depth of SMK and
PCK required to complete the tasks. This will triggered off a need to upskill their understanding ofmathematical ideas and function effectively to develop students mathematical competency. Thus,
solving tasks in this book will provide readers the opportunities to relearn the mathematics content
for teaching. Furthermore, it will also help them to identify (a) the objectives of teaching the content,(b) the gap between students’ prior knowledge and what is to be learnt, (c) what and how students
reorganise the content knowledge of their learning, (d) students’ difficulties in learning the content,and (e) what ideas will be developed through their new learning.
Readers may also choose to work with any task according to their interests. It is notnecessary to work out all the tasks according to the sequence in the book. However, for a deeper
understanding of the mathematical ideas embeded in a task, it is recommended that readers shouldsolve the task in the following manner: (a) solve the task by themselves and read the related
standards, (b) communicate their solutions with others to identify what is really new content forthem, and (c) paraphrase and summarise the communication with others based on the perpectiveof mathematical ideas and ways of thinking that align with the framework of SEA-BES CCRLS as
described in Chapter One.
This book is recommended for use in many ways and various contexts. Firstly, as all thetasks were designed based on school mathematics curriculum, so they can be used directly by
students as learning tasks. In addition, teachers can also use the book as a quick guide to create
similar mathematical tasks that incorporate mathematical thinking. Secondly, when the book isused in the context of in-service teacher education such as in lesson study, teachers can solve
the tasks in this book as a step to gain a deep understanding of the mathematical ideas in orderto prepare a unit of lesson plan based on the standard chosen. This may help to improve theeffectiveness of lesson planning and anticipate responses of students to the tasks. Thirdly, in thecontext of pre-service teacher education, the tasks in this book can serve as a mean to acquire
MKT which may be required for any teacher employment examination or entrace examination for an
education graduate programme. Fourthly, in the context of mathematics education research, thisbook can be used as a reference for MKT. Last but not least, when the book is used in the context
of curriculum reform and textbook revision, it could serve as a guide to formulate new objectives
and tasks which are not existing in their current curriculum and textbooks.
Mathematics Challenges for Classroom Practices at the Lower Primary Level is theoutcome of many contributions of educators and academia from different institutions. In order toensure good quality of content and streamline the presentation of the writings, many rounds ofediting and rewriting were unavoidable. It is our hope that the mathematical ideas and ways ofthinking promoted through this book will enhance the teachers’ capacities to develop their students’potentials in facing the challenging and demanding era ahead.

TEH Kim HongISODA Masami
GAN Teck Hock

Acknowledgement | xi

ACKNOWLEDGEMENTS
The authors/editors would like to thank and express our gratitude to the following contributors,
institutions and organisations in making this project and publication possible:
Dr Ethel Agnes Pascua-Valenzuela, Director of SEAMEO Secretariat, Bangkok, Thailand forproviding CRICED the opportunities to collaborate with RECSAM and other SEAMEO centres;
Dr Shah Jahan Bin Assanarkutty, the Centre Director of SEAMEO RECSAM for his encouragement
and support in realising the publication of the books;
Dr Johan Bin Zakaria, the ex-Centre Director of SEAMEO RECSAM for his confidence andgenerosity of granting permission to use the SEABES: CCRLS in mathematics to develop thisguidebook series;
The University of Tsukuba generously supported the funding of this guidebook series project. The
books will be disseminated to SEAMEO member countries and recommended for use by teachers,
teacher educators for the benefit of the SEAMEO community;
Assoc. Professor Dr Maitree Imprasitha, the Vice President of Khon Kaen University(KKU) andDirector of the Institute for Research and Development in Teaching Profession (IRDTP) for ASEAN,
Thailand, for providing the support in organising a workshop to lead the KKU and other affiliated
Universities lecturers in contributing writings to this guidebook series;
Dr Suppattra Pativisa, Senior Professional Expert of The Institute for the Promotion of Teaching
Science and Technology (IPST), Thailand, in lending the support and led the lecturers in contributing
writings to the guidebook series;
Dr Wahyudi, the former director of SEAMEO QITEP in Mathematics (SEAQIM) and currently the
Deputy Director of SEAMEO Secretariat, together with Dr Sumardyono, the current Director of
SEAQIM, provided the support to lead their mathematics lecturers in contributing the writings;
The specialists of APEC economies of the year September 2017 during the 13th APEC-Khon Kaen
University Conference, and the year February 2018, in providing suggestions and contributions ofresources in helping to conceptualise the content and outcome of the books;
All curriculum specialists and educators who attended the SEA-BES CCRLS in Mathematics
workshop in March 2017, held in SEAMEO RECSAM, contributed ideas and suggestions on
shaping the outcome of the guidebooks, as well as those who followed the sessions on how to write
the tasks and submitted the writings for considerations;
Pedro Lucis Montecillo Jr, mathematics specialist of RECSAM (until May 2018) for assisting thecoordination at the early stage of the project;
All staff of SEAMEO RECSAM who supported the production of these books by providing ideas,
coordination and manage the distribution of books to other institutions.
This work was supported by The Japan Society for the Promotion of Science (JSPS) Grants-in-Aid
for Scientific Research (KAKENHI) Grant Number 19H01662 and19K21743.

Table of Contents | xiii

Forewordv

Prefaceix

Acknowledgement

xi

Table of contents

xiii

Authors, Contributing Members and Reviewers

xv

Chapter 1: Guide to the SEA-BES CCRLS Framework in Mathematics

1

Chapter 2: Numbers and Operations•
Introducing Numbers up to 120•
Introducing Addition and Subtraction•
Utilising Addition and Subtraction•
Extending Numbers with Based-ten System up to 1 000 000 Gradually•
Producing Vertical Form Addition and Subtraction and Acquiring Fluency ofStandard Algorithms•
Introducing Multiplication and Producing Multiplication Algorithm•
Introducing Division and Extending It to Remainder•
Introducing Fractions and Extending to Addition and Subtraction of Similar
Fractions•Introducing Decimals and Extending to Addition and Subtraction

3

Chapter 3: Quantity and Measurement•
Comparing Sizes Directly and Indirectly Using Appropriate Attributes andNon-standard Units•
Introducing Quantity of Length and Expand to Distance•Introducing Quantity of Mass for Its Measurement and Operation•
Introducing Quantity of Liquid Capacity for Its Measurement and Operation•
Introducing Time and Duration and Its Operation•
Introducing Money as Quantity

113

Chapter 4: Shapes, Figures and Solids•
Exploring Shapes of Objects•
Characterising Shapes for Figures and Solids•
Explaining Positions and Directions

149

Chapter 5: Pattern and Data Representations•
Using Patterns Under the Number Sequence•
Producing Harmony of Shapes Using Patterns•
Collecting Data and Represent the Structure

169

Bibliography

189

Appendix A – Framework for CCRLS in Mathematics

191

Appendix B – Terminologies for Mathematical Thinking and Processes

197

Appendix C – Strand on Mathematical Processes-Humanity (Key Stage 1)

210

Appendix D – Learning Standards for Key Stage 2

213

Appendix E – Learning Standards for Key Stage 3

226

Authors and Contributing Members | xv

AUTHORS, CONTRIBUTING MEMBERS & REVIEWERS
Lead Authors:
Teh Kim Hong (SEAMEO RECSAM, Malaysia)
Isoda Masami (CRICED, Japan)
Gan Teck Hock (SEAMEO RECSAM, Malaysia)

Contributing Members
Book 1 and 2
Anake Sudejamnong (Suratthani Rajabhat University)
Ariya Suriyon (Nakhon Phanom University, Thailand)
Auijit Pattanajak (Khon Kaen University, Thailand)
Ausanee Wongarmart (The Institute for the Promotion of Teaching Science and Technology, Thailand)
Azizan bte Yeop Zaharie (Teacher Education Institute, Penang Campus, Malaysia)
Chanika Senawongsa (Lampang Rajabhat University, Thailand)
Chompoo Lunsak (Khon Kaen University, Thailand)
Fina Hanifa Hidayati (SEAMEO QITEP in Mathematics, Indonesia)
Jensamut Saengpun (Chiang Mai University, Thailand)
Jitlada Jaikla (Khon Kaen University, Thailand)
Kanjana Wetbunpot (Valaya Alongkorn Rajabhat University, Thailand)
Kasem Premprayoom (Thaksin University, Thailand)
Kanyakorn Anurit (Nakhon Si Thammarat Rajabhat University, Thailand)
Kanoknapa Erawan (Mahasarakam University, Thailand)
Khemthong Kotmoraka (Khon Kaen University, Thailand)
Leong Chee Kin, Daniel (Teacher Education Institute, Penang Campus,Malaysia)
Nisakorn Boonsena (Khon Kaen University, Thailand)
Narumon Changsri (Khon Kaen University, Thailand)
Nattawat Sadjinda (Valaya Alongkorn Rajabhat University, Thailand)
Pasttita Laksmiwati (SEAMEO QITEP in Mathematics, Indonesia)
Phattaraphong Kunseeda (Nakhon Phanom University, Thailand)
Pimlak Moonpo (Valaya Alongkorn Rajabhat University, Thailand)
Pimpaka Intaros (Lampang Rajabhat University, Thailand)
Pinyada Klubkaew (The Institute for the Promotion of Teaching Science and Technology, Thailand)
Phattharawadee Hadkaew (The Institute for the Promotion of Teaching Science and Technology, Thailand)
Rachada Chaovasetthakul (Prince of Songkla University, Pattani Campus, Thailand)
Rasidah binti Junaidi (Curriculum Development Department, Ministry of Education Brunei-Retired)
Russasmita Sri Padmi (SEAMEO QITEP in Mathematics, Indonesia)
Saastra Laah-On (Khon Kaen University, Thailand)
Sampan Thinwiangthong (Khon Kaen University, Thailand)
Sasiwan Maluangnont (The Institute for the Promotion of Teaching Science and Technology, Thailand)
Sirirat Chaona (Khon Kaen University, Thailand)
Sutharot Nilrod (The Institute for the Promotion of Teaching Science and Technology, Thailand)Thong-Oon Manmai (Sisaket Rajabhat University, Thailand)
Uki Rahmawati (SEAMEO QITEP in Mathematics, Indonesia)
Ummy Salmah (SEAMEO QITEP in Mathematics, Indonesia)
Wahid Yunianto (SEAMEO QITEP in Mathematics, Indonesia)

xvi | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Book 3Marsigit (Yogyakarta State University, Indonesia)
Nguyen Chi Thanh (Vietnam National University, Vietnam)
Pedro Lucis Montecillo Jr (SEAMEO RECSAM, Malaysia)
Pedro David Collanqui Diaz (Ministry of Education, Peru)
Ronnachai Panapoi (The Institute for the Promotion of Teaching Science and Technology, Thailand)
Siriwan Jantrkool (The Institute for the Promotion of Teaching Science and Technology, Thailand)
Toh Tin Nam (National Institute of Education, Singapore)
Wantita Talasi (The Secondary Educational Service Nakhon Ratchasima,Thailand)
Warabhorn Preechaporn (SEAMEO RECSAM, Malaysia)

Reviewers for Book 3
Mariam Binti Othman (SEAMEO RECSAM, Malaysia)
Murugan A/L Rajoo (SEAMEO RECSAM, Malaysia)
Warabhorn Preechaporn (SEAMEO RECSAM, Malaysia)
Wan Noor Adzmin Binti Mohd Sabri (SEAMEO RECSAM, Malaysia)Chapter 1: Guide to the SEA-BES CCRLS Framework in Mathematics |1

CHAPTER 1
Guide to the SEA-BES CCRLS Framework in Mathematics
The Southeast Asia Basic Education: Common Core Regional Learning Standards (SEA-BES
CCRLS) was developed and directed to create a harmonious SEAMEO Member community in the eraof artificial intelligence and data science through mutual understanding. In this respect, the CCRLS
framework in Mathematics (2017) outlines three basic components towards developing creative,competent and productive global citizens essential for achieving this aim. The comprehensive
illustration of the framework is attached in Appendix A. The revised framework in Figure 1 shows
the interconnection of the three components.

LivingFigure 1. Revised CCRLS Framework in Mathematics
This book is written to guide readers acquire a better understanding of this framework particularly
on mathematical thinking and processes which are embedded in all the tasks. The detailed
explanation of mathematical ideas, mathematical ways of thinking and mathematical activities can

2| Mathematics Challenges for Classroom Practices at the Lower Primary Levelbe found in Appendix B. The standards for the strand on Mathematical Processes-Humanity is
attached in Appendix C to provide readers with challenging activities to promote metacognitive
thinking at different level of arguments to make sense of mathematics. In order to understand thedevelopment and progression of learning from the primary level to the secondary level (Key Stage3), the learning standards of Key Stage 2 and Key Stage 3 can be referred in Appendix D andAppendix E, respectively.
The interconnection of the three components is shown in Figure 2. The ultimate aim of the CCRLS
framework is to develop mathematical values, attitudes and human characters which are theessence of a harmonious society. This component is closely related to the affective domain ofhuman character traits which correspond to soft skills that can be developed through appreciation.
In relation to this, acquisition of mathematics contents as hard skills and reflection on the thinking
processes are needed to inculcate the capability of appreciation. The reflection is necessary forlearners to recognise their cognitive skills derived through the contents. Even though contents
appeared to be learned independently through acquisition, the mathematical thinking and process,and the appreciation of mathematical values, attitudes and habits for human character is possible
to be developed through reflective experiences.
Mathematical Values, Attitudes
and Habits for Human Character
Mathematical Thinking
and Processes

Content

Figure 2. Interconnection of Components in CCRLS Framework in Mathematics
The three components will not be ideally operationalised without appropriate contexts. The tasks
in this book provide the contexts for developing the mathematical thinking and processes, which
are the key learning objectives. Completing the tasks correspond to gauging the readers’ acquiredmathematical knowledge for teaching. Thus, it is recommended that readers should constantly
reflect on the appropriateness of their solutions to the tasks. Other than this, comparing solutions
and discussion with others should always be done habitually in order to gain a deeper understandingof the mathematical processes. This may enable the readers to discover any hidden mathematicalideas and structures in the tasks with appreciation. The tasks are specifically designed to cater forthis purpose.
In a nutshell, an important target of solving the tasks in this book is to enable readers to acquire a better
insight of the learning standards. This insight will in turn help them to understand and appreciate theirnational mathematics curriculum from the perspective of SEABES CCRLS. Furthermore, since thelearning standards are developed based on the framework which emphasised on the componentsof contents, thinking and processes, as well as values of mathematics, ultimately, readers will beable to acquire mathematical teaching knowledge with appreciation.

Chapter 2: Numbers and Operations |3

CHAPTER 2
Numbers and Operations
Topic 1: Introducing Numbers Up to 120
Standard 1.1:
Enjoying counting orally and manipulatively with number names, without symbolic numeralsi.
Develop fluency of the order of number names and use it based on situations
ii. Set initial object for counting, direction of counting and recognise the last object with one-toone correspondenceiii. Distinguish the original situation with concrete objects and the representation of counting
chips, blocks or marbles
Sample Tasks for Understanding the Standards
Task 1: Counting and Comparing Sets of Objects

Diagram 1i.

Given the situation of both the birds and squirrels are looking for nests in Diagram 1.
•

If one nest is only for one animal, can all the nests be occupied? Why or why not?

•If there is a condition that only the birds can occupy the nests, can all the nests be
occupied? Why or why not?

•How is the idea of set embedded in this comparison task?

4| Mathematics Challenges for Classroom Practices at the Lower Primary Levelii.

Why is the condition of limiting the nests only for the birds significant in helping children
learn the idea of one-to-one correspondence?

iii. When asked to count the number of squirrels in Diagram 1, some children counted 5
squirrels and some counted other numbers such as 6, or 7. In this task, 5 squirrels is adenomination number. Children consider the squirrel as a unit for counting.
•

Explain the meaning of denomination number.

•

Based on Diagram 1, give some other examples of denomination numbers.

•

Suggest a possible reason why some children’s answers are more than 5.

•

How can you help children avoid the common error of double counting?

•

When asked “How many birds are there?”, a student said 6, another said 6 birds.
Which answer is more appropriate? Why?

Diagram 2iv. A teacher used same coloured chips to represent the nests and birds.
•

Can the chips for the birds and the nests be distinguished?

•

What does the quantity of chips represent?

•

Explain how the square grid in Diagram 2 can be used with the chips to help children
make sense of one-to-one correspondence.

•

What future learning can be followed after this task? Explain your answer.

Chapter 2: Numbers and Operations |5

Task 2: Counting Objects
Diagram 3 shows some picture cards used forchildren to learn counting the number “three”.
i.

Why are numbers “one” and “two” included in
the discussion of learning number “three”?

ii.

Counting the cups can begin with any one of
the three cups. As such, there are six counting
sequences. One example is “∆, , O”.
•

What are the other five possible counting
sequences?

•

Will the choice of first cup affect theoutcome of counting? Why or why not?

•How will the direction of counting the cups
affect the outcome of counting?

•

How the last cup counted is related to thetotal number of cups?

Diagram 3iii. Diagram 4 shows two sets of three interlocking
cubes. Two children counted the cubes
starting with a different cube and in different
directions as shown.
Direction ①

Direction ②

Diagram 4•

Which direction of counting should be
encouraged? Explain your reasons.

•Comparing the two situations, how will
direction of counting affect the outcome ofcounting the cubes?

iv. When guiding children to count, why
discussion of counting sequence could only
start from number 3, not 1 or 2?
v.

What are the requisite ideas for children tolearn counting?

6| Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 1.2:
Understanding and using the cardinality and ordinality of number with objects and numerals throughactivities of grouping, corresponding and ordering, and developing number sense and appreciate
its beautifulnessi.
Group objects for counting with conditions such as cups, flowers, and rabbits in situations andintroduce the numerals
ii. Obtain fluency of counting concrete objects, understand counting on, and recognise thenecessity of zeroiii. Compare different sets by one-to-one correspondence and recognise larger, smaller or equalwith appreciation in drawing paths between objects
iv. Compose and decompose numbers for strengthening the number sense and cardinality
v. Understand the difference of ordinal and cardinal numbers and use them appropriately insituations and challenge the mixed sequence
vi. Acquire number sense and appreciate the beautifulness in ordering numerals with and without
concrete objects
Sample Tasks for Understanding the Standards
Task 1: Grouping Objects for Counting

i.

ii.

Diagram 1 shows a task for children to learn“group objects for counting”.
•

What are the requisites for grouping
objects to count?

•Why question ③ should not be asked
before questions ① and ②?

Develop a story based on Diagram 1.
•① How many tortoises?
② How many fish?

③ How many animals in the pond?
Diagram 1

Explain how you will use the story toreinforce children’s abilities to groupobjects for counting.

Chapter 2: Numbers and Operations |7iii. Diagram 2 shows a task for children tolearn comparing quantities of two sets ofobjects.
• Using one-to-one correspondence,
explain how to find which set of animals
is larger or smaller.


Are there more puppies or ducklings?
Diagram 2• What are the challenges that may be
faced by children when completing thistask?
• Suggest how you would help children to
overcome the challenges.

Diagram 3iv. Diagram 3 shows two different counting strategies used by student A and B to count the

number of birds. A used the count-all strategy while B used the count-on strategy.
• Explain briefly the two strategies of counting.
• Which strategy of counting is more efficient? Explain your answer.

• Explain how the more efficient way of counting can help children learn addition of
numbers later.

8| Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Ordering Numbersi.

Diagram 4 shows three comparisons of
numbers using interlocking cubes.
•

How will you arrange the interlocking
cubes to help your students compare thetwo numbers?

•

How will you sequence the threecomparison tasks for your students?
 Explain your reasons.
 How does the transitive law affectyour choice of sequencing the tasks?

Comparison ③

4 or 6, which number is more?
How many more?

Diagram 4ii.

“4 is one more than 3.”“3 is one less than 4.”

The interlocking cubes in Diagram 5 are used
to show the numerical relationships between
the numbers 3 & 4 and 4 & 5.
•

Explain how you will use interlocking
cubes to help your students determine
the relationships between the numbers
 5, 3, 2, and 1.

“5 is one more than 4.”“4 is one less than 5.”

Diagram 5

Chapter 2: Numbers and Operations |9

Diagram 6
iii.

iv.

Diagram 6 shows ordering of numbers from zero to 10.
•

What number pattern do you observe in the ordering of these numbers?

•

What questions should you ask children in order to raise their awareness on thenumber pattern?

•

How would you use the diagram to help children learn counting forward and counting
backward?

•

Should zero be included in the diagram for children to learn ordering numbers up to
10? Explain your reasons.

Explain how the use of base-ten blocks would help children to appreciate the beautifulnessof ordering numbers up to 10?

10 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 3: Introducing Zero

Diagram 7i.

Diagram 7 shows a situation involving a sequence of decreasing quantity to introduce zero.
•

What type of counting is applied to introduce zero here?

•

Design a series of questions to help children learn the number zero with the use of thisdiagram.

•Describe briefly how you would introduce the numeral “0” with the situation in thisdiagram.

Diagram 8ii.

Diagram 8 shows a situation involving 4 balls and a basket.
•

Suggest an activity for children to explain the meaning of zero using the situation in
Diagram 8.

Chapter 2: Numbers and Operations | 11

Task 4: Composing and Decomposing Numbers
Task 4: Composing and Decomposing Numbers

Decomposing
DecomposingNumbers
Numbers

8
Diagram 9
Diagram 9i.

Diagram
9 shows
a asetpond.
i. Diagram
9 shows
setof
of 88 frogs
frogs atata apond.
• What are the subsets of frogs that can be formed from the 8 frogs?
What
are the subsets of frogs that can be formed from the 8 frogs?

•

students.
Explain
how the subsets can be used to introduce the idea of number decomposition
 yourDecompose
number 8 based on the subsets.
tostudents.

•

Decompose number 8 based on the subsets.



Explain how the subsets can be used to introduce the idea of number decomposition to your

Diagram 10

Diagram 10

Diagram 11

Diagram 11

ii.

Decompose number 6 based on thesituations in Diagram 10.

ii.

Decompose number 6 based on thesituations in Diagram 10.

iii.

Diagram 11 shows 7 interlocking cubes.
Explain how these interlocking cubes
can be used to help children find all thedecompositions of number 7.

iv.

Find all the decompositions of number 9.

iv.

Find all the decompositions of number
9.

iii.

Diagram 11 shows 7 interlocking
cubes. Explain how these interlocking
cubes can be used to help children find
all the decompositions of number 7.

12 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Making Teni.

Diagram 12 shows two decompositions of
number 10.
•

How many different decompositions
are there for 10?

•

What pattern is seen in all thedecompositions of number 10?

•

1 & 9 and 9 & 1 are two similar pairs ofdecompositions of 10. What are othersimilar pairs?

•

Explain how the ideas of decomposing
10 can help children learn number
operations at a later stage.

Diagram 12

Diagram 13ii.

Diagram 13 shows two arrangements of interlocking cubes showing different compositionsof 10.
•

Which arrangement do you think is easier for children to find all the decompositionsof 10 and enable them to appreciate the usefulness and beautifulness of ordering
numbers?
Explain your reasons.

Chapter 2: Numbers and Operations | 13

Task 5: Differentiating Cardinal, Ordinal and Nominal Numbers
A convesation between two children:
Rudy:

I live at the number 2 house on the rightside of the lift. The house number is 605.

Rudy:

I do not like this new house because I haveto travel six floors down to come to thegarden.

Din:

Wao! You are rich. You live in two houses.

Rudy:

I have only one house. There are six
houses on my floor. I actually live in thesecond house.

Din:

My friend, Raj, told me he lives in the firstfloor. Which button should I press to reachyour house when I get into the lift?

Rudy:

Don’t worry. Just call this telephone number
and I will come downstairs to meet you. My
number is 04-68818288

i.

With the help of examples, explain the
differences between cardinal, ordinal andnominal numbers.

ii.

Diagram 14 shows the conversation
between Rudy and Din.
•

iii.

How would you help children
understand the differences in themeaning of numbers used in thisconversation?

In this context, what are the words you
can use to describe order?

Diagram 14

Diagram 15
iv.

Diagram 15 shows Phirum and his friends after visiting a fun park. He was standing fourthfrom the front. He instructed the first 3 friends from the front to go into the first tuk-tuk, andanother 2 friends behind him to go into the second tuk-tuk. He was happy and said ‘The 5
of us can get into the 2 tuk-tuks’
From Phirum’s last instruction, he seemed to have some confusion on the situation.
•••

What was not accurate in his last instruction?
Why do you think he was confused?
How to help him to overcome the confusion?

14 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 1.3 :Introducing base-ten system with groupings of 10 and extend numbers up to 120i.
Extend numbers to more than 10 with base-ten manipulative representing numbers in ones
and tens, and appreciate the base-ten numeration system.
ii. Extend the order sequence of numbers to more than 10 in relation to the size of ones and tens
and compare numbers using numeral in every place value.
iii. Introduce number lines to represent the order of numbers and for comparison starting fromzero and counting by ones, twos, fives, and tens.
iv. Enjoy various ways of the distribution of objects with counting such as playing cards, andexplain it and enhance number sense
v. Draw diagram for representing the size of number with base-ten blocks such as a flat (square)for a unit of hundred and rectangular bar for a unit of ten
Sample Tasks for Understanding the Standards
Task 1:

Finding Easier Ways to Count Objects

Diagram 1i.

Diagram 1 shows 21 interlocking cubes given to three children, A, B and C. The threechildren counted the number of cubes in different ways.
•

Explain briefly the three ways of counting.

•

What are the advantages of these ways of counting?

•

The SEA-BES CCRLS framework in mathematics (refers Chapter 1) illustrates fivemathematical values. Which of the values are linked to these ways of counting?

•

Explain how these ways of counting can help children learn other mathematical ideas
in their later stage of learning numbers and operations?

Chapter 2: Numbers and Operations | 15

Task 2: Place Value
Numeration Systemi.

Explain the difference between number and numeral.

ii.

The universally used Hindu-Arabic numerals are formed by ten basic symbols, known asdigits, together with the idea of the base-ten place value.

iii.

•

What are the ten digits?

•

Why is the symbol for zero crucial in forming the numerals?

•

Explain briefly the idea of the base-ten place value system.

Other than the Hindu-Arabic numerals, another numeration system still in use is the Roman
numerals. This system uses seven Latin alphabets as basic symbols to form numerals:
I – 1, V – 5, X – 10, L – 50, C – 100, D – 500, and M – 1000.
•

Write in the Hindu-Arabic numerals, the numbers represented by each pair of the
Roman numerals.
(a) IV and VI		(b) IX and XI(c) XIV and XVI(d) XL and LX		(e) DC and CD(f) MMC and MCM•

Explain the major differences between the Roman numeration system and the HinduArabic numeration system.

Numbers Up to 20

10
Diagram 2i.

Diagram 2 shows a numeral-picture card with 10 dragonflies. At this early experience
with the numeral 10, children often see it as a representation of number ‘ten’, but do notunderstand the idea that the digit 1 represents 1 ten, and the digit 0 represents 0 ones.
•

Explain why this is so.

•

How would you help children to understand the idea.

16 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelii.

A teacher used some base-ten blocks tohelp children learn the idea of place value.
Diagram 3 shows the work of a child.
•

What is the misconception of the child?

•

What would you suggest to the teacher
in order to help the child correct the
misconception?

Diagram 3

Diagram 4
iii.

iv.

Diagram 4 shows how the numbers 9 to 20 are represented with base-ten blocks placed ona two-column chart.
•

Why are the retangular bars placed on the left column of the chart?

•

If the rectangular bars are placed on the right column, do they change the values of thenumbers? Explain your reasons.

•Some children find it difficult to tell whether the numerals 16 or 19 is larger. How would
you help the children overcome this difficulty?

What key ideas are children expected to develop when using base-ten blocks to represent
numbers?

Chapter 2: Numbers and Operations | 17

Numbers More than 20

Diagram 5i.

Diagram 5 shows four tasks involving counting of objects for Grade 1 students.
•

Explain why all four tasks are crucial in helping children to conceptualise base-ten
place value.

•

How will you sequence these four tasks?

•

Explain your reasons.

18 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram 6ii.

Diagram 6 shows another four tasks involving numbers up to 100.
•

Will you ask your students to do all the four tasks? Explain your reasons.

•

How will you sequence these four tasks? Explain your reasons.

Diagram 7iii.

Diagram 7 shows ‘adding one’ and ‘adding 10’ structure of number chart.
•

Fill eachwith a number.

•

Explain how this chart can be used to enhance children’s understanding of place value.

Chapter 2: Numbers and Operations | 19

Task 3: Saying Numbers in Different Languages
Saying numbers in different languages can be a daunting task. Table 1 shows how 1 to 20 isread in different languages. Do enjoy reading them.
Table 1
Saying Numbers in Different Languages

Khmer20 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

i.

ii.

Study and compare how the numbers 11 to 20 are said in the different languages.
•

Do you observe any interesting pattern in some of the languages? If yes, describe thepattern.

•

Some of the patterns are more helpful for children to learn numbers 11 to 20. Identify
the pattern and explain why it is helpful.

The Khmer language uses an interesting pattern in saying numbers 6 to 9.
•

Describe the pattern.

•

Do you think the pattern makes it easier for Cambodian children to learn numbers?
Explain your reasons.

Chapter 2: Numbers and Operations | 21

Task 4: Extending Numbers to 120

Diagram 8i.

Diagram 8 show four representations of regrouping involving ones, tens and hundreds.
•

Explain how these representations can help children develop the concept of base-ten
place value?

•

How will you order the sequence of tasks using each of these representations? Explain
your reasons.

ii.

Looking at the base-ten blocks in
Diagram 9, a student said: “The number
is 1 hundred and fourteen”.
But when asked to write the number,he wrote: 100104.

Diagram 9•

Why such misconception happened?

•

Explain how you can help the student
to arrive at the correct answer andcorrect his misconception.

22 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 5: Introducing Number Lines

Diagram 10i.

Diagram 10 shows two situations on two scaled bars.
•

Based on the scaled bars, a child says the kangaroo jumps 9 steps and the rabbit hops
18 steps. What is the child’s misconception•

Without an initial reference point, can you distinguish an interval and a scale ofmeasure? Explain your answer.

•

A number line is then aligned to the scaled bars.


What is a number line?

What can a number line be used for?



Why is it important to have ‘0’ at the begining of a number line?

•

How does the number line help children to understand the order of numbers?

•

How does the number line help children to determine the difference in the number ofsteps between the rabbit hop and the kangaroo jump?

Diagram 11
ii.

Between Line A, Line B and Line C in Diagram 11, which one cannot be considered as anumber line? Explain your reasons.

iii.

Must a number line begin with a zero point? Explain your reasons.

Chapter 2: Numbers and Operations | 23

Task 6: Different Types of Number Lines

Diagram 12i.

Diagram 12 shows four number lines.
•

Fill in eachwith a number for each of the number lines.

•

What are the diffrences in number lines (a) (b), (c) and (d) ?

•How do you explain the differences and similarities between number lines (a) and (c)?

Diagram 13ii.

A teacher prepares four tasks involving number lines in the sequence as shown in Diagram
13. Explain how this sequence of tasks can facilitate students’ understanding of drawing
number lines for different scales.

24 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 2: Introducing Addition and Subtraction
Standards 2.1:
Understanding situations for addition up to 10 and obtaining fluency of using addition in situations
i. Introduce situations (together, increase and combine) for addition and explain it with manipulativeand orally to define addition for operation.
ii. Develop fluency of addition expressions using composition of numbers for easier calculation
with number sense for composition of numbersiii. Apply addition fluency in learners’ life
Sample Tasks for Understanding the Standards
Task 1: Introducing the Meanings of Addition

Diagram 1

Diagram 2

Addition usually involves putting objects together, but it is not always so. Diagrams 1 and 2
show two addition situations inolving mangoesteens. The pictures of the blocks are used toillustrate the situations.
i.

ii.

iii.

Study the situation in Diagram 1.
•

Describe the situation.

•

How to represent this situation with a number sentence involving addition?

•

Make an addition story similar to the situation in Diagram 1.

Study the situation in Diagram 2.
•

Describe the situation.

•

How to represent this situation with a number sentence involving addition?

•

Make an addition story similar to the situation in Diagram 2.

Compare the two addition situations in Diagrams 1 and 2.
•

Explain the different meanings of addition in the two situations.

Chapter 2: Numbers and Operations | 25

Diagram 3
iv.

v.

Diagram 3 shows another story of addition. Compare this story with Diagram 1.
•

The story in Diagram 3 may be more difficult for children to perceive the idea of addtion
as compare to the story in Diagram 1. Explain why.

•

Make a similar addition story involving same category of objects, yet with different
attributes.

•

When learning addition, do you think it is necessary for children to do activity involving
both situations in Diagram 1 and Diagram 3? Why or why not?

Consider the following task for children:
Create a story for 3 + 4.
•

Explain how this task can help children conceptualise the meaning of addition.

26 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Addition with Sum Less than 10

Diagram 4i.

At the early stage of learning addition, many children rely on counting to find the sum of
two numbers. However, when the sum of two numbers is less than 10, such as 4 + 3,
addition of the two numbers can also be obtained without counting if children have learnt
composition and decomposition of numbers.
Diagram 4 shows how the addition is calculated with the help of interlocking cubes using
the decompose-compose strategy.
•

Explain the decompose-compose strategy.

•

Without the interlocking cubes, how will you use a diagramatic representation to
perform 4 + 3 with the strategy?

•

Is this strategy suitable for calculating 7 + 2? Explain your reasons.

•

What knowledge learnt previously is required to apply in this strategy?

Chapter 2: Numbers and Operations | 27

Task 3: Using Addition Cards to Develop Mathematical Thinking

Addition Cards

Answers on Opposite Sides
Diagram 5

Diagram 5 shows an arrangement of 25 addition cards with the answer for each addition written
on the corresponding opposite side of each card.
i.

Describe any pattern that you observe in these cards.

ii.

Suggest and explain how the pattern in these cards can be used to develop inductive
reasoning among primary school students.

iii.

Diagram 6

A teacher arrange 6 of the cards as shown in
Diagram 6 and intend to use them to promotemathematical thinking among his primary school
students.
•

What are the possible addition expressions
behind card “4”?

•Which addition expression should be in thiscard? Explain your reason.

•

What expression should be in the two cards
with “?”, respectively? Explain your reason.

•

Explain how you will use these cards todevelop a learning activity for promoting
mathematical thinking.

•Explain what mathematical thinking is being
promoted in your learning activity.

28 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 2.2:
Extending addition to more than 10 and obtaining fluency of using addition in situations
i. Extend addition situations and think about how to answer using the idea of making 10 withdecomposition and composition of numbers
ii. Explain the idea of addition with place value using base-ten blocks
iii. Develop fluency of addition expressions to more than 10 for easier calculation
iv. Apply addition fluency in learners’ life
Sample Tasks for Understanding the Standards
Task 1: Addition With Sum More Than 10i.

Diagram 1

ii.

iii.

Diagram 2

Diagram 1 shows a situation involving someeggs in two trays.
•

Why egg trays are used in this task?

•

How many eggs are there altogether?

•Write an expression to represent theaddition and explain how you find theanswer.

Without counting, addition with sum more than10, such as 9 + 3, can be calculated using thedecompose-and-compose stragety as follows:•

Describe the strategy briefly.

•

Do you think this strategy is easier or moredifficult for children in your class to learn
addition? Explain your reasons.

Diagram 2 shows another addition situation
involving 3 + 9.
•

Calculate 3 + 9 using the decompose-andcompose stragety by drawing the diagram.

•Some teachers argue that it is easier forchildren to learn 9 + 3 than 3 + 9. Do
you agree with this view? Explain yourreasons.

Chapter 2: Numbers and Operations | 29

8 + 6.

8+6

Diagram 3
Diagram 3

8

6

8+2

6–2

10

4

10 + 4 = 14

Diagram3
Diagram4iv. Diagram
3 shows3ashows
situation
involving
8 + 6. Diagram
4 shows
how David
used interlocking
iv. Diagram
a situation
involving
8 + 6. Diagram
4 shows
how David
used interlocking
cubes tocubes
calculate
8 + 6. 8 + 6.
to calculate

v.



What
did David
to maketo10?
• number
What number
diddecompose
David decompose
make 10?

What
did David
to get theanswer?
• numbers
What numbers
didcompose
David compose
to getthe answer?

Howrecordstrategies
withoutwithout
drawingdrawing
the interlocking
cubes?cubes?
• would
How you
would
you David’s
record David’s
strategies
the interlocking

Calculate
8 + 6 by8decomposing
a different
number number
from David
maketo10.
v. Calculate+ 6 by decomposing
a different
fromtoDavid
make 10.

vi. Between 8 and 6, which number would you decompose to do the addition? Justify your choice.
vi. Between 8 and 6, which number would you decompose to do the addition? Justify yourchoice.

30 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Using Addition Cards to Develop Fluency of Additon

Diagram 5i.

ii.

Diagram 5 shows 21 cards for basic facts of addition in disarray.
•

Rearrange the cards so that all cards with the same sum will be in the same horizontalrow.

•

What patterns do you notice in the horizontal rows and the vertical columns?

There are another 15 cards for basic facts of addition other than those shown in
Diagram 5.
•

Find all the missing 15 cards.

iii.

From the complete set of arranged cards, describe any pattern that can be used to help
children develop fluency in addition.

iv.

Explain how the discovery of patterns in the arrangement of cards can help children
appreciate the beautifulness of numbers.

Chapter 2: Numbers and Operations | 31

Standards 2.3:
Understanding situations for subtraction up to 10 and obtaining fluency of using subtraction insituations
i. Introduce subtraction situations (remaining, complement and difference) and explain orally withmanipulative to define subtraction for operation
ii. Develop fluency of subtraction expressions using decomposition of numbers for easiercalculationiii. Apply subtraction fluency in learners’ life
Sample Tasks for Understanding the Standards
Task 1: Introducing the Meanings of Subtraction

Diagram 1i.

Diagram 1 shows a situation involving subtraction. The picture of the interlocking cubes
are used to illustrate the situation.
•

How is the terminology “left” related to this situation?

•

How to represent this situation with a number sentence involving subtraction?


Explain the meaning of subtraction in this situation.

•

Two commonly used expressions in subtraction are “5 minus 2” and “take 2 awayfrom 5”. Explain the different contexts in using the two expressions.

•

When looking at the picture in Diagram 1, Amin explained the situation as“3 mangosteens plus 2 mangosteens is 5 mangosteens altogether”.
Give a possible reason for Amin’s explanation.

How will you guide Amin to extend his idea of addition to subtraction?

32 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

How many more rabbits are there than tortoises?
How many more rabbits are there than tortoises?

Difference

Difference

Diagram2
Diagram 2ii.

ii. 2Diagramshows anothersubtraction
situation involving
7 rabbits
and 5 andtortoises.
Diagramshows2anothersubtraction
situation
involving
7 rabbits
5 tortoises.

•How is the terminology “difference”
related
Diagram
2 to this situation?

How is theterminology“difference”
this situation?
How
to represent thissituation withrelated
a numbertosentence
involving subtraction.

ii.

•

Diagram 2 shows another subtraction situation involving 7 rabbits and 5 tortoises.
 Explain the meaning of subtraction in this situation.
 toHowis the terminology“difference”
relatedto this situation?
How
represent
this situation
witha numbersentence involving subtraction.

•

Explainthe themeaningofsubtraction
in situation.
thissituation.
8 children
a line. Thereare
3 boys.
 Explainmeaningofinsubtractionin this

How to represent this situation with a number sentence involving subtraction.

How many girls are there?

8 children in a line. There are 3 boys.
How many girls are there?

Diagram 3iii. Diagram 3 shows a third situation involving 8 – 3 = 5.
iii.

Explain the differences betweenDiagram
this subtraction
situation from situations in Diagrams 1 and 2.
3

Diagram 3

 Why is it not appropriate to read ‘8 – 3 = 5’ as ‘Take 3 away from 8 equals 5’ for this situation?
iii. Diagram 3 shows a third situation involving 8 – 3 = 5.
 Explain the meaning of subtraction in this situation.
 3Explain
theadifferences
betweeninvolving
this subtraction
Diagramshows
third situation
8 – situation
3 = 5. from situations in Diagrams 1 and 2.

••

iv. Why
is it notto readsituations‘8 – 3 = 5’inas‘Take 3 1,
away
from
Comparetheappropriate
three subtraction
Diagrams
2 and
3. 8 equals 5’ for this situation?

Explainthe thedifferences
between in
thissituation from situations in Diagrams
 Explainmeaning of subtraction
thissubtraction
situation.
1 and2. Which of these situations would you use as a first example to teach subtration to children?

Explain your reasons.
iv. Compare the three subtraction situations in Diagrams 1, 2 and 3.

Why isnotisappropriate
to read‘8subraction– 3 = 5’byasusing
‘Take
away
from
8 equals it Why
it not appropriate
to teach
the 3
‘take
away’
situation
only? 5’ for this Which of these situations would you use as a first example to teach subtration to children?
situation?
Explain your reasons.
vi. Consider the following task for children:•iv.

 Whytheis itmeaning
not appropriate
to ateachusing the ‘take away’ situation only?
Explainof subtraction
Createstorysubraction
for 9in– this
4. bysituation.
vi. Consider the following task for children: Explain
this task canhelp children
the2meaning
Compare thethree howsubtraction
situations
in conceptualise
Diagrams 1,
and 3. of subtraction.
Create a story for 9 – 4.

•

Which of these situations would you use as a first example to teach subtration to Explain how this task can help children conceptualise the meaning of subtraction.
children? Explain your reasons.

•

Why is it not appropriate to teach subraction by using the ‘take away’ situation only?

8v.

Consider the following task for children:8

Create a story for 9 – 4.
•

Explain how this task can help children conceptualise the meaning of subtraction.

Chapter 2: Numbers and Operations | 33

Task 2: Subtraction Less than 10

Diagram 4i.

ii.

Diagram 4 shows how 9 – 3 is calculated with the help of interlocking cubes.
•

Explain the strategy of subtraction.

•

Without the interlocking cubes, how will you use a diagramatic representation to
perform 9 – 3 with the strategy?

•

Use this strategy to calculate:


5–2



7–3

Do you think it will be easier for the children in your class to learn subtraction by thisstrategy? Why or why not?

34 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 2.4:
Extending subtraction to more than 10 and obtaining fluency of using subtraction in situations
i. Extend subtraction with situations and think about how to answer using the idea of 10 withaddition and subtraction of numbers (composition and decomposition of numbers)
ii. Explain the idea of subtraction in place value using base-ten blocks
iii. Develop fluency of subtraction expressions to more than 10 for easier calculation
iv. Apply subtraction fluency in learners’ daily life
Sample Tasks for Understanding the Standards
Task 1: Extending Subtraction to More Than 10i.

Diagram 1 shows 12 bird stickers to be used forteaching subtraction. The problem is as follows:
There are 12 stickers. 9 stickers are
given away. How many stickers are left?
A child calculated the answer as follows:

Diagram 1•
ii.

Diagram 2 shows some base-10 blocks usedwith a place-value chart to represent number 12.
•

Diagram 2

What is the misconception of the child?

Explain with the idea of place value, how
you will use the base-10 blocks to help thechild in (i) to overcome his misconception?

Chapter 2: Numbers and Operations | 35

iii.

iv.

Two children, A and B, used the decomposeand-compose strategy in different ways tocalculate 12 – 9 as shown in Diagram 3.
•

What are the differences in the two ways?

•

Which way do you think is easier for thechildren in your class? Explain yourreasons.

Consider the following task for children:Usethedecompose-and-compose
strategy to calculate:◊ 17 – 9◊ 15 – 6◊ 11 – 2

Diagram 3•v.

Explain how this task of proceduralisation
can enhance the computational fluencyof subtraction.

A teacher argue that the task “12 – 9” should be discussed before the task “12 – 3”.
Do you agree? Justify your reasons based on the idea of decomposition and compositionof numbers.

36 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Using Subtraction Cards to Develop Fluency of Subtraction

Diagram 4i.

ii.

Diagram 4 shows some subtraction basic fact cards and a few empty cards.
•

Rearrange the cards so that all cards with the same answers will be in a same horizontalrow.

•

What patterns do you notice in the horizontal rows and the vertical columns?

•

Explain why each pattern is formed in such a way.

•

Based on the patterns identified, write an expression on each of the blank cards tocomplete the pattern.

Explain how these patterns can be used to help the children in your class to develop
fluency in subtraction.

Chapter 2: Numbers and Operations | 37

iii.

Diagram 5
iv.

Diagram 5 shows a subtraction problem involving a 2-digit
number with a 1-digit number.
•

Find all possible solutions for the problem.

•

How many solutions are there altogether?

•

What strategy have you used in solving this problem?

Diagram 6 shows another subtraction problem.
•

Find all possible solutions for the problem.

•

How many solutions are there altogether?

Diagram 6v.

How many solutions are there altogether for the problem– = 2?

vi. Given another problem

–

= 7.

•

Make a conjecture on the number of all possible solutions
for the problem.

•

Test your conjecture.

vii. Given another problem

–

= 9.

•

How many solutions are there altogether?

•

How confident are you that your answer is correct?
Explain your reasons.

•

What about

–

= 8?

•

What about

–

= 10?

viii. What are the advantages of posing this kind of tasks for yourstudents?

38 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 3: Utilising Addition and Subtraction
Topic
3: UtilisingStandards
3.1: Addition and Subtraction
Standards 3.1:
Utilising addition and subtraction in various situations and understanding their relationships
i. Understandthe subtraction
difference between
andsubtraction
situations
with tape diagrams
Utilising
addition andin variousaddition
situations
andunderstanding
their relationshipsii.
Explainsubtractionasaninverseofadditionsituations
with
tape
diagramsi.
Understand the difference between addition and subtraction situations with tape diagramsiii. Explain
Understandaddition
three numbers,
withtape
threenumbers and combination ofii.
subtraction
aswithan inverseof additionsubtraction
situations withdiagrams
additionandsubtraction
situations
iii. Understand addition with three numbers, subtraction with three numbers and combination of addition
iv. andApply
addition andsubtraction in various situations such as in ordering numbers
subtraction
situations
iv. Apply addition and subtraction in various situations such as in ordering numbers
Sample Tasks for understanding the standards
Sample Tasks for understanding the standards
Task 1: Addition and Subtraction with Tape Diagram
Task 1: Addition and Subtraction with Tape Diagram
Teacher A’s Diagram:
7 boysTeacher B’s Diagram:
5 girls

7 boys5 girls
Altogether?

Altogether?

Diagram
Diagram1 1
Twoteachers,AAandandB,
B,drew
drew two
two different tapein in
Diagram
1 to1 help
theiri. i. Twoteachers,
tape diagrams
diagramsasasshown
shown
Diagramto help
pupilssolve solve
the following
addition
problem:theirstudents
the following
addition
problem:

girls in a group.
There areThere
7 boysareand7 5boys
girlsandin a 5group.
How manyHow
childrenthere altogether?
manyarechildrenare there altogether?
In what way do you think tape diagram is useful in helping your pupils undertand number
operations?
•
In what way do you think tape diagram is useful in helping your students understandoperations?
 number
Which ofthe two ways of drawing tape diagram do you think is easier for your pupils to•understand addition? Explain your reasons.
Which of the two ways of drawing tape diagram do you think is easier for your students
to understand addition? Explain your reasons.

8 children are playing in the park.
3 of them go back.
How
many children
are left?
8 children
are playing
in the park.
3 of them go back.
How many children are left?
6 children are playing in the park.
3 more children came to join them.
How
many children
are there?
6 children
are playing
in the park.
3 more children came to join them.
How many children are there?
Diagram 2

Diagram 2

ii.

ii.

Diagram 2 shows two word problems.

Diagram 2 shows two word problems.
 Draw a tape diagram for each of the
problems to help your pupils understand•
Draw a tape diagram for each of thethe problem situation.
problems to help your students understandthesituation.
 problem
Base oneach tape diagram, write a•mathematics sentence to represent theBase problem
on eachtape diagram, write asituation.
mathematics sentence to represent the
problem situation.

Chapter 2: Numbers and Operations | 39

There are 9 black marbles and 3
white marbles in a box.

iii.

How many more black marbles
are there than white marbles?

There are some black and whitemarbles in a box. Black marbles
are 3 less than white marbles. 9of the marbles are black.

Diagram 3 shows another two word problems.
•

Explain why many children find these two
problems difficult to understand.

•

Draw a tape diagram for each of the
problems to help your students understandthe problem situation.

•

Base on each tape diagram, write amathematics sentence to represent the
problem situation.

How many white marbles are
there?
Diagram 3
iv.

Diagram 4 shows a situation involving somemonkeys in a zoo.
•v.

Base on the diagram, create an addition
story and represent it with a mathematics
sentence.

Subtraction is the inverse operation of addition.
•

Create a subtraction story to show the
inverse property of addition.

•

Represent the story with a mathematics
sentence.

Diagram 4vi. Explain with tape diagrams, how you will help
chidren understand the inverse relationshipbetween addition and subtraction.

40 | Mathematics
Challenges
for Classroom
Practices at the Lower Primary Level
Task
2: Situationsof Combined
Operations
Task 2: Situations of Combined Operations
Task 2: Situations of Combined Operations
A combined operation in Diagram 5 may beAcombined
operation
in Diagram 5 may bedifficultfor children
to understand.
difficultforchildrentounderstand.
A combined operation in Diagram 5 may be
i. TwoDaniel
and Pearly explained
difficultforstudents,
children tounderstand.
i. Twostudents,theyDanielPearlythe methods
usedandto gettheirexplained
answersinDiagram6.
themethods
they
used
to
get
their
answersi. Two students, Daniel and Pearly
Diagram 5in
Diagram6.
explained the methods they used to get Explain
if boththe methods
are correct.
Diagram 5
theiranswers
in Diagram
6.
 Explain if both the methods are correct.
mathematics
story areto help• Create
Explain aif boththe methods
I add 5 to 7 to get 12, thenchildrenunderstand
7
+
5
–
8.
 Createamathematics
story
to help
correct.
Iminusadd 58toto7get
to getthen4.
the 12,
answerchildren understand 7 + 5 – 8.
minus 8 to get the answer 4.
what away
is the processin• In
Createmathematics
storyofto thinking
help
 In
what
way
is
the
processofthinking
increating
the
story
can
enhancethechildren understand 7 + 5 – 8.
creating
the
story
can
enhancetheconceptual understanding in combined
conceptual
operations ofunderstanding
your pupils. in combined
operations of your pupils.
I minus 8 from 5, but I am
Ishortminus
5, buttake
I amof 83,fromso I then
short
of
3,
so
I
then
take
away 3 from 7 to get 4.
away 3 from 7 to get 4.

7+5–8
7+5–8

Diagram 6
Diagram 6
6
DiagramDaniel’s solution:
Daniel’s solution:
9–3+2
9–3+2
=9–5
=9–5
=4
=4

Pearly’s solution:
Pearly’s solution:
9–3+2
9–3+2
=6+2
=6+2
=8
=8

ii.
ii.ii.

Diagram
Diagram 77
Diagram 7iii.

Diagram 7 shows how Daniel and Pearly
Diagramshows
Danielshows how
how
Danielandsolved the77combined
operation
9 and–3+Pearly2.
solvedthe
combined
operation
9
–
3
+ 2.
Pearly solved the combined operation
9+ 2. method is correct? Explain your – 3Which Whichmethod is correct? Explain yourreasons.
reasons.
•
Which method is correct? Explain Createa mathematics story to help
your reasons.
 Create
mathematics
children aunderstand9 – 3 story+ 2. to help
childrenunderstand
9
–
3
+ 2.
•
Create a mathematics story
to help
 In
what
way
is
the
processchildren understand 9 – 3 +of
2. thinking increating
theis story
can enhance Inwhat way
the processof thinkingthe
inconceptual
understandingincombinedcreating
the
story
can
enhancetheIn what way is the process of thinking
operations
ofunderstanding
yourpupils.
conceptualin combinedin creating
storiescanenhancetheoperationsofyourpupils.
conceptual understanding in combined
operations of your students.

Task 3: Making Addition and Subtraction Easier
Chapter 2: Numbers and Operations | 41

Task 3: Making Addition and Subtraction Easier
Task 3: Making Addition and Subtraction Easieri. Diagram 8 shows an addition involving three
Task 3: Making Addition and Subtraction Easier
numbers. Daniel explained the method hei. Diagramananswer.
addition involving threeused 8toshows
find thei. Diagram
8 shows
an addition
involving
numbers.
Danielexplained
the method
hethree
8+6+2
numbers.
Danielexplainedthemethodhei.usedDiagram
8 shows
an addition involving
theanswer.
 to findExplainDaniel’s method.
used
to
find
the
answer.
three numbers. Daniel explained the
8 + 86 ++ 62 + 2
heproperty
used method.
to of
find
the answer.
 method
ExplainDaniel’sWhat
addition
did Daniel use
It is easier to add up to 10
 Explain Daniel’s method.
in
hismethod?
first. So, I do like this …
Daniel’smethod.
 •WhatExplain
propertyof addition
did Daniel use
It is easier to add up to 10
 hisWhat
property of addition did Daniel useItiseasier
to
add
up
to
10
in
method?
first. So, I do like this …
in his method?
•Whatproperty of addition did
first. So, I do like this …
Daniel use in his method?

ii.
ii.
ii.
Diagram 8
Diagram 8
Diagram 88
DiagramDaniel used aamethod
addition
Danielmethodtotomake
makeaddition ofof threenumbers
easierfollows:
three
numbers
easieras asfollows:
Daniel used a method to make addition ofii. Daniel
useda method
to make addition of
three
numbers
easieras41)
follows:(628 ++ 59
three numbers easier as follows:
= 628
++
(41
(628
+ 41)
59+ 59)(628 + 41) + 59
= 628
++
100
= 628
+ (41
59)
= 628 + (41 + 59)
= 728
= 628
+ 100
= 628 + 100= 728
Explain=thethemethod.
728• Explain
method.

Explainthemethod.of addition was used?
What
property• Whatpropertyof addition was
Explainthe method.
used?
iii. What
property
addition
was used?
There
are ofalso
easier
way to do What property of addition was used?
subtraction
such
as
the
followingareiii.iii. There
There
are also
also easiereasier ways
way toto do
doexamples:
such
as
the
iii.subtraction
There areway to do
subtraction
suchalso
as easierthe following
following
subtraction
examples:56 – 16 –such
8 = 60as– 20the– 8 following
examples:examples:7 =–6056 –5616––18
8 =– 60
20––208 –9
56 – 16 – 8 = 60 – 20 – 8
6 =–6056 –5618––197 =– 60
20––209 –9
56 – 18 – 7 = 60 – 20 – 956 – 19 – 6 = 60 – 20 – 956 –why
19 – these6 = 60 subtractions– 20 – 9
 Explainarecorrect?
Explainwhy
why these subtractions
 • Explain
subtractionsarearecorrect?
Explain why these subtractions arecorrect?
In what way can these help to makecorrect?
the subtraction easier?
 • InInwhat
way
cancanthese
help help
to make
what
way
these
to
make
In what
way
can
these
help
make
the
subtraction
easier?
easier? of totheseiv.
What theis subtraction
the significancethe subtraction easier?
methods in developing computational
iv.iv. What
What
is thethe significancesignificance ofof these
theseis
fluency?
iv.methods
Whatin isdeveloping
the significanceof thesecomputational
methods in developing computational
methods
indeveloping
computational
fluency?
fluency?
fluency?
42 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 4: Applying Addition and subtraction

Diagram 9
Diagram 9 shows 9 birds perching on a telephone line. The rounded bird is at the middle of thebirds.
i.

What is the position of the rounded bird from the left side?

ii.

The third bird from the left side flew away, follow with the last two birds from the right.

iii.

iv.

•

What is the new position of the rounded bird?

•

How many birds are left on the telephone line? Write a mathematics expression to findthe answer.

Children are lining up in the school field. Adriel is at the 4th place from the front.
•

Babyrina is the 3rd child behind Adriel. What is the position of Babyrina from the front?
Write a mathematics expression to find the answer.

•

There are 8 children behind Babyrina. How many children are there altogether?

•

Explain how a number line can be used to help children solve this type of problem.

Why situations or contexts are crucial in helping children learn addition and subtraction
involving three numbers?

Chapter 2: Numbers and Operations | 43

Topic 4: Extending Numbers with Base-ten System Up To 1 000 000 GraduallyStandards 4.1:

Topic 4:

Extending Numbers with Base-ten System Up To 1 000 000 Gradually

Extending numbers using base-ten system up to 1 000
i. Experience counting of 1 000 by using various units and appreciate the necessity of the baseten numbers
system using base-ten system up to 1 000
Extendingii.
Extendthecounting
order of of
numbers
introduce
more
than
000 in relation
to the sizeof ones,tensi.
Experience
1 000 bytousing
variousunits
and1appreciate
the necessityof thebase-tenand hundreds
systemUse a thepartialnumberline totocompare
numberstranslation
sizetensof everyii. iii. Extend
order
of numbers
introducesize
moreofthan
1 000 throughin relation
to the sizeofofthe
ones,
and
hundreds
digit and select appropriate scaleiii. iv. Usea partialappropriate
number linediagram
to compare
size ofnumbersof the sizeof everydigitofRepresent
to showthesize of throughnumberstranslation
without counting
suchas threeandselectappropriate
scale
hundreds mean 30 of tens and visualise the relative size of numbers
iv.v. Represent
diagramto showbythe
size ofof numbers
Representappropriate
larger or smaller
numberssymbolinequalitywithout counting such as three ofStandards 4.1:

hundreds mean 30 of tens and visualise the relative size of numbers
v. Sample
Representor smaller numbers
symbol of inequality
Taskslarger
for Understanding
thebyStandards
SampleTasks
for Understanding
Task 1:Counting
Up to 1000 the Standards
Task 1: Counting Up to 1000

Grouping by 10s

by 10si.Grouping
Groupingby tens is a crucial process in the base-ten numeration system. Children need tounderstand this process in order to understand and appreciate the system.

i.

Grouping by tens is a crucial process in the base-ten numeration system. Children need tounderstand this process in order to understand and appreciate the system.

•


ii.

ii.

Explain the grouping process in the base-ten numeration system.

Explain the grouping process in the base-ten numeration system.

Various materials have been used to model the grouping-by-10 process. Diagram 1 showsmaterials
havesuch
beenmaterials,
used to model
the andgrouping-by-10
process. Diagram 1 shows aaVariouscomparisonof twopencils
blocks.
comparison of two such materials, pencils and blocks.
Material

100

10

1

Hundred-flat

Ten-barOne-cube

Pencils

BlocksDiagram11
DiagramWhat is a crucial difference between the nature of these two materials?

What are the advantages and disadvantages of these two materials in helping children
Whattheareideatheofadvantages
learn
regrouping?and disadvantages of these two materials in helping children••

What is a crucial difference between the nature of these two materials?
learn the idea of regrouping?

44 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Doyou
youthink
thinkthese
thesetwo
twomaterials
materialsare
areequallyequallyeffective
effectiveininhelping
helpingyour
yourpupils
pupilslearn
learnthethe ideaidea  Dogrouping
10s?
yes,why?
why?
no,whichwhichofofthese
twomaterials
materials
doyouthink
thinkmore
Do you
think
these
two
materials
areequallyeffective
inthesehelping
yourpupils
learn
the
idea
ofofgrouping
byby
10s?
IfIfyes,
IfIfno,two
doyou
isis more• of Do
you
think
these
materials
arewhichequally
helping yourstudents
effective?
Explainyourreasons.
grouping
by 10s?
Iftwo
yes,
why?
If no,
of effective
these twoinmaterials
do youthink islearn
moreeffective?
Explainyourreasons.
the
idea
of
grouping
by
10s?
If
yes,
why?
If
no,
which
of
these
two
materials
do youeffective? Explain your reasons.
is canmore
effective?
Explainyourbereasons.
How
caneacheach
thesematerials
beusedusedtotorepresent
representone
onethousand?
thousand?
 thinkHow
ofofthesematerials
 How can each of these materials be used to represent one thousand?
•How can each of these materials be used to represent one thousand?
iii.
Diagram2 2shows
showsthethenumbernumber230
230represented
representedby
bybundles
bundlesofofpencils.
pencils. 230
230 isis thethe sumsum ofof 22 sets
sets
iii.
Diagram
of
100
and
3
sets
of
10.
iii.
Diagram2
shows
thenumber230
representedbybundlesofpencils.
230
is
the
sum
of
2
iii.
Diagram
2 shows
the numberof 100and 3 setsof 10. 230 represented by bundles of pencils. 230 is the sum of 2 sets100
and 3ofsets
ofsets100 ofand
3 sets10. of 10.

Diagram22
Diagram
Diagram
Diagram22

Writea amathematics
mathematicssentence
sentencetotorepresent
representthe
thesum.
sum.
  Write•
Write a mathematics sentence to represent the sum.
Write
aHow
mathematics
sentence
torepresent
manysets
setsofof10
10arearethere
there
230?the sum.
  Howmany
inin230?
manysets setsof 10ofarein 230?
•HowHowmany
10 there
are therein 230?
Diagramshowssomebase-tenblocksiv.ii.ii.Diagram
3 showssomebase-tenblocksblocks
Diagram
33 shows
somebase-tenplacedonaplace-value
place-valuechart
torepresent
theplacedonon
a aplace-value
toto
represent
Diagram3showssome chartbase-tenblocks theplacedchart
represent
number423.
placedon a place-value
chart to represent thethenumber423.
number423.
number 423.
Explainthedevelopmentofof place
place
value
Explainthedevelopmentof• Explainthedevelopment
place valuebyusing
base-tenblocksthat
consistof
Explainthedevelopmentof blocks
place
by usingbase-tenblocksthat value
consistofvalue
by usingbase-tenthat
one-cubes,
10-barsand100-flats
tobyconsistusingbase-tenblocks that
one-cubes,
10-bars
andconsist
100-flatsof one-cubes,
10-bars
andof torepresent423.
one-cubes,
10-barsandrepresent
423.
100-flats
to represent
423.100-flats torepresent 423.
iii. Diagram 4 also shows 423 grouped as a sumiii. Diagram 4 also shows 423 grouped as a sumof 4 hundreds 2 tens and 3 ones, which caniii. Diagram
also shows
423 andgroupedas awhichsum canof 4 4hundreds
2 tens3 ones,also be written as 400 + 20 + 3. Diagram 7
of 4 also
hundreds
2 tensasandcanbe written
4003 +ones,
20 + which
3. Diagram
7
shows 423 regrouped as 4 hundreds 1 tens
also
bewritten
as
400
+
20
+
3.
Diagram7
shows
423
regroupedas4
hundreds
1
tens
v. Diagram
also shows 423 grouped as
and 13 3ones.
shows
423 regrouped as 4 hundreds 1 tens
a andsum13ofones.
4 hundreds 2 tens and 3 ones,
and 13
 ones.
Does
the beregroupingchangethewrittenwhich
can also
written as
400 + the20 +written Doesthe regroupingchangenumeral for 423? Explain your answer.
Diagram
4 shows423
regroupedas 4
 3.Does
the regroupingchangethenumeralfor 423?
Explainyourwritten
answer.
hundreds
1 tens13 ones.
numeral for
423?and
Explainyour answer.

ii.

Diagram 33
Diagram
Diagram 3
Diagram 3•

Diagram 4
Diagram 4
Diagram 4
Diagram 4

Does the regrouping change thewritten numeral for 423? Explain youranswer.

Chapter 2: Numbers and Operations | 45

Diagram 5 shows 423 regrouped asvi.iv. Diagram
5 shows 423 regrouped as 3
3hundreds512 showstens and423
3 ones.
iv.
Diagramregrouped asas
hundreds
tens and423
3 ones.
iv. Diagram
512 showsregrouped3
hundreds
12
tens
and
3
ones.
3 hundreds 12 tens and 3 ones.
 Does the regrouping change the value of•Does
the regroupingchange the423?
Explainyour answer.
Does
theregroupingchangethevalue  Doestheregroupingchange
value
of of
value
of
423?
Explainyourtheanswer.
423?Explain
Explainyouranswer.
423?
youranswer.
v. What changes and what remains unchanged
vii.v. Whatchangesand
andwhat unchanged
remains
after
achanges
numberis regrouped?
What
what
remainsv. Whatchangesandawhatremains
unchanged
unchanged
after
numberisregrouped?
after
a numberregrouped?
after
a numberis isregrouped?
Diagram 5
Diagram 5
Diagram
Diagram
5 5Counting Numbers More Than 100
Counting Numbers More Than 100
Counting
Numbers
More
Than
100
Counting
Numbers
More
Than
100

i.
Diagram 6 shows 422 pencils in bundles of 100,
10
and 1.66Base
on422
thispencils
example,how
to explain
i.i. Diagram
shows
422
pencils
inbundles
bundles
Diagram
shows
in
100,
i.
Diagram
6 shows
422
pencils
bundlesof of100,
to100,
childrenthe 1.
extensionofincountingby
10s to
of
10
and
Base
on
this
example,
10
and
1.
Base
on
this
example,howtoexplain10counting
and 1. Baseon
thisexample,howtoexplainbythe
100s?
to explain
toextension
childrenextensionof10s
tochildrenofcounting
tohow
childrenthe extension
ofthe
counting
byby
10s
to to
countingby
100s?
countingby
10s
tocountingby
100s?
counting by 100s?
 Explain why this example is important in
helping
your thispupilsto appreciate
Explainwhy
why
exampleisimportantinthe
inExplainwhy
this example
exampleimportant • Explain
thisis 10simportantextensionyour
fromcountingbyto counting
helping
pupilstoappreciatetheinhelping
yourstudents
toappreciate
helping
yourpupilstoappreciatetheby100s. from counting by 10s to counting
extension
extensionfrom counting
by 10sbyto10s
counting
theextensionfrom counting
to
by
100s.
bycounting
100s.
by 100s.

Diagram 6
Diagram 6
Diagram
Diagram
6 6ii.

Diagram 7 shows five lines of numbers arranged in some sequences. Fill in the missing

numbers.
Diagram
7
shows
five
lines
of
numbers
arrangedinsomesequences.
Fillin
missingii. ii. ii.
Diagram
7 7shows
fivelines
ofofnumbers
arrangedininsomesequences.
Fillmissing
Diagram
shows
five
lines
numbers
arrangedsomesequences.
Fillininthe
thethemissing
numbers.
numbers.
numbers.
Line ①
117
118
121
119
Line
Line
①①
Line ②

117
117
780

118
118
790

Line
Line
②②
Line ③

780
780
200

790
790

Line
Line
③③
Line ④Line
Line④④
Line ⑤
Line
Line
⑤⑤


200
200
650

121
121
820

119
119

400
400
400
750

650
650
854

852

854
854

852
852

840

820
820
600

700

600
600

700
700

750
750

846

Diagram 7

840
840

950
950
950

846
846

Diagram
Diagram
7 77
Diagram
In what ways is this task important in helping children understand larger numbers?

In
what
ways
is
this
task
importanthelpingchildrenunderstandlargernumbers?
•   InInwhat
ways
is
task
importanthelpingchildrenunderstandnumbers?
Createanothersimilartask
for yourpupils.
whatways
is thisthistask
importantinininhelpingchildrenunderstandlargernumbers?

• Create
Createanothersimilartask foryour pupils.
anothersimilartask
Create anothersimilartask forfor your
yourpupils.
students.

46 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram
Diagram88
iii.iii. Diagramshowspart
part
a numberlinescale.
with scale.
The number
665 iswith
markedwith anDiagram 88showsof aofnumber
line with
The number
665 is marked
an arrow.
arrow.


8 line with an arrow:
Mark each of the following numbers onDiagram
the number• Markof the658,each
676 and
699following numbers on the number line with an arrow:iii. Diagram 8 shows part of a number line with scale. The number 665 is marked with an arrow.



•



 704, 725, 737 and 743
 658,
699 numbers on the number line with an arrow:Markeach 676
of theandfollowing
What benefits will children acquire by doing this kind of task?
 658,676725,
and 737699 and 743
704,
 704, 725, 737 and 743

What benefits will children acquire by doing this kind of task?

What benefits will children acquire by doing this kind of task?
Diagram 9iv. Using the same number line, if 670 is replaced with 680 as shown in Diagram 9.
What is the new number indicated by thearrow?9Diagram
Diagram9Complete the new scale of this number line.
Using theif 670replacedwith 680
as680
shown
Diagram
9.
iv.iv. Using
the samesamenumber
numberline,
line,
but is670
is replaced
with
as in
shown
in Diagram
9.

What will be the new number indicated by each of the arrows in (ii)?
What is the new number indicated by the arrow?
•What is the new number indicated by the arrow?

•What will bethenew
newscalenumberindicatedby eachCompletetheof thisnumberline. of the arrows in (ii)?

•

What will be the new number indicated by each of the arrows in (iii)?

Complete the new scale of this number line.

Chapter 2: Numbers and Operations | 47

Task 2: Comparing Numbers
Task 2: Comparing Numbers

Task
Comparing Numbers
NumbersTask
2: 2:Comparingi. i.
Task 2: Comparing Numbers

Diagram 10
Diagram 10
Diagram 10

Diagram 10

Diagram 10

Diagram 11

Diagram 11

Diagram 11
11
Diagram
Diagram 11>
larger than>larger than>>larger than

or

or

<
<

smaller thansmaller than

Diagram 12

Diagram12
or 12
Diagramlarger than

or

<

<

smaller thansmaller than

Diagram 12
Diagram 12

Diagram 13

Diagram 13

Diagram 13

Diagram
Diagram 13
13Childrenoften
confusetheof values ofChildren often
confusewith thewith
valuesi. numbersChildrensuch
oftenasconfuse
withthe263.
values ofin 236
and
263.
such
as
in confuse
236
andi.numbersChildrenoften
withthe values ofnumbers such as in 236 and 263.
numbers
such
as
in
236
and
263.
 Mark each number with an arrow on the•  number
Mark
each
number
withanonarrowon theline inDiagram
10.
Mark eachnumber
with
an
arrow
the
 Markeach
numbernumberline
Diagram
10.an arrow on thenumberline
in in
Diagram
10.with Explain
why a number
line is helpful
numberline in Diagram
10. ini.
Childrenoftenchildren
confuse withthe values
helping
overcome
theirofin
Explainwhyanumberline
ishelpful• numbers
Explainwhyanumberline
is helpful in
such
as
in
236
and
263.
confusion
inreconigsing
valuesofhelpingchildrenovercome
their Explain
whyaovercomenumberline
is helpfulinhelpingchildren
theirnumbers.
confusionin reconigsing
valuesof confusion
helpingchildrenovercome
their
 in
Mark
each
number
with
an
arrow
on
the
recognisingvalues ofnumbers.
numbers.
ii. Diagram
11confusion
shows
numbers
in reconigsing
numberline inthe
Diagram
10.236 and 263values ofwrittenonaplace-value
chart.
ii. Diagram 11
shows the numbers 236 and 263
numbers.
ii. Diagram
shows
the
numbers
 Explainwhy
a numberline
is helpful236
in and 263
writtenon a11place-value
chart.
Betweena11
numberand a placeii. Diagramshows line
thenumbers
236 and 263
helpingchildrenovercome
theirwrittenon a place-value
chart.
value
chart,
which
do youis easieron aainplace-value
chart.
confusionreconigsing
valuesof written
Betweennumberlinethinkanda placetocomparevalues236
andis263?
numbers.
value
chart,thewhich
doof
you
think
easier• Explain
Betweennumber line and a place value
yourareasons.
totheof 236line
and263?
 compare
Betweenanumbers
numbera placeii. Diagram
11
shows
thevalues
andand263
chart,
which
do you 236
think
is easier toExplainyourreasons.
value
chart,
which
do
you
think
is
easierwrittenonaplace-value
chart.
 Whatabout thecomparingof 253comparevaluestheofvalues
236 ofand263? Explain
to compare236
and 510?
Explain the
yourvalues
answerwith and 263?
What reasons.
about
theandvaluesof 253your suitable
Betweena comparing
numberline
a placediagrams.
Explainyour reasons.
and 510?
answerwith
valuechart, Explain
which do your
you thinkis easieriii. Diagram
12 shows
two symbols
used263?
tosuitable
diagrams.
tocomparethevaluesof236
and•represent
What
about
comparing253
 Whatabout
comparingthe
the values
values ofof253
inequality.
Explainreasons.
iii. Diagram
12yourshows
twosymbolsused
to suitable
and and510?
Explainyouranswerwith
510?
Explainyouranswerwithinequality.
represent
Writesuitabletherelationshipbetween
theofpairdiagrams.
diagrams.
What
about
comparing
the values
253
of numbers 236 and 263 by using theand510?
Explainyouranswerwith symbols
Write therelationshipbetweentheiii.
Diagram
12
shows
two
symbols‘larger than’ or ‘smaller than’.pair used tosuitable
diagrams.
of numbers
236 and two
263 bysymbols
using the used toiii. Diagram
12
shows
represent
inequality.
(Note.
Whenandsymbols‘largerlearning
than’symbols
or positive‘smallerthan’.
iii. represent
Diagram
12
shows
two
used
to
inequality.
negative numbers in upper grades, therepresent
inequality.
Whenlearning
positiveand the pair(Note.
Write
therelationshipbetweenterms“greaterthan”
and “lessthan”
arenegative
numbersin
upper
grades,used.)
ofthe
numbers236
and263 bythe
using
theof
•  Write
relationshipbetween
the
pair
Write
the
relationshipbetweenthe pairtermssymbols“greaterthan”
and“lessorthan”
are than’.
‘largerthan’‘smallernumbers
236
and
263
using
the
iv. Diagram
13 shows
twonumbers
writtenona symbolsofnumbers236
and263 bybyusing
theused.)
place-value
chart.
symbols‘largerthan’orlearning‘smallerthan’.
‘largerthan’or‘smallerthan’.
(Note.
When
positive andiv. Diagram 13 shows two numbers written on a(Note.
When >numbers
learning
in upperand
grades, thechart.
 place-valueUsethenegative
symbolsor < to positive
represent
thenegative
numbers
in uppergrades,the(Note.
When
learning
andnegativeterms“greaterthan”
andpair“lessarenumerical
relationshipfor positive
theof than”
termsused.)“greaterthan”
and “less than”
are “greaternumbers
inand
upper
the terms712.
 numbers
Use
the483symbols> orgrades,< to represent
theused.)
numerical
relationshipforpair of
than”
and13
“less
than”
aretheused.)
iv.
Diagramshows
written on a
Sketchtheshows
positions
ofnumbers
thetwotwonumbers
numbersoniv.  Diagram
13
two
written onanumbers
483
and
712.
chart.line to show theira place-value
suitablechart.
numberplace-value
iv. Diagram
13relationship.
shows
two numbers written on a numerical
Sketch thepositions of the two numbers on
place-value
chart.
a suitableUse
thenumbersymbolsor< to represent
the
 Use
the symbols> orline<>
to torepresent
the
show
their
 Whena place-value
chart
toof pair ofnumerical
relationshipfor istheusedpair thenumerical
relationshipfor
numerical
relationship.
483
and
712,
which
numbers
483symbols
and
712.
483
and
• compareUsenumbersthe> 712.
or place< to value
represent theshould
we
look
at?
Why?
 Whena place-value
chart isused
to pair ofnumerical
relationship
for
the
 
Sketchthepositionsof thetwo numbers
on
483
and
712,
which
place
value
What
about
527
and
561?
compareSketchthepositions
of
the
two
numbersonnumbers483
and712.to show theirashould
suitablenumberline
we lookat?
Why?
asuitablenumber
line
to
show
their

What
about
309
and
604?
numerical relationship.
 Whatabout 527
and 561?
numerical
relationship.
What about
and 942?of the two numbers• Sketchthe 382
positions Whenaplace-valuechart
is used toabout 309number
and604?lineon Whata suitable
to show theircompare483 aandplace-value
712, which place Whenchartvalue
is used toWhat
382Why?
and 942?
numerical
relationship.
shouldwe about
lookat?
compare 483 and 712, which place value•

shouldwe527
lookat?561?
Why?
 Whatabout
and

When a place-value chart is used tocompare 483 and 712, which place value

What
about
382
and309942?
should
we
look
at?
Why?
 Whatabout
and 604?
 Whatabout 309
and527604?
 Whatabout
and 561?

 What about 382 and 942?

What about 527 and 561?
What about 309 and 604?
What about 382 and 942?

48 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 4.2:
Standards
4.2:
Extendingnumbers
using base-ten system up to 10 000
i. Visualise the 10 000 by using thousands, hundreds, ten and one as units
numbersbase-tenofsystem
up toto10introduce
000
ii. Extending
Extend theorderusing
sequence
numbers
more than 10 000 in relation to the size ofones, tens, hundreds and thousandsi.
Visualise the 10 000 by using thousands, hundreds, ten and one as unitsiii. ii.Use Extend
numberline with appropriate scale to show size of numbers and relative size of numbers
the order sequence of numbers to introduce more than 10 000 in relation to the size of ones,
whiletens,
focusing
on andthe thousands
scale
hundreds
iii.

Use number line with appropriate scale to show size of numbers and relative size of numbers while

focusingfor
on thescale
Sample Tasks
Understanding
the Standards
SampleTasks for Large
Understanding
the Standards
Task
1: Counting
Numbers
Task 1: Counting Large Numbers

Diagram 1i.

i.

Diagram 1

Diagram 1 shows some base-ten blocks used to represent a number.
Jess describe the number as follows:

Two pupils, Roy andDiagram 1 shows some base-ten blocks used to represent a number. Two students, Roy
and Jess
describetheas 4follows:
Roy: “Thereare 23setsnumberof hundreds,
sets of tens, and 6 ones.”
aresets
2 setsofofhundreds,
thousands, 43 sets
4 setsof tens and 6 ones.”
Roy: Jess:“There“Thereare 23setsofofhundreds,
tens, and6 ones.” “There
Are bothanswers
correct? Justify
explanation.
Jess:arethe2 setsof thousands,
3 setsyourof hundreds,
4 sets of tens and 6 ones.”••

What is a better way of representing 23 sets of hundreds? Why is it better?

Are both the answers correct? Justify your explanation.

What is a better way of representing 23 sets of hundreds? Why is it better?

1000s
Diagram 2

Jess: “There are 2 sets of thousands, 3 sets of hundreds, 4 sets of tens and 6 ones.”


Chapter 2: Numbers and Operations | 49
Are both the answers correct? Justify your explanation.

What is a better way of representing 23 sets of hundreds? Why is it better?

1000s
Diagram 2

Diagram 2
ii.
ii.

Diagram 2 shows 10 sets of hundreds being changed to 1 set of thousands. Why is this change
Diagram 2 shows 10 sets of hundreds being changed to 1 set of thousands. Why is thisimportant?
change important?
iii. How do you explain the new place value of 1000s using the place-value chart?
iii. How do you explain the new place value of 1000s using the place-value chart?

Diagram33
Diagram
iv.iv.

Diagram 33 shows
shows acountedthe number
Diagrama numbernumber represented
representedby
bysomesomebase-ten
base-tenblocks.
blocks.AApupil
student
counted
the
of
unit
cubes
as
follows:
number of unit cubes as follows:“One thousand, two thousand, three thousand, four thousand, five thousand ... five
“One thousand, two thousand, three thousand, four thousand, five thousand ... five thousand one
thousandone hundred,
five thousand
two hundred,
five thousand
three four
hundred,five five
hundred,five thousand
two hundred,
five thousand
three hundred,
five thousand
hundred,thousandfour
hundred,five
thousandfive
hundred,five
thousandsix
hundred,five
thousand five hundred, five thousand six hundred, five thousand seven hundred ... five thousand
thousandseven
hundred...
five
thousandseven
hundred
and
ten,
five
thousandseven
seven hundred and ten, five thousand seven hundred and twenty, five thousand seven hundred andand twenty,
thousandand thirty,
thousandsevenfive
hundredthirty,hundred
five thousand
sevenfive
hundredand seven
forty, hundred
five thousand
seven five
hundredand fifty,
thousand
forty, andfivesixty!”
thousand seven hundred and fifty, five thousand seven hundred and sixty!”
sevenandhundred•

Suggest aa moremore efficientefficient way
way ofof counting
counting thethe number
number ofof one-cubes.
one-cubes.
Suggest50 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Writing Large Numbers

k 2: Writing Large Numbers

Diagram
Diagram 43
Diagrama 4number
shows arepresented
number represented
by somebase-tenblocks.
student
wrote
Diagrami. 3 showsby somebase-tenblocks.
A Apupilwrote
thethenumber as
numberas
3000124.
3000124.
•

What was the error in the number written by the student?

•

What were the possible causes of the error?



What was the error in the number written by the pupil?

What werepossiblecauses of chart
the error?
• theExplain
how a place-value
can help the student to correct the error.



Explain how a place-value chart can help the pupil to correct the error.
13 sets of hundreds,
4 sets of tens and6 sets of ones.

13 sets of hundreds,
4 sets of Diagram
tens and
5ii.

6 sets of ones.

When given a task to write the number represented by the expressions in Diagram 5, a
student wrote it as 1300406.
•

Diagram 4

What was the error of the number written by the student?

When given• a task
to write the number represented by the expressions in Diagram 4, a pupil wro
What were the possible causes of the error?
it as 1300406.
•

Explain how the error can be corrected with a place-value chart.

What was the error of the number written by the pupil?

What were the possible causes of the error?



Explain how the error can be corrected with a place-value chart.

What skills need to be developed in order for students to acquire the fluency of reading an
writing large numbers?

•

What skills need to be developed in order for students to acquire the fluency of reading
and writing large numbers?

Chapter 2: Numbers and Operations | 51

Task 3: Using Number Line to Show Large Number
Task 3: Using Number Line to Show Large Number
Task 3: Using Number Line to Show Large Number↑↑

Diagram
Diagram4 4Diagram 6
i.

i. Diagram
6 shows
aanumberwithlarge
numbersi. Diagram
Diagram
4 shows
number
line
with
large
numbers
4 shows
a numberlinelinewithlarge
numbers
 Describe
Describe
thescalescaleof thenumberline.
ofnumberline.
 •Describe
the thescaleof thenumberline.
 WriteWrite
downthethenumbernumber marked
byarrow.
down
markedbythe
thearrow.
 •Writedown
the number
markedby thearrow.
ii.

Can 23 460 be marked accurately on the number line? Why or why not?

iii.

Explain how this number line can be use to compare the numbers 23 460 and 34 520.

23 460be marked
accurately
on thenumber
line?
Why
why
not?
ii. ii.CanCan
23 460be marked
accurately
on thenumber
line?
Why
or or
why
not?
Explainnumberbe useto compare
numbers
23 46034 520.
iii. iii.Explain
howhow
thisthis
numberlinelinecancanbe useto compare
thethenumbers
23 460andand34 520.

↑
a
iv.
iv.

↑
a

↑
b

↑
b
Diagram
Diagram 57

Diagram 7 shows another number line
without
Diagram
5 a scale.

iv.

Diagram 5 shows another number line without a scale.

•
What should numbers a and b be so that 23 460 can be marked accurately on the numbershouldnumbersa andline
b bewithout
so that a23scale.
460 can be marked accurately on the number
Diagram5Whatshows
anothernumberline?
line?

 •Whatshouldnumbers locationa and b beso 460that on23 this460 number
can be markedMarkthe accurateof 23
line. accurately on the number Mark the accurate location of 23 460 on this number line.
line?
v. v. Sketch
a numberscaletotocompare
comparevaluesof460
23 460Sketcha numberline
linewithwith appropriate
appropriate scale
thethevaluesof 23
and and 97 680.
 Markthe accurate location of 23 460 on this number line.
97 680.
•Explain how the comparison can be done.
v. Sketch a number
line thewithcomparison
appropriate
to compare the values of 23 460 and
Explain how
canscalebe done.
97 680.
•Is there a neccesity to compare large numbers with detailed scales such as ones?
 Is there a neccesity to compare large numbers with detailed scales such as ones? Explain
Explain your reasons.
yourreasons.
 Explainhow
the comparison can be done.

vi. vi.How
to draw
a lineof ofnumbers
586 needneedtotobe
bemarkedmarkedaccurately
accurately
How
to draw
a linenumberswith
withappropriate
appropriate scale
scale if
if 76
76 586
 toIsshow
thereits
a its
neccesity
to compare
large
numbers with detailed scales such as ones? Explain
relative
76705?
to show
relativeposition
position with
with 76705?
your reasons.
vi.

What
theimportant
importantelements
elements you
you needdrawing
andandreadingvaluesofvii.vii.What
arearetheneedtotoconsider
considerwhen
when
drawingreadingvaluesHow tonumbers
draw a lineof numberswith appropriate scale if 76 586 need to be marked accurately
line?
of numberson
onaanumber
numberline?
to show its relative position with 76 705?

vii. What are the important elements you need to consider when drawing and reading values ofnumbers on a number line?

52 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 4.3:

Standards 4.3:

Extending numbers using based-ten system up to 1 000 000
Extending numbers using based-ten system up to 1 000 000
i. Extend numbers up to 1 000 000 and learn the representation of the place value for grouping
3-digitnumeralsystemup tomillioni. every
Extendnumbers
up to
1 000 000
andlearn the representation of the place value for grouping every
3-digitnumeralsystemup tomillion of 3-digit numeral system such as thousand as a unit andii. Write
large
numbersusing
grouping
Standards
ii. compare
Write4.3:
large
numbers
using grouping
numbers
in relation
to it of 3-digit numeral system such as thousand as a unit andcomparenumbers
in relation
it larger and smaller based on comparison of place valuesiii. Develop
numbersense
suchtoas
Extendingnumbersusing
based-tensystemto smaller
1 000 000iii.
Developnumbersense
such aslargerup
based on comparison of place values throughthrough visualisation of relative size
ofand
numbers
visualisation of relative size of numbersi.
Extend numbers up to 1 000 000 and learn the representation of the place value for grouping every
SampleTasks
for
UnderstandingtheStandards
3-digitnumeralsystem up to million
SampleTasks
for UnderstandingtheStandards
ii.
Write large numbers using grouping of 3-digit numeral system such as thousand as a unit andcomparein relationPlacetoit ValueTask1:1:
BeyondnumberstheTask
Beyondthe Thousands
Thousands
Place
Value
iii.
Develop number sense such as larger and smaller based on comparison of place values throughvisualisation of relative size of numbers
The
3-DigitThe
3-DigitNumeral
Numeral System
SystemSample Tasks for Understanding the Standards③ the Thousands②
Task 1: Beyond
Place①Value Thousands

Hundreds

Tens

Ones

Diagram
Diagram 11 Hundreds
Thousands

Tens

Ones

The 3-Digit Numeral System③②①

Diagram11shows
shows aa base-ten
value
chart
up to
i. i. Diagram
base-tenplace
place
value
chart
upthousands.
to thousands.

•How many ones are regrouped to form tens?

How many ones are regrouped to form tens?

How many tens are regrouped to form hundreds?
Diagram 1
How many
regrouped
formhundreds?
thousands?
•  Howmanyhundreds
tens areareregrouped
totoform
i. Diagramshows bea base-ten
value
chart
thousands. Explain your reasons.
 What1should
the placeplacevalues①, ②
andup③,torespectively?
ii.

•  Howmany hundreds are regrouped to form thousands?
How many ones are regrouped to form tens?

Write the number “two hundred fifty two thousand three hundred and sixty seven” on the place-

How
many
tens
regrouped
to form
chart
in Diagram1.
• valueWhatshouldbeare
theplace values①hundreds?
, ② and ③ , respectively? Explain your reasons.


ii.

How
many
hundreds
areyouregrouped
to form
thousands?
Which
place
values dosee a repeatof ones,tens and hundreds?

Write the number “two hundred fifty two thousand three hundred and sixty seven” on the Whatshould be theplacetens
values①, ② and ③, respectively? Explain your reasons.
This repeatationof ones,place-value
chart in Diagram
1. and hundreds enable us form a 3-digit numeral system as shownii.

by the grouped place-value chart in Diagram 2.
Write the number “two hundred fifty two thousand three hundred and sixty seven” on the place• value
Which
place
valuesyoucansee
repeatof ones,tensreadandlarge
hundreds?
in Diagram
1. dochartExplain
how
this
chart
beausedto help
childrennumbers in the English
language.
 Which place values do you see a repeat of ones, tens and hundreds?

•This repeatation of ones, tens and hundreds enable us to form a 3-digit numeral
This repeatationof ones,and hundreds
enablechart
us form
3-digit numeral
system as shown
systemas shown
by thetensgroupedplace-value
in aDiagram
2.
Group of Millions
Group of Thousands
Group of Onesby the grouped place-value chart in Diagram 2.

②Hundreds
Tens to help
Oneschildren
Hundreds
Tens numbers
Ones in the① how Ones
Explainthis chart
can be usedread large

 Explain how this chart can be used to help children read large numbers in the English
English language.
language.
Group of Millions

②

①

Ones

Group of Thousands
Hundreds Diagram
Tens 2

Diagram 22
Diagram

Ones

Group of Ones
Hundreds

Tens

Ones

Chapter 2: Numbers and Operations | 53

iii.

iv.
v.

Write the number 756 238 on the grouped place-value chart in Diagram 2.
•

How would you guide children to determine the value of each digit?

•

Read and write the number in words.

What should be the place values ① and ② for the group of millions? Explain your reasons.

Write and read the following numbers in words.
•

63 502 000•

438 167 064

iii. Write the number 756 238 on the grouped place-value chart in Diagram 2.
vi. Given twolarge numbers: 43 816 764 and 438 167 064.


How would you guide children to determine the value of each digit?

Read and write the number in words.

•

Which number is larger?

•

What different methods can you used to compare the numbers?

iv. What should be the place values ① and ② for the group of millions? Explain your reasons.
v.

Write and read the following numbers in words.


63 502 000



438 167 064

Which number is larger?

Which method do you think is most efficient in comparing large numbers? Justify your answer.

•

Explainthe different methods.
vi. Given two large numbers: 43 816 764 and 438 167 064.

•

Whichmethod
do methods
you thinkisusedmostefficientin comparing large numbers? Justify your
What different
can you
to compare
the numbers?
answer Explain the different methods.

The Structureof Base-Ten Place Value System
The Structure of Base-Ten Place Value System
Name

Place Value

Pattern

Power of 10

Hundred millions

100 000 000

10?

Ten millions

10 000 000

10?

Millions

1 000 000

10?

Hundred thousands

100 000

10?

Ten thousands

10 000

10?

Thousands

1 000

10?

Hundreds

100

Tens

10

Ones

1

X10
X10

102
101
10?

Diagram 3i.

i.

Diagram 3 shows
the based-ten place values from ones to hundred millions which explain What pattern do you notice in the place values?
the comprehensive
of ofthebase-tennumeration
system. Study the structure Complete structure
the missing power
10 indicatedby each question
mark.
carefully.
ii.

ii.

Diagram 3

Diagram 3 shows the based-ten place values from ones to hundred millions which explain thecomprehensive structure of the base-ten numeration system. Study the structure carefully.

What do you think a base-five structure for place values should be?
Explain the structure.

•

What pattern do you notice in the place values?

•

Complete the missing power of 10 indicated by each question mark.

What do you think a base-five structure for place values should be?
•

Explain the structure.

54 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 5: Producing Vertical Form Addition and Subtraction and Acquiring Fluency ofStandard
Topic
Topic
5: Producing
5:Algorithms
Producing
VerticalVerticalForm Form
Addition
Additionand Subtractionand Subtraction
and Acquiringand Acquiring
Fluency
Fluencyof Standardof Standard
Algorithms
Algorithms
Standards 5.1:

Standards
Standards
5.1: 5.1:

Thinking about the easier ways for addition and subtraction and producing vertical form algorithms
i. Think about easier ways of addition or subtraction situations by using models with base-ten
blocksmeaningfully
forof
representing
base-tensystemi.
i.Think
Think
about
about
easiereasier
ways
ways
additionof additionor subtraction
or subtraction
situations
situations
by using
by using
modelsmodelswith base-ten
with base-ten
blocksblocks
ii. meaningfully
Produceandelaborate
efficient
ways
and
identify
meaningfully
for representing
for representing
base-tenbase-tensystemsystem the standard algorithms in relation to baseten system
withii.
ii.Produce
Produceand elaborate
andappreciationelaborate
efficient
efficientways ways
and identify
and identify
the standard
the standard
algorithms
algorithms
in relationin relation
to base-ten
to base-ten
systemwith
withappreciationiii. system
Explaintheappreciation
algorithms
of borrowing and carrying with regrouping of base-ten models
iii. iv. iii.
Explain
Explain
thefluency
algorithms
the algorithms
of borrowing
ofandborrowing
and carrying
and carrying
with regrouping
with regroupingof base-tenof base-tenmodelsmodelsAcquirein addition
subtraction
algorithms
Thinking
Thinking
about about
the easierthe easierways ways
for addition
for additionand subtraction
and subtraction
and producing
and producing
verticalverticalform algorithms
form algorithms

iv.

iv.
AcquireAcquirefluency
fluencyin additionin additionand subtraction
and subtraction
algorithms
algorithms

Sample Tasks for Understanding the Standards

SampleSampleTasksTasks
for Understanding
for Understanding
the Standards
the Standards

Task
1: 1:
Addition
ininVertical
Form
Task
Task
1: Addition
Additionin Vertical
Vertical
Form
Form
Addition Without Regrouping

Addition
Addition
Without
Without
Regrouping
Regrouping

Candy
Problem:
Rizalhas
has
13
candies,
Ricky
2424
candies.
CandyCandy
Problem:
Problem:
Rizal
Rizal
13has
candies,
13 candies,andand
Ricky
and
Ricky
hashas24has
candies.
candies.
How
manycandiesare
there
altogether?
How many
How many
candies
candies
are there
are there
altogether?
altogether?

Rizal’sRizal’s
candies
candiesi.i.

Ricky’s
Ricky’s
candies
candies

Diagram
Diagram
Diagram
1 11

Diagramshows
candy
problem.
Diagrams
2(a)
to
2(c)
show
three
stagesdeveloping
Diagrami. Diagram
11shows
1 shows
aacandya candy
problem.
problem.
Diagrams
Diagrams2(a)to2(a)
2(c)toshow
2(c)
show
three
three
stagesstagesof ofdevelopingof developing
analgorithm,
algorithm,
notarranged
arrangedorder,
solving
Candy
Problem.
anan algorithm,
notnot arranged
in inorder,in order,forforsolvingforsolvingthethe
Candy
the
Candy
Problem.
Problem.
Stage①
StageStage① ①

Grouping
tensandandonesones
inverticalform.
Grouping
Grouping
tens
tens
and
in
ones
verticalin vertical
form.
form.

Diagram 2(a)Diagram
Diagram2(a) 2(a)Chapter 2: Numbers and Operations | 55

Stage ②

StageGrouping②
tens and ones in horizontal form.
Grouping tens and ones in horizontal form.
3 sets of 10
3 sets of 10

Diagram
Diagram2(b)2(b)StageStage③ ③Grouping by tens, then add in sets of tens and sets of ones.
Grouping by tens, then add in sets of tens and sets of ones.

3 setsof 10
30.33setssetsofof 11 andand 44 sets
13 13
+ 24
3 sets
of 10
is is
30.
setsofof1 1isis7.7.So,So,+ =2437= 37

Diagram2(c)2(c)Diagram Howsequancethe
thestages
stages toto help
help yourthethevertical
formformof ofHow
will will
youyousequance
yourstudents
studentslearn
learn
vertical
addition?
Explainyourreasons.
addition?
Explainyourreasons.

•

Student A:

Student B:

Diagram 3
ii.

ii.

Diagram 3 shows how two students, A and B, adding two numbers in the vertical form.
After comparing their answers, Student
B told3Student A that the answer 65 was wrong.
DiagramHe then tried to explain his reason that could be seen in the following conversation.

Diagram 3 shows how two students, A and B, adding two numbers in the vertical form.
Student B: “You cannot write 4 on top of 2.”
After comparing their answers, Student B told Student A that the answer 65 was wrong. He
A: “Whynot?”then triedStudent
to explain
his reason
that could be seen in the following conversation.
Student B: “Because the two numbers on the right side must be on the same line, not the left

Student B: “Youside.”cannot write 4 on top of 2.”
Student A: “Why not?” WhatB:
does
Studentthe
B mean?
Student“Because
two numbers on the right side must be on the same line, not the left
side.”
 Why mustthe numbers be aligned on the right side?
•
•
•



How can base-ten blocks be used to help Student A correct his mistake

What does Student B mean?
Why must the numbers be aligned on the right side?
How can base-ten blocks be used to help Student A correct his mistake56 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Student A:

Student B:

Student C:

Diagram 4
iii.

Three students, A, B and C, calculated 12 + 25 in different ways as shown in Diagram 4.
•

Which of the ways do you think will be most difficult for your students? Explain yourreasons.

•

Which of the ways do you think can help your student understand vertical form addition
better?

•Student A explained how he got the answer as follows:“2 plus 5 is 7 and then 1 plus 2 is 3. So, the answer is 37”.


•

•

iv.

Did Student A represent each digit of 25 and 12 with the correct value in his
explanation? Explain your reasons.

Student B used some based-ten blocks to explain how he got the answer as follows:“I use 1 bar of 10s and 2 units of 1s, then another 2 bars of 10s and 5 units of 1s.
Altogether, I have 3 bars of 10s and 7 units of 1s which is 37.”


Draw a diagram of base-ten blocks to show the process explained by Student B.

How does the arrangement of the base-ten blocks help to explain the algorithmn
in vertical form?

Compare Student B’s way and Student C’s way of adding the two numbers.
What do you think is the difference in the two students’ thinking?

Is the difference significant? Explain your reasons.

Teachers in different part of the world practise slightly different ways of writing the verticalform of addition. For examples, while there are teachers who use an answer bar as in
Diagram 3, there are also teachers who do not use the answer bar as in Diagram 4.
•

Which of these practices do you prefer? Explain your reasons.

Chapter 2: Numbers and Operations | 57

Addition in Vertical Form with Regrouping

Diagram 5i.

Diagram 5 shows the sequencial steps in adding two numbers with regrouping in verticalform and pictures of base-ten blocks illustrating each of the steps. The pictures of base-ten
blocks are not arranged in the correct sequence.
•

Draw a straight line to match each step with the correct picture of base-ten blocks.

•

Explain how the regrouping of 10 ones as 1 tens is represented in the written vertical
form.

•

In what way can the use of base-ten blocks help children understand the regrouping
process when using addition in the vertical form?
[Note. The term “carrying” is also used to mean regrouping involving addition in somecountries.]ii.

A teacher discovered several ways her students
performed addition in vertical form as shown in
Diagram 6.
•

Diagram 6iii.

Another student showed an alternative way to
perform 28 + 27 as shown in Diagram 7.
•

Diagram 7

What are the advantages of using these
vertical forms to help children understandplace values and addition?

Comapring this way with those in Diagram 6,
do you consider it easier and more efficient?
Explain your reasons.
[Note. The line below the final answer is used toindicate the end of the calculation. Depending
on a country’s common practices, the line may
not be used.]58 | Mathematics Challenges for Classroom Practices at the Lower Primary Leveliv.

Diagram 8

As in the writing of the answer bar, teachers fromdifferent part of the world also practice different
ways to represent the ‘regrouped number’ whenusing the vertical form of addition. Diagram 8shows three such practices.
•

Which of these ways do you practice in yourclassroom?

•

Mathematically, is there any difference invalue between the three ways of marking the
regrouped ‘1’? Explain your reasons.

•

Can you say it is wrong if the method do notconform to your local practice? Why or why
not?

Chapter 2: Numbers and Operations | 59

Task 2: Subtraction in Vertical Form
Subtraction Without Regroupingi.

Diagram 9 and Diagram 10 show how two teachers, A and B, used base-ten blocks toperform 48 – 13.

Diagram 9

Diagram 10•

Compare the two ways of take away used by the teachers. Are there any conceptual
difference in the meaning of subtraction as shown by the two ways? Why or why not?
•Diagram 11 shows 48 – 13 in the vertical form.


Diagram 11

Which way of take away will you follow if you want tointroduce this vertical form subtraction to your students?
Explain your reasons.

60 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram 12
ii.

Two teachers, A and B, used base-10 blocks to show vertical form of subtraction 57 – 23 in
two different ways as shown in Diagram 12.
•

What meaning of subtraction does each of the ways show?
Diagram
Diagram
1212

DiscusstheA practicalityofeach blocks
way
intolearning
the procedureof subtraction?
ii.• ii. Two
Twoteachers,
teachers,
A
and
and
B,B,
used
used
base-10base-10
blockstoshowshowverticalvertical
form
form
of of
subtraction
subtraction
5757
––
2323
in in
two
two
different
different
ways
ways
asas
shown
shown
in in
Diagram
Diagram
12.12.

•  Whichofmeaning
the ways
is more does
efficient?
Explainyourreasons.
 WhatWhatmeaningof ofsubtraction
subtraction
does
each
each
of of
thethe
ways
ways
show?
show?
  Discuss
Discuss
thethepracticality
practicality
of of
each
each
way
way
in in
learning
learning
thetheprocedure
procedure
of ofsubtraction?
subtraction?

Subtraction With Regrouping  Which
Which
of of
thethe
ways
ways
is is
more
more
efficient?
efficient?Explain
Explainyour
your
reasons.
reasons.

i.

Diagram 13 shows an example of subtraction in vertical form that
requires regrouping.
Subtraction
Subtraction
With
With
Regrouping
Regroupingi. i. Diagram
Diagram
1313
shows
shows
anan
exampleexampleof ofsubtraction
subtraction
in inverticalvertical
•

form
form
that
that
requiresrequiresregrouping.
regrouping.

Use base-ten blocks to perform the subtraction.

  UseUsebase-tenbase-tenblocksblocksto toperform
performthethesubtraction.
subtraction.

•Diagram 13

tentenblocks.
blocks.

Think and show
different
algorithms
in verticalform
toverticalrepresent  Think
Thinkandandshow
showthethedifferent
different
algorithms
algorithms
in invertical
form
to to
represent
represent
thethesubtraction.
subtraction.
the subtraction.form

Diagram
Diagram
1313

ii.


Explain
Explainhow
how
thetheregrouping
regrouping
is is
done
done
using
using
basebaseExplain how
theregroupingisdone using
base-tenblocks.

ii. ii. Explain
Explainwhy
whyalgorithm
algorithmis isneccesary
neccesaryto tohelp
helpchildren
childrendodo

Explain why algorithm
is neccesary to help children do calculation?
calculation?
calculation?

Student
Student
A:A:

Student
Student
B:B:

1313
minus
minus
9 is
9 is
4. 4.

Diagram
Diagram
Diagram
1414
14

1010
minus
minus
9 is
9 is
1; 1;
1 plus
1 plus
3 is
3 is
4. 4.

iii.Diagram
Diagram
14
shows
shows
two
two
students’students’
ways
ways
of of
doing
subtraction
subtraction
in inverticalvertical
form
form
with
with
regrouping.
regrouping.
iii. iii.
Diagram
1414showstwostudents’ways
ofdoing
doing
subtraction
in verticalformwith regrouping.
  Explain
Explainthethedifferent
different
ways
ways
used
used
byby
thethestudents.
students.

•  Explainthedifferent
ways
used
by
thestudents?
students.
 Which
Which
way
way
dodo
you
you
think
think
is is
easiereasierforforyour
yourstudents?Explain
Explainyour
your
reasons.
reasons.
•

Which way do you think is easier for your students? Explain your reasons.

Chapter 2: Numbers and Operations | 61

iv.

Diagram
15 shows
howhow
the thesubtraction
iv. Diagram
15 shows
subtraction
136 – 136
57 is
done
by
using
base-tenblocksblocksin– 57 is done by using base-ten
the verticalform.
in the vertical form.
••

Diagram 15

Diagram 15

 Explainof thesubtraction
Explaineacheachstep stepof thesubtraction
with appropriate
writtenverticalformwith appropriate
writtenverticalform
of of
subtraction.
subtraction.
 What should you emphasise when
What using
shouldtheseyou emphasise
when tousing
concrete blocks
teachthesewritten
concreteblockstoteachwrittenvertical form to your students?
vertical form to your students?

62 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelv.

Children could easily get confused about written algorithm with abstract symbols. The

v.verticalChildrencould
get confused
writtenalgorithm withas
abstract
symbols.
The concrete
verticalformof easily
subtraction
is an about
abstract
representationcomparedto theform of subtraction is an abstract representation as compare to the concrete representation using
representation using base-ten blocks. Hence, special effort is needed to help children
base-ten blocks. Hence, special effort is needed to help children connect the two representations
connect
the fortwosubtraction
representations
especiallyinvolvingespeciallyregrouping.for subtraction involving regrouping.
Concrete Representation (in pictorial form)

Abstract Representation

Step ①

Step ②

Step ③

Step ④

Step ⑤

Step ⑥aDiagram 16

Diagram 16

Diagram 16 shows 6 steps in calculating 204 – 39 using the base-ten blocks. The vertical formof the first
steps
been
written for you.
Diagram
16 two
shows
6 had
steps
in calculating
204 – 39 using the base-ten blocks. The vertical
form
of
the
first
two
steps
had
been
writtenyou. steps.
 Complete the vertical form for each of theforremaining••When making connection between concrete and abstract representations, what do you think

Completethe verticalform
for each of the remaining steps.
are the important
pointsto consider?

When making connection between concrete and abstract representations, what do
you think are the important points to consider?

Chapter 2: Numbers and Operations | 63

Task 3: Common Errors of Addition and Subtraction

Diagram 17i.

Diagram 17 shows the work of four students with errors.
•

Identify the error in each case.

•

Explain the possible causes that children make such errors.

ii.

Explain how the error in each case can be corrected.

iii.

Collect a list of common errors of your students when doing addition and subtraction in
the vertical form. Analyse the errors.
•

Identify the difficulties faced by your students.

•

Suggest ways to help your students overcome the difficulties.

64 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 5.2:
Standards 5.2:
Standards
5.2: of standard algorithms for addition and subtraction and extend it up to 4-digit
Acquiring fluency
Acquiring
fluency of standard algorithms for addition and subtraction and extend it up to 4-digit
numbers
Acquiringnumbersfluency of standard algorithms for addition and subtraction and extend it up to 4-digit
numbers
theverticalverticalform
form
additionsubtraction
throughthe extensionof numbers
i. i. Extend the
addition
andandsubtraction
throughthe extensionof numbers
and andappreciate the explanation using base-ten block modeli. ii. Develop
Extend theverticalform addition
andupsubtraction
through theof numbers
andandextension
simple casefor 4- digit
numbersii.
fluencyof everyextension
to 3-digit numbers
digit
numbersappreciate the explanation using base-ten block modelii. Sample
Developfluencyof every extension
up to 3-digit numbers and simple case for 4- digit numbers
SampleTasks
forforUnderstandingtheStandards
Tasks
UnderstandingtheStandards
SampleTasks forUnderstanding
the Standards
Task
1:1:
Vertical
Form
Addition
Task
Vertical
Form
Additionforfor3-Digit
3-DigitNumbers
Numbers
Task 1: Vertical Form Addition for 3-Digit Numbers
Diagram1 shows
1 shows
additionsof 2-digit
i. i. Diagram
two two
additionsof 2-digit①②
numberswhich
whichthethe students
students havenumbers
have learnt
learntbefore
i. Diagramshows
two additions of 2-digit①②
additionof13-digit
numbers.
beforeadditionof 3-digit
numbers.
numberswhich
thestudents
havesuitablelearnt before Which of the task is mostto be
additionof
3-digitnumbers.
used toextend2-digit tonumber•Whichof thetaskaddition
is mostofsuitable Whichof thetask isnumbers.
most suitable to be
additionof 3-digit
beto used
to extendaddition of 2-digit
used to extend addition of 2-digit number
numberto additionof 3-digit
numbers?
to additionof 3-digit
numbers.
Diagram 1
Diagram 1
Diagram 1
ii. Diagram 2 shows five addition tasks involving

3-digit numbers.
Diagram22 shows
tasks
involvingii. ii. Diagramshows five
fiveaddition
additiontasks
3-digitnumbers.
involving
3-digitnumbers.
 How
do youstructure the sequence of
these four tasks to enhance the acquisition
How
youstructure
structure
sequenceof•  How
dodoyouthethesequenceof computational
fluency?
Justify your
these
four
tasks
to
enhancetheacquisition
ofdecision.
these five tasks to enhance theof computational
fluency? fluency?
Justify youracquisitionof computational
decision.
decision.
 Justify
Whichyourof thesetasks is most suitable to be
used to extend addition of 3-digit numbers
Whichthesetasks
tasksisismostmostyoursuitable
to be
to 4-digit
numbers?
Justify
decision.
•  Whichofofthesesuitableusedtoextendadditionof
3-digitnumbers
to be used to extend addition of 3-digit
to 4-digit numbers? Justify your decision.
numbers to 4-digit numbers? Justify
your decision.

Diagram 2
Diagram 2
Diagram 2

Chapter 2: Numbers and Operations | 65

Task
VerticalForm
Subtraction
3-Digit
NumbersTask
2: 2:
VerticalForm
Subtraction
forfor3-Digit
Numbers--

Diagram
3 shows
foursets
setsofof grid boxes
i. i. Diagram
3 shows
fourboxesfor
Createa asubtraction
subtractiontask
task
for subtraction. Create
involving
3-digitnumbers
in eachset ofinvolving
two two
3-digitnumbers
for eachof thegrid boxwith the following conditions:followingconditions:

•

-

•

-

OneOnetime regrouping
from thetime regrouping
from the hundreds
hundreds
place.
place.
One time regrouping from the tens
One time regrouping from the tens
place.
place.

The 3 digits in minuend are larger thanThe the3 digitsin minuend are larger thansubtrahend.
the subtrahend.

Two times regrouping from the tens•
Two andtimes
regrouping
from the tens andhundreds
place.
hundreds place.
ii. How do you sequence these tasks tocomputational
fluency?
Howdevelop
do you sequencethesetasks to develop•

Diagram 3
3
Diagramii.

computational fluency?

66 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 3: Representing a Subtracton Using Base-Ten Blocks
Task 3: Representing A Subtracton Using Base-Ten Blocks
125 – 86i.

i.

Diagram 4 shows six steps in using base-ten

Diagramsix steps
in –
using
blocks4toshows
calculate
125
86. baseten blocks to calculate 125 – 86.

which
stepdiddidthe
theregrouping
regrouping process • At At
which
step
processout thefor thetime?Explain the
carrycarryout forfirstfirsttime?
Explainthe regrouping.
regrouping.


ii.

ii.

iii.

At which step did the regrouping• processAt which
regrouping
process
carry stepout fordid
thethesecondtime?
Explainregrouping.
carrytheout
for the second time? Explain
the regrouping.

Write the corresponding vertical form
subtraction of each step.

Write

the

corresponding

verticalWhy
is it importantfor yourstudents to write
subtractionof eachstep.
the corresponding vertical form at each step?

formiii. [Note.
WhyTheis itterm
importantforisyourstudents
to write“borrowing”
also usedtomean
involvingverticalsubtraction
in at eachthe regrouping
corresponding
form
some
countries.]
step?
[Note. The term “borrowing” is also used tomean regrouping involving subtraction insome countries.]

Diagram 4
4
DiagramChapter 2: Numbers and Operations | 67

Standards
Standards
5.3: 5.3:
Standards
Standards
5.3:
5.3: number sense for estimation and using calculator judiciously for addition and subtraction
Developing
Developing
numbersense for estimation and using calculator judiciously for addition and subtractioni.

Develop number sense for mental arithmetic with estimation for addition or subtraction ofiii.

Appreciate the use of calculator in the case of large numbers for finding the total and the differenceStandards
5.3: sense
Developing
Developing
numbernumbersense forfor estimation
estimation andand usingusing calculator
calculator judiciously
judiciously forfor addition
addition andand subtraction
subtractioni.
Developnumber sense for mental arithmetic with estimation for addition or subtraction of numbers
numbersii. Developing
Identify
necessary
situations
touse
calculators
judiciously
in
real
life
numbersense for situations
estimation andusingcalculators
calculator judiciously
for addition
andsubtraction
Identify
necessary
to usejudiciously
in reali.i. ii.Develop
Developnumbernumbersense
senseforformental
mentalarithmeticarithmetic
withwithestimation
estimation
for
foraddition
additionororlifesubtraction
subtractionof
ofnumbers
numbersiii. iii.Appreciate
the useofuse
calculator
in the caseoflarge
numbers
for numbers
finding the fortotalfinding
and thethe
difference
Appreciate
theofcalculator
in
the
case
of
large
total and theii.ii. i. Identify
Identify
necessarynecessary
situations
situations
toto useuse
calculators
calculators
judiciously
judiciously
inin realrealorlife
life
Developnumber sense
for mentalarithmetic
with estimation
for addition
subtraction of numbers
differenceiii.
iii. ii. Appreciate
Appreciate
the
the
use
use
of
of
calculator
calculator
in
in
the
the
case
case
of
of
large
large
numbers
numbersfor
for
finding
finding
the
the
total
total
and
and
the
the
difference
difference
Identifyfor
necessary
situations tothe
useStandards
calculators judiciously in real life
Sample Tasks
Understanding

SampleSample
Tasks
Tasks
for
for Understanding
Understanding
thethe Standards
Standards
SampleTasks
forofUnderstanding
the Standards
TaskSample
1: Making
Estimationin Addition
and Subtraction of Numbers
TasksSense
for Understanding
the Standards

Task
Task 1:
1: Making
Making Sense
Sense ofof Estimation
Estimation inin Addition
Addition andand Subtraction
Subtraction ofof Numbers
Numbers

Task1: 1:
MakingSense
of Estimation
inand
Addition
andofSubtractionof Numbers
Task
MakingSense
of Estimationin Addition
Subtraction
Numbers

16 + 19 + 18 =
19+
18 =
16
1616+++19
19
++ 18
18
==
Diagram 1

Diagram 1

Diagram111
Diagram
Diagram

398 + 140
①①398
+ 140

387
56 + 43①①②398
398+++140140② 387 + 56 + 43
③ 52 + 365 + 48②
② 387
387 ++ 56
56 ++ 4343
③ 52 + 365 + 48
④ 91 – 28③③ 52
52 ++ 365
365 ++ 48
48④ 91 – 28⑤ 256 – 137④④ 91
91 –– 2828
⑤ 256 – 137

Diagram 2⑤⑤ 256
256 –– 137
137
Diagram 2

Diagram222
Diagram
Diagramⓐ 3096 + 5218
ⓑ 6274 + 2726

ⓐ 3096 + 5218

5128++–5218
2835ⓐⓐⓒ30963096
5218ⓑ 6274 + 2726
5000 – 1782ⓑⓑⓓ62746274++ 27262726ⓒ 5128 – 2835
ⓔ 966 – 28ⓒⓒ 51285128 –– 2835
2835
ⓓ 5000 – 1782
Diagram 3ⓓⓓ 50005000 –– 1782
1782
ⓔ 966 – 28ⓔⓔ 966
966 –– 28
28
Diagram3 3
Diagram
Diagram
Diagram 33

i. In real life, computational estimation can produce
answers that are close enough for making goodIn
In realreal
life, computational
computational
estimation
estimationcancan produce
producei. i.i.In real
life, life,
computational
estimation
can produce
decisions
without finding
the exact
calculation.
i. answers
Inanswers
real life,
computational
estimation
producethat thatareenoughenoughfor making
goodanswers
thatcloseareare close
close
enoughforforcanmakingmakinggood
goodanswers
thatarewithout
close
enoughmakingdecisions without
decisionsfinding
the forexact
calculation.
decisionsdecisions
without
without
finding
finding
the
the
exact
exactgood
calculation.
calculation.
 Whatknowledge is neccesary forfinding
the backgroundexact calculation.
childrento makeestimation
for the Whatbackgroundknowledge
is neccesary
for addition
children
What
Whattobackgroundbackgroundknowledgeknowledgeisis neccesary
neccesary forformake estimation
for the addition
task inDiagram1?
childrenchildren
to
to1?
make
make estimation
estimationforfor thetheaddition• taskWhatbackgroundknowledge
is addition
neccesary for
in Diagramtask
task inin Diagram
Diagram
1?estimation for the addition task in
to make1?
 children
A child estimatedthe addition sum to be larger A childestimatedtheadditionsum to be larger
Diagram1? less than 60.
than30,less
but
than
30,
but
than
60.
 AAchild
child estimatedestimated thethe addition
addition sum
sum toto be
be larger
largerthan
than 30,
30, but
but less
less than
than 60.
60.
How
did childthe the
child
come
up
with
• A
child
estimatedadditionsum
be thelarger than
How
did
the
come
up
with
theto
answer?
answer?
30,
but
less
than
60.

 How
How did
did thethe childchild comecome up
up withwith thetheanswer?
answer?

 2How
didfivetheexpressions
child comeinvolving
up with the answer?
ii. ii.Diagramshows
Diagram
2 shows
five expressions involving
addition and subtraction of numbers
addition and subtraction of numbers

Diagram
Diagram
22 shows
shows
five expressions
expressions
involving
involvingii. ii.ii.Diagram
2 shows
five fiveexpressions
involving
addition and addition
A child roundedup the two
numbers in
additionandandsubtraction
subtractionofofnumbers
numbers
subtractionofnumbers
 A child rounded up the two numbers in

expression ① and estimated the sum to be
500.expression
Explain ①
howchild didthethe
andtheestimatedsum to be
Achildchild
childroundedroundedrounded up
up
the
two
numbers
numbers expressioninin• estimation.
AAthethetwotwonumbers
500.
Explain up
howthechild did intheexpression
expression①① andandtheestimated
estimatedthesum
toExplainbebe①
and estimatedsum tothe
be sum500.tohowestimation.

500.
500. using
Explain
Explainverticalhow
how form,thethe find
child
child
did thetheWithout
an didthe childdidthethe estimation.
estimatedvalue for each of the remaining
estimation.
estimation.
epressions
Without② using
to ⑤. the vertical form, find an

estimatedvalue forverticaleach ofform,
the remaining•  Without
estimated
Without
Withoutusing
using
usingthethethe verticalvertical form,
form,findfind
findanananepressions②
to⑤.
valueExplainyourstrategy
used
in
the
for eachof theepressions
estimated
estimated
value
value
for
for remaining
each
each ofof thetheremainingremaining② to ⑤.
estimation of each task in Diagram 2?

epressions
epressions ②
②to
to⑤.
⑤.
Explain your
yourstrategy
strategyused
usedin estimationthe Explainin theofiii. Diagram 3 shows
someofexpressionsof Diagram
large
estimationeach task in2?
Explain
Explainyour
your
strategy
strategy
used
used
in
in
the
the
each
task
in
Diagram2?
numbers involving addition and subtraction.
estimation
estimation ofof eacheach task
task inin Diagram
Diagram 2?
2?
Diagramshowssome
some expressions
expressionsof large
large numbers
Which 3of3shows
theseexpressions
will you ofiii. iii.Diagram
numbers
additiontoandencourageinvolving
young children
usesubtraction.
mentaladditionandsubtraction.
iii.
iii.involving
Diagram
Diagram
33 shows
shows
some
some
expressions
expressions ofof large
largearithmatic even though the numbers appear
numbers
numbers
involving
involving
addition
addition andand subtraction.
subtraction.
large?
Explainyourreasons.
 Which of these expressions will you•Whichof these expressions will you encourage
encourage young children to use mental
 young
Which
Whichchildrenofof these
theseuse
expressions
expressions
will
will youyou
mentalarithmaticeven thougharithmatic
eventothough
the numbers
appearencourage
encourage
young
young
childrenchildren
to
to
use
use
mentalmentalthenumbersappearlarge? Explain your reasons.
large?
Explainyour reasons.
arithmaticarithmatic even
even thoughthough thethe numbers
numbers appear
appearlarge?
large? Explain
Explain your
your reasons.
reasons.

68 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Using Calculators at Right Situationsi.

Task 2: Using of Calculators at Right Situations

Calculation using algorithm can be challenging for some young children.
•i. Calculation using algorithm can be challenging for some young children.
Whattypes of situations will you recommend the use of calculators to help young
childrenlearn
additionand subtraction?
Give
to illustrate Whattypes
of situations
will you recommend
theexamples
use of calculators
to help youryounganswers.
children learn addition and subtraction? Give examples to illustrate your answers.

ii.

Diagram 4 shows some addition and subtraction problems in vertical form. Some of thedigits ii.in eachproblemspilled ink.
Diagram
4 shows are
somestained
addition withand subtraction
problems in vertical form. Some of thedigits in each problem are stained with spilled ink.

Problem ①

Problem ②

Problem ③

Problem ④

Diagram 4 4
Diagram

Find the missing digits in each problem.

If calculator is allowed to be used, do you think children will be able to solve these•

Find the missing digits in each problem.

•

problems by pressing the calculator keys without thinking carefully?
If calculator
is allowed to be used, do you think children will be able to solve these
problems
bycalculator
thinking
carefully?
 In whatpressing
ways canthechildrenbenefit keys
from without
the use of
calculator
when solving these•In what ways can children benefit from the use of calculator when solving these
problems?

problems?

Chapter 2: Numbers and Operations | 69

iii.

Diagram 5 shows some mixed addition and subtraction problems in vertical form with somemissing
numbers
as indicatedby each.and subtraction problems in vertical form with someiii. Diagram
5 showssome mixed
addition
missing numbers as indicated by each .



•
•



Problem ①

Problem ②

Problem ③

Problem ④

Find the missing numbers in each problem.

Diagram 5

Will you allow your students to use calculator when solving these problems?

Find the missing numbers in each problem.
Explain your reasons.

Will you allow your students to use calculator when solving these problems?
Explain your reasons.

70 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 6: Introducing Multiplication and Producing Multiplication Algorithm
Topic 6: Introducing
Multiplication and Producing Multiplication Algorithm
Standards
6.1:
Standards 6.1:
Introducing
multiplication and mastering multiplication table
i. Introducing
Understandthe meaningof multiplication
situations
multiplicationand mastering
multiplicationtable with models using the idea of addition anddistinguish from the common addition to find the total number
ii. i. Produce
multiplicationtableofinmultiplication
the case of counting
2 modelsand 5 witharray
diagrams,
pictures
Understandthe meaning
situations by
with
using
the ideaof additionand orfromextend
the common
find thetotal number
blockdistinguishmodels andit untiladdition
9 and to1 withappreciationof patterns
Producemultiplicationtable in thecase ofcountingby 2 and with
5 witharray diagrams, pictures oriii. ii. Develop
sense
for multiplication
throughmentalcalculation
fluencyblock
modelsandextend
it
until
9
and
1
with
appreciationofpatterns
iv. Use multiplication in daily life, differentiating the situations for multiplication in various situationsiii.
Develop sense for multiplication through mental calculation with fluency
understanding
any life,
numbercan be atheunit
for counting
in multiplication
iv.with Use
multiplicationthat
in dailydifferentiating
situations
for multiplicationin various situations
with understanding that any number can be a unit for counting in multiplication

Sample Tasks for Understanding the Standards
Sample Tasks for Understanding the Standards

Task 1: Situations of Multiplication
Task 1: Situations of Multiplication

Diagram 1
Diagram 1i.

Diagram 1 shows some plates of bananas, strawberries and doughnuts.
i•Diagram 1 shows some plates of bananas, strawberries and doughnuts.
Consider answering the following three sets of questions:
Consider answering the following three sets of questions:

Set
Set①
①
� How many plates of bananas are there?
How manyof bananas
are altogether?
there?
�o How
manyplates
bananasare thereoHow many bananas are there altogether?

oHow many strawberries are there altogether?

oHow many doughnuts are there altogether?

Set
②
Set ②
� How many plates of strawberries are there?
How manyof strawberries
are there?
�o How
manyplates
strawberries
are there
altogether?
Set ③
Set ③
� How many plates of doughnuts are there?
How manyof doughnuts
are altogether?
there?
�o How
manyplates
doughnuts
are there•Which of these sets of questions cannot be answered using a multiplication sentence?
Which of these sets of questions cannot be answered using a multiplication sentence?

Whyisis thethe setset ofin developing
the conceptof multiplication?
•  Whyofquestions
questionsimportant
importantin developing
the conceptof multiplication?



Chapter 2: Numbers and Operations | 71

Based on this task, how will you help your students to relate multiplication to addition?

Baseconsidermostmultiplication
important in developing
Basedononthe
thissituations,task, howidentify
will youtwo
helpideas
your youstudents
to relate
to addition?themeaning
multiplication
as yourepeatedaddition.
Based onofthis
task, how will
help yourstudents to relate multiplication to addition?
Base on the situations, identify two ideas you consider most important in developing theHow
doesthis task cultivate
the valueof seeking simpler ways to find answer inmeaningof multiplication
as repeatedaddition.
Base on the situations, identify two ideas you consider most important in developing
mathematics among your students?
the meaningof multiplication
repeatedHow
does thistask cultivateasthe
value addition.
of seeking simpler ways to find answer inmathematics among your students?
ii Whenasked
to findtotal
numbertheof doughnuts
in Diagram
1, fourstudents,
B, C and•How
does
thisthetask
cultivate
value of seeking
simplerways
to findA,answer
in D,
used mathematics
four different methods
to doso as shown in Diagram 2
among yourstudents?
ii When asked to find the total number of doughnuts in Diagram 1, four students, A, B, C and D,
used four different methods to do so as shown in Diagram 2
ii. When asked to find the total number of doughnuts in Diagram 1, four students, A, B, C andD, used four different methods to do so as shown in Diagram 2
Student A’s Method
Student B’s Method

•


•


Student A’s Method

Student B’s Method

Student C’s Method
3Student+ 3 + 3 C’s+ 3 Method+ 3 = 15
3 + 3 + 3 + 3 + 3 = 15
Student D’s Method
Student D’s Method

Diagram
2 2
Diagram

iii.
iii
iii

•


Diagram2
Explain how each student get the answerof 15.
Explain how each student get the answer of 15.

•

How does
does
skip
counting
connecttomultiplication?
How
skip
counting
connect
multiplication?
Explainhow
each
student
get theto
answerof 15.

 theHowdoes skip
counting 2connect
to multiplication?
Allsituations
in Diagram
represent
3 doughnuts in each plate, for 5 plates to make 15
doughnuts.
Different
countries
mayfollow
conventionsin writing
multiplication
All the situations in Diagram 2 represent 3different
doughnutsin each plate,for 5theplatesto make 15
sentence.
doughnuts. Different countries may follow different conventions in writing the multiplicationAll
the situations in Diagram 2 represent 3 doughnuts in each plate, for 5 plates to make 15
sentence.
Differentare
countries
may follow
different2conventions
multiplication•doughnuts.
In your country,
the situations
in Diagram
written as 5inx writing
3 = 15theor 3
x 5 = 15?
sentence.
Explainyouranswer.
 In your country, are the situations in Diagram 2 written as 5 x 3 = 15 or 3 x 5 = 15?

•Explain your answer.
In
yourdocountry,
are thethis
situations
written as 5inxwriting
3 = 15 theor 3multiplicationx 5 = 15?
How
you address
issue inof Diagram
different 2convention
Explainyouranswer.
sentenceinyourteaching?
How do you address this issue of different convention in writing the multiplication

sentence in your teaching?
How
doInyouof different
multiplication[Note.
thisaddress
book, 5this
x 3issuemeans
5 sets ofconvention
3 unless itiniswriting
statedthe
otherwise.
However,sentenceinyourteaching?
themethodofexpressing
multiplication
sentence
should
be
based
on
your
national[Note. In this book, 5 x 3 means 5 sets of 3 unless it is stated otherwise. However,thecurriculum.]method of expressing multiplication sentence should be based on your national[Note. curriculum.]
In this book, 5 x 3 means 5 sets of 3 unless it is stated otherwise. However, themethod of expressing multiplication sentence should be based on your national
curriculum.]72 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Problem
Problem
Problem
Problem
4 cm
4 cm4 cm
4 cm
A tape4 cm
long.
A tapeis
4 iscm4islong.
tape
cm
long.
AAtapeis 4 cmlong.
①
How
arepieces
3 ofpiecesof thein cm?
①①
How
longarelong
3 piecesthe thetape
in tapecm?cm?
How
long
are
tape
① How
long
are
3 3piecesofofthetapeinincm?
②
How
arepieces
5 ofpiecesof thein cm?
②②
How
longarelong
5 piecesthe thetape
in tapecm?cm?
How
long
are
tape
② How
long
are
5 5piecesofofthetapeinincm?
Diagram
Diagram
3 3 3Diagram
Diagram
Diagram
3 3

Rizal’s
MethodRizal’s
MethodRizal’s
MethodRizal’s
MethodRicky’s
MethodRicky’s
MethodRicky’s
MethodRicky’s
MethodDiagram
Diagram
4 4 4Diagram
Diagram44
Diagramiii.Diagram
Diagram
3 shows
a problem
involvingiii. iii.Diagram
3 shows
a problem
involving
3showsshows
problem
involving
Diagramaaproblem
involvingiv. iii.
Diagram
3 3several
shows
problem
involving
lengthsofseveralpiecesoftapes.
lengthsofpiecesof oftapes.
lengthsofseveralpiecestapes.
lengths
ofstudents,
severalpiecestapes.
lengthsof several
piecesof oftapes.
Two
Two
Rizal
and
Ricky,
TwoTwostudents,
Rizal
and
Ricky,
students, Rizal
Rizal andand Ricky,
Ricky,
Two students,
students,
Rizal
and
Ricky,
solvedquestionsolvedproblem①withwithdifferent
solvedproblem①①
withdifferent
solvedproblemdifferent
problem① with① solvedwithdifferent
methods
as different
showninmethods
asshown
inDiagrammethods
asshown
inDiagram4.
methods
asshownshownininDiagram
Diagram4.4. 4.
Diagram
4. as
methods
 Explain
Explaineach
method.
 Explaineach
method.
eachmethod.
 Explaineachmethod.
•Explain each method.
 SolveSolve
problem
②
using
each
 Solve
problem
②②
using
each
problemusing
each Solveproblem② usingeachmethod.
method.
•Solvequestion ② using eachmethod.
method.

Represent
thesituation
situationin method.
Representin
Representthe
thesituation  Represent
thesituation inin
problem②
a multiplication
problem② withawith
multiplication
problem②withwithmultiplication
problem②
aamultiplication•
Represent
thesituationin question
sentence.
sentence.
sentence.
② witha multiplication sentence.
sentence.
UsingUsing
yourmultiplication Using
yourmultiplication
yourmultiplication
 Using
yourmultiplication
sentenceasexample,sentenceasexample, Using
your assentence
as multiplication
example,sentence
example,explainthemeanings
explainthemeanings
ofsentence
example,explainexplainas
themeanings
meaningsof
of ofexplain
the
the
of
(a)
multiplier,(a)meanings
multiplier,(a)multiplier,(a)multiplier,(b)multiplicand,(b)multiplier,multiplicand,
andandand(b)multiplicand,(a)(b)
multiplicand,and(c)product.
(c) (c)product.
product.
(c) product.
(b) multiplicand, and WhichWhichmethodyou
prefer Which
methoddo doyoudo
preferfor for formethodyou
prefer Whichmethoddo toyou
preferfor
yourstudents
learn?
yourstudents
tolearn?
yourproduct.
studentstotolearn?
learn?
yourstudents(c)
Justify
yourchoice.
Justify
yourchoice.
Justify
yourchoice.
Justify
yourchoice.
Use
eachmethod
to
solve
the

Use
each
methodto prefersolve
thethe• Which
method
do
you
for
your
 Use
Use
eachmethod
methodsolve
each
totosolve
thefollowingproblem.
followingproblem.
students
to learn?
followingproblem.
followingproblem.
Justify
yourchoice.
“John
hasof green
ribbon
“John
has
18
cm18
of cm
green
ribbon“Johnhas
has18
18
cm
green
ribbon“Johncm
ofofgreenribbon6
cmofyellow
ribbon.
How
andand6 andcmofyellow
ribbon.
How
cmofofyellow
yellowribbon.
ribbon. HowHow
and many
66cm
times
the
length
of
•
Use
each
method
to
solve
the
many
times
the
lengthofthemany times
times thethe length
length ofof thethethemany
yellow
ribbonisequal
toyellow
ribbon
is
equal
to
the
followingproblem.
yellow
ribbon isis equal
equal toto thethetheyellow
ribbonlengthof
thegreen
ribbon?”lengthof
thegreen
ribbon?”lengthofofthe
thegreengreenribbon?”ribbon?”length“John
has
18
cm offuture
green
ribbonWhat
Whatlearningof Whatfuturelearningof andfuture learninglearning
 ofWhat
futureofofthis
6 cm
yellow
ribbon.
Howthismany
numberoperationisnumberoperationisnumber operation
operation isis this
thisnumbertimesproblem
thelengthoftheleadingyellow ribbon
problemyourleadingproblemleading your
your
problemleading
yourstudents
to?
Explainyouris equaltothelengthof
thegreen
students
to?
Explainyourstudents to?
to? Explain
Explain your
yourstudents
answer.
answer.
ribbon?”answer.
answer.



What future learning of number
operation is this problem leading
your students to? Explain youranswer.

Chapter 2: Numbers and Operations | 73

Task 2: Producing Multiplication Tables of 2 and 5

Diagram 5i.

Diagram 5 shows three sets of cherries, each set with two cherries. Students are taught toread the multiplicaton sentence as ‘3 times 2 is 6’.
••“3 times 2 is 6” (3x2=6) can also be read as “2 multiply by 3 equals 6”. In this case,multiplicand, multiplier, and product are three terms related to the multiplication
sentence.


Identify which number is the multiplicand, the multiplier and the product.



Explain the meaning of each term.

Concrete materials such as interlocking cubes can be used to help children understandmultiplication.
How do you use interlocking cubes to represent 3 x 2 = 6?

Draw the interlocking cubes that show 3 x 2 = 6.

Diagram 6ii.

In Diagram 6, the cherry sets are arranged starting from one set, and added a set to thenext row until the 9th row. The corresponding multiplication table 2 is also shown by theside of the arrangement of cherry.
•

As the number of sets increase by 1, what pattern do you notice in the total number ofcherries?

74 | Mathematics Challenges for Classroom Practices at the Lower Primary Level•

Explain how the pattern can help children find the answers for 10x2, 11x2 and 12x2?

•

What is the relationship of increasing the number of sets and the total number ofcherries?

Convention ①

Convention ②

Diagram 7iii.

Diagram 7 shows two different
conventions in writing the multiplicationtable of 5 which are practised in different
countries.
•

Which
convention
does
your
country’s national curriculum follow?

•

Will this different conventions cause
a confusion to your practice in theclassroom? Explain your answer.

iv.

Why are base numbers 2 and 5 usuallyusedfirst to introduce multiplication
before other numbers?

v.

A multiplication table is usually built from
1 to 9. Why is table 10 not included?

Chapter 2: Numbers and Operations | 75

TaskTask
3: Multiplication
and5 5
3: MultiplicationBeyond
Beyond Tables
Tables 22and
Task 3: Multiplication Beyond Tables 2 and 5

Diagram 6i.

i.

Diagram 8

Diagram 6 shows multiplication table 3 represented by interlocking cubes. Compare thiswithmultiplication
the arrangementof cherries
in diagramby
4. interlocking cubes. Compare thisDiagramdiagram
8 shows
table
3Diagram
represented
6
diagram withof cherries
in diagramcubes
6. different from the arrangement
In thewhatarrangement
way is the arrangementof interlockingof cherries?
i. Diagram
6 shows multiplication table 3 represented by interlocking cubes. Compare thisdiagramthe arrangement
arrangement ofof
cherries in diagram
4.
•
In whatwaywithis howthefromtothearrangementof Explain
this arrangement interlocking
of interlockingcubes
cubes different
can be usedshow
anothercherries?
multiplication
which is ofknows
as a multiplicationarray.
 In
what way issituation,
the arrangement
interlocking
cubes different
from the arrangementof cherries?
[Note. In this book, 5 x 3 means 5 rows by 3 columns unless it is stated otherwise.
•
Explainhowarrangement
cubescancanusedto basedshow
another
Oncethis
again,
method ofofexpressing
a multiplication
arrayshouldbeon Explainhowthisthearrangement
ofinterlocking
interlocking
cubes
bebe
used
to show
another
your situation,national
curriculum.]multiplication
which
is knows
asasaamultiplicationarray.
multiplication
situation,
which
is knows
multiplication array.
[Note. In this book, 5 x 3 means 5 rows by 3 columns unless it is stated otherwise.

[Note. In Once
this book,x method
3 means
5 rows bya 3
columns unless
it is statedotherwise.
again, 5theof expressing
multiplicationarray should
be basedon
yourthenational
curriculum.]
Once again,
methodof expressing a multiplication array should be based on yournational curriculum.]

Diagram 7ii.

ii.

Diagram 7 shows 5 rows or 4 columns of lorries.
Diagram
7 9
Diagram Write a multiplication expression to represent the array.

iii. Explain
how5theknowledgeof multiplicationarray can help children learn shapes andDiagram
9 showsrows
or 4 orcolumnsof lorries.
ii. Diagram
7 shows
5 rows4 columnsof lorries.

•

space in future grades.
 Write a multiplication expression to represent the array.

Write a multiplication expression to represent the array.

iii. Explain how the knowledge of multiplication array can help children learn shapes andspace in future grades.

76 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

iv.

iv. A teacher used an array of 9 by 4 lorries as shown in Diagram 8 and a rectangular card to
A teacheran arrayof 9 by the
4 lorries
as showntablein Diagram
10 and a rectangular card to
help hisused
students
developed
multiplication
4.
Ateacherteacher
used
array
9byby
4lorries
lorries
shown
inDiagram
8and
and
arectangular
rectangular
card
iv.
used
an
array
ofofof9the44lorries
asshown
ininDiagram
88andaarectangularcard
tototoiv.iv.A
Ateacher
used
anan
array
9by
asas
shown
Diagram
card
help
his
students
developed
multiplication
table
4.
help
his
students
developed
themultiplication
table
help
his
students
developed
themultiplication
table
4.4.4.
help
hisstudents
developed
themultiplication
table

iv. A teacher used an array of 9 by 4 lorries as shown in Diagram 8 and a rectangular card to
help his students developed the multiplication table 4.

Diagram
Diagram
88 8
Diagram

Diagram 8

Diagram 10

Diagram 8
He moved the rectangular card up and down and used appropriate questions to guideHe
rectangular
card
up
used
appropriatequestions
totoguide
Hemovedmoved
therectangular
cardupupand
anddown
downandandusedappropriatequestionsguide
He moved
thetherectangular
card
and
down
and
used
appropriatequestionsto guide
hisstudents
determine
the
basic
facts
for
multiplication
table
Diagram
9show
show
some
his
students
determine
the
basic
facts
for
multiplication
table
4.
Diagram99showsome
hisstudents
determine
the
basic
facts
for
multiplication
table
4.4.
Diagram
some
He
moved
the
rectangular
card
up
and
down
and
used
appropriatequestionsto guide
his students
determine
the
basic
facts
for
multiplication
table
4.
Diagram
11
shows
some
examplesofsuchfacts.
examplesofofdetermine
suchfacts.
examplessuchfacts.
hisstudents
the
basic
facts
for
multiplication
table
4.
Diagram
9
show
some
He
moved
the
rectangular
card
up
and
down
and
used
appropriatequestions
toguide
examples of such facts.

examplesof suchfacts.the basic facts for multiplication table 4. Diagram 9 show some
his students
determine
examples of such facts.

=
22x2x4x4=4==
33x3x4x4=4=

2x4=

2x4=

3x4=

3x4=

Diagram 11=
66x6x4x4=4=

Diagram
Diagram
99 9
Diagram••

6 x 4 =  the basic facts,
What questions will you ask not only to help your students determine
but also to develop their sense Diagram
for multiplication?
9
6x4=

What
questions
will
you
ask
not
only
help
your
students
determinethebasicfacts,
  Whatquestions
will
you
ask
not
only
tototohelp
yourstudents
determinethebasicfacts,
What
questions
will
you
ask
not
only
help
your
students
determinethebasicfacts,
butalso
develop
their
sense
for
multiplication?
but
also
tototodevelop
their
sense
for
multiplication?
but
also
develop
their
sense
for
multiplication?

Justify your choice of questions. Diagram 9

Whatquestions
willofofyou
ask not only to help your students determine the basic facts,
Justify
yourchoiceofquestions.
  Justify
yourchoicequestions.
Justify
yourchoicequestions.
but also to develop their sense for multiplication?
• Explain
how
this
task
can
help
your
students
learn
the
idea
of
area
infuture
futuregrades.
 Explain
Explain
how
this
task
can
help
your
students
learn
the
idea
area
infuture
grades.
how
this
task
can
help
your
students
learn
the
idea
ofofofarea
ininfuture
grades.
Explain
how
this
task
can
help
your
students
learn
the
idea
area
grades.
Justify
your choicequestions.
 Whatquestionswill ofyou
ask not only to help your students determine the basic facts,

also tohow
develop
their
forstudents
multiplication?
butExplain
this task
cansense
help yourlearn the idea of area in future grades.

Justify your choice of questions.



Explain how this task can help your students learn the idea of area in future grades.

Chapter 2: Numbers and Operations | 77v.

Diagram
12 showsa 9 rowcolumnarrays,a L-shaped
9v. Diagram
10 showsa 9 byrow9 by
9 column
array,
a L-shapedcard
cardandandan
anempty
empty 99 byby 9
multiplicationtable.
multiplication table.

Diagram
Diagram
12 10•

A teacher used the array together with the card to help children fill in the 9 by 9
 A teacher used the array together with the card to help children fill in the 9 by 9
multiplication table. Explain how this can be done.
multiplication table. Explain how this can be done.

•

Whattheareadvantages
and disadvantagesof this methodin helping
 are
What
the advantages
and disadvantagesof this methodof children
helping develop
childrenthe multiplicationtable?
develop the multiplication table?

78 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 4: Fluency in Multiplication Basic Facts
Patterns in Multiplication Tables

Diagram 13
Diagram 13 shows a chart with multiplication tables 1 to 9, leaving some products as blanks.
i.

Complete the table.

ii.

Explore number patterns in the chart.

iii.

iv.

•

What different patterns do you notice among the numbers?

•

Explain how and why each of the patterns is formed?

•

Is any of the patterns related to the commutative property of multiplication? Explain
your answer with examples.

•

Suggest how these patterns could help your students learn the basic facts ofmultiplication.

Look at the numbers along the diagonal from the top left to the bottom right and also the
diagonal from the top right to the bottom left. Explore these two lines of diagonal numbers.
•

What interesting patterns do you find in these two lines of numbers?

•

What about the difference between consequtive numbers?

•

Explain why each pattern exists.

Why is fluency with mental calculation important in developing sense for multiplication
among your students?

Chapter 2: Numbers and Operations | 79

PatternininRows
Rows
andColumns
PatternandColumns← Row 1

Pattern in Rows and Columns← Row 2
← Row 3

← Row 1
← Row 2
← Row 3

Diagram14
12
Diagram
Sum of Products in Multiplication Tables

Diagram 12

Sum of Products in Multiplication Tables

Sum of Products in Multiplication Tables← Row 5

Diagram 12 shows a complete chart withmultiplication
1 to 9. a complete chart with
Diagram tables
14 shows
multiplication
to29.and 3.
Study
the numberstables
in rows1 1,i.

Diagram 12 shows a complete chart withWhat
patterndoobserveabout theseii. i. Makea conjecture
about
relationshipmultiplicationtables
1 to
9.youthebetweenthreeof numbers.
threeanyrows
of rows
numbers?
Study the numbers in rows 1, 2 and 3.
 Is your conjecture always true?
i. ii.What
pattern
do you observe
about
Make
a conjecture
aboutthethese
relationship Justify
conjecture using athreerows ofyour
numbers?
between
any
three
rows
of
numbers.
property of multiplication.
ii. Make a conjecture about the relationship•Is yourconjecture
always true?
of numbers.
iii. between
Suggest any
howthreethe rows
relationshipcan help
students master basic facts ofyour•Is yourconjecture
alwaystrue?
Justify your conjecture using a
multiplication.
property
multiplication.
 Justify
your ofconjectureusing aproperty of multiplication.

iii. Suggest
the relationship
Diagram
13 shows how
a horizontal
box with threecan help
consecutive
10
and 12inhelpit.facts ofyour products
students
masterbasic
iii. Suggest
how
the 8,
relationshipcanAnotherverticalbox masterhas threeconsecutive
yourmultiplication.
students
basic
facts ofproducts
20, 24 and 28 in it.
multiplication.
i.

After studying the horizontal box
carefully, a student makes the following
Diagram
13 shows a horizontal box with threeconjecture:
Diagram products
15 shows
a andhorizontal
box withconsecutive
8, 10
12 in it.
“Theverticalsum of box
any rowofthreethree consecutive
consecutive
Anotherhas
threeconsecutive
products
8,
10
and
12 in
products
is andequal
the middle productproducts
20, 24
28 to
inbox
it.
it.multiply
Anotherverticalhas
threeconsecutiveby 3”.

i.

Diagram 13

Diagram 13
Diagram15← Row 5What pattern do you observe about theseStudy
in rows 1, 2 and 3.
three therowsnumbersof numbers?

products 20, 24 and 28 in it.

Afterstudying
thealways
horizontalbox
 Is theconjecturetrue?
carefully, a student makes the following
 After
How could
you thejustify
the conjecturei. conjecture:
studyinghorizontal
box carefully,
addingup theawithout
studentactually
makes thefollowing
conjecture:“The products?
sum of any row of three consecutive
products is equal to the middle product “TheIs theconjecture
to theof any extendable
row of threeconsecutive
multiplybysum
3”.
sum of any
of fiveconsecutive
products
is rowequal
to themiddle product Is
the conjecture
alwaysExplain
true? yourproducts,
respectively?
multiplyby 3”.
reasons.
 How could you justify the conjecture •without
Is theextendabletheaddingupto true?
theIs conjecture
theactuallyconjecture
alwayssum of any products in a vertical box
products?
such as 20, 24, and 28? Explain your
 •Is
the
conjecture
extendable
to theHow
could youjustify the
conjecture
reasons.
sum withoutof any row
of five consecutive
actually
adding up the
products,
Explainii. Diagram
14 respectively?
shows Column
5 of yourthe
products?
reasons.
products
in the chart. Another student
a conjecture
as extendable
follows:
makesIs theconjecture
to the•← Column 5
Diagram 14← Column 5
Diagram 14

Is the conjecture extendable to theofofanyinvertical“Thesum
sumsumallofproducts
products
in aofColumn5box
is
any row
five consecutive
such as 20, 24, and 28? Explain your9 x 25.”products, respectively? Explain yourreasons.
 Is thereasons.
student’s conjecture true?

ii.

Diagram
14 shows
Columnof the Other thanadding up
all the 5products• Column
Isin theconjecture
products
thechart.
Another
student
in5, how
could
youextendable
justify
the to the
makes
a sum
conjecture
follows: in a vertical box
student’s
conjecture?
of anyasproducts

20, 24, inand
28? Explain
your“The sumsuchof allasproducts
Column5 is
9 x 25.” reasons.
 Is the student’s conjecture true?
 Other than adding up all the products
in Column 5, how could you justify thestudent’s conjecture?

multiply by 3”.
Is the conjecture always true?

How could you justify the conjecture
without actually adding up the
products?

80 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram 13

Pattern in Rows and Columnsii.

Diagram 16 shows Column 5 of the products in the Is student
the conjecture
to thechart. Anothermakesextendable
a conjecture
as follows:
sum of any row of five consecutive
products, respectively? Explain your“The sum of all products in Column 5 is 9 x 25.”
reasons.

•••← Column 5iii.

 student’s
Is the conjecture
extendable
Is theconjecture
true? to thesum of any products in a vertical box
such as 20, 24, and 28? Explain your
Other thanadding up all the products in Column
reasons.

5, how could you justify the student’s conjecture?
ii.

Diagram 14 shows Column 5 of thein the chart.beAnotherstudent
Can products
the conjecture
extended
to othermakes Explaina conjecture
follows:
columns?
yourasreasons.
“The sum of all products in Column 5 is

9 x 25.”makes another conjecture as follows:
A third student Is the student’s conjecture true?

“The sum of all the 81 products is 25 x 81.”•

Diagram 14

Diagram 16•iv.

 Other than adding up all the products
in Column 5, how could you justify theIs the conjecturetrue?
student’s conjecture?

Justify the conjecture, without adding up all the
products.

With reference to the Common Core Regional
Learning Standards in Mathematics, in what way canthese tasks inculcate mathematical values among
your students?

Task 5: Multiplication in Daily Life

Diagram 17i.

Many things in grocery stores are packed in sets. Diagram 17 shows some grocery items
that come in packages.
•

The unit of counting for the package of oranges is 5.
What is the unit of counting for each of the remaining two items?

•

For each of the items, create a daily-life problem involving the item in a multiplication
situation.

•

Find other suitable items with different unit of counting. Create a dailly-life situation
involving the items.

Chapter 2: Numbers and Operations | 81

Standards6.2:
6.2:
Standards
Producingin inverticalform
andobtaining
fluency
Producingmultiplication
multiplication
verticalform
andobtaining
fluencyi.
Think about easier ways of multiplication in the case of numbers greater than 10 using arrayi. Think
aboutandeasierways
of multiplication in the case of numbers greater than 10 using array
diagrams
block
modelsblock models
ii. diagrams
Developand
multiplicationin vertical form using multiplication table, array, model, and base ten
ii. Develop
in vertical form using multiplication table, array, model, and base ten systemsystemmultiplication
with appreciation
withappreciationiii. Extend multiplication algorithm to 3-digit times 2-digit numbersiii. Extend multiplication algorithm to 3-digit times 2-digit numbers
iv. Obtain fluency for standard algorithm for multiplication
iv. Obtain fluency for standard algorithm for multiplication
Useestimation
estimation
withmultiplicationof tensor hundreds
in lifev.v. Usewithmultiplicationof tensor hundreds
in lifevi.
Comparethemultiplication
expressions
whichislarger,smalleror equivalentvi. Compare the multiplication expressions which is larger, smalleror equivalentvii.
Appreciate
the
use
of
calculator
sensibly
in
life
in
the
case
of
large
numbersvii. Appreciate the use of calculator sensibly in life in the case of large numbers

SampleTasks
TasksforforUnderstanding
UnderstandingtheStandards
SampletheStandards
Task
Numbers
GreaterThan
10 10
Task1:1:Multiplication
MultiplicationInvolving
Involving
Numbers
Greater
Than
Multiplicationbecomes
becomesdifficult
difficultfor
formany
manychildren
childrenwhen
whenititinvolves
involvesnumbers
numbers greater
greater than
than 10,10,i. i. Multiplicationsuchasas15
15x x8.8.
such Suggest and explain some easier ways to calculate 15 x 8.
•
Suggest and explain some easier ways to calculate 15 x 8.
ii. Diagram 1 shows 14 sets of interlocking cubes.
ii. Diagram 1 shows 14 sets of interlocking cubes.

Diagram 1
Diagram 1
Three students, A, B and C, used different strategies to find the total number of the cubes as
Three students, A, B and C, used different strategies to find the total number of the cubes
shown in Diagram 2.
as shown in Diagram 2.
Student A’s Strategy:

16

40
Diagram 2
Student B’s Strategy:

28

28

40

82 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

16

Student B’s Strategy:28

28Student C’s Strategy:

20

36
Diagram2
Diagram
2 (Contd.)

•
•
•
•
•


Explainthe strategy
of eachstudent.
Explainthe strategy
of eachstudent.

Write a mathematics sentence to represent each strategy.
Write a mathematics sentence to represent each strategy.
What property of multiplication was used in the students’ strategies?

What
propertyhow
of multiplication
was beusedin thestudents’ Explain
the property canused
to simplify
thestrategies?
multiplication 45 x 12 in future
grades.
Explain how the property can be used to simplify the multiplication 45 x 12 in future Which strategy will you encourage your students to use? Justify your answer.
grades.
Which strategy will you encourage your students to use? Justify your answer.

Task 2: Multiplication in Vertical Form
Diagram 3 shows
the 2:multiplication
3 x 23 done
ChapterNumbers and Operations| 83
inthe verticalform.the multiplication 3 x 23 done
Diagram
3 shows
in the vertical form.
Task 2: Multiplication in Vertical Form
i. Explain how the distributive property of
Task 2: Multiplication in Vertical Form
is usedto perform
i. multiplication
Explain how the
distributive
property theofmultiplication.
multiplication is used to perform the
Diagram
3 shows the multiplication33x x2323
done in the vertical form.
Diagram 3 shows the multiplication
multiplication. donein the vertical form.ii. Teacher A used the array situation as shown
i. Explain how the distributive
propertyof arraymultiplicationistoinDiagram
4 totheexplaintheverticalformofTeacher
Aproperty
usedsituation
asusedshown
i. Explain how ii.
the distributiveofperformthemultiplication.
multiplication,
whereasanotherteacher,
B,in
Diagram4
toexplaintheverticalformofmultiplication is used to perform the
Diagram 3
explained
it using
the anotherrepeatedaddition
multiplication.
multiplication,
whereasteacher,
B,
Diagram 3ii. Teacher A used the situation
asitshown
Diagram
5. addition
array situation
as in
shown
in Diagram
4 toexplainedusing
therepeatedii. Teacher A used the array situation as shown
situationasshown
in Diagram
5.
explainthe verticalform
multiplication,
whereasanotherin Diagram
4 to explain
the ofverticalform ofit usingWhichsituation
thinksituation
is easier as
toteacher,B, explainedtherepeatedaddition
multiplication,
whereasanotherteacher,B, do you
Diagram 3
explaintheverticalformofmultiplication
explained
it
using
the
repeatedaddition
Whichsituation
do
you
think
is
easiertoshown in Diagram 5.
Diagram 3
situation as shown in Diagram
5. students?
toyourreasons.
explainthe verticalExplain
form ofyour
multiplication
to your students? Explain your reasons.
 Which situation do you think is easier to
Teacher A’s explanation:
explain the vertical form of multiplication
Teacher A’s explanation:
to your students? Explain your reasons.
Teacher A’s explanation:

Diagram 4

Diagram 4
Diagram 4

Diagram 4

Teacher B’s explanation:
Teacher B’s explanation:

Teacher B’s explanation:

Diagram 5

Diagram 55
Diagram
Diagram 5•

Which situation do you think is easier to explain the vertical form of multiplication to
your students? Explain your reasons.

84 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task
3: Multiplication
Multiplication
2-Digit
Numberby 2-Digit
Numberand Beyond
Task 3:of of
2-Digit
Numberby 2-Digit
Numberand Beyond

Task 3: Multiplication of 2-Digit Number by 2-Digit Number and Beyond
Task 3: Multiplication of 2-Digit Number by 2-Digit Number and Beyond

Three teachers,
teachers, A,
different
methods
to perform
i. i. Three
A, BB andand C,
C, taught
taughttheir
theirstudents
studentsthreethree
different
methods
to perform
i. Three teachers, A, B and C, taught their students three different methods to perform
multiplication ininvertical
form
as as
shown
in Diagram6.
multiplication
vertical
form
shown
in
Diagram6.
multiplication in vertical form as shown in Diagram 6.

i.

Three teachers, A, B and C, taught their students three different methods to perform
multiplication in vertical form as shown in Diagram 6.

Diagram 6•

Diagram 6
Diagram 6Whichmethoddodo
you
think
is easierfor yourstudents
to learn?
your reasons.
Which method
you
think
is easierfor
yourstudents
to learn?
ExplainExplain
your reasons.
Diagram6
Which method do you think is easier foryour students
to learn? Explain your reasons.

ii. ii. AAstudent
calculated1313
x 24
using
a methodthatdifferent
was different
his teacher’s
method.
student calculated
24
using
a methodthat wasfrom hisfrom
teacher’s
method. The
Whichmethoddo 13
youxx thinkis easierfor yourto learn?
Explainyourmethod.
reasons.
ii.  A student
calculated
24 using
a methodthat students
was different
from histeacher’sThe
student’s
methodis shown
in Diagram
7.
The
student’s
methodisshowninDiagram7.
student’s method is shown in Diagram 7.

ii.

A student calculated 13 x 24 using a method that was different from his teacher’s method. The
student’s method is shown in Diagram 7.

Diagram 7
Diagram 77
Diagram Explain the student’s method.
 Explain the student’s method.
 How can the student’s method help to Diagram
explain eachof the teachers’ methods in Diagram 6?
7 of
can the student’s
help to explain eachthe teachers’ methods in Diagram 6?
• How
Explainstudent’smethod
method.
  Explain
thestudent’s
method.
Why methods in diagram 6 are usually taught to students instead of this method?
 Why methods in diagram 6 are usually taught to students instead of this method?
How
can
the
student’s
toeachexplaineachof theteachers’
methods• How
canarethethestudent’s
methodhelp to help
explainof theteachers’
methods
in Diagram
6? in
What
similarities
andmethod
differences
betweenthese
methods?
 What are the similarities and differences between these methods?
Diagram 6?
 Why methods in diagram 6 are usually taught to students instead of this method?
iii. Anotherstudent
calculated
13
x 24
using base-ten
blocksasshown in Diagramof8.this method?
What
the similarities
anddifferences
betweenthese
methods?
Whyaremethods
in diagram
6 aretaughttoas
studentsiii.•Anotherstudent
calculated
13
x 24
usingusuallybase-tenblocksshown ininstead
Diagram 8.

•

What are the similarities and differences between these methods?

iii. Another student calculated 13 x 24 using base-ten blocks as shown in Diagram 8.

iii.

Another student calculated 13 x 24 using base-ten blocks as shown in Diagram 8.

Diagram 8
Diagram 8







Explain the method.
Explain the method.



Diagram88of the teachers’ methods in Diagram 6?
How can this base-ten blocks help to explain
each
Diagram
How can this base-ten blocks help to explain each of the teachers’ methods in Diagram 6?
the method.
•Explain
Explainthe method.
How can this base-ten blocks help to explain each of the teachers’ methods in Diagram 6?

•

How can these base-ten blocks help to explain each of the teachers’ methods inDiagram 6?

Chapter 2: Numbers and Operations | 85

iv. iv.
Diagrams
9 shows
three
different
methods
to calculate
25 x2534.
Diagrams
9 shows
three
different
methods
to calculate
x 34.
Method ① : Using Table
25 x 34
x

20
5

30

600
150

4

80
20

Method ② : Using Base-10 Blocks

Method ③ : Using Block Diagram

DiagramDiagram
9
9•

areareto introduce
these
three
methods
to yourstudents,
howhow
will will
you yousequenceIf you
If youto introduce
these
three
methods
to yourstudents,
sequence thethe orderorder ofof teaching?
teaching? Justify your decision.

86 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Three
students
calculated
x 235
by threedifferent
methods
as shown
in Diagram
5 by three different
v.v. Threemethods
students
as calculated
shown
in Diagram
48 48
x 235
10.
by threedifferent
methods
as shown
in Diagram
10. 10.
Student A’s Method:

Student A’s Method:

Student C’s
B’s Method:

Diagram 10

Student C’s Method:

Diagram
1010
Diagramdisadvantages of •eachWhatof arethearethemethods?
advantages
andanddisadvantagesof eachof theofmethods?
Whattheadvantages
disadvantagesof eachthe methods?

be used to explain
 Student
How can
B’sStudent
and Student
A’s method
C’s methods?
be used to explain Student B’s and Student C’s methods?
•
How can Student A’s method be used to explain Student B’s and Student C’s methods?

Chapter 2: Numbers and Operations | 87

Task 4:Task
ProblemSolvingSolving
With Multiplication
4: Problem
With Multiplication
Task 4: Problem Solving With Multiplicationi.

Diagram
11 shows
a problem
involving
multiplying
a 3-digit
numberwith awith
2-digitnumber.
i. Diagram
11 shows
a problem
involving
multiplying
a 3-digit
numbera 2-digit
number.
Diagram 11 shows a problem involving multiplying a 3-digit number with a 2-digit number.

i.

Diagram11
11
Diagram

Diagram
11 number is at least a 4-digit number.
• product
The productof a 3-digit
numbera 2-digit
Theof a 3-digit
numberwith awith
2-digitnumberis at least a 4-digit number.

The
productofa 3-digit
numbera 2-digit
is atleast a 4-digit number.
 Dowith thestatement?
Why Whyornumberwhy you
Doagree
you
agree
with
the with
statement?
or not?
whynot?

 Do you agree with the statement? Why or why not?
What
3-digitand and2-digitnumbers,when
multiplied
together
willwillgivea aproduct•What
3-digit
2-digitnumbers,when
multiplied
together
give
productofof5-digit
5-digitnumbers?
Justify
your youranswers.
numbers?
Justify
answers.
 What 3-digit and 2-digit numbers, when multiplied together will give a product of 5-digit
numbers? Justify your answers.
ii. Diagram
12 shows
two multiplication
problems
in vertical
formformwithwithsomemissing
numbersasii. Diagram
12 shows
two multiplication
problems
in verticalsomemissing
numbers
indicated
by
each
.
as indicated
bytwo
each
.
ii. Diagram
12 shows
multiplication
problems in vertical form with some missing numbers as
indicated by each .
Diagram 12

Diagram
12
Diagram12
Find the missing numbers in each problem.
Use calculator
to help you solve the problems.
Find •the missing
numbers innumbers
each problem.
calculator
help you solve
the problems.
in promote
eachUseproblem.
Usetocalculator
to help
solve thethe
Does theFind
usetheof missing
calculator inhibit oryour thinking
processes
whenyou
solvingproblems.
problems? Explain your reasons.
Does the use of calculator inhibit or promote your thinking processes when solving the
problems?
Explain
your•Does
the useof reasons.
calculator inhibit or promote your thinking processes when solving the

problems? Explain your reasons.

88 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelii. Ben was given four digit cards:
iii. Ben
Benwaswasgiven
givenfour
fourdigit
digitcards:
cards:
ii.
ii. Ben was given four digit cards:

2
2
2

3
3
3

4
4
4

6
6
6

He used the four cards to make two multiplications by interchanging the digits as follows:
He
Heusedusedthethefourfourcards
cardstotomake
maketwo
twomultiplications
multiplicationsby
byinterchanging
interchangingthe
thedigits
digitsas
asfollows:
follows:
He used the four cards to makethe digits as follows:
3 2twox multiplications
4 6 and 2 by
3 interchanging
x 6 4 .
3 2 x 4 6 and 2 3 x 6 4 .
3 2 x 4 6 and 2 3 x 6 4 .
 Work out the answers for the two multiplications.
• Work
Work out
out thethe answers
answers forfor thethe two
two multiplications.
multiplications.
 Workout theanswersresults
for thedotwomultiplications.
 Whatinteresting
you
get?
What
interesting
results
do you• 
Whatinteresting
results
do youget?get?
 Try
Whatinteresting
results do you
get?

with
other combinationsof thefour cards to see if you can find similar results.

Try
with
other
combinationsofthe Try with other combinations of thefour
fourcards
cardstotosee
seeififyou
youcancanfindfindsimilar
similarresults.
results.
 Try with other combinations of the four cards to see if you can find similar results.

•


•




Diagram
Diagram 13
13
Diagram 13
Diagram
13 formUseininthetheverticalin Diagram
13 to
you youanalyseUsethethestructure
structureofofmultiplication
multiplication
verticalform
in Diagram
13helpto help
analysethethe
problem.
Useproblem.
the structure of multiplication in the vertical form in Diagram 13 to help you analyseUse
the structure of multiplication in the vertical form in Diagram 13 to help you analyse
the problem.
the problem.
Whatis theisrule
explainthe interesting
resultresult
you foundin theintwo
What
the torule
to explain
the interesting
you foundthe multiplications?
two multiplications?
 What is the rule to explain the interesting result you found in the two multiplications?
 UsingWhatisthethemakerule toexplain
theproblems
interesting
resultyou
foundin thethat
twomultiplications?
Using
the rule,
otherproblems
with fourother
different
digitscanproduce

rule,
make
other
with
four
other
different
digitsthat
can thesameinteresting
result.
Usingproduce
the rule,
makeproblems
with four other different digits that can produce thethesameotherinteresting
result.
Using
the rule, makeother problems with four other different digits that can produce thesame interesting
result.
same interesting result.

Chapter 2: Numbers and Operations | 89

Topic 7: Introducing Division and Extending It to RemainderStandards 7.1:
Topic 7: Introducing Division and Extending It to Remainder

Introducing
division
withDivision
two different
situations
andfind the answers by multiplication
Topic
and
Topic 7:
7: Introducing
Introducing Division
and Extending
Extending It
It to
to Remainder
RemainderStandardsdivision
7.1:
i. Understandwith quotative and partitive division for distribution situations
7.1:
ii. Standards
Think about
Standards
7.1:how to find the answer of division situations by distribution using diagrams,
Introducing division with two different situations and find the answers by multiplication
repeated
subtractions
and multiplication
Introducing division
twosituations
and findtheanswersbydivisiontowith
withtwo different
different
situationsfindtheanswers
by multiplication
multiplicationiii. Introducing
Obtainidentify
answersof division
throughinverseoperationof multiplicationi. fluency
Understanddivisionwith quotative
andandpartitive
divisionfor distribution
situations
ii.
Think
about
how
to
find
the
answerofdivisionsituations
bydistribution
using diagrams,
iv. i.i.Appreciate
the
use
of
multiplication
table
for
acquiringmentaldivision
Understanddivisionwithquotative
andpartitive
divisionfor
distribution
situations
Understand division with quotative and partitive division for distribution situations
repeated
subtractions
and multiplication
Think
about
how
find
the
of
situations byusing diagrams,
ThinkObtainabout fluency
how toto tofind
the answer
answerofofdivision
divisionby distribution
distribution
diagrams,iii.
identify
answers
divisionsituations
through inverse
operationusingof multiplication
repeated
subtractions
andmultiplication
repeated
multiplication
Sample Tasks
forsubtractions
Understanding
the Standards
iv.
Appreciate
the useandof multiplicationtable for acquiring mental division

ii.
ii.

iii.
iii.
iv.
iv.

ObtainObtain fluency
fluency toto identify
identify answers
answers ofof division
division throughthrough inverse
inverse operation
operation ofof multiplication
multiplication

Appreciate
the
use
of
for
SampleTasks
Understanding
thetableStandards
Appreciatetheforuseof multiplication
multiplication
table
for acquiring
acquiring mental
mental division
division
Task1: Quotative
andPartitive
Division
SampleTasks
forStandards
Task1:
Quotative
and Partitivethe
Division
SampleTasks
for Understanding
UnderstandingtheStandards

Task1:
Quotative
andPartitive
Meanings
of Division
Task1:Meanings
Quotative
Partitive Division
Division
ofand
Division
Meanings
Meanings ofof Division
Division

Diagram 11
Diagram
Diagram 1 show 12 tomatoes. Consider two division situations as follows:

Diagram1
Diagram 1 show 12 tomatoes. Consider two
divisionsituations as follows:
Diagram1Partitive Situation: Pack the tomatoes equally into 2 boxes.

Diagram1
tomatoes.
ConsidertwodivisionsituationsasDiagramSituation:
1 showshow 1212 Pack
tomatoes.
Considertwodivisionsituations
as follows:
follows:
Howtomatoes
many tomatoes
are there
in2 aboxes.
box?
Partitivetheequally
into

Quotative Situation:
Pack
2 tomatoesequally
into
box.
How
many
tomatoes
areathereina box?
Partitive
Pack
thetomatoesintoboxes.
Partitive Situation:
Situation:
Packthemany
tomatoes
into 2
2areboxes.
Howboxesequallyoftomatoes
there?
How
many
tomatoesarethere
How many tomatoes are there inin aa box?
box?
Quotative
Situation:
Pack
tomatoesinto
abox.
box.both representing 12 ÷ 2 = 6.
Quotative
Situation:
Pack
2
ai. Diagram
2 shows
the2twosituationsinto
graphically,
Quotative
Situation:
Pack
2 tomatoes
tomatoesintoa box.
How
many
tomatoesarethere?
Howboxesoftomatoes
there?
Howmany
many boxes
boxes
ofof
tomatoesare arethere?

i.

i.

Diagram 2 shows the two situations graphically, both representing 12 ÷ 2 = 6.

i. Diagram
2 shows
twosituations
situations graphically,
both
representing
12 ÷ 212= 6.
Diagram
2 shows
thethetwographically,
both
representing
÷ 2 = 6.

12 ÷ 2 = 6

ii.

•

Diagram 2

12 ÷ 2 = 6

Match each of the situations with the correct picture in diagram 2.

Explain the key difference between
the 2two situations.
Diagram

12
12 ÷
÷2
2=
=6
6

Diagram
2 2
Diagram

12
12 ÷
÷2
2=
=6
6

Illustrate the key difference between the two division situations with appropriate tape Matcheach
diagrams.
Matcheach ofof thethe situations
situations with
with thethe correct
correct picture
picture in
in diagram
diagram 2.
2.
Match each of the situations with the correct picture in diagram 2.
 iii. Explainthekeydifference
betweenthetwosituations.
Diagram3
shows
twonumberlines.
Explain the key difference between the two situations.

Illustratethekeydifference
situations• ii.
betweenthe
thetwotwodivisionsituations.
ii. Explain
Illustratethethekeykey difference
difference between
betweenthetwodivisionsituations with
with appropriate
appropriate tape
tape

ii.

diagrams.
diagrams.
iii.
Diagramshows
twolines.
Illustrate
the 33key
difference
betweenthe two division situations with appropriate tapeiii. Diagramshows
two number
numberlines.

diagrams.

Diagram 3
Draw on the number lines to show the key difference between the two division situations.

Diagram
Diagram 3
3
Draw
on
the
number
lines
to
show
the
key
difference
Draw on the number lines to show the key difference between
between thethe two
two division
division situations.
situations.

 Match each of the situations with the correct picture in diagram 2.
 Match each of the situations with the correct picture in diagram 2.
 Explain the key difference between the two situations.
 Explain
the keydifferencePractices
betweenatthe
situations.
90 | Mathematics
Challenges
for Classroom
thetwoLower
Primary Level
ii. Illustrate the key difference between the two division situations with appropriate tapeii. diagrams.
Illustrate the key difference between the two division situations with appropriate tapediagrams.
iii. Diagram 3 shows two number lines.
iii. Diagram
3 shows
twoiii. Diagram
3 shows
twonumbernumber lines.
lines.

Diagram 3
3
Diagram•iv.

Diagram 3
 Draw on the number lines to show the key difference between the two division situations.
Draw
on
the
number
lines
to
show
thedifference
key difference
betweenthe situations.
two division Draw on the number lines to show the key
between thetwo divisionsituations.

Diagram
4 shows
thethedividend,
divisorsentence.
iv. Diagram
4 shows
dividend,
divisorand
andquotientquotientin
in aa division
division sentence.

Diagram 4
Diagram4v.

•

 Whatdoes
divisormeaninineacheachofofthe
the division
division situations?
What
does
thethedivisor‘2’‘2’
mean
situations?

•

What does the quotient ‘6’ mean in each of the division situations?

v.

Division is the inverse of multiplication.

•

Which division situation is related to 6 x 2 = 12? Explain your answer.

•

Which division situation is related to 2 x 6 =12? Explain your answer.



What does the quotient ‘6’ mean in each of the division situations?

Depending on the contexts, different division

Divisionis themay
inverseofdifferent
multiplication.
Dependingon the contexts, different divisionsituations
relate tomultiplication
situations.
situations may relate to different multiplication situations.


v.



Which division situation is related to 6 x 2 = 12? Explain your answer.
Which division situation is related to 2 x 6 =12? Explain your answer.

Which division situation shows repeated subtraction? Explain your answer.

vi. Which division situation shows repeated subtraction? Explain your answer.
Basic Facts of Division
x

1

2

3

4

5

6

7

8

9

1

1

2

3

4

5

6

7

8

9

2

2

4

6

8

10

12

14

16

18

3

3

6

9

12

15

18

21

24

27

4

4

8

12

16

20

24

28

32

36

5

5

10

15

20

25

30

35

40

45

6

6

12

18

24

30

36

42

48

54

7

7

14

21

28

35

42

49

56

63

8

8

16

24

32

40

48

56

64

72

9

9

18

27

36

45

54

63

72

81

v.



Which division situation is related to 6 x 2 = 12? Explain your answer.



2: Numbers and Operations | 91
Which division situation is related to 2 x 6 =12? ExplainChapter
your answer.

Which division situation shows repeated subtraction? Explain your answer.

Basic Facts of Division
Basic Facts of Division
x

1

2

3

4

5

6

7

8

9

1

1

2

3

4

5

6

7

8

9

2

2

4

6

8

10

12

14

16

18

3

3

6

9

12

15

18

21

24

27

4

4

8

12

16

20

24

28

32

36

5

5

10

15

20

25

30

35

40

45

6

6

12

18

24

30

36

42

48

54

7

7

14

21

28

35

42

49

56

63

8

8

16

24

32

40

48

56

64

72

9

9

18

27

36

45

54

63

72

81

Diagram 5

Diagram 5i.

i.

Diagram 5 show the a 9 by 9 table of multiplication.

•

multiplication
can beexamples,used to findhow
basicthefacts
of divisionfrom his table.
Explainwith specificinverserelationshipbetween division andmultiplication can be used to find basic facts of division from his table.

Diagram 5 show the 9 by 9 table of multiplication.


Explain with specific examples, how the inverse relationship between division and92 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards
Standards7.2:
7.2:
Extending division into the case of remainders and use division for distribution in daily situations

Extending division into the case of remainders and use division for distribution in daily situations
Extendsituations
withremainders
and understandas a repeatedsubtraction
Standards
7.2:
i.i.
Extend division
divisionsituations
withremainders
and understanddivisiondivision
as a repeatedsubtraction
withStandards 7.2:
withremainders
remainders
Extending division into the case of remainders and use division for distribution in daily situations
ii.
Obtain fluency
fluency
division
andapply
it init daily
situations
Extendingfor
division
into thecaseof remainders
andusesituations
division for distribution in daily situations
ii. Obtainfor
divisionandapply
in dailyiii.
Understand simple
case
ofof
divisionalgorithmi.
Extenddivisionsituations
with remainders and understand division as a repeated subtraction withiii. Understandsimplecase
divisionalgorithmi.
Extend division situations with remainders and understand division as a repeated subtraction withremainders
remaindersii.
Obtain fluency for division and apply it in daily situations
Sample Tasksii. for Obtain
UnderstandingtheStandards
for division
andcaseapply
in dailyalgorithm
situations
iii. fluency
Understandsimpleof itdivision

Sample Tasks
UnderstandingStandards
iii. forUnderstandsimple case ofthedivisionalgorithm

Tasks for Understanding the Standards
Task 1: Division withSample
Remainder
Samplewith
TasksRemainder
for Understanding the Standards
Task 1: Division
Task 1: Division with Remainder
Task 1: Division with Remainder

3 in each box
3 in each box
3 in each box

5 boxes with 1 remaining
5 boxes with 1 remaining

4 boxes

5 boxes with 1 remaining

4 boxes

4 boxes

4 in each box
4 in each box

Diagram 1
Diagram
Diagram 1 1i.

i.

Diagram 1 shows 16 toy ducks. A pupil packed 3 toy ducks into each box. There are 5 boxes

4 in each boxi.
Diagram 1ofshows
16 toyducks.
A pupil packed
toy ducks
into eachThere
5 boxes
toy ducks
andaA
remainderof 1packed
toy 3ducks.
pupil box.
packedthe aretoy ducks
intoare 5
Diagram 1
shows
16 toyducks.
student
3Another
toypacked
ducks
into
each
box. equally
Thereof toy ducks
and a remainder
1 toyducks.
the toy ducks equally into
4 boxes.
There are of
4 Diagram
toy
ducks
in1aAnotherbox withpupil
no remainder.
boxes of toy 4ducks
remainder
1 toy
Another student packed the toy ducks
boxes. andThereaare
4 toy ducks inof
a box
withducks.
no remainder.
 Explain the similarities and differences between the two division situations.
into
4 boxes.
There
areAand4
toy
ducksbetween
a box
with
no
remainder.
Explainthesimilarities
differences
the twodivisionsituations.
i. equally
Diagram1 shows
16 toyducks.
pupil
packed3intoy
ducks
into
each
box. There are 5 boxes
 It is saidof
thatdivisible by4, but notpupil
divisible
by 3. Explain
meaningof divisible.
of toy ducks andIt isa saidremainder
1 16
toyisbyducks.
Anotherpackedtoytheducks
equally
intothat 16 is divisible
4, but notdivisible by 3. Explain
thethemeaningof divisible.
There
are 4 toy ducks
a box with no• 4 boxes.
Explainthe similarities
andindifferences
betweentwosituations.
ii. remainder.
A pupil the
carriedoutdivision
an investigation
on division with

•




ii.

A pupil carriedout an The
investigation
division withremainders.
result of oninvestigationis shown in

Thetworesultof investigationis shown in
Explain the similarities and differences remainders.
betweenthesituations.
Diagram
2. division

Diagram2.
It is said that 16 is divisible by 4, but notdivisible
by 3. Explain the meaning of divisible.
 Whatyoumeaning
ask to enable
your pupils
It is said that 16 is divisible by 4, but not divisible
by 3.questions
Explainwilltheof divisible.

What questions
will youwhether
ask to enable
yourispupils
to determine
a numberdivisible or not
to determine
whethera number is divisible or notdivisible
by 4?
ii.
A
pupil
carriedout
aninvestigationondivisionwith withdivisible
by4?
ii. A student carried out an investigation on division
 The
Explainthe patternof the remainders
inshown
Diagramin2.
remainders.
resultofinvestigationis
Explainthepatternoftheremainders
in
Diagram2.
remainders. The result of investigation is shown in
Diagram 2. One pupil said that 23 ÷ 4 = 4 remainder 7.
 One
Diagram2.pupil said that 23 ÷ 4 = 4 remainder 7.
 Will you accept this answer?
 Whatwillthisyou
ask to enable your pupils
 questions
Will you accept
answer?
 whether
How will you
fix
the matter?
todetermineanumberis divisible
orstudents
not
•
What
will
ask
to enableyour questions
How will you fix
theyoumatter?
divisibleby4?

Why a remainder is always smaller than thewhether a number is divisible or notto determine
Why a remainder
divisor? is always smaller than thedivisor?
bypattern
4? of the remainders in Diagram 2.
 divisible
Explainthe
If a divisor of a division expression is 7, What are
Diagram 2If a divisortheof possible
a divisionremainders?
expression is 7, What are
Diagram 2
 One
said
that 23 ÷ 4 = 4 remainder 7.
thepupilpossibleremainders?
•

Explain the pattern of the remainders in Diagram 2.

•Onestudentsaid
23 ÷ 4 = 4 remainder 7.
 How
will youfix that
the matter?


Diagram 22
Diagram Will you accept this answer?

Why a remainder is always smaller than theWill you accept this answer?
divisor?

If a divisor of a division expression is 7, What areHow will you fix the matter?
the possible remainders?

Chapter 2: Numbers and Operations | 93•

Why a remainder is always smaller
than the divisor?

•

If a divisor of a division expressionis 7, What are the possible
remainders?

Task 2: Calculate Division in Vertical Form

Task 2: Calculate Division in Vertical Formi.

Diagram 3 shows four stages ofcalculating 38 ÷ 4 in the vertical form.

What questions will you ask at eachstage to guide pupils perform thedivision?
Diagram 3 shows four stages of calculating 38 ÷ 4 in thevertical form.

Diagram 3 

i.

•
ii.

ii.

What other operations are also involved

What questions
will3you ask at each stage to in
guide
students perform the division?
Diagramcalculating division in vertical form?

What other operations are also involved in calculating
divisionvertical Whyandin how
is form?
each of the•identified operations involved with

Why and how is each of the identified operations division?
involved with division?

Diagram 4

Diagram 4
iii.

iii.

Diagram 4 shows how the calculations 16 x 42 and 672 ÷ 16 are done in the vertical form.

Diagram 4 shows how the calculations 16 x 42 and 672 ÷ 16 are done in the vertical form.


•
•

iv.

Why is addition involved in the multiplication algorithm?

Why is addition involved in the multiplication algorithm?



Why is subtraction involved in the division algorithm?

Why is subtraction involved in the division algorithm?

Three students, A, B and C, used three different methods to calculate division in the verticalform as shown in Diagram 5.

Diagram 5

iii.
iii.

Diagram
Diagram 4
4 showsshows how
how thethe calculations
calculations 16
16 x
x 42
42 andand 672672 ÷
÷ 16
16 areare donedone inin thethe verticalvertical form.
form.

94 | Mathematics
for Classroom
at the Lower
Primary Level
 Challenges
Whyinvolved
thealgorithm?
Why isis addition
addition
involved in
inPractices
the multiplication
multiplication
algorithm?


iv.

Why
Why isis subtraction
subtraction involved
involved inin thethe division
division algorithm?
algorithm?

iv.
Three
B
used
methods
todivisiontheThree
A, BA,
C,C,
threedifferent
different
methods
to calculate
in the vertical
iv. students,
Three students,
students,
A,and
B and
and
C,usedused threethree
different
methods
to calculate
calculatedivision in
indivision
the verticalvertical
as
in
5.
form
as shown
shown
in Diagram
Diagram5.
5.
form asformshown
in Diagram

Diagram
Diagram 5
55
Diagram

Which
Which ofof these
these methods
methods will
will you
you preferprefer to
to teachteach your
your pupils?
pupils?



Justify your choice.

•

Which
of these methods will you prefer to teach your students?
 Justify your choice.

•

Justify your choice.

Diagram 66
Diagram
Diagram66shows
showsthetheconnection
connectionbetween
betweenthe
theverticalverticalform
formofofdivision
divisionand
andmultiplication
multiplication
v.v. Diagram
performedinblock
diagram.
performed in block diagram.

•

Explain thethe connection
connection between
between thethe two
two forms.
forms.
Explain•

How will
will thisthethetwotwoforms
helphelp
you inguiding
your pupils
How
this idea
ideaofofconnection
connectionbetween
betweenforms
youin guiding
yourexplain the process of vertical form of division?
students explain the process of vertical form of division?

Chapter 2: Numbers and Operations | 95

Topic 8: Introducing Fractions and Extending to Addition and Subtraction of Similar Fractions
Topic 8: Introducing Fractions and Extending to Addition and Subtraction of Similar Fractions
Standards 8.1:
Standards 8.1:
Introducing simple fractions such as halves, quarters and so on using paper folding and drawing
Introducing simple fractions such as halves, quarters and so on using paper folding and drawing
diagrams
i diagrams
Introduce simple fractions using paper folding and drawing diagrams in the context of partwhole relationshipIntroduce
simplefoldingandsuchdrawing
diagrams
context of part wholeiii Use“a half of”
and fractions“a quarterusingof” inpapera daily
context
as half
a sliceinofthe
bread
relationshipiii Counta quarter for representing one quarter, two quarters, three quarters, and four quarters
Use “a halfand “asimple
quarterof” in a indailysuch asa sliceof breadivii Compare
andof”
explainfractions
thecontext
case where
thehalf
wholeis thesame
iii
Count a quarter for representing one quarter, two quarters, three quarters, and four quartersiv
Compare and explain simple fractions in the case where the whole is the same
Sample Tasks for Understanding the Standards
Sample Tasks for Understanding the Standards
Task 1: Representing Simple Fractions
Task 1: Representing Simple Fractionsi.
I also cut ½ of myribbon, but why is
my ribbon shorter?

I cut ½ of myribbon!

i.

John and Jason found a ribbon each attheir homes. They cut ½ of each ribbon
John and Jason found a ribbon each at theirand brought
theriibbon
next day.
homes.
They tocutthe½ schoolof eachandInschool,they
decidedtoexchangebrought to the school the next day. theInribbons.
school,they decided to exchange theribbons.
When they compared the ribbons, they
When they compared the ribbons, they got
got confused and Diagram 1 shows theconfused and Diagram 1 shows theconversation between them.
conversation between them.
 • Why
areare½ of½each
ribbonnot theWhyof eachribbonnotsamethelength?
same length?

Diagram 11
Diagramii.

What questions will you ask to adress• theWhatquestions will you ask toconfusion?
adress the confusion?

Diagram 2 shows three shapes used by a teacher to introduce ¼ to children.
ii Diagram 2 shows three shapes used by a teacher to introduce ¼ to children.

Diagram 2
Diagram 2•
How will you convince your students that all four parts of each shape are equal in size?
 How will you convince your students that all four parts of each shape are equal in size?
•
Explain how paper folding can be used to make the three shapes?
 Explain how paper folding can be used to make the three shapes?
iii.

Both the tasks in (i) and (ii) could be used to introduce simple fraction. What are theiii.similarities
Both the tasks
in (i) and (ii)between
could be theused
to introduce
and differencestwotasks? simple fraction. What are the similarities
and differences between the two tasks?

96 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

iv.

Shapes with shaded parts are commonly used to introduce simple fraction to children.
Diagram
3 shows
examplesof shapes
withtoshaded
parts.
iv. Shapes
withshadedfourparts
are commonly
usedintroduce
simple fraction to children.
Diagram 3 shows four examples of shapes with shaded parts.

Diagram 3•
•

Diagram 3

Which shapes are inappropriate for introducing ½?
Which shapes are inappropriate for introducing ½?

Explain your answer based on the following perspective:

Explain your answer based on the following perspective:Students’ previously learnt knowledge

Sizes of the partsThe shape of the ‘whole’ Students’ previously learnt knowledge
 Sizes of the parts

 The shape of the ‘whole’
Representation ①

Diagram44shows
shows three
three representations
v. v. Diagram
representationsofof¼.¼.
••

Representation ②

Representation ③

Diagram44
DiagramIf you want to discuss the concept ofwithstudents,
how willyou
Iffraction
you want
to your
discussthe conceptof fraction
order the sequence of discussion?
with your students, how will you order the
discussion?
 sequence
Justify yourofdecision.

Justify your decision.

Chapter 2: Numbers and Operations | 97

vi. Various representations of fractions are presented to children in order to help them learn
simpleat different
Task
2: fraction
Comparing
Simplecomplexity.
Fractions Diagram 5 shows some representations of ¼.
i. A student used a tape to measure the length ofa table. The measurement was 1 m and aremaining part as shown in Diagram 7.
In order to answer the question in Diagram 8,
the student cut off the remaining part and used
it as a unit to measure the length of 1 m. The
result is shown in Diagram 8.

Diagram 7

 How many metre is one remaining part?

Question How many metre is two remaining parts?

“How many pieces of the remaining partare equal to 1 m?”

 How many metre is three remaining parts?
 How could this task be used to compare
fractions?
 Design another similar task for ½.

Diagram 5•

Which representations will you use for each of the following purposes?
Introducing simple fraction to your students.
Diagram 8

Learning simple fraction by paper folding.
ii. Diagram 9 shows four identical strips of Promoting mathematical reasoning among yourstudents.
paper.
Each strip of paper is 1 m long. One
strip is divided into two equal parts while theFor each of the purposes, explain how you will use
the threerepresentations
to achieve
otherstrips are divided
intoit.four equalparts. The strips of papers are then shaded
Diagram
9. lines withvii. Diagramas6 shown
showsin two
numberdifferent scales. One number line represent a
Explain whereas
how you thewouldusenumberthese strips oflength of 2 metres,
other
paper
to
make
the
followingcomparisons.
line represents a length of 1 metre.


•

•
•
Diagram 6
Diagram 9
•

1

1

2

3

is longer,
mofor m?
Use ↑ to Whichmark thepositionm on
24each of the two number lines.
 Which is longer, m or m?
4
4
How do you explain the different
positionof the ↑?

How will you use paper folding to help
your students solve this problem?

98 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Comparing Simple Fractions
Task
2:
Comparing
SimpleFractions
Task
Task 2:
2: Comparing
Comparing Simple
Simple Fractions
Fractions
Task
Task 2:
2: Comparing
Comparing Simple
Simple Fractions
Fractions

Diagram 7
Diagram
Diagram 7
7
Diagram
Diagram 7
7
Question
Question
Question“How many pieces of the remaining part
“How
many
Questionareequal
to pieces
1 m?” of“How
many
piecesof thethe remaining
remaining part
part
Question
are
equal
to
1
m?”
are equal
tomany1 m?”pieces of the remaining part
“How
“How many pieces of the remaining partareare equal
equal toto 11 m?”m?”

Diagram 8
Diagram 8
Diagram
Diagram 8
8
Diagram
Diagram 8
8

Diagram 99
Diagram
Diagram
Diagram 99Diagram
Diagram 99i. A student used a tape to measure the length ofi.i. AAstudent
student useda tape measure
to measure
the ofalengthtable. used
The measurement
was the1 mand ofai. aA student
used
a tapetape toto measurethelengthlengthofatable.
The
measurement
wasatable.
The
measurement
was
1
m
and
a
remaining
part
as
shown
in
Diagram7.
ai.table.
Themeasurement
was
1
m
and
a of
student
used
a
tape
to
measurethe
1i. mAanda remaining
part
as
shown
in length
remaining
part
as
shown
in
Diagram
7.
A
student
used
a
tape
to
measurethelengthofremaining
part
as
shown
in
Diagram7.
atable.
The
measurement
was
1
m
and
a
Diagram7.
atable.
The
measurement
was
1
m
and
In orderto answer
theshownquestionin Diagram
8, aremaining
part
as
in
Diagram
7.
In
to
the
questionin8,
remainingasshown
in Diagram
thestudent
cutpartoff the
part
and7.
used
In orderorder
to answer
answertheremaining
questionin Diagram
Diagram
8,
Inasstudent
order
to measureanswerthelengthquestioninusedcut
offthe
remaining
part
and
itthe
a
unit
to
the
of
1
m.
The
the student
cut
off
the
remaining
part
and
used
In
order
answerthequestion
Diagram8,
Diagram8,tothestudent
cut ofoff 1initresult
as
a
to
the
lengthm.
The
order
tomeasureanswerthe 8.
question
inthe
Diagram8,
shown
in
Diagramitremaining
as In
aisunitunit
tomeasurethelengthofunit1part
m.
Theusedthestudent
cut
off
the
remaining
and
part
and
used
it
as
a
to
resultisshowninDiagram8.
thestudent
cut
off the remaining
part and used
resultisshowninDiagram8.
itit as
a
unit
to
the
lengthofofone
1 m.remainingThe
resultmeasure
How
metre
is
part?
asmanyathe
unitlength
to measure
measurethelengthofis1
1 m.
m. TheTheshown
How
many
metre
is
one
remainingpart?
resultisshowninDiagram8.
in
Diagram8.
How
many
metreinisDiagram
one remaining
part?
resultisshown
8.
 How many metre is two remaining parts?
 How
many
metre
is
remainingparts?
 How
How
many
metre
one
remaining
part?
How
many
metre
is twotworemainingparts?
How
many
metre
is oneone
remaining
part?
many
metre
isis
• How
many
metre
is threeremainingparts?
part?
 How
many
metre
is
three
remainingparts?

How
many
metre
is
two
remainingparts?
How
many
metre
is threetwo
remainingparts?
 How
manyremainingparts?
 How
could
this metretask beis used
to compare How
could
this
task
be
used
to
compare How
How
many
metre
three
remainingHow
could
thismetretask betocompareparts?
• fractions?
many
isis
two
How
many
metre
is usedthreeremaining
remainingparts?
fractions?
fractions?
parts?

How
could
this
task
be
used
to
compare Design
anothersimilar½. to compare How
could thistasktask
be forused Designanother
fractions?
Designanother similar
similar task
task forfor ½.
½.
fractions?
•
How many metre is three remaining
 parts?
DesignDesign another
another similar
similar task
task forfor ½.
½.

ii.
ii.
ii.
ii.

•

How could this task be used tocompare fractions?

•

Design another similar task for ½.

Diagram 9 shows four identical strips ofDiagram9four
stripsofpaper.
Each
strip of paperis 1 m long.
Diagram
9 showsshows
four identical
identical
stripsOneofpaper.
Each
strip
of
paper
is
1
m
long.
One
Diagram
9
shows
four
identical
stripsofstrip
is Eachdividedinto
equal
whilestrips
the ofpaper.
strip
oftwo
paper
is 1parts
m long.
One
ii.
Diagram
9
shows
four
identical
strip
is
divided
two
equal
while
the
ii.paper.
Diagram9into
showsfour isparts
identical
strips
of
Each
strip
of
paper
1 mfourlong.
otherstrips
are
into
equal
strip
isthreedividedinto
twodivided
equal
parts
while
the Onepaper.
Each
strip
of
paper
is
1
m
long.
other
three
stripsaredivided
into
four
equal
paper.
Each
strip
of
paper
is
1
m
long.
One
One
strip
is
divided
into
two
equal
parts
parts.
The
strips ofpapersarethenotherstrip
three
are
dividedinto
fourshaded
equal
is divided
into
two
parts
while
parts.
The
stripsofpapersarethenshadedstrip
dividedinto
two equal
equal
parts
while thethewhile
theisinotherthree
stripsaredividedasshown
Diagram9.
parts.
The
stripsofpapersarethenshadedother
threestripsaredivided
into
four
equal
as
shown
in
Diagram
9.
other
three
stripsaredivided
into
four
equal
into
four
equal
parts.
The
stripsofpapersas shown
inThe
Diagram9.
parts.
stripsofareshadedparts.
The you
stripsof papers
papersare thenthen
thenshadedasshown
in Diagram
9. shadedare
Explain
how
would
use
these
strips
of
as
shown
in
Diagram9.
 paperExplainwould
use
stripsas shown
inyou
Diagram
9.
tohow
make
the
followingcomparisons.
Explain
how
you
would
use these
these
strips of
of
to
make
following• paper
Explainhowtheyou
would comparisons.
use these
paper
to
make
the
followingcomparisons.
 Explain
how
would
use these
1make
1 use
Explainhow youyou
wouldthese strips
strips ofofofistopaper
to thefollowing strips
Whichlonger,mfollowing
or the
m?
paper
make
comparisons.
1
1
2
4
paper
to
make
the
followingcomparisons.
1 m or 1 m?

Which
is
longer,comparisons.
 Which is longer, 2 m or 4 m?
2
4
1 3
1
2
11 m m?
11 m?

Which
is
longer,
or

Which
is
longer,
m
or
2
3 or

Which
is
longer,
m
or
2
4 m?
Which
is
longer,m
4
42

22 3 m?44
 WhichWhich isis longer,longer, 4 m
m or
or
4 m?
4
4
2
3
22
Which
is
longer,or 33 m?
m?
 WhichWhich isis longer,longer, m
m or
or
m?
4
44

4
44

Chapter 2: Numbers and Operations | 99

fractiona fraction
numeratoriii. iii.
UnitUnitfraction
a fraction
withiii. Unitisisfraction
is awith
fractionwith 1numerat
1.
Two
examplesofunit
fraction1.
Two
examplesofunit
fractionare1
numerator
1. Two examples
unitare2
How manyHow
setsmanyofiii. msets
areof
there
1 therem?
Unitfractionisa fraction
withnumeratorm inarein 1iii.
m?
1
4
Unit
fractionisafractionwithnumerator
1
1
4
and arewhich
commonlycalled 11onecalled on
and arewhich
which
arecommonlycommonly
1. Two examples
of unit fraction
1
fractionare2
4 of unit fraction are
2
1. Two4 examples
11of 1 m are thereof m are thereHow
in 1 m?
1
2
many
sets
m?
22
half
and
one
quarter,
respectively.
1 inm1in
4
How manysets
of many
m4are
there
m?
How
sets
of
are
there
in
m?
4are commonly
one quarter,
respectively.
11 onecalledhalfandand
one quarter,44
and whichcalledone half
4
4
4
and
which
are
commonlycalledone
12
44
11of 1 are
223 m?
of m are thereHow
in m?
1 one
3
half
and
quarter,
sets
there
4
4 manyUnitfraction
is a shows
fractionnumerator
Diagram10
a shows
sequenceof
How manysets
of many
m are
there
How
sets
of ininarem?
there
in respectively.
m?iii. respectively.
half
and
one quarter,
respectively.
Diagram
10 witha sequence
44 4
4 444
4
11
1
3
questionsinvolvingunit
fractions1.
Two
examplesofunit
fractionare1
1
2
3 33 2questionsinvolving
unit22 fractionof are there How
in So,
m?
3
many
sets
of11 are
m is
are
there
in10
1 m?
which
is longer,morin m?
Diagramof byDiagram10shows
aabysequence
44 manyHow
setsofwhichthere
designeda teachertohelpofhis help hSo,longer,m
orshowsm? a sequence
11
44 44 m?
Diagram10shows
sequenceof todesignedateacher
44 44
4
4
and
which
are commonly
calledonequestions 22 involving
unitquestions
fractions23
involvingunit
fractionsstudents
comparesimplefractions.
441122
questionsinvolvingunit
fractionsstudents compare simple fractions.
33
ger, m or m?
Why?
How
many
of mmarem?a teacher designed
designed
tohalfhelp
Why?
and hisone
respectively.
4
4 So,
which
is sets
longer,or therem? in bybyaa teacher
to
4444
44
designedby quarter,
teacherto help
help his
his
44
students
comparesimplefractions.
students
comparesimplefractions.
 How
will
the
use
of
unit
fraction
11
33
students
comparesimplefractions.
 How will the use of unit fractio
How many sets of are there in m?
Why?
Diagram
10 shows
a sequence
44
44
promoteyour students’abilityof to ability
promoteyour students’
Diagram10
Diagram10
1 2
3 unitDiagram
10

How
will
the
use
of
unit
fractionquestionsinvolving
fractions
1
2
3
22
33
• How
will
compare, use
and of?unit
Howwill thetheuseofunit fraction
fraction? his
So, which is longer, m or promotem?
4compare4teacher
4 4 , 4toandyourstudents’abilitytodesigned
by
a
help
4
4
4
4
4
promotetoagram 10promote your
your students’students’ ability
abilitytoDiagram 10 compare 1 , 2 and 3 ?
students compare
simplefractions.
11 22
33
Why?
4 4
4
compare , and ??
compare
44 44
44
1

will
the unit
useof unit
fraction
iv. Anotherthat
it is notimportantto use How
theto
unit
metre
in
askingthequestions.
iv. teacher
Anotherargues
teacherarguesthat
it is not important
use
the
metre
in askingthe question
1in2 asking
3 1theiv. Anotherteacherargues
that
it
is
not
important
to
use
the
unit
metre
questions.
promoteyourstudents’abilityto2
3
It will be easierfor
students
toanswer totheanswer
questionsin terms of1in,anddirectly.
Diagram
10
It
will
be
easierfor
students
thequestions
terms
of
,
and
directly.
1 422 433 4
her argues that it Itiswill
notbe
important
to students
use the unit
metre inaskingthe questions.
4 44compare,and?
easierfortoanswerthequestions
in
terms
of
and
directly.
iv. Another teacher argues that it is not1 importantin askingthe questions.
2
3 to use the unit metre
44 44
44
11 22
33
er for students to answer
theinagreetermsof the, anddirectly.
 beDo
you
agree
with
the
Why
or whynot?
questions
youwithteacher?
Whyor whyofnot?
4 4 the4
It willeasier
forDo
students
toteacher?
answerquestionsin terms, and directly.
44 44
44
•
Do you agree with the teacher? Why or why not?
gree with the teacher? Why or why not?
 Do you agree with the teacher? Why or why not?
iv. Another teacher argues that it is not important to use the unit metre in asking the questions.
11 22
33
It will be easier for students to answer the questions in terms of , and directly.
Do you agree with the teacher? Why or why not?

44 44

44

Standards
8.2:
Standards
8.2:
ending fractions using tape diagram and number line to one, and think about how to add orStandards 8.2:
tract
similarfractionsfor
producing
simplealgorithm
Extending
using
tape
diagramand Practices
number
line
one,
and
think
about
to addExtendingfractions
using
tape
diagramand atnumberline
to one,andthinkhow
about
howor
to add or100|fractions
Mathematics
Challenges
forClassroom
the toLower
Primary
Level
Standards
Standards
8.2:
8.2:
subtract
similarfractionsfortapeproducing
simplealgorithm
subtract
similarusing
fractionsfor producing
simpleline
algorithm
Extendingfractionsdiagramand number
to one, and think about how to add or
Extendfractionsmore
than
one unit
quantity
forlinerepresenting
theremaining
part
(unit
fraction)subtract
similarto
fractionsfordiagram
producing
simplealgorithm
Extending
Extendingfractionsfractions
using
using
tape
tape
diagramandandnumbernumberlinetotoone,
one,and
and
think
think
about
abouthow
howto
toadd
add
oror
Standards
8.2:
Standards
8.2:
such
as
measuringthelengthoftape,recognisingtheremaining
parts
as
a
unit
measureoflength,i
Extendfractions
to
more
than
one
unit
quantityforrepresenting
theremaining
part
(unit
fraction)i
Extendfractions
to
more
than
one
unit
quantityforrepresenting
theremaining
part(unit fraction)
subtract
subtractsimilar
similarfractions
fractionsfor
forproducing
producingsimple
simplealgorithm
algorithm
Extendfractions
to
more
than
one
unit
quantity
fortherepresenting
theremainingpart
andi understandproperandimproperfractionssuchassuchmeasuringthelengthtape,
recognisingremainingparts
aspartsaaboutunit
measureof length,
as measuring
theoflengthoftape,recognisingremainingasa(unitunit
measureExtendingfractions
using
tape
diagramandnumberline
to the
one,
and
think
how
tofraction)add
or of length,
Extendingfractions
using
tape
diagramandnumberlineistoa one,
and
think
howoftoadd orsuch
as measuring
theandlengthtape,
recognisingtheremainingparts
asa unitaboutmeasurelength,fractionswithquantities
inofandtwoways;
firstly,
whole
unit
of
quantity
andandunderstandproperimproperfractionsandunderstandproperimproperfractionssubtract
similarfractionsfor
producing
simplealgorithm
i i Appreciate
ExtendExtendfractionsfractions
to
to
more
more
than
than
one
one
unit
unit
quantityquantityforforrepresenting
representingthetheremainingremaining
part
partsecondly,(unit(unitfraction)
fraction)andunderstandproperandimproperfractionssubtract
similarfractionsfor
producing
simplealgorithm
based
on
the
numberofunit
fraction
ii
Appreciate
fractionswithquantities
in
two
ways;
firstly,
whole
is
a
unit
of
quantity
andsecondly,ii asAppreciate
fractionswithquantities
ways;
firstly,
whole
aunit
unit
of quantity
and secondly,
diagramsuch
andnumberline totheone,
andofthink
about
howin
totwo
add
or
such
asmeasuring
measuringthelengthlength
of
tape,
tape,
recognising
recognisingthetheremainingremaining
parts
parts
asas
aisaunit
measuremeasureofoflength,
length,ii based
Appreciatewithquantities
in twoways;
firstly,
wholeis a unitofquantity
andsecondly,i understand
Extend
to
more
than
one
unit
quantityforrepresenting
remaining
part
(unit
fraction)
Comparefractionsinfractions
thethe
case
where
the
whole
is
the sameandexplainitthewithtape diagram
oron
thefractions
numberofunit
fraction
basedonnumberofunit
fraction
ducingsimplealgorithm
iand
Extendfractions
to
more
than
one
unit
quantityforrepresenting
theremaining
part
(unit
fraction)andunderstandproperproperandandimproperimproperfractionsfractions
based
onasthenumber ofthe
unitlength
fractionof tape, recognising1
suchmeasuringtheremaining
parts
as
a
unit
measureiii
Comparefractions
in
the
case
where
the
whole
is
the
same
and
explain
it
with
tape
diagramor of oriii
Comparefractions
in
the
case
where
the
whole
is
the
same
and
explain
it
with
tape
diagramwithquantities
and
so
on
line,
and
developfraction
number
sense
such
as
such
as
measuringthelengthoftape,recognisingtheremaining
parts
as
a
unit
measureofii ii number
Appreciate
Appreciate
fractionsfractionswithwithquantities
quantities
in
in
two
two
ways;
ways;
firstly,
firstly,
whole
whole
is
is
a
a
unit
unit
of
of
quantityquantity
andandsecondly,
secondly,iii
Compare fractions in the case where the whole is theit with tape diagram or length,
10 same1 and explain
1
length,andunderstandproperandimproperfractions
than
one
unit
quantityforrepresenting
theremaining
part
(unit
fraction)andunderstandproperand improper
fractionswithquantities
andsoproduceon
numberline,
anddevelopfraction
numbersense
such
as1such
based
based
on
on
the
the
numbernumber
of
of
unit
unit
fractionwith quantities
and so on
numberline,
anddevelopfraction
numbersense
asquantities
Think
about
how
to
add
or
subtract
similarfractionswith suchtape
diagramnumber
line
andon
and
so
numberline,
anddevelopfraction
numbersense
as10withor
10
iialgorithm
Appreciate
fractionswithquantities
intwofirstly,wholeisisa aunitofquantity
andsecondly,
10 andngth
tape,recognisingtheremainingparts
asa unitmeasureoflength,iiCompare
Appreciate
fractionswithquantities
infractions
twoisways;
ways;
firstly,
whole
unit
of
quantity
andsecondly,iiiiiisimple
fractionsfractionsininthethecase
case
where
where
the
the
whole
whole
is
the
the
same
same
and
explainexplainitnumberitwith
with
tape
tape
diagramdiagramoror produce
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ivofivCompare
Think
about
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similarwithtapediagramorline
andproduceivThink
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number
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and
about
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similar fractions with tape1 1 diagram or number line and produce
based
on
the
numberunit
fractionnd improper
fractionswithwithquantities
quantities
and
and
so
so
on
on
numbernumber
line,
line,
and
and
developdevelopfraction
fraction
numbernumber
sense
sense
such
such
as
as
simplealgorithm
withfluency
simplealgorithm
withfluency
simplealgorithm
withfluency
1010 same and explain it with tape diagram orquantities
two
ways;
firstly,wholeis caseacase
unitwhereof quantity
and secondly,
Comparefractionsintheisistheiii iiiinfor
fractionsin thethewherethewholewhole
the same and explain it with tape diagram or
mple
Tasks
UnderstandingtheStandards
iv
iv
Think
Think
about
about
how
how
to
to
add
add
or
or
subtract
subtract
similarsimilarfractionsfractions
with
with
tape
tape
diagramdiagramorornumber
numberline
lineandandproduce
produce1unit fraction number line, and develop fraction number sense such
withquantities
quantitiesand
andso
soon
on
as 10 with
SampleTasks
for
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forfluency
Understanding
the Standards
SampleTasks
for
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the Standards
Standards
simplesimplealgorithm
algorithm
withwithfluencycase where the whole is the same and explain it with tape diagram or
iv
Think
about
how
to
add
or
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numberline
1 add or subtract similar fractions with tape diagram or numberandsk 1: Proper
iv Fractions
Think
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andwithquantities and so on
p fraction number
sense
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fluency
10
SampleSample
Tasks
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thetheStandards
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Task
1: Proper
FractionsTask
1:
ProperFractions
Task
1:for
ProperFractionsproduce
simplealgorithm
withfluencyr subtract similar fractions with tape diagram or number line and produce
encyfolding can be easily
SampleTasks
SampleTasksforforUnderstanding
Understandingthe
theStandards
Standards i. Paper
Task
Task1:1:
Proper
ProperFractions
Fractionsi. i. Paperbeeasilyused
toPaper
represent
½can
and
¼.be
foldingcaneasilyi. folding
Paperfoldingcanbe easilyusedtorepresent
½
and
¼. ¼.
ding the Standards
usedtorepresent
½
and
used
to
represent½
and ¼.
Task1:1:Proper
ProperFractions
FractionsTask
 What other unit fractions

What
other
unit
fractionsi. i. Paper
Paper
foldingfolding
can
can
be
be
easilyeasilycanrepresented
 alsoWhat
unit
fractionsbeotherWhat
otherunit fractions
canalso
be represented
used
used
tocan
torepresent
represent
½also
½and
andbe
¼.¼.represented
easily
using
the
same
also
be
represented
cani. Paper i.foldingcanbeeasilyusedeasily
toeasily
using
the
same
Paper using
folding thecan samebeeasilyeasilyusing
the samemethod?
 What
What
other
other
unit
unit
fractionsfractions
method?
represent½
and¼.
used to represent ½ and ¼.
method?
can
can
also
alsomethod?
beberepresented
represented
i. Paper folding can be easilyWhatfractions
easilyeasily
using
using
the
the unit
same
same
What otherunit otherfractionsalso
used to represent ½ and ¼. •
1 1 1 can1ii. Unit
fractions
such
as
and
can
also
be
represented
method?
method?
ii. Unit
such easily
be fractions
represented
3as 3 and51using1 the
5
11 What other unit fractions
easily
using
the
Unit
such
asby
andassameii.fractions
Unit
fractionssuchandareii. noteasilyrepresentedsame
method?
arenoteasilyrepresented
by
3
5 3
5
can also be represented
method?
paperpaper
folding.
Diagram
1 1 byfolding.
Diagramarenoteasilyrepresentedarenoteasily1 1representedby11easily using the same
1 1
ii.
Unit
Unit
fractionsfractions
such
such
as
as andandpaperfolding.
Diagram
1 are 1ii. ii.
Unit
fractions
such
as
and
paper
Diagramshows
a method
tofolding.
representshows
a methodtorepresent
3 35 5 5 5 1
method?
1showsamethodtorepresent
are
are
not
not
easilyeasilyrepresented
represented
by
by
on
a
fractionbar
usingparallelshowsamethodtorepresent
notrepresented
by paper folding.
ona easilyfractionbar using parallel
15
1 5
ii.
Unit
fractions
such
as
and
lines.
paper
paper
folding.
Diagram
Diagram
1 13 parallel
5lines.
Diagramshows
methodtoparallel
represent
on a1folding.
fractionbar
using
on a afraction
bar
using
11
are
not
easilyrepresentedby1
1 showsshowsaamethodmethodtotorepresent
representlines.
lines.
Describehowtheparallelii. Unit fractions such as and ona fraction
bartheusing
parallel
5 lines.
5
howparallelpaperfolding.
1
3
5 Describe
lines
in Material
Sheet
1Diagram
at
on
bar
bar
using
using
parallelparallel1
lines
infraction
Material
Sheet1ata afraction
Describehowtheparallelare not easily represented by on
Describehowtheparallelshowsamethodtorepresent
the end of this topic can belines.
Describehow
theinparallellines
theendlinesof this
topic
canin Material
Sheet
1Sheetat in 15 at
paper folding.
Diagram •1 lines.
lines
1Material
1beused, ,using
andonto
a represent
fractionparallel
1at1bar
1
Sheet
1end
end
of
this
8parallel
end
of
this
topic
can
be
the
of
this
topic
can betothe
represent
, the
,3the
and
shows
a method to represent usedMaterial
Describe
how
how
the
parallelDiagram
1
1Describe
3 8lines.
5
1
1
1 1,
respectively
onaDiagram 1
1 topic
usedtorepresent
,
,
and
lines
inin
Material
Material
Sheet
Sheet
1 1atat , , andtorepresent
canbeused torepresent
10
ona fraction bar using parallel,linesrespectively
on
3a 8
3 8
Diagram
Diagram11
10 theDiagramfractionbar.
1Describehow
the
1 this
the
end
end
of
of
this
topic
topic
can
can
bebeparallellines. 1
respectivelyona on aand 10bar.
, , respectivelyonafraction, respectively
fraction
1
1
1
1
10 in Material
lines
used
usedtoto
represent
represent, , , and, Sheetand 1 at
3bar.
38 8
bar.
thefraction
end of this
topic can be
Diagram
Diagram1 1 Describe how the parallel bar.1 1fraction, , respectively
respectively onon a1a 1lines in Material Sheet 1 at
1010
used to represent, , and1
iii.
Diagram2
a fraction 3 8
the
end
of
this
topic
can
be
fraction
fractionshowsbar.
Diagram 1
1 bar. 16
1 1, respectively on a
Diagramiii.
Diagramshowsbar. 22 shows
used to represent ,iii.
, and
10 a 6 fraction bar.
1
1
3 8
1
ram 1
fraction
iii. Diagram
2 showsbar.
fraction1
iii. Diagram
2 ashows
a fraction
bar.
6
6

List
all
properfractionsthat,respectively
on
a
6
1
10
•
List
all beproperfractionsusingthat can be
bar.
bar.
can
represented
111
 Listall proper fractions
thatfraction bar.
1
6
represented
usinga thisbar.
iii.iii. Diagram
Diagram
2 2shows
shows
a fraction
fraction
this bar.
Diagram 22
Diagram66
canrepresentedusing beListthatall proper
List allfractions
properfractions
that
6
6
bar.
1
this
bar.
can
be
represented
using
bar.
ExplainthemeaningofDiagram 2
canberepresented
using
11iii. Diagram
2 shows ofa properfraction•
Explainthe meaning
6
properfraction.
thisthisfractions
bar.
DiagramDiagram2
2
Explainfraction.
List
List
all
allbar.
properproperfractions
that
that
66

the
meaningofbar.
1
1iii. Diagram 2 shows a fraction proper
canbebe
represented
using
using
fraction.
 canExplainthe meaningof represented
Explainthemeaning
of
6

List
all
properfractions
that
this
this
bar.
bar.
Diagram
Diagram2 2bar.
6
proper fraction.
proper fraction.
can be represented using  Explain
Explainthethemeaningmeaning ofof
 Listthisbar.
Diagram
2 all proper fractions thatproperproperfraction.
fraction.
can be represented using Explain the meaning of
this bar.
ram 2
proper fraction.
 Explain the meaning ofproper fraction.

Chapter 2: Numbers and Operations | 101

Diagram3
Diagram 3

Diagram 3

ii. John
poureda intojugof twowater
into1-litre
two
identical
1-litreThe
amountJohn poured
aajug
ofofwater
two
identical
containers.
Thecontainers.
amountof wateris
1 ℓ andiv.ii. John
poured
jug
water
into
identical
1-litrecontainers.
Theamountof wateris 1ofℓ water is 1 ℓa remainingapartas shown
in Diagram
3. in Diagram 3.
remaining
part
as
shown
and a remaining part as shown in Diagram 3.

Diagram 4

Diagram 4v.

Diagram 4Hethen compared
comparedthethe
remainingof waterthecontainer
1-litre container
asDiagramshown 4.
in
He thenremainingpart partof waterwith thewith
1-litreas shown in
Diagram 4. He then compared the remaining part of water with the 1-litre container as shown in Diagr

How many remaining parts are there in 1 ℓ of water?

• litreHow
remainingpartsinare
in 1 ℓ of water?
Howis many
one remaining
How many
many
remainingparts
arepart?
there
1 ℓ thereof water?

• How many litre is one remaining part?
How many litre is one remaining part?

iii. Fractions with quantities can be understood in two ways as shown in the following questions
and answers:

vi. Fractions
quantities
be understood
two ways asinshownin theasfollowing
questionsiii. with
Fractionswithcanquantities
can beinunderstood
two waysshown inthe following quest
and answers:
Question①:answers:
How many litre are two remaining parts?
and2

Answer ①: 2 remaining parts are ℓ .

3
Question ①: How many
litre are two remaining parts?
1

Question ②:Answer
How manyare 2 setspartsof ℓare? 2ℓ.
①: 2litreremaining
13

2

Answer ②: 2 sets of ℓ are ℓ ”.



•
•

3

3

3

1

Question ②: How many litre are 2 sets of ℓ ?
3What are the similarities and differences
betweenthe two ways?
1
2

Answer ②: 2 sets of ℓ are ℓ ”.
33

Which way do you think is easier for your students to understand fractions?

 What are the similarities and differences between the two ways?
What are the similarities and differences between the two ways?
 Which way do you think is easier for your students to understand fractions?
Which way do you think is easier for your students to understand fractions?

102 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram 5

Diagram 5
Diagram 5
Diagram 55
Diagram
Diagram
Diagram555
Diagram

Diagram 6
Diagram 6
Diagram 6
Diagram 66
Diagram
Diagram
Diagram666
Diagramvi.
five 1-metre
1-metre tapes
tapes
vi. Diagram
Diagram 55 shows
shows fivevi. divided
Diagram
5
shows
five
1-metre
tapes
into
5
equal
parts.
into
5showsequal parts.
vi. divided
Diagramshows
five 1-metre
1-metre tapes
tapes
vi.
Diagram55 5
fivedividedinto
equal
parts.
vi.
Diagram
5
shows
five
vi.
Diagram
5
shows
five 1-metre
1-metre tapes
tapes
vi. divided
Diagramfive
1-metretapes
dividedinto555shows
equalparts.
1
into
equal
parts.
•dividedHow
many
Howintomany
m arethere in555equalparts.
1
dividedinto
equal
parts.
dividedinto
equal
parts.
 How many 511 m are there in
How1manymany
m are
are there
there ininm? 551111m
 
How
 
How

11 manym?
How
many55 m
marearethere
thereinin
in
m?
How
many
m
are
there
5

1
m?
3
 1 m? 55

m?
311

m?

m?
m?
 5331m?
vi. Diagram 5 shows five 1-m
 5333m?
m?

5
dividedfractions
into 5 equal parts.
5 m?
 Comparethefollowingm?
 55 m? the following Comparefractions
55Diagram 5.
using
1
Comparethe 5.following
following
fractionsComparefractions• Comparethefractionsusing
Diagram How
many m are there
Comparethefollowing
fractionsComparethefollowing
fractions5 using
Comparethefollowing
fractionsusing
Diagram5.
2
3
using Diagram
Diagram 5.
5.

Which
is longer,
or 31m?
2 m
m?
using
Diagram5.
using
Diagram5.
using
Diagram5.
55
 Which is longer, 22 m or 33 m?
 WhichWhich isis longer,longer, 5422m
m or
or 5333m?
m?


or

m?

Which
isis
longer,m
 WhichWhichisislonger,longer,455522m
m
or1555533m?
m?

or

Which
longer,mor
m?
5 or 1555m?
 Which is longer, 54455m
 WhichWhich isis longer,longer, 544m
m or
or 11 m?
m?
5544mCompare
 
How
will
you
comparethefractions

Which
isis
longer, WhichWhichislonger,longer,morm? the followin
is
or
111m?

Which
longer,
m
or
m?
555
5the
 How
will
you
comparefractionsusing
Diagram
5.
using
unit
fraction?

How
will
you
comparethefractions using
How willyou
compare the fractions
unit
fraction?
How
will
you
comparethefractions
How
will
you
comparethefractions
2
How
will
you
comparethefractions
using
unit
fraction?
•  Howwillyou
comparethefractionsusing unit fraction?
 Which is longer, m
using
unit
fraction?
usingunit
unitfraction?
fraction?
5
using
unit
fraction?
using
4 Which is longer, m
5vii. Diagram66 shows
shows a fraction
fraction number
line.
vii.
number line.
line.
vii. Diagram
Diagram 6 shows aa fraction number
vii. Diagram
Diagram 66 showsshows
a fraction
fraction
number
line.
1 a
vii.
numberline.
thereHow
you compare t
How manyarein willline.
1 mvii.
666shows
aaafraction
numbervii.Diagram
Diagramshows
fraction
number
line.
vii.
shows
fraction
numberline.
10• Diagram
How many
m
are
there
in
1
using
unit
fraction?
How many
many 10111m
m are
are there
there inin How
10
11
10
1 manym?
Diagram 5
 
How
How
many10 m
marearethere
thereinin
many
10 m are there in
How
11 m?
10

m?
 1
1
m? 10
4 m?


411m?
 10
m?
m?

41m?
 10444m?
m?
vii. Diagram 6 shows a fraction n1044 m? the following fractions
 Compare

 10
m?

m?
10
10
 Comparethe 6.following fractions 1
10
10
using
Diagram How
many m are ther
Comparethe following
following
fractionsComparethefractions10• Compare
thefollowing
fractionsusing
DiagramComparethefollowing
fractionsComparethe6.
following
fractions usingComparethefollowing
using
Diagram6.
5
6 fractions
Diagram6.
using
Diagram6.
usingWhichis more,5 m or
6 1m?
 10
m?
Diagram6.
using
Diagram
6.10
Diagram6.
 usingWhichis more,55 m or 10
66 m?
 WhichWhich isis more,more, 10
m
or
m?
5
6

m
or
m?
5
64
9 55
2 66
10
10
Diagram 6

m?

Which
more,
m
or
m?
 WhichWhichisisis
ismore,more,10
mor
or10
m?

Which
is
more,
m
or
m?

m
m?
910
210

Which
more,
m
or
m?
10
10
10
10
10
10
10
10
 Which is more, 10
m
or
9
2
9
2 m?
10
10
 WhichWhich isis more,more, 39 9m
m or
or 722m?
m?

101022 m?
99m
Compare

Which
more,
mm
or
 WhichWhichisisis
ismore,more,10mor
or10m? the followin

m?

Which
more,
or
310
710

m?
10
10
10
10
10
10
 Which is more, 3310
m or
m?
710
using
Diagram6.
7
 WhichWhich isis more,more, 1033m
m oror 1077m?
m?

1033 m or 101077 m?
10

Which
is
more,
 WhichWhich isis more,more,10 m
m or m?
m?
5
10 or1010
Which
is more, m
10
10
10viii. What rules on comparing fractions doviii. What
Whatexpect
rules ononcomparingfractions
do
9
viii.
fractionsdoyourstudents 
toconclude
Whichdois more, mviii. you
What rules
rules onon comparing
comparingfractions
do
viii.
What
rules
comparingfractions10
youexpectyourstudents
toconclude
viii.
What
rules
on
fractions
do
viii.
What
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from these tasks?

Task 2: Introducing Improper Fractions

Chapter 2: Numbers and Operations | 103

Task 2: Introducing Improper Fractionsi.

Diagram 7
Diagram 7
Diagram 7 shows a 1-metre tape divided into 4 equal parts and four other tapes.

i.

7 shows
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•


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 Express each length as a fraction in metres.
 Express each length as a fraction in metres.
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What isis thethe difference
difference between
between proper
proper andand improper
improper fractions?
fractions?
4

Is proper or improper fraction? Explain your answer.
Is4104 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

3: Addition
and Subtractionof Similar
FractionsTask Task
3: Addition
and Subtractionof Similar
FractionsTask 3: Addition and Subtraction of Similar Fractions
Task 3: Addition and Subtraction of Similar Fractions
Task
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Problem

Chapter 2: Numbers and Operations | 105
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how
you
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use
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12
to
Show
how
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Diagram1212
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solve
solve
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the
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problem.
solve
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solve the problem.
Diagram 12ii. Diagram 13 shows a 1-metre tape divided into
Diagram
Diagram 12
Diagram
12 12
5 equal parts and a number line.
Diagram
Diagram
12
Diagram
12
Diagram1212
v. Diagram 13 shows a 1-metre tape divided
5 equalparts
andnumberline.
•intoHow
many
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v. Diagram
13
shows
a 1-metretape
divided
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 Diagram
How many
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oftapethetape?
Diagram
13
shows
shows
a
aone
1-metre1-metretape
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Fill in 13
theshows
missing
fraction tapein each
 of
into
into
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5
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parts
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and
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andafraction
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the
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 How
Fill
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each
 of
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thenumber
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How
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one
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ofthe
the
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How
many
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the
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the
tape?
 Fillinthemissing
fractionineach
 tape?
of• theUsethetapediagram oror thethe number
numberline
line
number
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Use
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diagram
Fill
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missing
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fraction
ineacheachofof Fillinthemissing
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each

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to
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to
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line.
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the thenumberline.
 Usetape diagramor the number line
1
2
to
find

m
+
m
Use
Use
the
tape
tape
diagramdiagramorthe
thenumbernumberline
line
 
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thenumberline
Use
5the tape
5 diagram or the number line
to
to
find2
1 find
to
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to
find
 4m + 3m
5 m
 5115 m
11 −225 22
Diagram 13

m
m
++

+
m

m
+
m mm
4 m
 55Between
m55− 355m55tape diagram and number
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line,
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Diagram 13
calculations?
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line,
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
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numberDiagram 13
Diagram 13
vi. Calculate
calculations?
Justify
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line,
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which
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iii.vi. Calculate
Diagram1313
calculations?
calculations?
Justify
Justify
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yourchoice.
choice.
2
3calculations?
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5
5
vi.
vi.
Calculate
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Calculatevi. •Calculate
5+ 2
 5 −
252337 33++
 5227++
55−52555 will•  755How
you guide your students to do
7
55 5522 22−−
calculations
without using tape 7these−− 77 you guide yourHow
students totododo
7 7777will• How
will youguide yourdiagramor
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thesecalculations
without
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

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you
to
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How willwill
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your students
students
totapedodiagramor
numberline?
vii. Findasmanysolutionsaspossible
for
these
these
calculations
calculations
without
without
using
using
tape
tape
these
calculations
withoutusing
tapediagramor number line?
these calculations
without using tapeeach ofthe following
problems.
diagramornumbernumber
line?
line?
or
numberline?
diagramor or
numberline?
vii. Find diagram
asdiagrammany
solutionsaspossible for
8iv. vii.
Find
as
many
solutionsas possible
for each
each
of
the
followingproblems.

+
=
vii.
Find
Find
as
as
many
many
solutionssolutionsaspossiblepossibleforvii.
solutions
possibleforvii. Find
Find10 as
as10many
many
solutions as
as aspossible
forfor10problems.
ofeachthefollowing
each
each
of
of
the
the
following
followingproblems.
problems.
ofthefollowingproblems.

8of the following problems.
 each+  = 3
10 =
10 88
 10
−



88
9 +
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+=
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10
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10

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9 =
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 9
+
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 959+92999= 998 =8 
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  8Howwill
problems help your
7 575 7 these
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
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
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7777will
77 these77 −77 77 problems help your Howalgorithm
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How
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
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will
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
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of
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How will these
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for
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students
studentstotodevelopdevelopfluency
fluencyininstudents
todevelopfluency
instudents toto develop
develop fluency
fluency inin algorithm
students
offractions?
algorithm
algorithm
forfor
addition
additionandandsubtraction
subtraction
algorithm
foradditionandsubtraction
algorithm
foradditionandsubtraction
forofofadditionand subtraction of fractions?
fractions?
fractions?
offractions?
of fractions?

106 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Material
Fraction
ScalesMaterialSheet
Sheet1:1:EquidistantEquidistantParallel
ParallelLines
LinesforforConstructing
Constructing
Fraction
Scales

0

1

2

3

4

5

6

7

8

9

10

Chapter 2: Numbers and Operations | 107

Topic 9: Introducing Decimals and Extending to Addition and Subtraction
Standards 9.1:

TopicTopic
9: Introducing
9: Introducing
Decimals
Decimalsand Extending
and Extending
to Addition
to Addition
and Subtractionand Subtraction

Introducing decimals to tenths, and extend addition and subtraction into decimals
i. Introduce simple decimals to tenths by remaining part such as using tape diagram withappreciation
Introducing
Introducing
decimalsdecimals
to tenths,
to tenths,
and extend
and extend
addition
additionand subtraction
and subtraction
into decimals
into decimals
ii. Compare size of decimal numbers on a number line with the idea of place value
i.iii. i.Extend
Introduce
Introduce
simplesimple
decimals
to tenths
to tenthsbyremainingby remaining
part
such
part the
as
such
using
as using
tape
diagramtapediagramwithinwith
additionand decimals
subtractionof decimals
utilising
place
value
systemvertical form
appreciation
appreciation
untilup
to tenths
ii.iv. ii.
CompareComparesizeappropriate
ofsizedecimalof decimal
numbers
numbers
on a numberonnumberline withline
thewithidea
theand
of
idea
place
of placevalue value
Think
about
place
value
foraapplying
additionsubtraction
in lifeStandards
Standards
9.1: 9.1:

iii.

iii.
ExtendExtendaddition
additionand subtraction
and subtractionof decimals
of decimals
utilisingutilising
the placethe placevalue value
systemsystemin verticalin verticalform until
form until
up to uptenthsto tenths
SampleTasks
for
Understanding
iv.
iv.
Think
Think
aboutabout
appropriate
appropriate
placeplacevaluethevalue
for Standards
applying
for applying
addition
additionand subtraction
and subtraction
in life in life
SampleTasks
for Understanding
for Understanding
the Standards
the Standards
TaskSample
1:Tasks
Decimal
Numbers
Task Task
1: Decimal
1: Decimal
Numbers
Numbersi.

i.

A task is used to help children learn the concept of 0.1 using measurement of water as
Ai. taskA is
task
is usedto help
to children
help children
learn learn
the concept
the conceptof 0.1ofusing
0.1 usingmeasurement
measurementof waterof water
as as
shown
inusedDiagram1.
shownshownin Diagramin Diagram
1.
1.

The amountThe amountof waterof inwatera clayin jara clayis measured
jar is measured
with 1-litre
with containers.
1-litre containers.
Each container
Each container
is divided
is divided
1
1
into 10intoequal
One equalof partthe container
will containℓ or 0.1ℓ ℓorof0.1
water.
ℓ of water.
10 parts.
equal parts.
One part
equalof the container
will contain
10

10

There are
There
exactly
are exactly
3 containers
3 containers
and a remaining
and a remaining
part of partwaterof as
water
shown
as shown
in the diagramin the diagram
above.above.
◊ How◊many
0.1 ℓofof0.1
water
there
1 ℓ ofinwater?
How setsmanyofsetsℓ of arewater
areinthere
1 ℓ of water?
◊ How◊many
0.1 ℓofof0.1
water
there
the remaining
part of partwater?
How setsmanyofsetsℓ of arewater
areinthere
in the remainingof water?
◊ What◊ isWhat
the totalof waterof inwaterℓ ? in ℓ ?
is theamount
total amount
Diagram
Diagram1
11
Diagram•In what
In whatway does
does
the use
ofuse
clay
jar
clayandjar measurement
and measurementof waterof waterchildrenhelp children
to
In
what
waywaydoes
thetheuse
ofofclayjar
andmeasurement
ofhelpwater
help tochildrentoconcretise
concretisethe concept
the conceptof 0.1?
of 0.1?
concretise the concept of 0.1?
ii.

Diagramii. Diagram
2 shows
2 shows
anotheranothertask, task,
also usedalso used

ii. to help
Diagram
2 shows
anothertask, alsousedtotochildren
help children
understandunderstandthe meaning
the meaning
of
of
help
childrenunderstandthemeaningof
0.1.
0.1. 0.1.
CompareComparethis task
this withtask thewithtask
the intask inCompareDiagram
Diagram
1. this
1. task with the task in Diagram 1.
 • Which WhichWhichtasktask
task
do do
you
do
you
think
isismoreis moreyouthink
think
moreeffective
effective
effectivein helping
in helping
your yourstudents
students
in helping your students understand theunderstandunderstandthe concept
the conceptof 0.1?
of 0.1?

◊ fraction
What fractionof the squareis shaded?
◊ Whatof the squareis shaded?
is the fraction◊ How◊isHow
the fraction
relatedrelated
to 0.1?to 0.1?

Diagram
22
Diagram
Diagram2

concept of 0.1?

•Justify Justify
your decision.
your decision.

Justify your decision.

108 | Mathematics Challenges for Classroom Practices at the Lower Primary Leveliii.

A teacher intends to introduce decimaliii. numbers
A teacherby
intends
introduce
decimalusing toDiagram
3 tohelp numbers by using
Diagram understand
3 to help students
understand the relationshipstudents
the relationship

11 mm
mm =

1

10

cm == 0.1
0.1 cm.
cm.

•

How
the teacher
teacherexplain
explainthe
therelationship?
How could
could therelationship?

•How could
discussion
be extended
to other decimalHow
couldthethe
discussion
benumbers
such
as
0.2
and
0.3.
extended to other decimal numbers
such as 0.2, and 0.3.

Diagram 3
Diagram 3

ther teacher intends to extend the learning of decimal numbers using a 1-metre
iv. Another teacher intends to extend the learning of decimal numbers using a 1-metre tape.
e. The 1-metre tape is divided into 10 equal parts as shown in Diagram 4.
The 1-metre tape is divided into 10 equal parts as shown in Diagram 4.

Diagram 4

What numerical relationship is expected to be learntDiagram
through 4the diagram?
•

What numerical relationship is expected to be learnt through the diagram?

To what extend do you think the diagram is effective in helping your students
understand the relationship?
•
To what extend do you think the diagram is effective in helping your students understandthe relationship?

Based on the diagram, a teacher asks the following questions:diagram, a teacher asks the following questions:“How many m is•oneBasedof the on
10theparts?”“How many sets of 0.1
m are
there
1 m?”
“How
many
m isinoneof the 10 parts?”

manyof 0.1m are
there in 1 m?”
 What is the main“How
purposeofsetsaskingthesequestions?

 In what way is 
the useterminology
usefulin helping
students
Whatofis thethe mainpurpose “set”
of askingthesequestions?
conceptualise the relationship?
In what way is the use of the terminology “set” useful in helping students
conceptualise the relationship?

Chapter 2: Numbers and Operations | 109v.

Diagram 5 shows two tasks involving decimal numbers.
iv.
•

Diagram 4 shows two tasks involving decimal numbers.
How will you order the sequence of these tasks?
 How will you order the sequence of these tasks?

•

youryourdecision.
Justify
Justify
decision.
Task ①The diagram shows a tape of length more than 2 m.
Each metre of tape is divided into 10 equal parts.
◊

How many sets of 0.1 m are there altogether?

◊

What is the length of the tape in m?

Task ②The diagram shows a decimal number line.
◊

Write down the decimal number that each ↑ is pointing at.

◊

How many sets of 0.1 are there in each of the decimal numbers expressed in Ⓐ to Ⓔ?

Diagram 4
Diagram 5
vi.

Diagram 5

Diagram 6

v. Diagram 5 shows a decimal number 234.5
place-value
chart.
Diagram 6writtenshowsona adecimal
number234.5 written on

a place-value chart.
 Explain how the place-value concept
is extended fromnumbers
to to introduceiii. wholeA
teacherintends•
Explain how the place-value
concept
is extended
decimal numbers. numbers by using Diagram 3
from whole numbers to decimal numbers.
1
students
understandthe rela Explain why the placevalue of 
is 10.
1•
Explain why the place value1ofmm
 is= .cm = 0.1 cm.
10 How will you guide your students
topredict the next placevalue
to thethe teacher exp

How
could
•
How will you guide your students to predict theright of ☺?
relationship?
next placevalue to the right of ☺?

How could the discussvi. How will you sequence the following tasks based on difficulty levels?
extended to other decimal nvii. How will you sequence the following tasks based on difficulty levels?
suchsets
as 0.2, and 0.3.
Task ①: 234.5 is the sum of ____ sets of hundreds ____ sets of tens ____
Task ① : 234.5
is
the
sum
of
____
sets
of
hundreds
____
sets
of
tens
____
sets
of onesof ones and ____ sets of tenths.
and ____ sets of tenths.
Diagram3
Task ②: 234.5 = 200 + ____
+ 4 + ____

TaskTask
② :③:
234.5
+ ____+ 4 + sets____of 0.1.
2.8 =is200
the sumof ____

Anotherteachersum
of of
12____
setsof
0.1ofisintends____. to extend the learning of decimal numbers using a
TaskTask③ :④:
2.8Theisiii.
the
sum
sets
0.1.

tape. The 1-metre tape is divided into 10 equal parts as shown in Diagram 4

Task 2: Addition and Subtraction of Decimal Numbers

Task ④ : The sum of 12 sets of 0.1 is ____.

110 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Addition and Subtraction of Decimal Numbers

Task 2:
Additionand Subtractionof Decimal
NumbersTask
2: Addition
and Subtractionof Decimal
Numbersshows
theworkofofa student
when
adding25.3
and 14.
Task 2: Addition andi. Subtraction
of77Decimal
Numbersi. Diagram
Diagram
shows
the
work
a student
when
adding25.3 and
14.andi.
Diagram
7 shows
the workof a student
when adding
25.3
14.
What
isWhattheerror?
•  What
thestudent’s
student’s
error?
 isisthe student’s
error?

i.

Diagram 7 shows the work of a student when adding 25.3 and 14.

How
will
you
help
student
correctthe error?
How
willyouthishelp
thiserror?
student
correcttheerror?
error?
•  Howyou
thisstudent
correctthe willWhatishelpthestudent’s

Diagram
Diagram77
Diagram7

How will you help this student correct the error?

ii.
8 shows
two 1-litre
containers
each divided
into 10into
equal
The
two
containers
ii. Diagram
8 showstwo
1-litrecontainers
each divided
divided10
equalparts.
The
two
containers
ii. Diagram
Diagram
8 shows
1-litrecontainers
each
into
10parts.
equalparts.
The
two
containers
Diagram
7ℓ oftwo
have 0.8ℓ
and
0.5
water,
respectively
as
shown
in
Diagram7.
have 0.8 ℓ and 0.5 ℓ of water, respectively as shown in Diagram 7.

have 0.8 ℓ and 0.5 ℓ of water, respectively as shown in Diagram 7.
ii.

Diagram 8 shows two 1-litre containers each divided into 10 equal parts. The two containers
have 0.8 ℓ and 0.5 ℓ of water, respectively as shown in Diagram 7.

Diagram 8
Diagram 88
Diagram

iii.

•

Explain how you will use Diagram 8 to find the answers for Explain how you will use Diagram 8Diagram
to find the
8 answers for

Explain how you will use Diagram 8 to find the answers for 0.8 ℓ + 0.5 ℓ
 0.8 ℓhow
+ 0.5
ℓ will use Diagram 8 to find the answers for Explainyou 0.8 ℓ + 0.5 ℓ
 0.8 ℓ – 0.5 ℓ
 0.8
0.8 ℓℓ +– 0.5
0.5 ℓℓ
0.8 ℓ0.8– 0.5ℓ in the vertical form, two teachers, A and B, used different questions
When
teaching+ 0.5
iii. When
0.8
vertical form, two teachers, A and B, used different questions
 teaching
0.8ℓ – 0.5ℓ + 0.5 in9theto guide theirstudents.
Diagramshows the two sets of questions.
to guide their students. Diagram 9 shows the two sets of questions.

iii.

iii.

When teaching 0.8 + 0.5 in the vertical form, two teachers, A and B, used different
When teaching 0.8 + 0.5 in the vertical form, two teachers, A and B, used different questions
TeacherA’s Questions
questionsto students.
guide theirstudents.
Diagram9sets
shows
the two sets of questions.
to guide their
Diagram
9 shows
theA’stwoQuestionsof questions.
Teacher
What is 8 plus 5?
What is 8 plus 5?
How doTeacher
we write A’s
the Questions
13?
How do wewrite the 13?
What is 0 plus 0?
What isis 0 plus 5?
0?
What
What about
the81plusthat we carried over?
What
about
the
1 the
that13?
we carried over?
How
So, where
dodo
wewe
putwritethe decimal
point?
So, where
do we
What
is 0 plus
0?put the decimal point?
What about the 1 that we carried over?
Teacher
Questions
So,B’swhere
do we put the decimal point?
Teacher B’s Questions
How many sets of 0.1 are there in 0.8?
Howsetsmanysetsareof 0.1
areinthere
How manyof 0.1
there
0.5? in 0.8?
Teacher
B’s
Questions
How
many
sets
of
0.1
are
there
in 0.5?
So, how many sets of 0.1 are there altogether?
So, howmanysets
of 0.1
there
altogether?
many
sets
of of
0.1
arearethere
in 0.8?
How doHow
we write
13
sets
0.1?
How
do
we
write
13
sets
of
0.1?
How many sets of 0.1 are there in 0.5?
So, how many sets of 0.1 are there altogether?
Diagram 9How
do we write
13 sets of 0.1?
Diagram9Which set of questions do you think will help your students develop a better Which setof questions
do youthink will
yourstudents develop a better
Diagram
Diagram
99 helpyour
understandingon additionof decimal
numbers?
Explainreasons.
understanding on addition of decimal numbers? Explain your reasons.

• Whichofquestionsdowillyou
yourstudents
developa betterWhichsetofsetofquestions
do
you
think
willhelp
help
your
students
a better
Designa setquestionsthat youusethink
whenwillguiding
yourstudents
to do
0.8develop– 0.5the vertical
Design form.
a set ofon
questionsthat
you willnumbers?
use when Explain
guiding your
yourreasons.
students to do 0.8 – 0.5
understanding
additionofof decimal
decimalinunderstandingon
addition
numbers?
Explainyourreasons.
in the vertical form.

•

Designa verticalset of questions
that you will use when guiding your students to do 0.8 – 0.5
in theform.
in the vertical form.



Design a set of questions that you will use when guiding your students to do 0.8 – 0.5

Chapter 2: Numbers and Operations | 111

iv. Diagram 9 shows a measuring tape and two tapes of length 0.8 m and 1.5 m, respectively.

Diagram9showsshowsaameasuring
measuring tape
tape andand two
tapes
of
0.8
mmand
1.5
respectively.
iv.iv. iv.
Diagram
10
two
tapes
oflengthlength
0.8
and
1.5m,
m,
respectively.
iv. Diagram
9 shows
a measuringtapeandtwotwotapesof
0.80.8m and
1.5 m,
respectively.
Diagram
9 shows
a measuring
tape
and
tapes
oflengthlengthm and1.5m,
respectively.

Diagram 910
Diagram
Diagram 9Diagram 9

9 to find the answers forShow and explain how Diagram Diagram
9 can be usedand explain
how Diagram
10 canbebe usedtoto
find theanswers forShowShow
Showand
and
explainhow Diagram
Diagram 99 cancan be used
used to find
find thethe answers
answers forfor
1.5explain
m + 0.8 howm Show and explain how Diagram 9 can be used to find the answers for

1.5
m
+
0.8
m

1.5
m
–
0.8
m
 1.5 m +1.50.8mm+ 0.8 m
 1.5 m + 0.8 m

1.5–– 0.8mv. A teacherguiding
1.5 m
mwas
0.8
m a student to add 1.5 and 0.8. The following is the conversation between
 the
m –m
0.8
and m
the student.
 1.5
m1.5
–teacher
0.8
v.
v. AAteacher
teacher was
was guiding
guiding aa student
student toto addadd 1.5
1.5 andand 0.8.
0.8. TheThe following
following isis thethe conversation
conversation between
betweenteacherandthestudent.
v. A theteacherwas
guidingastudent
to
add
1.5
and
0.8.
The
followingistheconversation
between
“Try
to
think
about
how
many
sets
of
0.1
are
there.
First,
howmany
Teacher:
the teacher
the student.
v. A teacher
wasandguidinga student to add 1.5 and 0.8. The following is the conversation
sets of 0.1 are there in 1.5?”
the teacher and the student.
•

between the teacher and the student.

sets.” about how many sets of 0.1 are there. First, how many
Student : “15toTeacher“Try tothink
think about how many sets of 0.1 are there. First, how many
Teacher :: “Try
sets
of
are
there
in
1.5?”
“Then,
how
many
sets
0.1 arethere
in 0.8?”Teacher
:
“Try
to
think
about
how
many
sets
of 0.1
are there. First, how many
Teacher :sets of0.1
0.1
are
there
inof
1.5?”
sets
of
0.1
are
there
in
1.5?”
sets.”
Student : “8sets.”
Student“15 sets.”
Student :: “15“So, how many sets of 0.1 are there altogether?”: sets.”StudentTeacher: “15Teacher“Then,howhowmany
manysets
setsofof0.1
0.1 arearethere
thereinin 0.8?”0.8?”
Teacher :: “Then,.. 15
sets andsets… are
23 sets
altogether?”Student: “Emmhow
many
sets8of0.1
there
in 0.8?”Teacher: “Then,Student: “8 sets.”
sets.”
Student
what number is 23 sets of 0.1?”
Teacher: :“8“So,Student : “8 sets.”
“So,
how
many
of
there
Teacher
:
“Oh!
I got
it … sets
23
sets
0.1areis 2.3.
So,altogether?”
1.5 plus 0.8 is 2.3!”
Student
:
“So,
how
many
sets
ofof0.1
0.1
are
there
altogether?”
Teacher :Teacher : “So, how many sets of 0.1 are there altogether?”15
and…
altogether?”Student :: “Emm
“Emm....thinking
15 sets
setsprocesses
and 88sets
setstriggered… 23
23 sets
sets
altogether?” Student
Explain the student’s
by theteacher’s questioning.
Student : “Emm .. 15 sets and 8 sets … 23 sets altogether?”
“So,
what
number
is
23
sets
of
0.1?”
Teacher:numberdesigned
is 23 sets ofa0.1?”
Teacher: “So,
vi. Diagram
10 shows
a setwhatquestionsteacher to guide students do 1.5 + 0.8
whatofnumber
is 23 sets ofby0.1?”
Teacher: “So,
inStudent
vertical form.
The
questions
are
not
inis
the
correct
sequence.
“Oh!
I
got
it
…
23
sets
2.3.
So,
:
setsof
of0.1
0.1
is
2.3.
So,1.5
1.5 plus
plus0.8
0.8 isis 2.3!”
2.3!”
Student : “Oh! I got it … 23 arranged
Student : “Oh! I got it … 23 sets of 0.1 is 2.3. So, 1.5 plus 0.8 is 2.3!”
 Explain
Explain thethe student’s
student’s thinking
thinking processes
processes triggered
triggered byby thethe teacher’steacher’s questioning.
questioning.
Explainthe
the student’s
student’s thinking
processes
triggeredbytheteacher’squestioning.
•  Explain
thinking
processes
triggeredbytheteacher’squestioning.
① What is the sum of (10 + 10) sets of 0.1?
vi.
Diagram
10
shows
a
set
of
questionsdesignedbyateachertoguide
students② How do designed
we write theby
13asetsof 0.1?to guide students do
vi. Diagram 10 shows a set of questions
teacherdo 1.5
1.5 ++ 0.8
0.8
in
form.
questionsarrangedinthecorrect
sequence.
Diagram
10 shows
showsaof
designedateacher
to toguide
students
do 1.5+ 0.8③areHownotdowewrite by
the
20a
sets
of 0.1?
vi.vi. Diagram
11
a set
set
of questions
questionsdesignedbyteacherguidestudents
do 1.5+ 0.8
in verticalverticalform. The
The
questionsarenotarrangedinthecorrect
sequence.
vertical form.
form. The
The questions
questions arenotarrangedthesequence.
④
What
the sum ofin(5+the8)correct
setsof 0.1?
ininvertical
are
notis
arrangedincorrect
sequence.
⑤ How many sets of 0.1 are there altogether?

Diagram
10 of
①
the
①What
What isis
the sumsumof (10(10 ++ 10)10) sets
sets ofof 0.1?
0.1?
①②
What
isdo
thewe
sum
of the
(10
+
10)
sets
of 0.1?
How
write
13
sets
of
0.1?
②
How
do
we
write
the
13
sets
of
0.1?
 How will you sequence the questions?
②③
How
dodo
wewe
write
the
1320
sets
ofof
0.1?
How
write
the
sets
0.1?
③
How
do
we
write
the
20
sets
of
0.1?
 Design a set of questions
to
guide
your
students
to
do
1.5
–
0.8
in vertical form.
③④
How
do
we
write
the
20
sets
of
0.1?
of
④What
What isis thethe sumsum ofof (5(5 ++ 8)8) sets
sets
of 0.1?
0.1?
④⑤
What
is
the
sum
of
(5
+
8)
sets
of
0.1?
How
many
sets
of
0.1
are
there
altogether?
⑤ How many sets of 0.1 are there altogether?
⑤ How many sets of 0.1 are there altogether?
Diagram
Diagram10
Diagram10
11
Diagram 10
 How
the
questions?
How will
will you
you sequence
sequencethequestions?
questions?
•  HowtheHowwill
will you
you sequence
sequence thequestions?
 Design
Design aa set
set ofof questions
questions toto guide
guide your
your students
students to
to do
do 1.5
1.5 –– 0.8
0.8 inin verticalvertical form.
form.
Designaa set
set ofof questions
questions to
to to
do do
1.51.5– 0.8
in verticalform.form.
•  Design
toguideguideyour
yourstudents
students– 0.8
in vertical112 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

vii.
Fractionsanddecimalnumbers
aredifferent
forms
to
representpartsofwhole.
vii.
vii.
Fractionsanddecimalnumbers
aredifferent
forms
forms
to represent
represent parts
parts ofof aaa whole.
whole. Two
Two
Twovii. Fractions
Fractionsand
anddecimal
decimalnumbers
numbersare
aredifferent
different
formsto
11
44to represent parts of a whole. Two
1
4
examplesofsuch
representationsareandexamples==3.5
3.5==2.8.
2.8.
examplesofofsuch
suchrepresentations
representationsare
are33322 =3.5andand22255 =
2.8.
2

5

• Perform
Perform
andcomparethefollowing
additionandsubtraction.
Perform
andcomparethefollowing
additionandsubtraction.
Performand
andcompare
comparethe
thefollowing
followingaddition
additionand
andsubtraction.
subtraction.
11

44

1+ 2
4and 3.5


++2.8
2.8

 33322 ++2255 andand3.5
3.5+
2.8
2
11

5
44

1− 2
4and 3.5

2.8
 33322 −−2255 andand3.5
3.5–––2.8
2.8

2

5

 WhatWhatisisthetheadvantageof decimal
decimalnumberascomparecomparetototosimple
simplefractionsfractionsininindoing
doing
Whatistheadvantage
advantageof
decimalnumber
numberasascomparecompareto
simplefractions
fractionsin
doing• computation?
Whatis theadvantage
ofofdecimal
numberas
simpledoing
computation?
computation?
computation?

Chapter 3: Quantity and Measurement | 113

CHAPTER 3
CHAPTER
Quantity33and Measurement
CHAPTER
Topic 1: Comparing
Size Directly and Indirectly Using Appropriate Attributes and NonQuantityand Measurement
standard
Units
Quantity and Measurement

Topic 1: Comparing size directly and indirectly using appropriate attributes and non-standard
units
Topic
1: Comparing
Standards
1.1:size directly and indirectly using appropriate attributes and non-standard
unitsStandards 1.1:
ComparingStandards
1.1: and describing quantity using appropriate expression
Comparingand describing
quantity using
appropriate
expression
i. Compare
two objects
directly
by attributes
instead of stating in length and amount of water
Comparing
and asdescribing
quantity
usingandappropriate
expression
suchlongeror
shorter
less
or
more
i.
Compare two objects directly by attributes instead of stating in length and amount of water such
two objects
non-standard
units
appreciateunification of units
asCompare
longertwo
or shorterandlessindirectly
or attributes
more using
i. ii. Compare
objects
directlybyinsteadof stating in length
andtoamountof waterthesuch
ii. iii. as
Compareobjectsindirectly
using non-standard
to appreciatetheof
unificationof unitsUseof quantityunits(such
as numbercups) forcounting and appreciate
longerappropriate
ortwo
shorteranddenomination
less or more
Use appropriateof quantity(such asunitsof cups) for
andofappreciate
ii.iii. Compare
two objects
indirectlyusing
non-standard
to appreciatethecounting
unification
units
the
usage
ofdenomination
units
for quantity
forcontext
suitablenumber
context
the appropriate
usage of units
for quantity of
forquantity
suitable(suchiii.
Usedenomination
as number of cups) for counting and appreciate
the usage of units for quantity for suitable context
SampleTasks
for Understanding
the Standards
SampleTasks
for Understanding
the Standards
Sample Tasks for Understanding the Standards
Task 1: Introduce Measurement Using Attributes
Task 1: Introduce Measurement Using Attributes
Task 1: Introduce Measurement Using Attributes
Direct Comparison
DirectComparison
DirectComparisoni.

A teacher showed pencil A and pencil B in a container
A teacher showed pencil A and pencil B in
as ini. Diagram
1 toasherclass. 1One
student claimed that
container
in pencil
Diagramher class.
i. Aateacher
showedA andtopencil
pencil B
islongerthan
pencilA,
butBB inanother
student
student
claimedisaOnecontainer
as inDiagramthat
1 topencil
her class.
longerthan
pencilA,butanother
student
disagreed.
One student claimed that pencil B is••

disagreed.
longerthan pencil A, but another student
disagreed.
What made them disagree?
What made them disagree?

What
you
doyou
todisagree?
help
these two students settle
 should
Whatthem
Whatmadeshoulddo to help these
theirdisagreemnet
overthelengththeirof the pencils?
two should
students
settleWhat
you do tohelp thesedisagreemnet
over thelength ofthetwostudents
settletheirpencils?
disagreemnet
over the length of thepencils?

Diagram 1

Diagram
Diagram 11

ii.

Two threads
P and
are placedside
by sideTwo ii.threads
P and
Q Qareplacedside
by side as in
as inDiagram2 andstudents
are
asked
ii.
Two
threadsPandQ
areplacedside
bysidetoDiagramcompare
2 and thestudents
are
asked
to
comparethe twolengths.
as in Diagramtwo
2 andstudents are asked tolengths.compare the two lengths.

•

What must the students first do in orderWhat
the students
firstorder to compare
to compare
thestudents
two lengths?
 must
What
must
the
firstdodo inin orderthe twolengths?
toExplaincomparethetwolengths?
your answer.
 Explain
Length yourcan answer.
be stated in different forms
 Length can be stated in different forms
forms
length.State and explain othersuchasofheight.
Lengthforms
canof bestated in different forms such aslength.

Explainsuch
yourasanswer.
height. State and explain other•
Diagram 2

Diagram 2
Diagram 2height. State and explain other forms of length.

1
1

114 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Indirect Comparison
IndirectComparison
IndirectComparison
Indirect Comparison

Indirect Comparisoni.

Diagram 3 shows a situation of comparing
Diagram
3 showssituationa situationof comparing
i. i.theDiagram
3 shows
height ofa metalatableand ofthecomparing
breadththeheightofa metal
table
and
the
breadthi.oftheDiagram3
showsasituationheight
of
a
metal
table
and
the
breadtha photo frame which is fixed on of
thecomparing
wall.
of
a
photo
frame
which
is
fixed
on
the
wall.
the
height
of
a
metal
table
and
the
of a photoframe awhich
is fixed
the breadth
wall.
i. Diagram
3 shows
situationof oncomparingof a photo frame which is fixed on the wall.
 height
Why isof
directcomparison
notthesuitableintheatable andbreadth Why
Why
is metaldirectcomparison
notsuitablein
isdirectcomparison
notsuitablethiscase?
of a photo
frame
whichcomparison
is fixed onnot
thesuitablewall.in incase?
thisthisWhy
is direct
case?
this case?
Without
acomparison
ruler or a measuring•  Why
is using
directsuitableWithoutusing
a ruleranotmeasuring tape,Withoutusing
a ruleror ora measuringhowindirectcomparison
can beintape,
this
case?
tape,
how
indirectcomparison
canWithout
using
a
ruler
or
a
measuringindirect comparison canbebedone in how
this case?
done
in
this
case? comparison can be
tape,
how
indirectdonein thiscase?
doneusingin thisacase?
•Without
ruler or a measuringtape, how indirect comparison can bedone in this case?

Diagram 3
Diagram
Diagram
3 3
Diagram
Diagram33
ii.

Diagram 4
Diagram
Diagram
Diagram
4 44
Diagram 4

Diagram 4 shows a pen and a pencils ofii. Diagram 4 shows a pen and a pencils of

Diagram
shows
pen
and a
aZaw
pencils
lengths.
Two
students,
Zawof
ii. ii. different
Diagram44shows
aa penandpencilsofdifferent
lengths.
Two
students,
Zaw
Zaw
ii.different
Diagram
4 shows
a students,pen
anda pencils
lengths.
Two
Zaw
Zaw
and
Khin,
used
two
different
methods
to ofdifferent
lengths.
Two
students,
Zaw
Zaw
and
Khin,
used
two
different
methods
tolengths.
Two
students,
Zaw
Zaw
anddifferent
Khin,
used
two
different
methods
comparethese
twolengths
as methods
shown
into
and
Khin,
used
two
different
toincomparethese
twolengths
as
shown
and
Khin,
used
two
different
methods
compare5 these
two lengths as shown in to
Diagrams
and 6.twocomparethese
lengths as shown in
Diagrams
5
andcomparethese
Diagrams 5 and 6. 6.two lengths as shown in
Diagrams
5 and
6. 6.
Diagrams
5 andExplain the similarities and differences
 Explain
Explain thesimilarities
anddifferences
 ofsimilarities
anddifferences
the two themethods.
ofthetwo
methods.

Explainthesimilarities
anddifferences•
Explainthemethods.
similarities and differencesof the two
of
the
two
methods.
 Whichofthesemethods
will
you
the
two methods.
ofWhichWhich
ofthesethesemethods
methodswillwillyouyou recommendof foryour students?
Explainrecommend
foryourstudents?
Explain

Which
of
these
methods
will
recommend
for your students? Explainyou
reasons.
• your
Which
of
these
methods will you
yourreasons.
recommend
your reasons. for your students? Explain
recommend
for your students? Explain
your reasons.
Zaw Zaw’s Method
Zaw
Zaw’s
MethodZaw
Zaw’s
MethodZaw Zaw’s Method

your reasons.

iii.

The paper clips used by Khin is an example
of non-standard unit for length.
•

Diagram
5 5Diagram
Diagram
Diagram
5 5Diagram5
Khin’s Method
Khin’s
MethodKhin’s Method
Khin’s Method

Diagram
6 6
Diagram
Diagram
Diagram
6 6
Diagram 6

Using these paper clips as units formeasurement, what is the length of thepencil and the pen, respectively?

iii. The
paper
clipstwo
usedby Khinis an example•iii.The
Suggestothernon-standard
units
The
paper
clips
used
by
Khin
is an
exampleiii. of
paper
clips
used
Khin
is an
examplenon-standard
unit
for bylength.
unit
forfornon-standard
lengththatyouuse. is an exampleiii.of ofThe
paper clips
usedbylength.
Khin
non-standard
unit
forcanlength.
 Usingthese paper
as units forof non-standard
unit clips
for length.
 Using
Usingthese
thesepaper
paperclips
clipsas asunits
unitsforformeasurement,
what
is
the
lengthof the• Whatmeasuring
aspects
shouldyoumeasurement,
whatisclipsthelength
of
Using
these
paper
as
units
for
measurement,
what
is
the
lengththethepenciland thepen, respectively?
considerwhenchoosing
the ofnonpencil
and
the
pen,
respectively?
measurement,
what
is
the
lengthof thepencil and the pen, respectively?
standard
unit?
pencilandotherthe pen,respectively?
 Suggest
two
non-standardunits Suggest
Suggestothernon-standard
units
 for
twotwoother
units
length thatyou
cannon-standard
use.
length
that
you
can
use.
 forfor
Suggesttwo
othernon-standard
units
length that you can use.
length that aspects
you can use.
 Whatformeasuring
should you WhatWhatmeasuring
measuringaspects
aspectsshould
shouldyouyou consider
when choosing
the nonconsiderwhenchoosing
the
non
What
measuring
aspects
shouldconsiderunit?
when choosing the non-you
standard
standard
unit?
considerwhen choosing the nonstandard unit?
standard unit?

2

2 2
2

Chapter 3: Quantity and Measurement | 115

iv.

Diagram 7
Diagram
7

iv. The pen in Diagram 4 is measured using

The pen in Diagram 4 is measured using different
different
paper
clips aspaper clips
as shown
in Diagram
7. shown in Diagram 7.

What is the length of the pen?

How can this task•

What is the length of the pen?

•

How can thistask help thestudents
appreciateofthestandard
appreciate
importanceimportanceofstandardunits
inmeasurement?
units in measurement?

help students

Diagram 7v.

Diagram 7 shows two cups with different shapes,Diagram
but can7 hold same amount of water.
Diagram 8
 What could you ask your students to do in order to convince them that both the cups hold thev. Diagram 7 shows two cups with different shapes, but can hold same amount of water.
same amountof water?
v. Diagram
shows
cups
with
different
shapes,but canthem
holdthatsameof water.
 8
What
could two
you ask
yourstudents
to do inorder to convince
both amountthe cups holdthe
 What conclusioncanyourofstudents
sameamountwater? make from this task?
•
What
could
you ask
students
to from
do in Whatconclusion
canyour
your students
makethisorder
task? to convince them that both the cups
hold the same amount of water?

Diagram 7•

What conclusion can your students make from this task?

Diagram 7 shows two cups with different shapes, but can hold same amount of water.
What could you ask your students to do in order to convince them that both the cups hold thesame amount of water?

What conclusion can your students make from this task?

Diagram 8
vi. Diagram 8 shows two glasses with same amount of orange juice and a scoop. However, mostDiagramchildren will choose glass A thinking
that it has8more juice since its level of juice is higher.


Diagram 9

Design a task using the scoop to compare the amount of juice in the two glasses.

 9Explain
howtheglasses
idea of denominationof quantity
for counting
is usedyourtaskaforcomparingvi. Diagram
shows
two
withamounta sameofjuiceorangeandscoop.
However,vi. Diagram
8 shows
two
glasseswithsameofamountorangeandjuice
ain scoop.
However,mostvolume of the two glasses.
mostchildrenwillchooseglass
Athinking
that
it
has
more
juice
since
its
level
of
juice ischildren will choose glass A thinking that it has more juice since its level of juice is higher.
higher.

Design a task using the scoop to compare the amount of juice in the two glasses.



• how
Designa taskthe scoop
to compare
the amountof juicein theExplainthe ideaof using
denominationof quantity
for counting
is usedin yourtasktwofor glasses.
comparingvolume of the two glasses.
•
Explain how the idea of denomination of quantity for counting is used in your task forcomparing volume of the two glasses.
3

Diagram 8

116 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 2: Introducing Quantity of Length and Expanding It to DistanceStandards 2.1:
Introducing centimetre for length and extend to millimetre and metrei. Compare
length of different objects and introduce centimetre with calibrated tape of one
Topic 2: Introducing Quantity of Length and Expanding It to DistancecentimetreStandards 2.1:
ii. Demonstrateequivalent length with addition and subtraction such as part-part wholeiii. Extend
centimetre
to millimetre
toextend
represent
remainingparts with ideas of equally dividing andIntroducing
centimetreforlength andto millimetre
andItmetre
Topic 2: Introducing
Quantityof LengthandExpanding
to Distanceidea of making tens
i.Standards
Compareof different objects and introduce centimetre with calibrated tape of one
2.1: length
iv. Extend
centimetre
to metreto measure using metre stickcentimetre
v. Estimate
lengthofobjectsandselectappropriate
toolsormeasuring
unit for measurement with
ii.Introducing
Demonstrateequivalent
lengthwithadditionand subtractionsuch
as part-part whole
centimetre
for lengthandextendto millimetre
and
metre
iii.
Extend centimetre to millimetre to represent remaining parts with ideas of equally dividing andfluencyidea
of makingtensof different objects and introduce centimetre with calibrated tape of onei.
Comparevi. Convert
andlengthcommonof length
for comparison
iv. mixed
Extendcentimetre
to metreunits
to measureusing metre
stick
centimetre
v.ii. mixed
Estimate
lengthof objectslengthand selectappropriate
toolsadding
or measuring
unit
for measurementwithvii. Convert
andcommonunitsof addition
lengthwhenorassubtractingin acquiring
the sense
Demonstrateequivalent
withandsubtraction
suchpart-partwholeiii. fluency
Extend centimetre to millimetre to represent remaining parts with ideas of equally dividing andfor quantityvi.
Convert mixed and common units of length for comparison
idea of making tensvii.
and commonof length
when
addingiv. Convert
Extend mixed
centimetre
to metre units
to measure
using
metre
stickor subtracting in acquiring the sense forquantityv.
Estimate
length of objects andselectappropriate tools or measuring unit for measurement with
Sample Tasks
for UnderstandingtheStandards
fluencySampleTasks forUnderstandingStandardsvi.
Convertmixed
and common theunitsof length for comparison
Task 1: Representing
Length
Using
Units
vii.
Convert mixed
and common
unitsVariousof length when
adding or subtracting in acquiring the sense for
Task 1:quantity
Representing Length Using Various Units
Sample Tasks
for Understanding the Standards
Comparing
LengthComparing Length

Task 1: Representing Length Using Various Units

Comparing Length

Paper

Scissors

Win

RewardWin

RewardPaper

Scissors

Rock
Diagram 1

Diagram 1i.

i.

Rock-paper-scissors is a game for two players. Diagram 1 shows the rules of the game. A win

Rockwith the rock, the paper
the scissors
be awarded
a one-unit,1two-unit,
three-unit
Rock-paper-scissors
is a andgame
for twowillplayers.
Diagramshowsandtherules cardof the game.
respectively. The cards won by each player
are arrangedin front of each player on the opposite
Diagram1
A win with
therock,table.
the Atpaper
thescissors
willbetheawarded
one-unit,istwo-unit,
andside ofa samethe endandof thegame,
the player
withlonger cardaarrangement
thei. winner.
Rock-paper-scissors
is a gametwo players.
Diagramrulesarrangedof the game.
win of each
three-unit
card respectively.
Theforcards
won byeach1 shows
playerthearein Afront
withandthe Maria
rock, thepaper
scissors
will beareand three-unit cardBen
playedtheandgame
the results
shown
Diagram
2. game,player ontheoppositeside
oftheaandsame
table.awardedAt
theainone-unit,
endof two-unit,
thethe player with therespectively. The cards won by each player are arranged in front of each player on the opposite
longer cardarrangement
isthe
theendwinner.
side ofa same table. Atof the game, the player with the longer card arrangement is thewinner.

and Maria
playedgame andresultsare shownin Diagram
Ben andBen
Maria
playedthethegame
andthethe
resultsare shownin2.Diagram 2.
Diagram 2 Without moving the cards, explain how Ben and Maria could use the one-unit card to determine
the winner.

•

Diagram
Diagram 22
 How will you relate the unit cards to the learning of centimetre?
 Without moving the cards, explain how Ben and Maria could use the one-unit card to determine
the moving
winner. the cards, explain how Ben and Maria could use the one-unit card to
Without
determine the winner.

1 How will you relate the unit cards to the learning of centimetre?

•

How will you relate the unit cards to the learning of centimetre?

1

Chapter 3: Quantity and Measurement | 117

Using Calibrated Tape to Introduce Centimetre and Milimetre

Diagram 3i.

ii.

Diagram 3 shows the scales on calibrated tapes which are marked 1 centimetre apart.
•Why is cm a better unit than other arbitrary units for measurement?

•

The pencil is 11 cm in length, but the stick is a little longer than 7 cm. How can you
use this situation to introduce millimetre as a remaining part of one centimetre?

A teacher placed a ruler callibrated with mm beside the stick as shown in Diagram 4.

Diagram 4•

How many mm are there in 1 cm?

•

The length of the stick is 7 cm and a remaining part. How many mm is the remaining
part?

•

What is the length of the stick in cm and mm?

118 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Extending Centimetre to Metre
ExtendingCentimetre
Metre
Extending
ExtendingCentimetre
Centimetre
toto
toMetreMetre

Diagram 5
Diagram
Diagram
333 track measured using a 30 cm ruler. The
Diagram 5 shows the length of a toyDiagram
car racing
total length of the track can be expressed as:i.
i.i.i.

Diagramshows
thelengthtoy
carracingtrack
measured
using
30
cm
ruler.
The
total
Diagram
Diagram333
shows
showsthethelengthlength
ofof
ofaaa
toy
toy
car
carracingracingtrack
trackmeasured
measured
using
using
aaa30
30cm
cmruler.
ruler.The
The
total
total

30 length
cm
+ of
30cm
+
30
cm
+expressed
25
cm =as:115
length
of
the
track
can
be
expressed
as:length
ofthe
the
track
track
can
can
be
be
expressed
as: cm
30
cm
30cm
30
cm
25
cm
115
cm
30
30
cm
cm+++
30cm
30cm+++
30
30
cm
cm+++
25
25
cm
cm===
115
115
cm
cm

•

Why is a metre stick with a length 100 cm more appropriate to measure the track?

Why
metre
stick
with
length100
cm
more
appropriate
measurethe
track?
 WhyWhyisisisaaa
metre
metre
stick
stickwithwithaaalength
length100
100
cm
cmmoremore
appropriate
appropriate
toto
tomeasuremeasurethethe
track?
track?

• Howtoexplaintherelationshipbetweenmetre
and centimetre?
Howtoexplaintherelationshipbetweenmetre
andcentimetre?
How
How
to
toexplain
explainthethe
relationshiprelationshipbetweenbetween
metre
metre
and
and
centimetre?
centimetre?

Diagram 6

Diagram
Diagram
Diagram444

ii. ii.ii.ii.Diagram
6 shows
partofameasuring
measuring
tape.
Three
students
gave
theirasreadings
as follows:
Diagram
shows
part
tape.
Three
studentsgave
theirreadings
follows:
Diagram
Diagram444
shows
showspart
part
ofof
ofaaameasuring
measuring
tape.
tape.
Three
Three
students
studentsgave
gavetheir
theirreadings
readingsas
asfollows:
follows:
Dave
Dave
Dave

Mary
Mary
Mary

Alan
Alan
Alan

ⓐ
ⓐ
ⓐisisis444mm
m

ⓐ
cm
ⓐ
ⓐisisis333mm
m777cm
cm

ⓐ
87
cm
ⓐ
ⓐisisis444mm
m87
87cm
cm

ⓑ
ⓑ
ⓑisisis888mm
mⓑⓑⓑisisis888mm
m

ⓑ
30
cm
ⓑisisis555mm
m30
30cm
cm
ⓑ

ⓒ
ⓒ
ⓒisisis777mm
m

ⓒ
cm
ⓒ
ⓒisisis777mm
m555cmcmⓒ
15
cmⓒisisis666mm
m15
15cm
cm
ⓒ

Which
sets
readings
arewrong?
 Which
Which
sets
sets
ofof
ofreadings
readingsare
arewrong?
wrong?

•

Which sets of readings are wrong?

What
are
the
difficulties
faced
by
the
students
who
made
the
wrong
readings?
 What
What
are
are
the
thedifficulties
difficultiesfaced
faced
by
by
the
the
students
studentswho
who
made
made
the
thewrong
wrong
readings?
readings?

•

What are the possible difficulties faced by the students who made the wrong readings?

•

How would you help the students overcome the difficulties in reading of measuring
scale?

How
would
you
help
the
students
overcomethedifficulties
readingmeasuring
scale?
 HowHow
would
wouldyouyouhelp
help
thethestudents
studentsovercome
overcomethethedifficulties
difficultiesinininreading
reading
ofof
ofmeasuring
measuringscale?
scale?

333

Chapter 3: Quantity and Measurement | 119iii.

Why is a metre stick not suitable for measuring
the size of the tree trunk in Diagram 7?
iii. Whyiii. isWhy
a metreis a stick
metrenotstick
suitablenot suitable
for measuring
for measuring
the
size
the
of
size
the
tree
of
the
trunk
tree
in
trunk
Diagramin
Diagram5?
5?
•Explain briefly how you could measure
the size of the trunk with a suitable Explain Explainbrieflytool.
briefly
how you
how
could
you measure
could measure
the themeasuring
size ofsize
the trunkof thewith
trunk
a withsuitablea suitablemeasuringmeasuring• tool.
Whattool.
units will you use? Justify yourchoice.
 What units
Whatwillunits
youwilluse?
youJustify
use? Justify
your yourchoice.choice.
•
Which of the following lengths could be
acceptable estimates for the size of the Which Whichof the followingof the following
lengthslengths
could becould betree trunk?
acceptable
acceptableestimates
estimates
for theforsize
theofsizethe of thetree trunk?
tree trunk?
30 to 50 mm
50 3030
– to
5030
mm
–cm
50 mm
2
to
3
m
 30– 5030cm– 50 cm
2 to 3 km
 2 –3 m2 – 3 m
 2 –3 km
2 – 3 km
Justify your answers.

Diagram 7

Diagram
Diagram5
5Justify Justify
your answers.
your answers.

Task 2: Conversion of Units for Length
Task 2:
Task
Conversion
2: Conversionof Unitsof forUnits
Lengthfor Length
Adding and Subtracting Length
AddingAddingand Subtracting
and Subtracting
LengthLengthi.

Diagram 8 shows a white board withmeasurement given by two students, Ary andi. Diagram
i. whereDiagram
6 theshows
6 shows
ais 325
white
acmwhite
board
with
Mimi
lengthand theboard
width withmeasurement
measurement
given
by
given
two
by
students,
two
students,
Ary
and
Ary andis 1 m 40 cm.
Mimi where
Mimi where
the length
the lengthis 325iscm
325
and
cmthe
and thewidth
iswidth
1 Mimim is
401foundcm.
m 40 differentcm.
Ary andanswers when

adding the total length of the four sides of the
Ary
and
Ary
andfoundMimi
different
found different
answers
when whenboard
asMimishown
in Diagram
9.answers
addingadding
the totalthelengthtotal lengthof the fourof thesides
fourofsides
the of the
board
as
board
shown
as
shown
in
Diagramin
Diagram7.
7.
•
Explain the differences between the two

Diagram 8
Diagram
Diagram6
6

•ways of calculating the answers.
Explain Explain
the differences
the differences
betweenbetweenthe twothe two
ways ofways
calculatingof calculating
the answers.
the answers.
Which answer is correct?

Which Whichansweranswer
is correct?
is correct?
How will you help the student with thewrong answer?
 How willHow
youwillhelp
youthehelp
student
the studentwith thewith theAry Ary
Mimi Mimi
wrong
wrong
answer?
answer?
ii. How will you guide students in the following
involving m, cm and mm?
325 cm325 cm
325 cm325 cm ii. calculations
Howii. willHow
youwillguide
you students
guide students
in the following
in the following
325 cm325 cm
325 cm325 cm
calculations
calculations
involving
involvingm, cm m,andcm
mm?
and mm?
•
7
cm
2
mm
+
5
cm
1 m 401 mcm 40 cm
140 cm140 cm+ 1 m+ 40
1 mcm 40 cm+ 140+ cm140 cm
 20
27mm
cm++2 25mm
cm+ 5 cm
• 73 cmm
cmm
652 m 652
80mcm 80 cm
930 cm930 cm
 3m
 203cm
m 20
+ 2cm
m+2m
•
23 cm 8 mm – 8 cm
Diagram 9 23cm 23
8 mm
cm –8 8cm
mm – 8cm
Diagram
Diagram
7
7


•

•


8 m 25 cm – 5 m 20 cm
8m
 258cm
m 25
– 5cm
m 20
– 5cm
m 20 cm

120 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 2.2:
Introducing distance for the extension of lengthi. Introduce kilometre to measure distance travelled using various tools and appreciate theexperiences of measuring skills
ii. Distinguish distance travelled and the distance of two places on the map
iii. Compare mixed units of length with appropriate scale on number line
Sample Tasks for Understanding the Standards
Task 1: Visualising a Distance of 1 Kilometre
Measuring Distance with km

Diagram 1i.

ii.

Diagram 1 shows a standard 400-metre with 8 running tracks in a school field.
•

Which track will exactly give a distance of 400 m in a complete round?

•

How do you use the running track to help students visualise and understand a distanceof 1 km? Explain your answer.

A group of children is going for a 1000-metre race.
•

How many rounds should they run?

•

Which track the children need to take for their advantage? Explain your answer.

•

To what extend can this task help children understand the relationship between km
and m?

Chapter 3: Quantity and Measurement | 121

Task 2: Distance Travelled and Distance Between Two Places on the Map
Adding and Subtracting Length

Diagram 2i.

Diagram 2 shows a location where Ali and Adam live.
Ali has to travel by car through junction A, junction B and to reach the grocery shop.

ii.

•

What is the distance travelled from Ali’s house to the grocery shop?

•

If Ali walks through a straight path from his house to the grocery shop, what is the
distance he walked?

What is the distance travelled by Adam from his house to the school?
•iii.

If Adam walks through a straight path from his house to the school, what is the distance
he walked?

How can you use this task to help your students learn the relationship between kilometre
and metre?

122 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 3: Compare Mixed Units of Length

Diagram 3i.

ii.

Diagram 3 shows a number line marked with km and m. Locate and mark the following
distance on the diagram
•

865 m

•

1280 m•

1 km 50 m•

1 km 500 m

Draw a number line to compare the following pairs of distance.
•

Which is longer, 3 km 60 m or 3080 m?

•

Which is longer, 8 km 6 m or 7 km 920 m?

•

Which is longer, 2580 m or 2 km 58 m?

Chapter 3: Quantity and Measurement | 123

Topic 3: Introducing Quantity of Mass for Its Measurement and Operation
Standards 3.1:

Topic 3: Introducing Quantity of Mass for Its Measurement and Operation
Topic 3: Introducing Quantity of Mass for Its Measurement and Operation

Introducing
gram for mass and extend to kilogram and tons
Standards 3.1:
Standardsmass
3.1: of different objects directly using balance and introduce grami. Compare
Introducing gramfor mass andextendto kilogram
tonssubtraction such as part-part wholeii. Demonstrateequivalent
mass
with
additionandandIntroducing gramfor mass andextendto kilogram
andtonsiii. Extend
gramtokilogram,measurewithweighingscalei.
Compare mass of different objects directly using balance
and introduce grami.
Comparemassofdifferent
directlyusingbalanceandintroduceii.
Demonstrateequivalentmasswith additionand subtractionsuch
as part-partiv. Extend
kilogramtometrictonobjects
through
relativemeasure(such
asgram25 whole
children, each weigh 40
ii.
Demonstrateequivalent
mass
with
additionandsubtraction
suchaspart-part
whole
iii.
Extend gram to kilogram, measure with weighing scale
kilogram)iii.
Extend kilogram
gram to kilogram,withweighing
scale (such as 25 children, each weigh 40
iv.
Extendto metric measure
ton throughrelativemeasurev. Estimate
massof objects
and
tools
or measuring
forweighmeasurement
withiv.
Extendkilogramto metric
tonselectthroughappropriate
relative measure(such
as 25 children,unit
each
40
kilogram)
fluencykilogram)mass of objects and select appropriate tools or measuring unit for measurement withv.
Estimate
vi. Convert
mixedmassand ofcommonunits
of appropriate
mass for comparisonv.
Estimate
objects andselecttools or measuring unit for measurement withfluency
fluencyvi.
Convertmixed
commonunits
units offorforcomparisonvii. Convert
mixed
andandcommonofmassmass
addition and subtraction in acquiring the sense
vi.quantity
Convert mixed
mixed andand common
common units
units ofof mass
mass forfor addition
comparisonvii.
Convertand subtraction in acquiring the sense for
for
vii.

Convert mixed and common units of mass for addition and subtraction in acquiring the sense forquantityquantity

Sample Tasks for Understanding the Standards
Sample Tasks for Understanding the Standards

SampleTasks for Mass
Understanding
the Standards
Task 1:Comparingof Objects
Task 1: Comparing Mass of Objects
Task 1: Comparing Mass of Objects

Whichobjecttheheaviest?
heaviest?
Whichobjectis istheWhich object is the heaviest?

Diagram 1i.

Diagram 1

Teacher Lim displayed a whiteboard eraser,a 1small piece of rock, a paper puncher and a
Diagrami. Teacher
whiteboard
eraser,a 1.
small
piece askedof rock,his
a paperpuncher
and a whichspongeblock Limon adisplayed
table asashownin Diagram
He thanstudents
to guessstyrofoam block on a table as shown in Diagram 1. He than asked his students to guess which
wasi.theheaviest
object.
Some
studentssaid
thesponge
block
was
the
heaviest.
Teacher
Lim displayed
whiteboard
a styrofoam
small pieceof rock,a paperpuncher and awasthe heaviest
object. aSome
studentseraser,
said theblock
was theheaviest.
••ii.

styrofoam block on a table as shown in Diagram 1. He than asked his students to guess which

the whyheaviest
object.
Some
students
said the styrofoam block was the heaviest.
wasExplain
why
answercouldbeExplainthetheanswercould
bewrong.
wrong.
Without
usingbalance,will you do to help the students correct their wrong
Explain why
theanyanswercouldwhat
be wrong.

perception,
any?balance, what will you do to help the students correct their wrong
Without
using ifany
 Without ifusing
any balance, what will you do to help the students correct their wrong
perception,any?
ii. Diagram 2 show a simple balance used for comparing mass. At any one time, the balance canperception, if any?
compare the mass of two objects.
Diagram
2 showa simpleusedforforcomparing
comparing
mass.
any
one
thecanbalanceii. Diagram
2 showa simplebalancebalance usedmass.
At anyAtone
time,thetime,balancecompare thethe massof of
twotwoobjects.
can compare
massobjects.

Diagram 2
What is the minimum number of times you need to compare the four objects for ordering them
Diagram 22
Diagramfrom the heaviest to the lightest object?
Explain your answer with diagrams.
 What is the minimum number of times you need to compare the four objects for ordering them
Whatfromis theminimumof times you need to compare the four objects forthe heaviest
to thenumber
lightest object?
ordering
them
from
the
heaviest
to the lightest object?
Explain your answer with diagrams.
•

1

Explain your answer with diagrams.

1

124 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram
Diagram 33iii.

iii. Diagram 3 shows some stacks of Malaysian coin.

Diagram 3 shows some stacks of Malaysian coin.
Diagram How could you use these coins witha simple3balance to order the four objects in Diagram 1.
•
How could you use these coins with a simple balance to order the four objects in Whatneed1?
tobe consideredwhen choosingiii. Diagram
3 showssomestacks of Malaysiancoin. the types of coins in the comparisons?
Diagram How will you use this situation to help your students learn measuring mass with non-standard • How
could
you usethesecoins withwhen
a simplebalancethe
to orderobjects
Diagram 1.
units?
What
need
to be
consideredchoosing
typestheof fourcoins
in theincomparisons?

 What need to be considered when choosing the types of coins in the comparisons?
iv. Diagram 4 show two faces of a Japanese 1-yen coin. Interestingly, one piece of 1-yen coin•
How1 willyou
use this situation to help your students learn measuring mass with nongramuse(g).this situation to help your students learn measuring mass with non-standard
 weighs
How willyou
standard
units?
units?
iv. Diagram
4 show
twotwofaces
of a Japanese
1-yen 1-yencoin. Interestingly,
one pieceof 1-yeniv.
Diagram
4 show
faces
of a Japanesecoin. Interestingly,
one
piececoinof 1-yen coinweighs(g).(g).
weighs1 1gram
gram

Diagram 4

Diagram 5

 As shown in Diagram 5, the eraser is balanced by 24 pieces of 1-yen coin.
44
Diagram55
Diagram
Diagram HowDiagram
many g is
the whiteboard eraser? Explain your answer.
shownIf theinnumberof coins
increase
to 48 pieces,
will happen As
Diagram5, the iseraseris balanced
by 24predictpieces whatof 1-yencoin. to the left pan of the•
Asbalance?
shown in Diagram 5, the eraser is balanced by 24 pieces of 1-yen coin.
 How many g is the whiteboard eraser? Explain your answer.
 Whatwillmany
happenthe whiteboard
balance if anothereraseris added
to answer.
the left pan?
Howg istothe
eraser?
Explainyour If the number of coins is increase to 48 pieces, predict what will happen to the left pan of thebalance?
 
FindIfany
item ofthat
weighs1 g.
thesmallnumbercoins
is increase
to 48 pieces, predict what will happen to the leftpan
the balance?
 Whatwill ofhappento yourthe balanceanother
eraseris added
to thepan?the item.
Designa task forstudentsif tomeasuremassof objects
in gleft
using



 any
What
will
happento the1 balance
if another eraser is added to the left pan?
Find
small
item
that weighsg.

Designa taskforitemyour thatstudents
to measure•  Findany
small
weighs
1 g. mass of objects in g using the item.



Design a task for your students to measure mass of objects in g using the item.

22

Chapter 3: Quantity and Measurement | 125

Task 2: Addition and Subtraction Involving Mass
Task 2: Addition and Subtraction Involving Mass

Diagram 6i.

Diagram 6

Diagram 3 show two situations with a balance scale of 1 kilogramme mass capacity.
 Some students recorded the mass of oranges as 850 g.

i.

 What is the error?

Diagram 6 show two situations with a balance scale of 1 kilogramme mass capacity.
 How will you help the students to correct the error?

• A recorded
90 g apple is added
basket withoranges.
Some students
theto themassof oranges
as 850 g.
 What is the new reading on the balance scale?

What is the error?

•

How will you help the students to correct the error?

•

A 90 g apple is added to the basket with oranges.
What is the new reading on the balance scale?

3126 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 3: Extend
Units of Mass to Larger Units
Task 3: Extend Units of Mass to Larger Units

Task 3: Extend Units of Mass to Larger Units

Diagram 4i.

i.

Diagram 4 shows two situations involving a beaker on a 1-kg weighing scale.

Diagram
7 shows
involving a beaker on a 1-kg weighing scale.
 Whatis the twomasssituationsof 500 ml of water?
•i.

Diagram 7What is the mass of 1000 ml of water?
Diagram 4

What
is themassof 500ml toofintroduce
water?kilogramme to your students?
 How
will youuse thissituation

Diagram 4 shows two situations involving a beaker on a 1-kg weighing scale.
•What Whatis themassof 1000 ml of water?
is the massof 500ml of water?
What is the mass of 1000 ml of water?

ii.kilogramme
Diagram 5 shows
the reading
for the mass•How willHow
yousituation
use thissituation
to introduce
to yourstudents?
you will
use this
to introduce
kilogramme
to your students?
of three identical books.
What is the mass of each book?

ii. Diagram 5 shows the reading for the mass
identical books.
ii.of three
Diagram
8 shows the reading for the mass of

three
identical
books.
A teacherhis students to find theWhat iii.
is themass of asked
each book?
mass of their respective school bag using•

a weighing scale. However, the students
Whatis the mass of each book?
are asked to estimate the mass of theirbags before the actual weighting.
iii.iii.
A teacherasked hisstudents
to findthe
A teacherasked
his
students
to find the mass
mass of their respective
using an estimation
 How school
does bagmakingof
theirrespective
schoolbag
using
a weighing scale. before
However,actual
the students
weighing help ain weighing
are asked
to estimate
the
mass
of
their
scale.
However,thestudentsaredeveloping a sense of quantity amongasked tobags before the actual
students?
estimate yourthe weighting.
mass of their bags before theDiagram 5

actualweighting.
How
doesmaking an estimation
before actual weighing help indeveloping a sense of quantity among• students?
How does making an
yourestimation before
actual weighing help in developing a senseof quantity among your students?
4

Diagram 8
Diagram 5

4

Chapter 3: Quantity and Measurement | 127

iv.

Diagram 9 shows another 10-kg weight scale.

iv. Diagram 6 shows another 10-kg weight
Label the appropriate graduation marks on thescale. Label the appropriate graduation
marksscale.
on the scale.

iv. Diagram 6 shows another 5-kg weightscale. Label the appropriate graduation
marks on the scale.

Diagram 6 9
Diagram

Diagram 6

v. Diagram 7 shows a pencil, a can of drink,a pair of boots, a boy and a list ofmeasurements for mass.

v.

Diagram 10 shows a pencil, a can of drink, aMatchitem witha its
acceptable
pair7eachofboots,
boy
and a list of mass.
v. Diagramshows
estimatedmass.a pencil, a can of drink,a pair of boots, a boy and a list of mass.


1 to 2 kg– 22 kg
1 to
kg

35 to 40 kg



35 to
40 kg
kg
– 40



5 to 10 g



5 to
10 gg
– 10



300 to 350 g



300 to350 gg
– 350



How
this task help
in developing• doesMatcheachitem with a Matcheachitem withacceptablesense ofkg among
youritsstudents?
estimated
mass.
estimatedmass.



its acceptable

How does this task help in developingHow
does yourthisstudents?
task help in developing aa•senseof kg among

sense of kg among your students?

Diagram 7

5
Diagram 7

Diagram 10
5128 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram 11

Diagram 8

vi. Diagram 11 shows a man standing on a bathroom scale. The reading shows 80 kg.

vi.

Diagram 8 shows a boy standing on a bathroom scale. The reading shows 80 kg.

What
is massthe mass
10 men?
 • Whatis theof 10 ofmen?

Diagram 8

Diagram8
 Approximately,
men willaweigh1000scale.
kg?
Diagram
8 shows ahow
boy many
standing
bathroom•
Approximately,howmanyonmen
will weigh1000Thekg?reading shows 80 kg.
A busispassengers.
whatTheis themaximumload
Whatthetakemassof
10 men?onApproximately,vi.  Diagram
8can
shows
a42
boy
standinga bathroom scale.
readingshows
80 ofkg.the bus in kg?
•
A
bus
can
take
42
passengers.
Approximately,
what
is
the
maximumload of the bus inHowyoumass
use these
to introduce metric tonne?
Whatwillis theof many
10 situationsmen?
  Approximately,
how
men will
weigh 1000 kg?
kg?
 How does this task
help
in mendeveloping
a sensemetric tonne among your students?
howpassengers.
many
will
weigh
1000ofkg?
  AApproximately,
bus can take 42
Approximately,
what is the maximum load of the bus in kg?
•
How
will
you
use
these
situations
tointroduce
tonne? load of the bus in kg?
A bus
take
passengers.
Approximately,
ismetric
the maximum  How
willcanyou
use42thesesituations
to introduce what
metrictonne?

vi.

vii. In Malaysia, the capacity of a lorry isHowwill doesyou
these
to introduce
tonne?
  •How
does
thisusetask
helpsituations
in developing
a senseof metric
tonne
among
your
students?
measuredbased
onamong
the goods
carries
How
this
task
help
in developing
ametric
sense
of metric
tonne
youritstudents?
and tonne
expressed
in metric
tonne.
How does this task help in developing a sense of metric
among
your students?

vii. Invii.Malaysia,
the it meancapacityof“a aof
lorry
is is
 InWhatdoes
to say2 -metric
Malaysia,
thecapacitya lorrymeasured
based
on
goods
it acarries
and
tonne
lorry”? thevii. In measured
Malaysia,
thecapacity
lorry
is
based
on
theofgoods
it carries
expressed
in metricmeasured
based tonne.
onthe goods
it carries
and expressedin metric
tonne.
••
Diagram 9

Diagram 12
Diagram
Diagram 99

 Diagrama lorry with theandexpressed9inshows
metric tonne.
following
information:What
does doesit meanto say“a 2“a-metric Whatit meanto say2 -metric
tonneWhat
does
it meantonnelorry”? to say “a 2 -metric
lorry”?
BDMlorry”
2500KG
tonne?
(Permisible Laden Weight)

 BTM
Diagram
9 shows
a lorry
the
1810KG
12
lorry
withwithDiagram
Diagram
9 showsshowsweightaa lorrywiththethe(Curbweight=actualof lorry)following
information:
following
information:
following
information:
 BDM
What
is2500KG
the actual mass of theBDM
2500KG(Permisible
Laden
Weight)lorry?
(Permisible
Laden
Weight)
BTM
1810KG
 BTM
How
many
kg
of
goods can the
1810KG(Curb
weight=actual
weightof lorry)
lorry
carry?
(Curb
weight=actual
weightof lorry)

What
What
is theactualof theis theactualmassof lorry?
the Whatis theactualmass
ofmassthelorry?
lorry?
many
of
can
the
How
How
many
kg goodsof cangoods Howmany
kg kg
of goods
thecanlorrythelorry
carry?
lorry
carry?
carry?

6

Task
Conversion
Units
Task
4: 4:
Conversion
ofof
Units

Chapter 3: Quantity and Measurement | 129

Task 4: Conversionof Units
Comparing
Mass
Comparing
Mass
Comparing
Mass
Which
objectheavier?
Explainyouranswers.
i. i.Whichobjectis isheavier?
Explainyouranswers.
i.  OneWhich
object ofisofheavier?
Explainyourofanswers.
One
kilogramcottonone
kilogramofiron?
kilogramcottonor oronekilogramiron?
Onecottonone
kilogram of iron?
1 kg
100
g of
iron
or
1200
gor
of
cotton?
 1• kg
100
gkilogram
of
iron
orof
1200
g of
cotton?
1 200
kgg100goroforiron
orcotton?
g of cotton?
1 kg
g iron1100
g1200
cotton?
 1• kg
200
iron
1100
g
•

1 kg 20 g iron or 1100 g cotton?

Diagram 13
Diagram
Diagram1010
ii. Diagram 13 shows a number line marked with kg and g. Locate and mark the following
Diagramshows
a numberline
markedwith
andLocateandmarkfollowingmass.
ii. ii.Diagramshows
a numberline
marked
with
kgkg
and
g. g.Locateandmarkthethefollowing
mass.
mass.1010
736
 • 736
g gg
736
2480
  2480gg•
2480 g1 kg
  1 kg
6060
gg•
1 kg 60 g
2 kg
600
  2 kg
600
gg•
2 kg 600 g
Draw
a numberline
comparefollowing
pairs
distance.
iii.iii.
Draw
a numberline
to tocomparethethefollowing
pairs
of of
distance.

iii. Draw
a numberline5tokgcompare
the following
pairs of mass.
Whichlighter,g or4580
 Whichis islighter,
5 kg 2020
g or
4580
g?g?
Which
isheavier,lighter,
kg
20
orkg
g?
Which
95 kg
ggor
84580
kg
120
 • Whichis isheavier,
9 kg
1515
g or
8
120
g?g?
•

Which is heavier, 9 kg 15 g or 8 kg 120 g?

AdditionandSubtraction
Involving
Mass
AdditionandSubtraction
Involving
Mass

Addition and Subtraction Involving Mass

350
+ 85
  350
kgkg
8080
g +g 85kgkg

Diagramshows
twocalculations
involvingunits
Diagram
1111
shows
two
calculations
involving
units
of
mass.
of mass.
Diagram 14 shows two calculations involving units ofmass. i. i. What
What
arecommonerrorsyourstudents?
arethethecommonerrorsof of
yourstudents?

2 tonne245– 786
  2 tonne
245
kgkg
– 786
kgkg

i.

Diagram14
Diagram
Diagram
1111

ii.

How
you
help
your
correctWhat
thecommonerrorsofstudents
your students?
ii. ii. areHowwillwill
you
help
your
students
to tocorrect
thetheerrors?
errors?
How will you help your students to correct theerrors?

77

130 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 4: Introducing Quantity of Liquid Capacity for its Measurement and Operation
Topic 4: 4.1:
Introducing Quantity of Liquid Capacity for its Measurement and Operation
Standards
Standards 4.1:

Introducing litre for capacity of liquid and extend to millilitre
for capacityof liquidand extend
to millilitreand introduce litre with measuring cups of 1
i. Introducing
Compare litre
amountof waterin different
containers
litre
i.
Compare amount of water in different containers and introduce litre with measuring cups of 1ii. Demonstrateequivalent capacity with addition and subtraction such as part-part wholelitreiii. ii.Extend
litre
bydecilitre/100
millilitre
cup
for representing
remainingparts withideas of equally
Demonstrateequivalent capacitywithadditionand subtraction
such as part-part
wholedividing
andlitremaking10, and extend
untilformillilitreiii.
Extendby decilitre/100
millilitre cup
representing remaining parts with ideas of equallydividing
and making
10, and extend
until millilitre
iv. Estimate
capacityof containers
and selectappropriate measuring unit
Estimate
capacityof containers
appropriate
measuring unit
v. iv.Convert
mixed
and common
unitsandof select
capacity
for comparisonv.
Convert mixed and common units of capacity for comparison
vi. vi.Convert
mixed and common units of capacity for addition and subtraction in acquiring theConvert mixed and common units of capacity for addition and subtraction in acquiring the sense
sense
for
quantityfor quantity
SampleTasks
forUnderstanding
Understanding the
SampleTasks
fortheStandards
Standards
Task 1: Measuring Amount of Liquid in Litre and Decilitre

Task 1: Measuring Amount of Liquid in Litre and Decilitre

My bottle fills 8 cups.
Mine is more than yours

My bottle
fills 7 cups.
John

Jason

Diagram 1
Diagram 1i.

Diagram1 shows
Jason
eachwithwitha bottleof orange
juice.theOver
the phone,i. Diagram
1 shows
Jasonandand John,
John, eacha bottleof orange
juice. Overphone,
JasonJason
and andJohnJohn
arearediscussing
which
of
their
bottle
can
hold
more
juice.
discussing which of their bottle can hold more juice.


Can Jason claims that his bottle can hold more juice? Why or why not?

How can you compare the amount of juice accurately?

•

Can Jason claims that his bottle can hold more juice? Why or why not?

•How can you compare the amount of juice accurately?

1Chapter 3: Quantity and Measurement | 131

ii.

1ℓ
1ℓ

Diagram
2 shows
containerthat cancan hold
1 litre aofbig
water.
The waterin onehold
litre
water.
The
water
in thatone
big
Diagram
2 of
shows
a big
can
big1container
canfillcontainer
10
small
containers.
container
canfill 10 small
containers.
hold
1 litreof water.
Theiswaterin one One
big
One
small
container
1 decilitre.
smallcontainer
1 decilitre.
container
can fillis10
small containers. One
small container is 1 decilitre.
 • Whatis theisrelationshipbetween between
litre
What
the relationshipdecilitres?
 and
What
isand
the relationshiplitre
decilitres? between litre
and decilitres?

iii.

Jason and John use these containers

Diagram 2
Diagram
2 2
Diagram

Jason’s bottle
Jason’s bottle
1ℓ
1ℓ

Diagram 2 shows a big container that

ii.
ii.

iii. Jason and John use these containers toto measure
theoftheir
juice toin theiramountof juiceiniii. measure
Jason
andtheJohn
useamountthese
containers
respective
bottle.
The
resultsare shownrespective
bottle.
Theofresultsshowninmeasure theamountjuice inare
theirinDiagram3.
Diagramrespective3.bottle. The results are shown in
Diagram 3.
 What is the amount of orange juice inWhat
is amount
the amountof orange
bottle?
 • Jason’s
Whatis theof orange
juice in juice
inJason’sbottle?
Jason’s bottle?
 What is the amount of orange juice in John’s
What isbottle?
the amount of orange juice in•What is the amount of orange juice
John’s bottle?

in John’s bottle?

John’s bottle
John’s bottle

1ℓ
1ℓ

Diagram
Diagram 33
Diagram 322

132 | Mathematics Challenges for Classroom Practices at the Lower Primary Leveliv.

Diagram 4 shows a task given as an exercise
for students.
•

What is the purpose of the task?

Diagram 4

Diagram 5v.

Diagram 5 shows the amount of water in a kettle measured using 1-litre cups.
•

What is the amount of water in the kettle?

Chapter 3: Quantity and Measurement | 133
Task 2: Measuring
Amountof WaterMillilitre
Task 2: Measuring
Amountof WaterininMillilitre
Task 2: Measuring Amount of Water in Millilitre

11dldℓ11dldℓDiagram 6

Diagram 6i.

Diagram 6

vi. 6Diagrama one-decilitre
containerandand aaten-millilitre
medicinecup. Water
the
Diagramshows6ashows
one-decilitre
container
ten-millilitre
medicine
cup.inWaterin theone-decilitre container can fill 10 ten-millilitre medicine cups.
one-decilitre
container can fill 10 ten-millilitre medicine cups.
Diagram
6 shows
a one-decilitre
containera ten-millilitre
vi. Whatis therelationshipbetweendecilitreandandmillilitre? medicine cup. Water in the
one-decilitre container can fill 10 ten-millilitre medicine cups.

•

is the relationshipbetweendecilitre
litre and millilitre?
What is Whattherelationshipbetweenand millilitre?
What is the relationship between decilitre and millilitre?

•

What is the relationship between litre and millilitre?

What is the relationship between litre and millilitre?

1 dℓ

1 dℓ

Diagram 7
Diagram 7

Diagram 7ii.

vii. vii.
Water
from
aa
jug
Theresultresultisisshown
shown
DiagramWater
fromjugisispoured
pouredinto
intothese
these containers.
containers. The
in in
Diagram
7. 7.

Water fromjug isisthe
pouredthese
containers.
The result is shown in Diagram 7.
 aWhatamountinto
of waterin thejug?
•What is the amount of water in the jug?

What is the amount of water in the jug?

44134 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram
8 88
Diagram
Diagram

Diagram 8
viii.
Diagram
8 8shows
theamountofofwater
ininainabottle
measuredusing
1-decilitre
cups.
viii.
Diagramtheamountmeasuredusing
1-decilitre
cups.
viii.
Diagram8showsshows
theamountofwaterwater
abottlebottlemeasuredusing
1-decilitre
cups.
iii.

Diagram 8 shows the amount of water in a bottle measured using 1-decilitre cups.
   Whatisistheamountofofwater
ininthebottle?
Whatamountbottle?
What
isthe
the
amountofwater
water
inthe
the
bottle?
•What is the amount of water in the bottle?
iv.ix.ix.Many
Many
types
of of
drinksarepackedandmeasured
ix.
types
are
and
Many
types
ofofdrinks
drinksarepacked
and
Many
types
drinksarepacked
packedandinmeasured
mililitre.
For
examples,abottleofmineralmeasuredinmililitre.
For
examples,a abottle
measuredininmililitre.
mililitre. For
Forexamples,examples,abottlebottlewater
is 500mℓ
anda500
packageorange
juice
of
water
isis500mℓandaofapackage
ofofofofmineral
water
mℓ
and
packageofmineralmineral
water
is500
mℓ
and
a
packageorange
juice
mℓ
as
shown
ininDiagram
8.8.8.
juice
is200
mℓ
asas
shown
isorange
200 mℓasisshown
inmℓDiagram9.
orange
juice
is200
200
shown
inDiagram
Diagram •How
will
you
use
these
toto
help
How
will
you
use
these
items
help
How
will
you
use
items
to
help
How
will
you
theseitems
items
to
help
students
estimate
volumeofofofsomecommonstudents
estimate
volumecommonstudents
estimate
volumeofsomesomecommonstudents
estimate
somecommoncontainers
suchasasascontainers
such
containers
such
containers
such
as

500
mℓmℓmℓ500
500
Diagram
8 88
Diagram
Diagram
Diagram 9200mℓmℓmℓ
200
200

cup,
adrinkingcup,adrinkingdrinkingcup,•aa drinking
cup,a abowl,
andandabowl,bowl,
and•a bowl, anda abottle
ofoffruitjuice?
juice?
abottlebottleoffruit
fruit
juice?

•

a bottle of fruit juice?

v. Diagram 10 shows a teaspoon and a tablespoon.
x.x.x.Diagram
Diagramshows
teaspoonandDiagram9 9 9shows
showsa a ateaspoon
teaspoonand
anda a atablespoon.
tablespoon.
• tablespoon.
Why are these spoons usually used tomeasureliquidcooking
ingredientsin daily   Whyarethesespoonsusually
used
tototo
Why
are
these
spoonsusually
used
Why
are
these
spoonsusuallyused
life?
measureliquidcooking
ingredients
inindaily
measureliquidcooking
ingredientsmeasureliquidcooking
ingredients
indailydaily

Diagram
9 99
10Diagram
Diagram

life?
life?
life?
•
Is there a standard capacity for each of   IsIsthesethere
a astandard
capacityfor
each
ofofofthere
capacity
for
each
Is
there
astandard
standard
capacityforeachspoons?
these
spoons?
these
spoons?
these
spoons?

Chapter 3: Quantity and Measurement | 135

Task 3: Conversion of Units

Task 3: Conversion of Units

Comparing Liquid Capacity

Comparing Liquid Capacity

100 mℓTask 3: Conversion of Units

1 dℓ

200 mℓ2 dℓDiagram 10

300 mℓ3 dℓDiagram 11

Comparing Liquid Capacityi.

Diagram 10 shows a number line marked with decilitre and millilitre. Locate and mark the
Diagram
11 shows
a number line marked with decilitre and millilitre. Locate and mark thefollowingliquid capacity.

i.

following liquid capacity.


•
•
i.

ii.

iii.

100 mℓ200 mℓ300 mℓ35 mℓ
280 mℓ

1 dℓ 50 mℓ



2 dℓ 8 mℓ

280 mℓ

1 dℓ

2 dℓ
Diagram 10

3 dℓ

a numberline to compare the following pairs of liquid capacity.
•ii. Draw
1 dℓ 50mℓDiagram 10 shows a number line marked with decilitre and millilitre. Locate and mark the Which is more, 7 ℓ or 72 dℓ?
• following
2 dℓ 8 liquidmℓ capacity.
 Which is more, 5 dℓ 28 mℓ or 485 mℓ?
 35 mℓ aWhich
is more,2 ℓ compare
40 mℓ or 380
Drawnumberline tothemℓ?
following pairs of liquid capacity.
 280 mℓ

•
ii.

35 mℓ

Which
more, 7 ℓ or 72 dℓ?
1 dℓ 50ismℓ

2 dℓ
8 is
mℓ
AdditionandSubtraction•  Whichmore, 5 dℓInvolving
28 mℓ or Liquid
485 mℓ?
Capacityi. pairs
Diagram
11 shows
a problem involving
Draw a number line to compare the followingof liquidcapacity.
liquid capacity. A student gave the answer•
Which is more, 2 ℓ 40 mℓ or 380 mℓ?
 Which is more, 7 ℓ or 72 dℓ?
as 5 ℓ.
There are 7 ℓ of waterin a bottle.
1

A centilitre,cℓ,isdrank.
of 28
a litre.
2Whichdℓ has is
been
more,
5 dℓ
mℓ or 485 mℓ?
100
•
•

How much is left?

Which
is more,2 ℓ140How manymℓ iscℓ?mℓ or 380 mℓ?
Diagram 11

How many cℓ is 1 dℓ?

 Why is there an error?
 How will you help the student overcome
the error?

ii. Based on your explanation in (i), how will
you help students who give 8 dℓ as the
Addition and Subtraction Involving Liquidanswer for (7dℓ + 1ℓ)?
Addition
Capacity and Subtraction Involving Liquid Capacity

i. Diagram 11 shows a problem involving

There are 7 ℓ of water in a bottle.
2 dℓ has been drank.
How much is left?
Diagram 12
11
Diagramcapacity.
A student
gave the answeriii. In liquid
what ways
are measuringof length,
i. mass
Diagram
12 capacity
shows thea problem
same? involving
as 5andℓ. liquid

liquid capacity. A student gave the answerℓ. is there an error?
as 5
Why

Whywillis therean error?
• How
you help
the student overcome
the error?
•
How will you help the student
overcome
error? in (i), how willii. Basedon your the
explanation
you help students who give 8 dℓ as theii. Basedin (i), how willansweronforyour(7dℓexplanation+ 1ℓ)?
you help students who give 8 dℓ as the(7 aredℓ +measuring
1 ℓ)?
iii. answer
In what for
ways
of length,
mass and liquid capacity the same?
iii. In what ways are measuring of length,
mass and liquid capacity the same? 6

136 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 5: Introducing Time and Duration and Its Operation
Standards 5.1:

Topic 5: Introducing Time and Duration and Its Operation

Introducing analogue time and extend it to duration
Standards 5.1:
i. Tell and write analogue time of the day corresponding with different activities in daily life suchasIntroducing
morning, analoguenoon, afternoon,
and
night
time and day
extendit toduration
ii. Show time by using clock face with hour hand and minute handi.
Tell and write analogue time of the day corresponding with different activities in daily life such asiii. Understandthe relative movement of clock hands
morning, noon, afternoon, day and nightii.

Show time by using clock face with hour hand and minute hand

Samplefor Understanding
the Standardsiii. Tasks
Understandthe relative movementof clock hands
SampleTasks
Understanding
the Standards
Task 1:
Tellingand forWriting
Time
Task 1: Telling and Writing Time

① Going to school② Going home from school③ Dinner time

④ Bedtime
Diagram 1

Diagram 1

Chapter 3: Quantity and Measurement | 137i.

Diagram 1 shows four activities in a day.
•

Draw the hour and minute hands to show your time to do each activity.

•

How does this task help your students to understand time?

•

Discuss the common difficulties faced by students in showing and telling time.

Diagram 2
ii.

Diagram 2 shows a clock face marked with hours and minutes.
•

How will you use this clock face to help your students understand and tell time?

•

How do you explain the relative movement of the minute hand and the hour hand?

•

How many rounds will the hour hand travel if the minute hand travels one complete
round?

•How many rounds will the minute hand travel if the hour hand travel one complete
round?

138 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 5.2:
Extending clock time to duration of one day
i. Introduce duration in hours and minutes based on beginning time and end time of activities
ii. Express
Standards time
5.2: and duration on time line, and understand duration as difference of two
Standards 5.2:
distinguished
times
Extending clock time to duration of one day
Extending andclocksubtraction
time to duration
one day and timeiii. Addition
of ofduration
iv. Extend
timeand
durationtosecondsi.
Introduce
durationin hours
andminutes based on beginning time and end time of activitiesi.
Introduce duration in hours and minutes based on beginning time and end time of activities
ii.
Expresstimeanddurationunits
on timeline, and forunderstandduration as difference of two
v. Convert
mixed
and
commonof durationcomparisonii.
Express time and duration on timeline, and understandduration as difference of two
distinguished times
distinguishedtimes
vi. Estimate
duration
of
time
and
selectappropriate
measuring
unit for measurement with fluency
iii.
Addition and subtraction of duration and timeiii. appreciate
Addition andthe
subtractionof durationand timeandsignificance
of
time
and
duration
in
life
iv.
Extend time and duration to seconds
iv.
Extend time and duration to secondsv.
Convertand common
units
of duration forvii. Appreciatethemixed
differencein timedependingoncomparison
the area (time zone) and the seasonsv.
Convertmixed
and common
units
of duration forcomparison
vi.
vi.

Estimate duration of time and select appropriate measuring unit for measurement with fluency
Estimate duration of time and select appropriate measuring unit for measurement with fluency
and appreciate the significance of time and duration in life
and appreciate
the significance of timeand duration in life
Samplefor UnderstandingStandards
vii. Tasks
Appreciate
the difference in timethe
dependingon the area (time zone) and the seasonsvii.
Appreciate the difference in time depending on the area (time zone) and the seasons
Tasks forUnderstanding
the Standards
TaskSample
1: A Sense
of Hourand MinuteSampleTasks forUnderstanding
the Standards
Task 1: A Sense of Hour and MinuteTask 1: A Sense of Hour and Minute

7:00
7:00
7:00

::

::

::

::

Ⓐ Jason left home.
Ⓐ Jason left home.

Ⓑ Jason arrived atⒷ Jason arrived at
the bus stop.
the bus stop.

Ⓒ The bus departedⒸ The bus departed
from the bus stop.
from the bus stop.

Ⓓ The bus arrived atⒹ The bus arrived at
the school gate.
the school gate.

Ⓔ Lesson started.
Ⓔ Lesson started.

Diagram 11
Diagram
Diagram 1i.

thetimes
times
Jason
to travel
from
i.i.Diagram
Diagram11shows
shows thefor forJason
to travelfrom home
to homeschool. to school.
Diagram 1 shows the times for Jason to travel from home to school.
Read and write the time for each activity.

How many minutes was the bus journey?
How many minutes was the bus journey?

Read andand write
time
for eachactivity.
•  Read
writethethe
timefor eachactivity.

How many minutes was the bus journey?

•

Diagram
Diagram 22
Diagram 2

ii.

ii.
Diagram22shows
shows a timeband
between
7 o’clockto 8 o’clock
intodivided
60 intervals.
Diagramtime
scale
between
7 o’clockto 8divided
o’clock
into 60 intervals.
ii.
Diagram 2 shows aatime
band
between
7 o’clockto 8 o’clock
dividedinto 60 intervals.
What is the duration for 1 interval?
What is the duration for 1 interval?

•  What
durationinterval?
Mark isthethedurationof thefor
bus1journey
on Diagram 2.
••

 Mark the duration of the bus journey on Diagram 2.
 Some students faced difficulties in finding the duration of the bus journey. How will you use MarkSomethestudents facedofdifficulties
finding the durationof the bus journey. How will you use
businjourney
Diagram 2duration
to help thesethe
students
overcomeon
theDiagram
difficulties?2.
Diagram 2 to help these students overcome the difficulties?

Some students faced difficulties in finding the duration of the bus journey. How will you
use Diagram 2 to help these students overcome the difficulties?
33

Chapter 3: Quantity and Measurement | 139
One Dayin Hours
Task 2:Task
One2:Dayin Hours

Task 2: One Day in Hours

Diagram
Diagram 33

i. Diagram
3 shows
time
scalewhere
where 11day
is divided
into 24into
intervals.
Diagram
3 shows
time
band
day
is divided
24 intervals.

i.

What is the duration for 1 interval?

 Johnwent
for a fieldfor
trip.1 The
trip started at 7 o’clock in the morning and ended at 3 o’clock in
Whatis thedurationinterval?
the afternoon. Mark the duration of the trip on Diagram 3.

•

 Whatthe adurationof the The
trip intrip
hours?
John
wentisforfield trip.
started at 7 o’clock in the morning and ended at 3•

ii.o’clock
How many
completeroundsMark
will eachthe clock hands
moves
in a Diagram
day?
in theafternoon.
theofdurationof thetrip on3.
iii.

Ali says he will visit his best friend, Ahmad at 8 o’clock tomorrow.

What isdurationof thein Diagram
hours? 3
 the
Whyis the statedtime trip
confusing?

•

How to get rid of the confusion?



i.ii. Diagram
3 shows
time band
where
day isintohands
24 intervals.
How many
completeroundswill1eachofdivided
the clockmoves in a day?
iii.





ii.

What is the duration for 1 interval?

Task 3: Time Duration in Seconds

Ali says he will visit his best friend, Ahmad at 8 o’clock tomorrow.

John went for a field trip. The trip started at 7 o’clock in the morning and ended at 3 o’clock inthei. afternoon.
Mark
the trip on Diagram 3.
Stand on one
footthewithduration
your eyesof
closed.

Why is the stated time confusing?

•

 the
How
many seconds
balance yourself?
What isdurationof the cantripyouin hours?
ii.

Hold your breath.

• manyHow toget rid of
the confusion?
Howcompleteroundswill each of the clock hands moves in a day?
How many seconds can you hold?



iii.

Some examples of units for measuring time duration are day, hour, minute and second.

iv.

Three students competed in a 400 m race. The result is recorded in Table 1.

Task 3: Time Duration
inisSeconds
 What unit
appropriate to measure the duration in (i) and (ii)? Explain your answer.
Taski. 3: StandTime Durationin Seconds
on one footwith your eyes closed.
Table 1
Running Timei.

•StandHow
longfoot
canwithyouyourbalanceon oneeyes yourself?
closed.

ii.

 How
many
seconds can you balance yourself?
Hold
yourMira
breath.
1 minute 50 seconds

ii.

Student

Hold your breath.
JoeTime taken to run 400 m

2 minutes

How long can you hold?

•

How Judy
many seconds
can you hold?
65 seconds

iii. Some
Some examples
time
duration
areareday,day,hour,
minuteand andsecond.
iii.
examplesof
ofunits
unitsforformeasuring
measuringtime
durationhour,
minutesecond.
iv.

Who ran the fastest?

What unit is appropriate to measure the duration in (i) and (ii)? Explain your answer.

What
the differenceto
in measuretime between
theduration
fastest andinthe
student?
What unit
is isappropriate
the(i)slowest
and (ii)?
Explain your answer.

•

Three students
competed
a 400themfinishingrace. The
resultis recorded Anotherstudent inreachedline 37
seconds
after Judy.in Table 1.


Table 11
Table
RunningTime
Time
Running
Student

What is the time taken by this student
to complete
the run? competed in a 400 m race.
iv. Threestudents

The result is recorded in Table 1.

Time taken to run 400 m

Mira

1 minute 50 seconds

Joe

2 minutes

Judy

65 seconds

4•

Who ran the fastest?

•

What is the difference in time between
the fastest and the slowest student?

•

Another student reached the finishingline 37 seconds after Judy.



Who ran the fastest?

What is the difference in time between the fastest and the slowest student?



Another student reached the finishing line 37 seconds after Judy.



What is the time taken by thisstudent to complete the run?

140 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 4: Estimation Involving Time

Task 4: Estimation Involving Time
Task 4: Estimation Involving Time

CampingCampingOO


 40
40ss

Sweeping the floor O

Sweeping the floor O 1 hour 1 hour 5 minutes

 5 minutes

 2 days 2 days

Gardening O

Gardening O

Running a 200 m O

race

i.

Running a 200 m O

Diagram 4 shows four sets of activities and time durations.


i.

i.

Diagram 4

race
DiagramMatch each activity with theappropriate
duration.
Diagram
4 4 Determine an appropriate tool for measuring the time for each activity.
Diagram
4 shows four sets of activities and time durations.

Diagram 4 shows four sets of activities and time durations.
Match each activity with the appropriate duration.

•  Matcheachanactivity
with theduration.
Determine
appropriate
toolappropriate
for measuring
the time for each activity.
•

Determine an appropriate tool for measuring the time for each activity.

55Chapter 3: Quantity and Measurement | 141

Task 5: Additon and Subtraction Involving Duration of Timei. Adam’s father sent him and his friends to the bus station to take a bus to the Youth Park.
bus andleft thestation atInvolving
7:30 in themorningofand
it took 80 minutes to reach the Youth
Task 5: The
AdditonSubtraction
Duration
Time
Park.
i.

Adam’sfatherhim andhishisfriends
to thebusatstation
to takea bus to the Youth Park. The•
Whattimesent
did Adamandfriends
arrivethe YouthPark?
bus left the station at 7:30 in the morning and it took 80 minutes to reach the Youth Park.

If Adam’s
fatherdrove
20 friends
minutesto thebus
stationfrom his home and arrived
What
time didAdamand hisarriveat theYouthPark?
5 minutes before the bus departed, what time did they start from home?

If Adam’s father drove 20 minutes to the bus station from his home, what time did they startfrom home?
A group of students left the school at 9:25 in the morning and arrived at the National
Monument at 3:10 in the afternoon. Three students recorded the duration of the journey as
follows:
Agroup of students left the school at 9:25 in the morning and arrived at the National Monument
at 3:10 in the afternoon. Three students recorded the duration of the journey as follows:
Student A: 12 hours 35 minutes
Student A: 12 hours 35 minutes
Student B: 6 hours 15 minutes
Student B: 6 hours 15 minutes
•

ii.
ii.

Student
StudentC:
C: 55 hours
hours 45
45 minutes
minutes
•

•


Which
answer?
Whichstudent
studentrecorded
recordedthe
thecorrect
correct
answer?
How many different ways can you use to solve the problem?
How many different ways can be used to solve the problem?
Which of the ways do you think is easier for your students to learn? Why?

•

Which of the ways do you think is easier for your students to learn? Why?

Diagram5
Diagram 5iii. Diagram 5 shows a time line from 12:00 noon to 12:00 midnight.
iii. Diagram 5 shows a time line from 12:00 noon to 12:00 midnight.
 Why is each hour on the time line divided into 6 intervals?
•
Why is each hour on the time line divided into 6 intervals?
 Mark the following times with  on the time line.

•
9 o’clockin the eveningMark
the following
times with  on the time line.
 10 minutes past 2 in the afternoon
 9 o’clock in the evening 20 minutes to 6 in the evening Half
10 minutes
in the afternoonpast 10 inpast
the 2eveningUse the time line to help you solve the following problem.
 20 minutes to 6 in the evening
Over the phone, Kapono told his best friend what he did on new year day.
 Halfpast
10 toin thethecinema
evening“Afterlunch,
I wentfor a show. Two hours later, I returned home. While at home, I hada shower for 10 minutes. Then I did my school work. It took me 80 minutes to complete the work. Then
I chat with my father and sisters for 40 minutes. After that, I spent another 1 hour to read my favourite
story book. Then, I played chess with my younger brother for 50 minutes. At that time, my mothercalled us for dinner, I looked at my watch, and the time was exactly 7 o’clock in the evening.” What time did Kapono start doing his school work?
 What time did Kapono start playing chess?

142 | Mathematics Challenges for Classroom Practices at the Lower Primary Level•

Use the time line to help you solve the following problem.
Over the phone, Kapono told his best friend what he did on new year day.
“After lunch, I went to the cinema for a show. Two hours later, I returned home. While at home, I
had a shower for 10 minutes. Then I did my school work. It took me 80 minutes to complete thework. Then I chat with my father and sisters for 40 minutes. After that, I spent another 1 hour
to read my favourite story book. Then, I played chess with my younger brother for 50 minutes.
At that time, my mother called us for dinner, I looked at my watch, and the time was exactly
7 o’clock in the evening.”What time did Kapono start doing his school work?

What time did Kapono start playing chess?

What time did Kapono finish his lunch?

Chapter 3: Quantity and Measurement | 143

Task 6: Different Time Zone
Task 6: Different Time Zonei.

Sirichailives
in inBangkok
andhishisfriendin However,Tokyo. However,does noti. Sirichai
lives
Bangkok andfriendKenjiKenji
lives inlives
Tokyo.
Sirichai doesSirichai
notunderstand
why
Kenji
is
always
in
bed
when
he
calls
at
10.00
o’clock
atnight.
understand why Kenji is always in bed when he calls at 10.00 o’clock at night.
[Source: https://www.timeanddate.com/time/map/]Tokyo

Bangkok

X W V

U

T

S R

Q

P O

N M Z

A

B

C

D E

F

G

H

I

J

K L

Diagram
Diagram 66
Diagram 6 shows the locations of Bangkok and Tokyo in different time zones.
Diagram
6 shows the locations of Bangkok and Tokyo in different time zones.
How do you use Diagram 2 to help Sirichai understand the situation?



Why are there different time zones in the world?

Bangkok is in zone G and Tokyo is in zone I. What is the time in Tokyo when Sirichai calls
at 10.00 o’clock at night?

How
do youuse
Diagram
6 to between
help Sirichai
understandthe situation?
 Whatis thedifferencein hours
consecutive
time zones?

•

What is the difference in hours between consecutive time zones?

•

Why are there different time zones in the world?

•

Bangkok is in zone G and Tokyo is in zone I. What is the time in Tokyo when Sirichai
calls
at extend
10.00cano’clockat night?
To whatthis task
cultivate your students’ appreciation of time zone towards the

ii.

•
ii.



•



What is the best range of time Sirichai should call Kenji if he wishes to talk to Kenji at night?

usefulness of mathematics in daily life?

What is the best range of time Sirichai should call Kenji if he wishes to talk to Kenji atnight?

To what extend can this task cultivate your students’ appreciation of time zone towards theusefulness of mathematics in daily life?

7

144 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 6: Introducing Money as Quantity
Standards 6.1:
Standards
Standards6.1:
6.1:

Introducing money as quantity and use it as the model of base ten system
Introducingmoney
money
quantityquantity
andanduseuse
it itasasthethemodelmodelofand
ofbasebasetentensystem
systemi. Introducing
Introduce
unit
ofasas
money
using
notes
and
coins
determinethe correct amount of money
ii. Use counting by fives and so on for base-10 systemi. i.
Introduce
Introduceunit
ofofmoney
money
using
usingnotes
notesandandcoins
coinsand
anddetermine
determinethecorrect
correctamountamountofofmoney
moneyiii. ii.
Appreciatetheunitfluencyforandcalculationof moneywithall the thefouroperations
ii. Use
Usecounting
countingbyby
fives
fivesandsosoononforforbase-10
base-10systemsystemiv. iii.Appreciate
numbersense
forconversion
andwithtransaction
ofoperations
money in lifeiii. Appreciate
Appreciatethethefluency
fluency
forforcalculation
calculation
ofofmoney
money
withallallthe
thefourfour
operations
iv.iv.

Appreciate
Appreciatenumber
numbersense
senseforforconversion
conversionand
andtransaction
transactionofofmoney
money
ininlife
life

Sample Tasks for Understanding the Standards
SampleSampleTasks
TasksforforUnderstanding
Understandingthe
theStandards
Standards

Task 1: Learning about Money in SEAMEO Countries
Task
Task1:1:Learning
Learningabout
aboutMoney
Money
ininSEAMEOSEAMEOCountries
Countries
Set
Set①①: Malaysian: MalaysianRinggit
RinggitSet
Set②②: Thai: ThaiBaht
Baht

RM
RM9292Set
Set③③: Singaporean: SingaporeanCent
Cent

Set
Set④④
: Philippines: PhilippinesPiso
Piso

Diagram
Diagram
Diagram111
i.

i. i. Determine
Determine
theamountamountofmoneymoney
inindifferent
differentcurrencies
currencies
asasshown
shown
ininDiagram
Diagram
1.1.
Determine
the theamountofofmoneyindifferent
currencies
asshown
in Diagram
1

11

Chapter 3: Quantity and Measurement | 145

Brunei Dollar

Indonesian Rupiah

Vietnamese Dong

Lao’s Kip

Diagram 2

ii.

ii.

Diagram 2

Diagram 2 shows a few sets of currencies of some SEAMEO countries.

Diagram
2 shows
few
sets of currencies
ofvalue.
some SEAMEO countries.
 Matchthe setsa of
currencies
with the same•

Match the sets of currencies with the same value.
2146 | Mathematics Challenges for Classroom Practices at the Lower Primary Leveliii.

Different systems of currency use different units of money to denote the values of coins andiii. Different
systemsof currency
use ofdifferent
of in
money
denote thecountries.
values of coins andnotes. Table
1 shows
the units
moneyunits
usedsometoSEAMEO
notes. Table 1 shows the units of money used in some SEAMEO countries.

Table 11
Table
MoneyUnits
UnitsofofSome
Some
SEAMEOCountries
Money
SEAMEOCountriesiii. Different
systemsof currency
use different
units of money to denote the values of coins andnotes. Table 1 shows the units of money used in some SEAMEO countries.

Country

Unit of Money

SymbolTable 1
RinCountries
ggit Malaysia
Malaysia
Money
Units of Some SEAMEOSingaporeCountry

RM
Symbol S$

ore Dollar
SingofapMoney
Unit

BruneiMalaysia

Ringgit Malaysia

RM

Cambodia
Singapore

Singapore Dollar

S$

Indonesia
Brunei
Lao
Cambodia
Indonesia
Myanmar
Lao
Philippines
MyanmarThailandPhilippines

Timor Leste

Thailand

Vietnam

Timor Leste• Vietnam
Complete
Table
1. 1.
Complete
Table
CompleteTable
iv. Table
2 shows
the1.different denominations of coins of Malaysia and Singapore.

iv.

Table 2 shows the different denominations of coins of Malaysia and Singapore.

iv. Table
2 shows
the different denominations of coins of Malaysia and Singapore.
Table
2

Table
2 and Singaporean Coins
Malaysian

Table
2
Malaysianand Singaporean Coins
Malaysian and Singaporean CoinsCountryCountry

Coin Denomination
Coin Denomination

Malaysia

Malaysia

5 sen

10 sen20 sen50 sen

5 sen

10 sen20 sen50 sen5 cents
5 cents

10 cents
10 cents

20 cents
20 cents

50 cents50 cents

Singapore
Singapore

•



Why
is thereno denominationfor 30 cents?
Whyis thereno denominationfor 30 cents?

What will happen if there is no denomination for 20 cents?

What will happen if there is no denomination for 20 cents?

•

S$1

S$1

Why is there no denomination for 30 cents?

What will happen if there is no denomination for 20 cents?
33

Chapter 3: Quantity and Measurement | 147

v.

Table 3 summarises the denomination of Indonesian currency notes.

v. Table 3 summarises the denomination of Indonesian currency notes.

Table
Table 33
Denomination
Indonesian
RupiahDenomination ofof
Indonesian
RupiahCurrency Unit

Denominationv. Table 3 summarises the denomination of Indonesian currency notes.

Rupiah1 000
2 000 5 000 10 000 20 000 50 000
v. Table
Table3 3summarises the denomination of Indonesian currency notes.

•

100 000

Denomination of Indonesian Rupiah
Table
Preparea3asummary
of ofthethedenominationof currency
notesnotes
in your
and and compare
Preparesummary
denominationof currency
incountry
your country
Denomination of Indonesian Rupiah
compareit withthe denominationof rupiah. Denomination
CurrencyUnit
it withthedenominationof rupiah.

Currency
Denomination
 Whatare Unit
the similarities
differences
betweenthe000
two currency
Rupiah1 000 and
2 000
5 000 10000 2050 000 notes?
100 000••What are the similarities and differences between the two currency notes?



Explain
how the currency
are aligned
system.
Rupiah1 000 notes
2 000
5 000 to10the000base-10
20 00050 000100and000
Prepare a summary of the denomination of currency notes in your country
Whydenominations
involving
the numerals
1, 2 and5 arebase-10
commonly
found incompareit withthe denomination
rupiah.
Explain
how thecurrencynotes
areofaligned
to thesystem.
 Preparea summaryof thedenominationof currency
notes in yourcountry andcurrency?

What
are
the
similarities
anddifferences
betweenthetwocurrency
notes?
compare it with the denomination of rupiah.
Whydenominations
involvingthenumerals
1,
2
and
5
are
commonly
found in currency?
the similarities
currency notesaligned tobetween
the base-10
system.
Explain
What how
are theandaredifferences
the twocurrency notes?
 Explain
Why denominations
involvingthenumeralsand5 are commonly
how the currency
notes
are
aligned 1,
to 2thebase-10system. found in
Task 2: Operations Involving
Moneycurrency?

Why denominations involving the numerals 1, 2 and 5 are commonly found in

Task
Operations
Involving Moneyi. 2:
Calculate.
currency?
 Task
S$8
S$5
2: +Operations
Involving Moneyi.

Calculate.



RM23.50 – 65 sen•

3 RM23.50
x ₭2500– 65 sen• 2:S$8+ S$5Involving Money
Task
i.Operations
Calculate.

ii.

ii.

S$8 + S$5

i. xCalculate.
• ₫3
45000
÷RM23.50
5₭2500– 65 sen S$8÷+ 5
S$5•₫45000 3 x ₭2500

If 1 piece of S$1 can exchange for 3 pieces of RM1 as shown in Diagram 3.


RM23.50 – 65 sen

₫45000 ÷ 5

 ₫45000 ÷ 5
In a particularday, 1 piece of S$1 can exchange for 3 pieces of RM1 as shown in 3 x ₭2500
ii.
1 piece of S$1 can exchange for 3 pieces of RM1 as shown in Diagram 3.
DiagramIf 3.

ii.

If 1 piece of S$1 can exchange for 3 pieces of RM1 as shown in Diagram 3.

Diagram 3
How many pieces of RM1 can you get for 5 pieces of S$1?
Diagram 3

How many pieces of S$1 can you get for Diagram
75 pieces of3 RM1?

•

Diagram3pieces
How many
pieces
S$1 canget forof RM1?of S$1?
How manypiecesof ofRM1
canyouyou
get75for5 pieces•How
piecesofofS$1
canyouyou
forpieces
75 pieces manyHow many
pieces
S$1 cangetgetfor 75of RM1?of RM1?





How many pieces of RM1 can you get for 5 pieces of S$1?

How many pieces of RM1 can you get for 5 pieces of S$1?

Rp 15 000
Rp 15 000

Rp 27 000
Diagram 4

Rp 27 000

Rp 60 000
Rp 60 000

Diagram 4vi. Diagram 4 shows the prices of three types of fruits sold at a supermarket in Jakarta. Mr Wahyudi
Rp 15 0004ofshows
Diagramthe prices
threeof fruits
at a supermarket in Jakarta. Mr Wahyudi
Rpsold27 000boughtvi.2 bunches
bananasand aofbox
of typesmangoes.
Rp 60 000iii.

bought 2 bunches of bananas and a box of mangoes.
Diagram
4 4Diagram How much did he pay for the fruits?
Diagram
4 shows
themangoes
prices of inthreetypes offor
fruitssold at aofsupermarket
Jakarta.
Mr Wahyudi
 vi.
If he
return thebox ofexchange
2 packets
apples, howin muchbalancewill If2hereturn thebox of mangoes
in exchange
for 2 packets of apples, how much balance willboughtbunchesof bananas
and
a box
oftypesmangoes.
he getthe seller?
Diagram
4from
shows
the
pricesofthreeoffruits.
Mr
Wahyudibought2 bunches ofhe get from the seller?
 andHowamuch
pay for the fruits?
bananasboxdidofhemangoes.

How much did he pay for the fruits?

•

he get fromthe seller?
How much
did hepay for the fruits?

•

If he return the box of mangoes in exchange for 2 packets of apples, how much balance4
will he get from the seller?

If he return the box of mangoes in exchange for 2 packets of apples, how much balance will44

148 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Chapter 4: Shapes, Figures and Solids | 149

CHAPTER 4
Shapes, Figures and Solids
Topic 1: Exploring Shapes of Objects
Standard 1.1:
Exploring shapes1 of objects for finding their attributes
i. Roll, fold, stack, arrange, cut, draw, and trace objects (blocks such as boxes, cans and so on)
for knowing their attributes
ii. Use attributes of blocks for drawing pictures by tracing shapes on the paper and explain how
to draw it with the shapesiii. Create patterns of shapes (trees, rockets and so on) by using the attributes and recognise thecharacteristics of shapes
iv. Appreciate the functions of shapes of objects in life
v. Appreciate the names of shapes in daily life by using mother-tongue
Sample Tasks for Understanding the Standards
Task 1: Exploring Attributes of Shapesi.

Diagram 1

Diagram 2

1

Shape is not referred as figure.

Diagram 1 shows two Grade 1 students
doing a task of stacking objects given by theirteacher, Mr Botum. The challenge of the task
is to stack the objects to the tallest possibleheight.

Another teacher, Mr Udin gives a runway
and some objects to his Grade 1 student asshown in Diagram 2. The task of the student
is to identify and group together objects that
could roll down the runway smoothly.
•

What attributes of objects does eachof the teachers expect the students to
identify through the activity?

•

What questions will you ask the students
in order to guide them understand theattributes of objects?

•

What questions will you ask the students
in order to promote their mathematicalthinking?

150 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Trace

Draw

Diagram
Diagram33
i.ii.

DiagramaDiagram33shows
showsthethework
workofofaaGrade
Grade11student
studenttracing
tracingshapes
shapesusing
usingdaily
dailyobjects
objectstotocreate
createpicture.
The
windows
andandthethekitekitein thepictureweretracedusing
the samesquaredbox. box.
The
a picture.
The
windows
in thepictureweretracedusing
the samesquaredfollowingis the isconversation
betweenthe student
and hisandteacher.
The following
the conversation
betweenthe student
his teacher.

Chapter 4: Shapes, Figures and Solids | 151

Teacher : “What shapes are your windows?”
Student : “Squared face of the box.”
Teacher :Teacher“What :other
do youryou windows?”
have in your drawing?”“Whatshapes
shapes are
Student :Student“Round: “Squared
shapes, facetriangularface of the chocolate box, rectangular face of the tissue box,of the box.”
and
…
mmm
…
so
many!”
Teacher : “What other shapes do you have in your drawing?”
Teacher :Student
Do you: useanyshapes,
other triangularshape that
is ofthethesameas thewindow?”“Roundface
chocolatebox,
rectangular face of the tissue box,
and … mmm … so many!”

Student : “Mmm … no!”Teacher : Do you use any other shape that is the same as the window?”

Teacher : “What about the flying kite?”
Student : “Mmm … no!”Student :Teacher“Mmm:…
no, about
it is not
same
“What
the theflyingkite?”as the window.”
Student : “Mmm … no, it is not the same as the window.”

Teacher : “Why not?”: “Why not?”Student :Teacher“Because
they don’t look the same. It is just a kite.”
Student : “Because they don’t look the same. It is just a kite.”

Why does the student see the window and the kite as different shapes?

•
ii.



Why does the student see the window and the kite as different shapes?

Why does the student see the window and the kite as different shapes?

and
cut
out
many
piecesof oftriangularfaceswithdifferent
different
sizes.
He thenA tracedstudent
tracedandcutoutoutmany
many
piecestriangularfacesfaces
withsizes.
Hesizes.
theniii.A student
Aii.student
tracedandcutpiecesoftriangularwithdifferent
He thenpastedthe cut-out
to createa sailingboatasasshown
shown in4. 4.
pasted thecut-out
piecespieces
to createa sailingboat
in Diagram
Diagrampasted the cut-out pieces to create a sailing boat as shown in Diagram 4.

Diagram 4



Diagram 4

Describe the shapes used in drawing the boat.

•

Describe
thecanshapesin drawing
the4boat.
 How
this taskused
be extended
for students’future learning?
Diagram•How
canprimary
thisshapes
taskbeusedextendedfor students’children?
Describe
thein drawing
the boat. future learning?

•How canbe extended
students’learning?
How
can this
thistask
activity
developfor
a senseof futureappreciation
towards mathematics among
primary
children?
How can this activity develop a sense of appreciation towards mathematics amongHow can this activity develop a sense of appreciation towards mathematics among

primary children?

152 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 2: Characterising Shapes for Figures and Solids
Standard
2.1:
Topic
2: Characterising
the Shapes for Figures and Solids
Standard
2.1: Describingwith characters
Describing
figures withfigurescharactersof shapes of shapes
i. i. UseUse
characteristics
of ofshapesforforunderstanding
characteristics
shapesunderstandingfigures
figures(quadrilaterals,(quadrilaterals,square,square,rectangle
rectangle andtriangle,
right
angle,
same
length)triangle,
right
angle,
same
length)ii. ii.Introduce
lineline
andand
right
angle
with
relationsIntroduce
right
angle
with
relationstotoactivities
activitiessuch
suchasaspaperpaperfolding
folding andand useuse it for
describing
figures
withwith
simpleproperties(such
as triangle
hashas3 sides)
describing
figures
simpleproperties(such
as triangle
3 sides)iii. iii.Classify
triangles
by byspecificcomponents,suchtriangle,
Classify
triangles
specificcomponents,
suchasasside,
side,vertex
vertexandand angle
angle (right-angled(right-angled triangle,
equilateral,
isosceles)
and
then
know
the
properties
ofeachclassification
equilateral, isosceles) and then know the properties of each classification
iv. iv.Reorganising
rectangularshapeandandsquaredshape
as asfigures
by by
using
right
angle
and
lengthof
Reorganising
rectangularshapesquaredshape
figures
using
right
angle
and
lengthsides
of sides
Sample Tasks for Understanding the Standards
Sample Tasks for Understanding the Standards
Task 1: Exploring Shapes
Task 1: Exploring Shapes
i.i.

AAsheetfolded to
toformformaaright
rightangle,
angle,a as
shown
in Diagram
1. Thispapersheetofof paperpaper is folded
a as
shown
in Diagram
1. Thispaper-folding
folding
activity
can
be
used
to
introduce
right
angle
to
primaryschoolstudents.
activity can be used to introduce right angle to primary school students.

Diagram 1
Diagram
1 to reinforce your students’ understanding
How will you extend this paper-folding
activityof right angles?
 How will you extend this paper-folding activity to reinforce your students’ understandingof right angles?
•
Draw a shape that has 2 right angles only.

•

Draw a shape that has 2 right angles only.
Is it possible to have a shape with 3 right angles only? If yes, draw the shape. If no,
why not?
 Is it possible to have a shape with 3 right angles only? If yes, draw the shape. If no,
why not?
•
A shape has 4 right angles only, but it is not a rectangle neither a square. Draw the
shape.
 A shape has 4 right angles only, but it is not a rectangle neither a square. Draw the
shape.
•
A closed shape has 4 equal sides, but it is not a square. What shape can it be?
•

A closed shape has 4 equal sides, but it is not a square. What shape can it be?

Chapter 4: Shapes, Figures and Solids | 153

ii. A piece of square paper is cut along its two diagonals to get four right-angled triangles
A pieceof squarepaper
is cutalong
diagonals
to getright-angled
trianglesii. ii.A pieceof squarepaper
is cutalong
its its
twotwodiagonals
to getfourfourright-angled
triangles
as shown
in Diagram
2.
ii. A pieceof squarepaper iscut along its two diagonals to get four right-angled triangles as
shown
in Diagram
as as
shown
in Diagram
2. 2.
shown in Diagram 2.

Diagram 2
Diagram
Diagram
2 22
Diagram How will you convince your students that the four triangles are identical?
Howyouconvince
yourstudents
thatfourtriangles
identical?
willyou
convince
yourstudents
that
the
four
triangles
areidentical?
• HowHowwillwillyou
convince
yourstudents
thatthethefour
triangles
areareidentical?
 How will you guide your students to determine whether a, b, c and d are right angles?
How
guide
your
students
to determine
whethera,cb,and
c and
d are
right
angles?
 How
willwillyouyouguide
yourstudents
to determine
whether
a, b,
d are
right
angles?
•
How will you guide your students to determine whether a, b, c and d are right angles?
•

Diagram 3
Diagram
Diagram3
Diagram
3 3

 The four pieces of triangles can betriangles
canbe be to four
TheThefourfourpieces
of of
can
The
piecesof piecestriangles
canbe rearranged
rearranged
toform
atriangles
rectangle
as shown
rearranged
to formaDiagram
rectangle
as
shown
to
form
a
rectangle
asshown
form arearranged
rectangle
as
shown
in
3.
in Diagram 3.
in
Diagram3.
in Diagram 3.
Arrangethe fourthe
piecesof triangles
in another Arrange
four piecesof triangles
inArrangethepieces trianglesof triangles
 toArrangefourfourin inanotherway
topiecesform aofrectangle.
way
form
athe
rectangle.
anotherway
to forma rectangle.
anotherway
to forma rectangle.

 Arrange
fourofpiecesof triangles
toArrangethe four thepiecestriangles
to form figure
Arrangepiecesof triangles
 Arrange
thethefourfourpiecesof triangles
to toform figureand figureⒷ in DiagramⒶ and figureⒷ in Ⓐ
Diagram4, respectively.
form
figurefigurein DiagramformⒶⒶ
andandfigureⒷⒷ
in Diagram
4, figurerespectively.
4, respectively.
4, respectively.

Diagram 4
Diagram
Diagram
Diagram4 4
• Whatmentalabilityabout
shapesandfigurescanstudents
developfrom
thisthis
activity?
Whatmentalabilityabout
shapesandfigurescanstudents
developfrom
activity?
Whatmentalabilityabout
shapesfigures
students
developfrom
activity?
  Whatmentalabilityabout
shapesandandfigurescancanstudents
developfrom
thisthis
activity?
Explainyouranswer.
Explainyouranswer.
Explainyouranswer.
Explainyouranswer.

154 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Diagram
5 shows
twotwotasks
designedbyforprimaryschoolstudents
students
learniii. Diagram
5two
shows
tasks
designedbyaateacher
teacherfor hishis school
primaryschoollearn
iii. iii.
Diagram
5 shows
tasks
designedby a teacher
for
his primary
students
tolearn to to
triangles
andquadrilaterals.
triangles
andquadrilaterals.
triangles and quadrilaterals.

②a: straight
Draw a line
straight
line connecting
pair ofwith
pointsthe same
Task ② Task
: Drawconnecting
each paireachof points
the withsame
Colourthe triangles
and quadrilaterals
formednumbers.numbers.
Colour the
trianglesand quadrilaterals
formed by
the by thestraight lines.
straight lines.

Diagram 5 5
DiagramDiagram
5•

What is the difference between figure and plane figure?

•

Compare Task Ⓐ and Task Ⓑ.

What
is the difference
figureandplane figure?
What is thedifference
betweenbetween
figure andplane
figure?

 Compare① andCompareTask ① Task
and Task②. Task ②.

arethethesimilarities
andanddifferencesin theof triangles
and quadrilaterals

What
are
similarities
differences
inideasthe ideasof triangles
and quadrilaterals
 What
areWhatthesimilarities
and differencesinthe ideasoftriangles
andquadrilaterals
learnt learnt
throughthesetwothesetasks?
through thesetwotasks?
learntthrough
two tasks?

Chapter 4: Shapes, Figures and Solids | 155

iv. Diagram
Diagram66shows
showssome
somefigures.
figures.
iv.

•
Diagram 6
Diagram 6
Determine which of these figures are not triangles.
Determine which of these figures are not triangles.

•For
Foreacheachfigure
figurethat
thatisisnot
notaatriangle,
triangle,explain
explainwhy
whyititisisnot
nota atriangle.
triangle.

•What is a triangle?
What is a triangle?

v.
of thethe following
following figures.
figures.
v. Identify
Identifyand
andexplain
explainbriefly
brieflythe
thecharacteristices
characteristics of

•

•


•

Rectangle
Rectangle
SquareSquareDetermine whether each of the following statements is true or false. Explain yourreasons.
Determine whether each of the following statements is true or false. Explain yourreasons.
All rectangles are not squares.
 All squares are rectangles.
 All rectangles are not squares.
 Some rectangles are squares.
 All squares are rectangles.

Some rectangles are squares.

Mathematics Challenges for Classroom Practices at the Lower Primary Level
Task156
2:| Classifying
Triangles
Task 2: Classifying Triangles

Task 2: Classifying Triangles

Constructing Triangles with Straws
Constructing Triangles with Straws
Constructing Triangles with Straws

Diagramshows howhowa atriangle
Diagram 7 shows
trianglecancanbe beDiagram 7using
shows
how
a triangle
can 2-legbeconstructedthreestrawsandthreeconstructed using three straws andconstructed
using
three
strawsandjoints.
Diagram
8 shows
the construction
three 2-leg
joints.
Diagram
8 shows theof athree
2-leg joints. Diagram 8 shows the
2-legjoint.
construction of a 2-leg joint.

construction of a 2-leg joint.

i.i.

Diagram 7
Diagram 7
Diagram 7i.

The following
following isTheis aaset
setofof33straws.
straws.
The following is a set of 3 straws.

Explain whyof straws
cannotExplain
why thisthissetsetof straws
 makeExplain
why thisset of straws
acompletetriangle?
cannotmakeacompletecannotmakeacompletetriangle?
triangle?
•
Whatis the condition, in term of•

Diagram 8
Diagram 8
Diagram 8

lengths,
for condition,
three straws
to form
Whatis thein termof a complete
What is triangle?
the condition, in term oflengths, for three straws to form alengths, for three straws to form acomplete triangle?
complete triangle?

Chapter 4: Shapes, Figures and Solids | 157

Triangles
Triangles
Type
Type
① ①

ii.ii. ii.AAteacher,
A teacher,Ms Ms
Malee,
asked
herherstudents
students
teacher,
Malee,
asked
her
students
tototoconstructconstructtriangles
triangles
using
strawsof ofconstructtriangles
usingusing
strawsofstraws
different
different
different
lengths.
lengths.
All constructed
Allthethetriangles
triangles
lengths.
Allthetriangles
wereconstructed
constructed
were
were
pin
pin
on
the
on
the
class
class
bulletinbulletinpin on the class bulletin board as a display
boardaasdisplaya displayof theof From
thestudents’students’
work.
work.
of board
theasstudents’work.
the display,
From
From
the
the
display,
display,sheshediscovered
discoveredthat
she discovered that there are 3 types that
of
there
there
areare3
3 typesof intrianglesofDiagramtriangles
as9.
shown
as shown
in intriangles
astypesshown
Diagram
Diagram
9. 9.

Type ① are triangles that always have a
Type
Type
① are
①base
are
triangles
trianglesthat
always
have
have
a a
horizontal
at anythat
point
ofalwayshanging.
horizontal
horizontal
base
base
at anyat anypoint
pointof hanging.
of hanging.

 Triangle① hangingat vertices
A, BA,andC. C.
 Triangle① hangingat vertices
B and
 All
horizontal
bases.
 All
horizontal
bases.
Triangles
Type
TrianglesType② ②Type ② are triangles that have a horizontal
Type
Type
②are
are
triangles
triangles
thatthat
have
have
a a
base
at②onlyone
pointof hanging.
horizontal
horizontal
base
base
at at
onlyonly
oneone
point
point
of of
hanging.
hanging.
Type③ are triangles that always have a Triangle
 Triangle②②
hanging
hanging
at vertices
at vertices
D, ED,and
E and
F. F.
 Onlyhorizontalbase.
 Onlyoneonehorizontalbase.
Triangles
Type
TrianglesType③ ③slanted base at any point of hanging.
Type
Type
③ are
③ are
triangles
triangles
thatthat
alwaysalways
have
have
a a
slantedslanted
base
base
at
any
at
any
point
point
of
hanging.
of
hanging.
•
Explain how the three types oftriangles are related to classification
 Explain
howby
how
the
the
three
three
types
types
of of
of Explain
triangles
lengthsofthesides.
triangles
triangles
arearerelatedrelatedto classification
to classification
trianglesof triangles
by lengths
byyou
lengthsoftheof thesides.
• ofExplainhow
will
usesides.
thisconstruction of triangles from straws
 Explain
Explainhowhow
youyouwill
willuseusethisthis
to help
yourstudents
learnequilateral,
construction
constructionof trianglesof triangles
from
from
strawsstrawsisosceles and scalene triangles.
to tohelphelpyour
yourstudents
studentslearn
learn
equilateral,equilateral,
isosceles
isosceles
andandscalenescalenetriangles.
triangles.

 Triangle
 Triangle③ hanging③ hangingat vertices
at vertices
G, H
G,and
H and
I. I.
 All
slantedbase.
 Allslantedbase.
Diagram
9 9Diagram

Diagram
10 10
shows
anan
isosceles
triangleiii. iii.
Diagram
Diagram
10
shows
shows
an
isosceles
isosceles
triangle
triangle
cutcutoutoutaa piece
of
cut
out
from
from
piece
a piece
ofpaper.
paper.
of paper.
Explain
how
you
can
fold
thethepaper• Explain
Explain
how
how
you
you
can
can
fold
fold
the
paper
paper
triangleto showshow
that:triangle
triangletoto show
that:
that:
Angle
same
size
angle
 Angle
Angle
b bisis
bsame
is
same
size
size
asasangle
asangle
c.c c.
Angle
notnotthethe
same
size
asas
 Angle
Angle
a a isais not
is
the
same
same
size
size
as
angle
b
angle
angle
b. b.

Diagram
Diagram
Diagram1010
10

iv. Diagram 11 shows an equilateral

158 | Mathematics Challenges for Classroom Practices at the Lower Primary
Level
triangle
cut out from a piece of paper.

iv.

Diagram 11

 Explain
you canthe paper
iv. Diagram
11 howshows
an fold
equilateral

Diagramtriangle
11 shows
triangle
cut out
triangle
toequilateralshowa that:cut an
out
from
piece
of paper.
from a piece of paper.
 Angle a is the same size as Explain
howb.you can fold the paper
angle•
Explaintriangle
how you
fold
the paper triangle
to canshowthat: Angle b is same size as angle c.
to show that: Angle a is the same size as Withoutfolding the triangle, how Angle aangle
is theb.same size as angle b.
do you reason that angle a and Anglesame
as angle c.
angleb cisarethesizesame?
 Angle b is same size as angle c.
 Without folding the triangle, how
you reason
that angle
 Withoutdofolding
the triangle,
how adoand
you
angle
c
are
the
same?
reason that angle a and angle c are thesame?

Diagram
Diagram 11
11

Diagram
Diagram1212
v.

v.

v.

Diagram 12 shows two equilateral triangles of different sizes. A student claimed that the
Diagram 12 shows two equilateral triangles of different sizes. A student claimed thatsizes of angles a, b, c in triangle ① are smaller than the sizes of angles d, e, f in triangle ②.
the sizes of angles a, b, c in triangle① are
Diagram
12 smaller than the sizes of angles d, e, f in②. you use paper cutting to convince your students that angles a, b, c, d, e and f• triangle
How will
Diagram12same?
shows two equilateral triangles of different sizes. A student claimed thatare the sizes
Howofwill
you useto convince
your students
thatanglesa, b, c,d,d, e,e andtheanglesa,paperb, c cutting
in triangle① are smaller
than thesizesof angles
f in fare the same?
triangle ②.


How will you use paper cutting to convince your students that angles a, b, c, d, e and f
are the same?

Chapter 4: Shapes, Figures and Solids | 159

Standards 2.2 : Describing Solids With Characteristics of Shapes
Standards: Describing Solids With Characteristics of Shapes
Standards2.2
2.2:
Standards
2.2: Describing Solids
WithtoCharacteristicsof Shapesi.
Use thecharacteristics
of shapes
understand solids
such as boxes can be developed by sixi.
Use the characteristics of shapes to understand solids such as boxes can be developed by six
rectangular
parts
with
simpleproperties
i.Describing
Use thesolids
characteristics
of shapes
to shapes
understand solids such as boxes can be developed by six
withwithcharacteristics
ofrectangularparts
simpleproperties
parts
with
simpleproperties
ii.i. rectangular
Developwiththeproperties
Use the boxes
characteristics
of shapes
to understand solids such as boxes can be developed by six
ii. Develop boxes with the properties
rectangular
parts
with
simpleproperties
ii.
boxes
with
the
properties
iii. Develop
Appreciate
the solids
arounddaily life through considering the function of the solidsiii.
the solids
arounddaily life through considering the function of the solids
ii. Appreciate
Develop boxes
with thepropertiesiii.
iii. Appreciate
Appreciatethe
thesolids
solidsaroundarounddaily
dailylife
lifethroughthroughconsidering
consideringthe
thefunction
functionof
ofthe
thesolids
solidsSample Tasks for Understanding the Standards
Sample Tasks for Understanding the Standards
SampleTasks
TasksforforUnderstanding
Understandingthe
theStandards
Standards
SampleTask 1: Properties of a Box
Task 1: Properties of a Box
Task1:1:Properties
PropertiesofofaaBox
Box
Task

i.
i.

Diagram 1
Diagram 1
Diagram
Diagram11

Diagram 2
Diagram 2
Diagram22
Diagram

Diagram 1 shows a rectangular box
Diagram 1 shows a rectangular box

i. i. Diagram
1 shows
a rectangularbox③withwith
6 faces.
Faces①,②, andare
Diagram1 shows
rectangularwith
6 faces.
Facesa①,②, and ③ box
are
6 faces.
Faces
①
,
②
,
and
③
are
shown
shown
in
the
diagram,whereasthewith 6 faces.
Faces
①, ②,
and ③ are
shown
in
the
diagram,whereasthein otherthe diagram,whereasthe other three
three
faces
are
hidden.
shown
in
the
diagram,whereastheotherfaces are hidden.
faces
arethreehidden.

other
three
faces
are hidden.
 How
many
edges
and vertices does
 How many edges and vertices does
the
box
have?
•  How
edges
Howmany
manyedgesandandvertices
verticesdoes
does
the
box
have?
the
box
have?
the
box
have?

ii. Each face of the box is traced on paper
Each
face
of theis tracedon paperii. ii. Eachfaceof theboxboxis tracedon paperas
shown
Diagramii. asEachfaceinthe box 2.
is traced on paper
asshown
inofDiagram
2.
shown
in
Diagram2.
in Diagram
2. face?
as shown
What shapeis each What shape is each face?
What
shape
each
face?
•  What
shape
is is
each
face?
iii. Faces ④, ⑤, and ⑥ are opposite to facesiii. Faces ④, ⑤, and ⑥ are opposite to facesiii. iii.Faces
④ and
, ⑤⑤,
,③,
and
⑥
tofacefaces①,②,
respectively.
Each
is
Faces
and
⑥are
areopposite
opposite
faces
②,,④,
and
respectively.
Eachtoface
①①,
, ②
and③,
③
, correspondingly.
Eachisjoined
with
4
other
faces
and
hence
①,
②, andEachfaceisjoinedwith③,
4 respectively.
other
faces
and
hence
face
is joinedwith
4 other
faces
and
hence
forming
4 adjoiningedges.
For
example,joined
with
4
other
faces
and
hence
forming
4adjoining
adjoiningedges.
edges. For
Forexample,
example,formingface①4 is
joined withedges.
faces ②,
⑤ andforming
4 adjoining
For③,
example,face①
isjoined
with
faces
②,
③,
⑤and
and
face
① is joined with faces ② , ③ , ⑤⑥.
face① is joined with faces ②, ③, ⑤ and⑥.
⑥.
⑥.
 Explain the condition in terms of Explain the condition in terms of
oppositefaces
and adjoiningedges•  Explain
condition
inin termsofExplainthethecondition
terms
of
oppositefaces
and adjoining
edges
for
the
6
faces
to
form
the
box.
oppositeadjoining
opposite
facesandadjoiningedges
forthe 6faces
faces
toand
form
the box.edges
forfor
the
66
faces
toto
form
the
box.
the
faces
form
the
box.

iv. iv.Diagram
Diagram3 3shows
showsaaset
set ofof faces.
faces. IfIf thetheiv.faces
Diagram
3 and
shows
a set
of
faces.
If thearearecutjoinedbyby
theiredges.
faces
cut
and
joinedtheiredges.
iv. faces
Diagramshows
a set byof faces.
If theare 3cutand joined
their edges.
faces are cut and joined by their edges.
•  CanCanthethesetsetofoffaces
facesform
form aa proper
proper Can the set of faces form a proper
box?
Why
or
why
not?
box?
Why
or
why
not?
 box?
Can thesetorofwhy
faces
Why
not?form a properbox? Why or why not?

Diagram 3
Diagram
Diagram33
Diagram 3

160 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelv.

Diagram 4 shows another set of faces.
v. Diagram 4 shows another set of faces.
•  Canthethesetsetof faces
formforma properbox?
Canof facesa proper
Diagram4shows
another
setoffaces.
Why
or
why
not?
box? Why or shy not?

v.

Can the set of faces form a properbox? Why or shy not?

Diagram 4

Diagram
Diagram
4 4Diagram
Diagram 55
iii. Diagram
5 shows
types
of cards
withdifferent
different measurements.
measurements. AAproper
be be
vi. Diagram
5 shows
fourfourtypes
of cards
withproperbox
boxcancanconstructed
6 piecesof these
cards.
examples,6 6pieces
piecesofof cardcard Ⓐcan make
constructedusingusing
6 piecesof these
cards.
ForForⒶ canmake aDiagram
5examples,cube,
and 2 pieces
cardⒶ4 with
4 piecesof cardⒷ makecan makea cuboid.
cube, aand2 piecesof cardofⒶ
withpiecesof cardⒷ cana cuboid.

iii. Diagram 5 shows four types of cards with different measurements. A proper box can be
 How
many
different
boxes
are
able
to
constructusing
using these cards?
cards?
•constructedHow
many
different
boxes
youyouable
toFor
constructusing
6 piecesof arethesecards.
examples, 6thesepieces of card Ⓐ can makea cube,2 piecesof cardⒶ constructwith 4 pieces
card Ⓑ can make a cuboid.
 andPhotocopy
Diagram
5 and
all theof
boxes.
•

Photocopy Diagram 5 and construct all the boxes.

How many different boxes are you able to construct using these cards?



Photocopy Diagram 5 and construct all the boxes.

Chapter 4: Shapes, Figures and Solids | 161

Standards 2.3: Drawing circle and recognising sphere based on circle
Standards
2.3:
Standards
2.3:
Drawing circle and recognising sphere based on circle
i. Think about how to draw a circle and find the centre and radius
i. Drawing
Think about
to draw a circle
and basedfind thecircle how
and recognisingsphereoncentre
circle and radius
ii. Draw a circle with instrument such as compass
i.
Think
about
how
to
draw
a
circleandfind
theii. Draw a circle with instrument such as compass centre and radiusiii.
picturesusing circles
such
as Spirographii. Enjoy
Drawdrawing
a circle with
instrument
such ascompassiii.iii.Enjoy
drawingpicturesusing
circlessuch
as asSpirographEnjoydrawingpicturesusing
circlessuch
Spirographiv. Find the largest circle of the sphere with diameterand identify the sphere by its centre and radius
iv.
Find
the
largestcircleofthespherewithdiameter
identify
the sphereits centre
andiv. Find the largest circle of the sphere with diameter and andidentify
the sphereby itsby
centreand radius
v. Appreciate
circles and spheres in daily life such as manhole, and the difference between soccer
radiusv. v. Appreciate
circles
as manhole,manhole, andandthethedifference
differencebetween
betweensoccerball
and rugby
ballandAppreciate
circles
andspheres
spheresinindaily
dailylife
life such
such assoccerballball
and
rugby
ball
and rugby ballSampleSampleTasks
Tasks forfor Understanding
Understanding thethe Standards
Standards
Sample Tasks for Understanding the Standards
Task 1: Properties of a Circle
Task
Properties
CircleTask
1:1:
Properties
ofof
aa
CircleDiagram
Diagram1 1
Diagram 1i. i. A cow
is tiedto toa treewitha 2-metrerope
in afield.
A cow
is tieda treewitha 2-metrerope
ingrassa grass
field.
i. A cow is tied to a tree with a 2-metre rope in a grass field.
 What is the furthest distance the cow could move away from the tree?
• WhatWhat isis thethe furthest
furthest distancedistance thethe cow
cow could
could move
move away
away fromfrom thethe tree?
tree?
[Note. Taking the mouth of the cow as the reference point.][Note. Taking
Taking thethe mouth
mouth ofof thethe cow
cow asas thethe reference
reference point.]
point.][Note.
 After a day, the cow has eaten all the grasses within its’ reach, left with a patch of bare
 After
has eaten
allpatch
the grasses
land.a day,Whattheis cow
the shapeof theof barewithinland?its’ reach, left with a patch of bare•Afterday, thecow
has eaten
the grasses
within its’ reach, left with a patch of bareland. aWhat
is theshapeof theallpatchof bare land?
land. What is the shape of the patch of bare land?

Diagram 2
Diagram 2
Diagram 2ii. A compass is a simple instrument useda circle.
2 shows how
ii. toA draw
compass
is a Diagram
simple instrumentusedthetaizhia symbol
can instrument
be constructed
ii. A
compass
is a simple
used
to
draw
circle.
Diagram
2 showshow
from3taizhicircles.
todraw
a circle.
Diagram
2 showshowthesymbolcan beconstructedthetaizhisymbol can be constructedfrom
3 circles.
from
3 circles.
•Use
a compass to construct the taizhi
Use symbol.
a compass to construct the Usecompass to construct the
taizhia symbol.
taizhi symbol.

162 | Mathematics Challenges for Classroom Practices at the Lower Primary Leveliii.

iii. Diagram 3 shows a bicycle with circulariii. Diagram 3 shows a bicycle with circularwheels,3 andanotherbicyclewithDiagrama bicycle
with circularwheels, shows
and anotherbicyclewithsquaredwheels.
wheels,andanotherbicyclewithsquaredsquared wheels.
wheels.
iii. Diagrama bicycle
circularWhy is3ashows
circularwheelwith
betterthan awheels,
Why andis a circular
betteranother wheel
bicyclewiththan asquaredwheel?
• squared
Whysquared
iswheels.
a circularwheel
betterthan awheel?
squared wheel?
  Whatpropertiesof a circle
makeis
aproperties
circular wheel
thana it a Why
Whatof abettercirclemakeit abetterchoicethanasquare?
squaredwheel?
•
What
propertiesof aa circle
make it abetterchoice thansquare?
better choice than a square?
 What properties of a circle make it abetter choice than a square?

Diagram 3
Diagram 33
Diagram
Diagram 3

Diagram
4 showa manhole.
iv. iv.Diagram
4 showa manhole.
iv. Diagram
4 showa manhole.
iv. Diagram 4 show a manhole.

Whyis manhole
the manhole
in theof•  Why
is themade madein madethe shapeWhyisthethemanholein the shapeWhyisofmanholein theothera circleandmadenotany
a circle
and
not
any
other
shape
such
as
shape
of
a
circleandnotanyshape suchof a circle
and not any other othershape
as square.
square?
shapesuch
as square.
shape
such
as square.

Diagram4
Diagram4
Diagram
Diagram 44

Diagram
Diagram4 5v.

Diagram
5 shows
plate.
iv. Diagram
4 showsa abroken
broken circular
circular plate.
Diagram
4 find the centre of the circle.
 Photocopy the diagram and explain
how toDiagram•Photocopy the diagram and explain
how 4to find the centre of the circle.


Explain how you can determine the diameter of the circular plate.

iv. Diagramshowshowa brokencircularplate. the diameter of the circular plate.
• 4
Explainyou candetermine
iv. Diagram
4 shows a broken
circularplate.
 Photocopy the diagram and explain how to find the centre of the circle.
 Photocopy the diagram and explain how to find the centre of the circle.
 Explain how you can determine the diameter of the circular plate.
 Explain how you can determine the diameter of the circular plate.

Chapter 4: Shapes, Figures and Solids | 163

Diagram 56
Diagramvi. Diagram 5 shows a picture of two lamp poles, A and B.

There is a treasure hidden

vi. Diagramsomewhere
6 showsinathepictureof two lamp poles, A and B. There is a treasure hidden
diagram.
somewhere in the diagram.
• Isthethetreasure
information
to locate
the A.
treasure? Explain your reasons.
Given thatisenough
8 cm from
point


•

Given that the treasure is 8 cm from point A.

Also given, the treasure is 6 cm from point B.

Is the information enough to locate the treasure? Explain your reasons.
 Is the information enough to locate the treasure now?

 How
are you
the locationof the treasure?
Also given,thecertain
treasureis 6now
cmonfrom
point B.
 Use a compass to determine and mark the possible locations of the treasure on5.
Is the Diagram
information
enough to locate the treasure now?



How certain are you now on the location of the treasure?

Use a compass to determine and mark the possible locations of the treasure on
Diagram 5.

164 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 2: Properties of a Sphere
Task
Task
2: 2:
Properties
Propertiesof ofa Spherea SphereTask 2: Properties of a Spherei. Diagram 6 shows a rugby ball and threei. i. Diagram
76 shows
ballandandthree
i. Diagram
Diagramshows
6 shows
aa rugby
arugby
rugby
three
three
directionsof viewing
atballit.balldirections
ofof
viewing
atat
it.at
directions
directionsofviewingviewing
it. it.
 Sketch the shape you see from each•  Sketch
shape
you
each
of
SketchSketchthetheshapeshapeyouyou
see
see
from
from
each
each
of thethedirections
ofseeviewing.
theof viewing.
of directions
ofthethedirections
directionsof ofviewing.
viewing.

ii.

Diagram 8 shows a football and three
directions of viewing at it.
•

What shape will you see from each of thedirections of viewing?

What can you conclude about the shapeii. ofDiagram
7 shows
a footballthree
a sphere
when view
from anddifferentii. ii.Diagram
Diagram
7 shows
7 ofshows
a footballa football
and
three
three
directions
viewing
at it. anddirections?
directions
directionsof ofviewingviewing
at at
it. it.
•Diagram6
Diagram
Diagram
Diagram
6 76

iii.

 why
What
will canyou rollsee smootherfrom each
Explaintheshape
football  What
What
shape
shape
willwillyouyou
see
see
from
from
each
each
of
the
directionsofviewing?
than the rugby ball.
of ofthethedirections
directionsof ofviewing?
viewing?

iv.

 how
What
you
concludeabout
theExplainyoucanguide
young
childrento find  What
What
can
can
you
you
concludeconclude
about
about
thetheshapeofasphere
when
view
from
the diameter of the football.
shape
shape
of of
a sphereadirections?
sphere
when
when
view
view
from
from
different
different
different
directions?
directions?
iii. Explain why the football can rolliii. iii.Explain
Explainwhy
why
thethefootballcancanrollrollsmoother
thanthefootball
rugby
ball.
smoothersmootherthan
thanthethe
rugby
rugby
ball.
ball.

Diagram 7
Diagram
Diagram
7 7
Diagram 8iv. Explain how you guide young children toiv. iv.Explain
Explainyouyou
guide
guide
young
childrenchildrento tofind howthehow
diameterofyoungthefootball.
find
findthethediameterdiameterof ofthethefootball.
football.

Diagram 9
Diagram 8
Diagram
Diagram8 8
v. iv. Diagram
Diagram9 8shows
showstwo
twosolids,
solids,a aspheresphereandanda acube,
cube,cutcutinina averticalverticaldirection.
direction.
iv. iv.Diagram
Diagram
8 shows
8 shows
twotwosolids,solids,a spherea sphereandanda cube,a cube,cutcutin ainverticala verticaldirection.
direction.
•  Sketch
Sketchthetheshape
shapeofofthe
thecrosscrosssection
sectionfor
foreacheachofofthe
thesolids.
solids.
  Sketch
Sketchthethe
shape
shape
of of
thethecross
crosssectionsectionforfor
each
each
of of
thethesolids.
solids.

Chapter 4: Shapes, Figures and Solids | 165

Diagram109Diagram
Diagram 9v. Diagram 9 shows the same sphere and cube, but cut in a horizontal direction.
vi. v.Diagram
10 9showsDiagramshowsthethesamesamespheresphereandandcube,cube,but
butcutcutininaahorizontal
horizontaldirection.
direction.
 Sketch the shape of the cross section for each of the solids.
•  Sketch
the
shape
ofof
the
cross
sectionforfor
each
ofof
the
solids.
Sketch
the
shape
the
cross
sectioneach
thesolids.

Diagram 10
Diagram 10
Diagram 11
vi. Diagram 10 shows two solids cut in a slanting direction.
vi. Diagram 10 shows two solids cut in a slanting direction.
vii. Diagram 11 shows the same two solids cut in a slanting direction.
 Sketch the shape of the cross section for each of the solids.
 Sketch the shape of the cross section for each of the solids.
•
Sketch the shape of the cross section for each of the solids.
vii. What will you conclude on the cross-sectional shape of a sphere?
vii. What will you conclude on the cross-sectional shape of a sphere?
viii. What will you conclude on the cross-sectional shape of a sphere?
 When will you get the largest cross-sectional circle from a sphere?
 When will you get the largest cross-sectional circle from a sphere?
•
When will you get the largest cross-sectional circle from a sphere?

166 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic 3: Explaining Positions and Directions

Topic 3:

Explaining Positions and Directions

Standards 3.1:

Standards: Exploring how to explain a position and direction

to positions
explain aandpositionand ofdirectioni.Exploring
Identify how
simpledirections
an object accurately using various ways such as in myi. perspective,
Identify simple
positionsanddirectionsof an andobjectusing
various
in your perspective in the classroom,
theaccurately
left, right, front,back,west,ways
east, such
north,as inmy perspective,in your perspective in the classroom, and the left, right, front, back, west, east,south
and with measurement

north, south and with measurement
Draw the map around the classroom with consideration of the location
ii. Draw the map around the classroom with consideration of the location
iii.iii. Design
a gameto appreciate
the changing
of positions
and directions
in a classroom
Designa gameto appreciate
the changing
of positions
and directions
in a classroomii.

Sample Tasks for Understanding the Standards

Sample Tasks for Understanding the Standards

Task 1: Position and Direction of an Object

Task 1: Position and Direction of an Object

Row
5
Row
4
Row
3
Row
2
Row
1
Column1

Column2

Column3

Column4

Column5

Diagram 1

Diagram 1

Diagram 1 shows a wall shelf with 25 compartments labelled with its rows and columns. Soft toys

are placed1inshowssome ofthe compartments.
A class of children
are givento choose Softa
Diagrama wallshelf with 25 compartments
labelled
withtheits opportunity
rows and columns.
favourite
fromintheshelf.of the compartments. A class of children are given the opportunity totoys
are toy
placedsomechoose a favourite toy from the shelf.
i.

Alfonso says: “My favourite toy is in the fourth row from the bottom.”i.

Alfonsosays: “My favourite toy is in the fourth row from the bottom.” Is “fourth row from the bottom” the same as “fourth row from the top”?
•Is the“fourthrow from
the bottom”
thelocatesametheasboy’s“fourthrow from
top”?
Is
information
enoughfor you tofavourite
toy?theWhyor why not?

•Is the information enough for you to locate the boy’s favourite toy? Why or why not?

Chapter 4: Shapes, Figures and Solids | 167

ii.

Dylan says: “I like the toy in Column 5.”•iii.

Can you locate the toy? Why or why not?

ii. toyDylansays: “I likethe toyin Column
5.”
What
is locatedat thefollowingcompartments?
Can you locate the toy? Why or why not?

• iii. First
column from the left and fifth row from the bottom.
What toy is located at the following compartments?

iv.

•

 Firstcolumnfrom thethe left
andandfifth second
row from the
Fourthcolumnfromrightrowbottom.
from the top.

•

Column 2 and row 3•

Column 5 and row 4•

What
information
locatethewall shelf?
 minimum
What minimum
informationisis neededneeded toto
locatea toya intoy
thein
wall
shelf?



Fourth column from the right and second row from the top.

Column 2 and row 3Column 5 and row 4

iv.
Designa method
to identify
a location inin theshelf
by using
only numbers.
Designa method
to identify
a location
thewallwall
shelf
by using
only numbers.


Explain your method.



How will your method differentiate the location of Pikachu and tiger?

•

Explain your method.

•

How will your method differentiate the location of Pikachu and Tiger?

Right

Left
Diagram 22
Diagram

168 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelv.

Diagram 2 shows a game board which is a plan of table arrangement in a classroom. The

v. Diagram 2 shows a game board which is a plan of table arrangement in a classroom. The rules
rulesof thegame
shown
Diagram
3. of table arrangement in a classroom. The rules
v. ofDiagram
2 shows
a aregameboardinwhich
the game
are
shown
in
Diagram
3. is a planof the game are shown in Diagram 3.
Game Rules:
Game Rules:① The game board is displayed in the front of the class.
① The game board is displayed in the front of the class.
② The class is divided into two teams.
② The class is divided into two teams.
③ Members from each team will take turn to choose a table on the game board by giving instruction③ toMembersfrom(taking
each teamwillastaketurn to choose
table on thegame board
by giving instruction
the teacher
the boy
referencepoint). aExamples
of instructionare:to the teacher (taking the boy as reference point). Examples of instruction are:
“3 tables to the front.”“3 tables to the front.”“3 tables to the front and 4 tables to the left.”“3 tables to the front and 4 tables to the left.”④ If an instruction is given correctly, the corresponding table will be claimed by the team and thewill mark isthegiven
tablecorrectly,
for the team
the game board.
④ teacher
If an instruction
the oncorresponding
table will be claimed by the team and theteacher will mark the table for the team on the game board.
⑤ The first team that claims 4 tables consecutively in a horizontal, vertical or diagonal line, is thewinner.
⑤ The
first team that claims 4 tables consecutively in a horizontal, vertical or diagonal line, is thewinner.

•

Diagram
Diagram3 3
Diagram 3
Diagram 4 shows the instructions given by members from two team, A and B, during a game
Diagram 4 shows the instructions given by members from two team, A and B, during asession.
Diagram 4 shows the instructions given by members from two team, A and B, during a game
game session.
session.
Table Marked
Table1st
Marked

1st
2nd
2nd
3rd
3rd

Team AA
 2 tables toTeam
the right

Team BB 2 tables to 1 table to theTeam
front and

 2 tables to the right 1 table to the right and 1
to the table1 tableto backthe right and 1
table
to
the
 2 table to thebackback 2 table to the backleft to the front and 2 tables to the
1 tabletheleft to the left and 1 table to 3 tables
front to the left and 1 table to the
3 tablesthe front

Diagram 4

Diagram
4 4Diagram Mark the tables according to the instructions given, using different marks for different teams.
Mark the tables according to the instructions given, using different marks for different







Mark the tables according to the instructions given, using different marks forteams.
different
teams.
 Aftermarking
their 3rd table, the members of Team A shouted: “Yeah! We win! Wewin!”
Do you agreewith table,
them? Explain
your reasons. A shouted: “Yeah! We win! We
Aftermarking After markingtheir
their3rd
3rd table,thethemembers
membersofofTeam
Team A shouted: “Yeah! We win! We
win!” Do you agree with them? Explain your reasons.
win!”
Do
you
agree
with
them?
Explainyourreasons.
To what extend can this game help to inculcate appreciation towards the ideas of positions
anddirections
among
your
students?
To whatextendcan thisgamehelp to inculcate appreciation towards the ideas of positions• andTo directions
what extend
canyourthisstudents?
game help to inculcate appreciation towards the ideas ofamongvi. Compare
the twosystemsof determining
locationson a plane in Diagram 1 and Diagram 2.
positionsanddirections
among yourstudents?
vi. Compare
the thetwosimilarities
systems ofand
determining
locationsonthe
a planein Diagram 1 and Diagram 2.
What aredifferences
betweentwo systems?

vi. Compare the two systems of determining locations on a plane in Diagram 1 and WhichWhat arethe similarities
and differences
betweenthe twosystems?
systemwill you discuss
first with yourstudents?
Explainyour reasons.
Diagram2.
Which system will you discuss first with your students? Explain your reasons.

•

Which system will you discuss first with your students? Explain your reasons.

•

What are the similarities and differences between the two systems?

CHAPTER
5 5
CHAPTER

Chapter 5: Patterns and Data Representations | 169

CHAPTER
5 Representations
Patterns
andData
Patterns
andData
Representations
Topic
1:Patterns
Using
Patterns
Under
the Number
Sequence
andData
Representations
Topic
1: Using
Patterns
Under
the
NumberSequence
Standard
1.1:1:
Arrangingobjects
forbeautifulpatterns
under
the number
sequence
Standard
1.1:
Arrangingobjects
for thebeautifulpatterns
under
the number
sequence
Topic
Using
Patterns
Under
NumberSequencei. i.Know
the beautifulnessof patterns
in casesof arranging
objects
based
on numbersequenceKnow
the beautifulnessof patterns
in casesof arranging
objects
based
on numbersequence
Standard 1.1:
ii. ii.Arrange
objects
accordingtonumbersequencetofind
simplepatterns
Arrangeobjects
to numbersequence
to findsequence
simple patterns
Arrangingobjects
for according
beautiful patterns
under
the number
iii. iii.i.Arrange
expressions
suchas
additionandsubtractiontofindsimplepatterns
Knowthebeautifulnessofpatterns
in
cases
of
arrangingbased
on number sequence
Arrange expressions such as addition and subtraction to objects
find simple
patterns
ii. Arrange objects according to number sequence to find simple patterns
iv. iv.Express
the representationof patterns
usingusing
placeholders(emptybox)box)Express
the representation
ofaddition
patterns
placeholders(emptyiii. Arrange
expressions
such asand subtraction
to findsimplepatterns
v. v.iv.
Enjoythearrangementofobjects
basedonnumbersequence
in
daily
life
Express
therepresentationofpatterns
using
placeholders(emptybox)
Enjoy the arrangement of objects based on number sequence in daily life
the onarrangementof objects
based
on number sequence in daily life
vi. vi.v.FindEnjoy
patterns
numbertablessuchsuchasinas
calendars
Findpatterns
on numbernumbertablesin calendars
calendarsvi. Findpatterns
ontablessuch asin
SampleTasks
for Understanding
the Standards
SampleTasks
for forUnderstanding
thetheStandards
SampleTasks
UnderstandingStandards
TaskTask
1: Patterns
fromfrom
NumberSequence
1: Patterns
NumberSequence
Task 1: Patterns from Number Sequencei.

1 shows
10 coloured
chipschips
in a row.
i.Diagram
11 shows
10ini. Diagram
Diagramshows
10 coloured
colouredchips
in aa row.
row.

Diagram
Diagram
1 11
Diagram
Diagram
shows
the
10
rearranged
aapatternthat
represents
a numbersequence.
Diagram
2 shows
the 10rearranged
intointoapatternthat
represents
a numbersequence.
Diagram
22 shows
thechips
10chipschips
rearranged
intopatternthat
represents
a numbersequence.



Diagram
2 2
Diagram
Diagram 2
Rearrange
the chips
to form
moremorepossiblepatterns
and andwritewrite
the corresponding
number Rearrange
the chips
to form
possiblepatterns
the corresponding
number•sequence.
Rearrange
the
chips
to
form
more
possiblepatterns
and
write
the
corresponding
sequence.
number sequence.
How
doesdoesthe formationof patterns
promotechildren’s
learningof numbersequence?
 How
the formationof patterns
promotechildren’s
learningof numbersequence?
•How
How
does
the
formationofpatterns
promotechildren’s
learningofnumbersequence?
does
thenumbersequence
lead
tolearningofaddition?
 How does the number sequence lead to learning of addition?
What number
patterns
can leadto multiplication?
Explain youryour
answer with examples.
numbercan leadtoleadmultiplication?
• What
How doesthe patterns
number sequence
to learning ofExplain
addition? answer with examples.

•

What number patterns can lead to multiplication? Explain your answer with examples.

170 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelii.ii. Twocubestotorepresent
representthe
the
first
numbers
Twostudents,
students,AAandandB,
B, usedused multi-linked
multi-linked cubes
first
fivefive
oddoddnumbers
as as
shown
in
Diagram3.
shown in Diagram 3.
Student A’s Representation

Student B’s Representation

Diagram 3
Diagram3

Which
representation
do youris betterin helping
studentslearn
the property • Whichrepresentation
do yourthinkthinkis betterin helping
students
learn thepropertyof odd of
odd numbers?
your choice.
numbers?
Justify Justify
your choice.
 • How
willwillyouyouuseusemulti-linked
cubes
to represent
the first
How
multi-linked
cubes
to represent
thefivefirstevenfive numbers?
even numbers?

iii.

iii.

In what way can this task develop primary students’ appreciation toward the beautifulness
what way can this task develop primary students’ appreciation toward the beautifulness ofofInpatterns
in number sequence?
patterns in number sequence?

Chapter 5: Patterns and Data Representations | 171

Task
2:
Patterns
from
Addition
Subtraction
Expressions
Task
Task
2:
2:Patterns
Patterns
from
from
Addition
AdditionandandandSubtraction
Subtraction
Expressions
Expressions
Task 2: Patterns from Addition and Subtraction Expressionsi.

Diagram 4 shows two sets of expressions

ii.ii.

Diagram
Diagram 55 shows
shows another
another two
two sets
sets of
of

iii.
iii.

of
ofDescribe
expressions.
expressions.
•  Describe
thepatterns ininthesethese
the patterns
twotwosets
sets
of
expressions.
of expressions.
How
How
will
will
these
these
patterns
patternshelp
helpyouryourstudents
students

i.i. involving
Diagram
Diagramaddition
44 shows
showsandtwo
twosubtraction.
sets
sets ofof expressions
expressions
involving
additionandandsubtraction.
subtraction.
i. involving
Diagramaddition
4 shows
two
sets of expressions
involving addition and subtraction.
Describe
Describe thethe patternpattern inin eacheachset• Describe
thepatterneach set
set ofofofexpressions.
expressions.
 expressions.
Describe the pattern in each set of
expressions.

Diagram
Diagram44
Diagram
Diagram 4ii. Diagram 5 shows another two sets ofexpressions.
expressions.
ii. expressions.
Diagram 5 shows another two sets of expressions.
Describe
Describethe
thepatterns
patternsininthese
thesetwo
twosetssets

learn
additionandandsubtraction
subtractionofnumbers?
numbers?
iii. learn
Howaddition
will thesepatterns
helpofyourstudents
iii. Howwill
thesepatterns
help
yourlearn addition and subtraction of students
numbers?
learn addition and subtraction of numbers?

Diagram
Diagram55
Diagram
iv.
iv.

iv.

iv.

Diagram 5
Diagram
Diagram66shows
showsaapatternpatternof
ofexpressions.
expressions.

Diagram 6 shows a pattern of expressions.
Diagram 6 shows a pattern of expressions.
Diagram
Diagram66

Diagram6
Diagram
Fill
Fillininall
allmissing
missingnumbers
numbersininthe
thepattern.
pattern.
Justify
Justifyeach
eachofofyouryouranswers.
answers.

Fillallmissing
numbers
pattern.
Justify
each
of
your
answers.
Fillininallmissing
numbers
in inthethepattern.
Justify
eachofamong
youranswers.
• In
In
what
what
way
way
can
canthisthis
task
taskpromote
promotemathematicalmathematicalthinking
thinking
among
primary
primary
students?
students?
•

InInwhat
thistask
promotemathematicalthinking
among
primary
students?
whatway
waycan
can
this
task
promotemathematicalthinking
among
primary
students?

172 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Task 3: Patterns in Daily Life

Task 3: Patterns in Daily Life
Tablet Counter

Tablet Counter

Diagram 7 shows a triangular tablet counter usedby pharmacists to automatically calculate the total
Diagram
7 shows
a triangularcounternumberof tablets
accordingto thetablet
numberof rows
usedby onpharmacists
to automaticallyfilledprintedthe counter itself.

calculate

the

total

numberoftabletsi.according
What is tothethenumberof tablets
eachof thenumberof rowsfor
filledprintedrows?
onfollowing
the counter
itself.

i.

Row 3



What is the number of tablets for each
 ofRow5
the following
rows?


•

Row 10Row 3

ii. Table 1 shows the total number of tablets for5the• firstRow
3 rows.

Diagram
Diagram 77

ii.

•
Row
Table
1 10
Total Number of Tablets for Each Row
Totalnumber ofTable Row
1 shows the total
1 the first 3 rows. 1
tablets for2
3
3
6
Table 1
4
Total Number of Tablets for Each Row
5
6
Row
Total
1 1.
 Fill in the1missing totals in Table
2

3

3

6

 Describe the number pattern in the table.
 What mathematical
attitude can be4
developed through this task? Explain youranswer. 5

6
Fill in the missing totals in Table 1.
•

Describe the number pattern in thetable.

•

What mathematical attitude canbe developed through this task?
Explain your answer.

Chapter 5: Patterns and Data Representations | 173

Calendar Patterns
CalendarCalendarPatterns
Patterns

Diagram
Diagram8 8Diagram 8 shows the calendar for January 2021.
Diagram
Diagram8 8shows
showsthethecalendar
calendarforforJanuary
January2021.
2021.
i. Choose any horizontal row of numbers from the calendar such as 3, 4, 5, 6, 7, 8, and 9.
i. i. Choose
Chooseany
anyhorizontal
horizontalrowrowofofnumbers
numbersfrom
fromthethecalendar
calendarsuch
suchasas3,3,4,4,5,5,6,6,7,7,8,8,and
and9.9.
•Describe the number pattern in each row.
  Describe
Describethe
thenumbernumberpatternpatterninineacheachrow.
row.
•
Choose three consecutive numbers from any of the rows. What pattern can you find  Choose
three consecutive
numbers
fromChoosenumbers
fromany
anyofofthe
therows.
rows.What
Whatpatternpatterncan
canyouyoufindfindfrom
fromfrom thethree
threeconsecutive
numbers? Explain
why.
thethethreethreenumbers?
numbers?Explain
Explainwhy.
why.
Chooseanyverticalcolumnnumbersfromthecalendar
such
as
12,
19,
and
26.
ii.ii.
ii. Choose
Chooseany
anyverticalverticalcolumn
columnofof
ofnumbers
numbersfrom
fromthethecalendar
calendarsuch
suchasas5,5,
5,12,
12,19,
19,andand26.
26.
Describe
thenumberpatternineacheachcolumn.
 • Describe
thenumberpatterninineachcolumn.
Describe
thenumberpatterncolumn.
 • Choose
threeconsecutive
numbersfromany
ofofofthecolumns.
Choosethreeconsecutive
numbers
from
any
the
columns.What
Whatpatternpatterncan
canyouyoufindChoosethreeconsecutive
numbers
from
any
the
columns.
find
from
the
three
numbers?
Explain
why.
find
from
the
three
numbers?
Explainwhy.
from the three numbers? Explain why.
iii.
iii. Choose
Chooseany
anysquaresquarebox
boxofofoffourfournumbers
numberssuch
suchasasiii.
Chooseanysquarebox
fournumberssuchas• What
Explainwhy.
why.
Whatpatternpatterncan
canyouyoufindfindfrom
fromthethefourfournumbers?
numbers?Explain
Explainwhy.

174 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Topic
2: 2:
Producing
Patterns
Topic
ProducingHarmony
Harmonyof
ofShapes
Shapes Using
Using Patterns
Topic
Topic 2:
2: Producing
Producing Harmony
Harmony ofof Shapes
Shapes Using
Using Patterns
Patterns

Standard
2.1:
Standard
2.1: Arranging tiles of different or similar shapes in creating harmony

Standard
Standard 2.1:
2.1: Arranging
Arranging tiles
tiles ofof different
different oror similar
similar shapes
shapes inin creating
creating harmony
harmonyi.
Know the beautifulness of patterns in cases of arranging the objects based on shapes,
Arrangingtiles
ofdifferentorsimilarshapesincreating
harmony
i.i.
Know
the
beautifulnessofpatterns
in
cases
of
arrangingtheobjects
Know
beautifulnessof patterns in cases of arranging the objects based
based onon shapes,
shapes,coloursthe
and
sizes
and
sizes
i. Knowcolours
thebeautifulnessofpatterns
in
cases
of
arrangingtheobjects
basedonshapes, colours
colours and sizes
ii. sizes
Arrange objects according to shapes, colours and sizes to show patterns
andii.
Arrangeii.
Arrange objects
objects according
according toto shapes,
shapes, colours
colours andand sizes
sizes toto show
show patterns
patterns
ii. iii.
Arrangeobjects
accordingto shapes,
colours
Arrangeboxes
accordingto shapes,
colourand
andsizes
sizes toto show
createpatterns
structureiii.
Arrangeboxes
accordingtoshapes,colourandsizes
tocreatestructure
Arrangeaccordingto shapes,
coloursizes
to create
structureiii. iii.
Arrangeboxesboxes
accordingto shapes,
colourandandsizes
to create
structureiv.
Arrange circles and spheres for designingiv. iv.
Arrangecirclesandspheresfordesigning
Arrangecirclesandspheresfordesigningiv.
Arrange circles and spheres for designingv.
Enjoy
the creation
based
on different
shapes,colour
sizes
daily
life
v. Enjoy
the creation
based
on different
shapes,colourandand
sizes
in in
daily
life
v.
Enjoy
v.
Enjoy thethe creation
creation based
based onon different
different shapes,
shapes, colour
colour andand sizes
sizes inin dailydaily life
life

SampleTasks
for
SampleTasks
forUnderstanding
Understandingthe
the Standards
Standards
SampleSample Tasks
Tasks forfor Understanding
Understanding thethe Standards
Standards
Task
1: 1:
TilingPattern
Task
TilingPattern
Task
Task 1:
1: Tiling
Tiling Pattern
Pattern
Task ①: Extend the design.
Task
Extend thethe design.
design.
Task ①:①: Extend

Task ②: Copy the design.
Task
Task ②:②: Copy
Copy thethe design.
design.

Task ③: Use the three types of blocks to create your own design.
Task
three types
types ofof blocks
Task ③:③: UseUse thethe threeblocks toto createcreate your
your own
own design.
design.

Task ④: Extend the design.
Task
Task ④:④: Extend
Extend thethe design.
design.

Diagram
1 1
Diagrami.

Diagram
Diagram 1
1i.
Diagram1 shows
on designing
patternsusing
using geometrical tiles
tiles for
Diagram1 shows
four fourtaskstasks
on designing
patterns
for primaryprimaryi.i.
Diagram1 shows
four
tasks
on designing
patterns usinggeometricalgeometricalfor
primary
Diagramchildren. 1 shows four tasks on designing patterns using geometrical tiles for primarychildren.
children.
children.


How will you sequence the tasks for your students?
How will you sequence the tasks for your students?



Explain
Explain your
your reasons.
reasons.

•

 willHow
willsequence
you sequence
the tasks
for yourstudents?
Howyouthe tasks
for yourstudents?
 Explain
your reasons.

•Explain your reasons.

Chapter 5: Patterns and Data Representations | 175

Diagram
Diagram 2
2
Diagram 2ii. Diagram
Diagram
2 shows
twodesigns
createdbytwo
primary
studentsusingcirculartiles.
2 shows
twodesigns
createdtwo
primary
studentsusingcirculartiles.
ii. ii. Diagram
2 shows
twodesigns
createdbyby
twoprimary
studentsusingcirculartiles.
 Based on each of the designs, pose a task involving number pattern for primary • Based
on on
each
of thedesigns,pose
a taskinvolving
numberpatternfor primary
Based
each
of thedesigns,pose
a taskinvolving
numberpatternfor primarymathematics.
mathematics.
mathematics.
 How could you help your students relate the designs to number pattern?
 How could you help your students relate the designs to number pattern?
•
How could you help your students relate the designs to number pattern?

Diagram 3
Diagram 3
iii.
Diagram 3 show a part of a floor tiling design which consists of six complete and oneiii.
Diagram 3 show a part of a floor tiling design which consists of six complete and one
incomplete
arrangements
darktilingtiles.design which consists of six complete and oneiii. incomplete
Diagram
3 showa partofofdark
aoffloorarrangements
tiles.
incomplete arrangements of dark tiles.
 How many dark tiles will you predict to complete the 7th arrangement?
 How many dark tiles will you predict to complete the 7th arrangement?
Whatmathematicalthinking
and processes
could
through this task?
•  Howmanydark tiles will
you predict
to complete
the be7th promoted
arrangement?
 What mathematical thinking and processes could be promoted through this task?

•

What mathematical thinking and processes could be promoted through this task?

176 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

iv.
iv.
iv.

Diagram
Diagram
4 4Diagram 4
Diagram 4 shows another part of a floor tiling design which consists of four complete
Diagrampart
consistsof offour
completeandDiagram4showsshowsanother
anotherpartofofa afloor
floortilingtilingdesign
designwhich
which
consists
fourcompleteand one4incompleterhombus.
one
rhombus.
andincomplete
one incompleterhombus.
 How many white tiles are inside each of the complete rhombus?
 How many white tiles are inside each of the complete rhombus?
• Howeach ofcomplete
Howmany
manywhite
whitetiles
tilesarewill inside
you predict
to thecompletethe rhombus?
fifth rhombus?
 How many white tiles will you predict to complete the fifth rhombus?
Whatmanymathematicalin futuregrade
are embedded
this task?
• Howwhite tilesideas
will youpredict
to complete
the fifthinrhombus?
 What mathematical ideas in future grade are embedded in this task?

•

v.
v.
v.

What mathematical ideas in future grade are embedded in this task?

Diagram 5
Diagram
Diagram55
Diagram 5 shows a design formed by 10 circles.
Diagram 5 shows a design formed by 10 circles.
Diagramshows
designwith
formedby 10ofcircles.
 5
Draw
theadesign
the help
a compass.
 Draw the design with the help of a compass.
Draw
Based
on thewith
design,whatofembedded
mathematical ideas can be discussed with•the design
the help
a compass.
 Basedon the design, what embedded mathematical ideas can be discussed withyour students?
your students?
•
Basedon willthe you
design,embeddedmathematicalideas canbe discussedwith your How
use whatthe design
to prepare
your students
for learning
the embedded
students?
 How
you usethe design to prepare your students for learning the embedded
ideaswillin higher
grades?
ideas in higher grades?
•
How will you use the design to prepare your students for learning the embedded ideas
in higher grades?

Chapter 5: Patterns and Data Representations | 177

Task 2: Creative Design in Daily Life
Task 2: Creative Design in Daily Life
Task
Task
2: 2:
Creative
Creative
DesignDesign
in in
Daily
Daily
Life
Life

Diagram 5
Diagram
Diagram
5 5Diagram
Diagram66
Diagram
Diagram6 6

i. i. Diagram
5 shows
a basic
piece
of of
flooring
tile.tile.
These
basic
piecesof tilescancanbe bearrangedtoDiagram
5 shows
a basic
piece
flooring
These
basic
piecesof tilesarrangedform
variouscreative
designs.
Diagram
6 shows
an example
of design
createdcreatedusing 8using
piecestoformvariouscreative
designs.
Diagram6
shows
anexampleofdesign8i. i. Diagram
Diagram
5 shows
5 shows
a basic
a basic
piece
piece
of of
flooring
flooring
tile.tile.These
These
basic
basic
piecespieces
of of
tiles
tiles
can
can
bebe
arrangedarranged
to toofpiecesthe tiles.
ofthetiles.
form
form
variousvariouscreativecreative
designs.
designs.
Diagram
Diagram
6 shows
6 shows
anan
exampleexampleof ofdesigndesigncreatedcreatedusing
using
8 pieces
8 piecesof ofthethetiles.
tiles.

•

Based on this design, what embedded mathematical ideas can be discussed with yourBased on this design, what embedded mathematical ideas can be discussed with yourprimary
students?
  Based
Based
onon
thisthisdesign,design,what
what
embeddedembeddedmathematicalmathematical
ideas
ideas
can
can
bebe
discussed
discussed
with
with
your
your
primary
students?
primary
primary
students?
students?

Use
basic
flooring
in Material
Sheet
2 atendtheofend
thistotopic
to anothercreate anotherii. ii. Usethethebasic
flooring
tilestiles
in Material
Sheet
2 at thethisoftopic
createflooring

ii. ii. Use
Use
the
the
basic
basic
flooring
flooring
tiles
in Material
in Material
Sheet
Sheet
2from
at
2 at
the
end
end
of of
thisthis
topic
to to
createcreateanotheranotherflooring
flooring
flooring
designwhich
isfromdifferent
Diagram
6.topic
designwhich
istiles
different
Diagram6.thedesigndesignwhich
which
is different
is different
from
from
Diagram
Diagram
6. 6.

•

In what ways can this task cultivate mathematical habits of mind?

Inways
what
ways
can
this
taskmathematicalcultivatemathematical
habits of mind?
  In In
what
what
ways
can
can
thisthistask
task
cultivatecultivatemathematical
habitshabits
of of
mind?
mind?

• what
In
what
ways
can
this
task
help
to develop
appreciation
towards
mathematics?
Inways
what
ways
can
this
task
help
to
developappreciation
towards
mathematics?
  In In
what
ways
can
can
thisthis
task
task
help
help
to to
developdevelopappreciation
appreciation
towards
towards
mathematics?
mathematics?
iii.

iii. iii. Diagram
Diagram
7Diagram
shows
anan
empty
squaredsquaredtile.tile. squared tile.
Diagram
7 shows
an empty
squarediii. 7 shows
7emptyshows
an tile.
empty

•
•

Diagram
Diagram
77 7
Diagram
Diagram 7

Diagram
Diagram
Diagram
88 8
Diagram 8iv.

  Create
Createa basic
flooring
flooring
designdesignonon
thisthisCreatea basic
flooringon
thissquaredtile.
 a basic
Createadesignbasic
flooring
design on
thissquaredsquaredtile.tile.
squared tile.
What
do
you
need
to
considerwhen creating the  What
What
do
do
you
you
need
need
to to
considerconsiderwhen
whenthetile
What
do
you
need
consider whenbasiccreating
flooring
inorder
totilemakeitorder
attractive?
creating
the
basic
basic
flooring
flooring
tile
in order
into
to to
make
make
it attractive?
it creating
attractive?the basic flooring tile in orderto make it attractive?

Diagram
8 shows
anotherbasic
tile.
iv. iv. Diagram
Diagram
8 shows
8 shows
anotheranother
basic
basic
tile.
tile. Three floor tilingdesigns
were
formedbydesigns
arranging
32
pieces
Three
Three
floor
floor
tilingtilingdesigns
were
were
formedformedbyby of this basic
iv. in32
Diagramshows
anotherarrangingarranging
32
piecespieces
of9.8
of
this
this
basic
basic
tiletile
asas basic tile.
tile as shown
DiagramThree9.floor
shown
shown
in Diagramin Diagram
9. tiling designs were formed byarranging 32 pieces of this basic tile asshown in Diagram 9.

178 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Arrangement ①

Arrangement ②

Arrangement ③

Diagram9
Diagram 9•

Use theUsebasic
flooring
tile tilein Material
at thetheendendof oftopic
to duplicate
the basic
flooring
in MaterialSheet
Sheet 33 at
thisthis
topic
to duplicate
the thethree designs.
three designs.

•

Study arrangements①,arrangement② and ③ carefully.
 Describe how① is changed to arrangement ②?

Study arrangements ①, ② and ③carefully.

 Describe
arrangementchangedto
to arrangement
arrangement ③?
Describe
how howarrangement①①
is ischanged②?

What mathematical ideas in future grade are embedded in this task?

  Describe
how arrangement ① is changed to arrangement ③?
Interpretation in art refers to attribution of meaning to an art work. What is yourpersonal interpretation on each of the geometrical art in Diagram 8?

•

What mathematical ideas in future grade are embedded in this task?

•

Interpretation in art refers to attribution of meaning to an art work. What is your personal
interpretation on each of the geometrical art in Diagram 9?

•

How could interpretations of these designs help your students to appreciate theharmony of geometrical art?

How could interpretations of these designs help your students to appreciate theharmony of geometrical art?

Chapter 5: Patterns and Data Representations | 179

Material Sheet 2: Basic Flooring Tiles – Design I

Material Sheet 2: Basic Flooring Tiles – Design I180 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Material
Sheet
3: BasicFlooring
Tiles
DesignIIII
Material
Sheet
3: BasicFlooring
Tiles– –DesignChapter 5: Patterns and Data Representations | 181

Topic 3: Collecting Data and Represent the StructureStandard 3.1:
Topic 3: Collecting Data and Represent the Structure
Collecting
data through categorisation for getting informationi.Standard
Explorepurpose of why data is being collected.
Collecting data through categorisation for getting information
3.1:theii. i. Groupeddata
by creating
similarattributeson the denomination of categories and count them
Explore the purposeof why data
is being
collected.
ii. (check
Groupeddataby count)
creating similar attributes on the denomination of categories and count them (check markmark
andandabout
count) what information is obtained from the tables with categories and how to use itiii. Think
iii.

Think about what information is obtained from the tables with categories and how to use it

SampleTasksforforUnderstanding
UnderstandingStandards
Sample Tasks
thetheStandards
Task 1: Collecting
Task
CollectingData
Data

i.

i.

Every student from a class was asked to choose a picture card to represent his/her favourite

Every student from a class was asked to choose a picture card to represent his/her favourite
fruit. Then, everyone placed their favourite fruit on the board as shown in Diagram 1.
fruit. Then, everyone placed their favourite fruit on the board as shown in Diagram 1.

Favourite Fruits

Diagram11
Diagram•  Whatthemostmostpopular
popularfruit?
What isis thefruit?
••

How many students choose pineapple as their favourite fruit?

How many students choose pineapple as their favourite fruit?

What can we do with the data collected in order to make it easier to answer thesequestions?
What
can we do with the data collected in order to make it easier to answer thesequestions?

182 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelii.

With the purpose of answering the questions in (i), three students, A, B and C, counted thestudents
for each categoryof fruitsdifferentA,
methods
shownthein Diagram
ii. number
With theof
purposeof answering
the questionsin (i), using
three students,
B and C,ascounted2.number of students for each category of fruits using different methods as shown in Diagram 2.
Student A’s Method (Using Check Marks)
DurianPapayaPineapple

Mango

Starfruit

Guava











8

7

4

5

2

4

Student B’s Method (Using Tally Marks)
DurianPapayaPineapple

Mango

Starfruit

Guava

8

7

4

5

2

4

Student C’s Method (Line up the Cards)

8
7
4
7
5
7

2
7
4
7
Diagram 2
2
Diagram•Compare the three methods of counting. Which method will be easiest for yourstudents? Explain your reasons.

•

Why is tally mark more efficient than check mark in counting the number of students ineach category?

Chapter 5: Patterns and Data Representations | 183

Standards 3.2:
Organising the data collected and represent using pictogram for easy visualisationi. Produce the table and pictograms from collected data under each categories
ii. Interpretation of tables and pictograms as a simple conclusion about the data being presented.
Standards
3.2: Organising
thecollected
data
collected
and represent
using
pictogramforvisualisationeasy visualisation
Standards
3.2:
the
and
using
pictogramforforeasy
Standards
3.2:Organising
Organisingthedatadatacollected
andrepresent
represent
using
pictogrameasyiii.
pictograms
collecting
dataand
addingdatavisualisationin daily activities in learners’ life
i.Produce
Producethe andtable
andthroughpictograms
fromcollected
data
under
each
categories
i. i. Appreciate
Produce
the
from
data
under
each
categories
thetable
table
andpictograms
pictograms
fromcollectedcollected
data
under
each
categories
ii.ii.

ii.
Interpretationof tablesand pictogramsasa simpleconclusion
about
thebeing
data
being
presented.
Interpretation
ofoftables
andasasa asimple
conclusion
about
the
data
presented.
Interpretationtablesandpictograms
pictograms
simpleconclusion
about
the
data
being
presented.

Appreciate
pictograms
throughcollecting
data
and adding
inactivities
daily
activitiesin learners’
iii.iii. iii.
Appreciate
pictograms
throughcollecting
data
and
data
indatadaily
in in
learners’
lifelife life
Appreciate
pictograms
throughcollecting
data
andaddingadding
data
in
daily
activities
learners’
SampleTasks
for
UnderstandingtheStandards

SampleTasks
for Understanding
the Standards
Sample
Tasks
for
the
SampleTasks
forUnderstanding
Understanding
theStandards
Standards

Task 1: Representing Data

Task
1: Representing
Task
1:1:Representing
Data
Task
Representing
DataData

i.

i. i.

Table 1 shows the data collected on students’ favourite fruits.
i.TableTable
1 shows
the collected
data
collected
on students’favourite
fruits.
Table
1 1shows
theononstudents’favourite
fruits.
shows
thedatadatacollected
students’favourite
fruits.

Table
1
Table
Table
11 1
Table
Students’
Favourite
FruitsStudents’
Favourite
FruitsStudents’
Favourite
FruitsStudents’
Favourite
FruitsFavourite
FruitsFavourite
FruitsFavourite
FruitsDurianPineapple
Mango
Guava
DurianPapaya
PapayaPineapple
MangoStarfruit
Starfruit
Guava
DurianPapayaPineapple
Mango
Starfruit
Guava

NumberofofStudents
Numberof Students 8 8
NumberStudents

8

77

7

44

4

55

5

22

2

44

4

studentdrew
a apictogram
asasshown
ininDiagram
3 3totorepresent
the
Table
1.1. 1.
A studenta pictogramasshownshown
in
3represent
totheindatain
Table
student
drewdrew
shown
Diagramthedata
data
in
Table
AAAstudent
drew
apictogram
pictogramasinDiagram
Diagram
1 represent
to represent
thedata
in Table 1.
Students’
Favourite
FruitsStudents’
Favourite
FruitsStudents’
Favourite
FruitsDurianPineapple
Durian Papaya
PapayaPineappleMango
MangoStarfruit
StarfruitGuava
Guava
DurianPapayaPineapple
Mango
Starfruit
Guava
Diagram
3 31 3
Diagram
Diagram
DiagramWhat is the most popular fruit?

What
What
is mostthemostpopularfruit?
ispopularfruit?
•  Whatis thethemostpopularfruit?

  Howmore
students
likethanHowmany
How
many
more
students
likedurianthan
starfruit?
many
more
students
likedurian
durianthanstarfruit?
starfruit?

•  IsHow
many
more
students
likedurianthan
starfruit?
totoshow
the
data
table
or
the
pictogram?
ExplainyourIsit itbetterIs
it betterto show
the in
data
in
the
table
orthe
in
the
pictogram?
Explainyourreasons.
better
show
the
data
inthe
the
table
orininpictogram?
Explainyourreasons.
reasons.


Whatcouldcould
be doneto thetographin orderto makeit easierto read?
 Whatbe donethe graphin orderto makeit easierto read?

could tobeshowdone tothedatagraphinintheorder
to makeeasierto read? Explain your reasons.
•  IsWhat
it betterthetable
or initthe
pictogram?
  Threestoresnext
eachdurian,papayaandrespectively
tototheThreeThree
storesnext
toschool,the school,eachsellingdurian,papayaandmango
respectively
to thestoresnexttotothe
theschool,eachselling
sellingdurian,papayaandmango
mango
respectively
thestudents.
Another
store
sells
other
fruits.
If Ifeach
student
isisbuying
one
favourite
fruit
from
students.
Anothersells
otherfruits.
If toeachstudent
is buyingone
favourite
fruit
students.
Anotherstorestore
sells
other
fruits.
each
student
buyingone
favourite
fruit
fromfrom
What
could
be
done
to
the
graph
in
order
make
it
easier
to
read?
these
stores,
which
store
will
numberofoffruits
thesestores,whichwill thehave
the highest
numberof sold?
fruitssold?
thesestores,which
storestore
willhave
have
thehighest
highestnumberfruitssold?

•• Whatisistheofofchanging
the categories
ofofthis
set
data?

What
isconsequence
the consequenceof changing
the categories
of
this
set
of data?
Whattheconsequencechanging
categories
this
setofofdata?
Three
storesnext
to the school,
eachthesellingdurian,papayaandmango respectively

to the students. Another store sells other fruits. If each student is buying one favourite
fruit from these stores, which store will have the highest number of fruits sold?
What is the consequence of changing the categories of this set of data?

184 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelii.

A class of 28 students is investigating their birthday in 2021. Each of them is given a smileyii.
A classof 28
theirhimself/herself.
birthday in 2021. Each of them is given a smiley
card
as
shown
in students
Diagramis 2investigating
to represent
ii.
ii.

card as shown in Diagram 4 to represent himself/herself.
A class of 28 students is investigating their birthday in 2021. Each of them is given a smiley
A class of 28 students is investigating their birthday in 2021. Each of them is given a smileycard as shown in Diagram 4 to represent himself/herself.
card as shown in Diagram 4 to represent himself/herself.

Diagram 4
Diagram 4
Diagram 4
Then, the students were asked to tell which day of the week their birthdays in 2021 will be byThen,placing
thestudents
were
asked
to
tell
which
day
birthdays
theirsmileycards
onasquaregridon theaweek
bigweekpiece
of cardboard
as in
shown
in will beThen, the students were asked to tell which drawn
day
of of
the
their theirbirthdaysin 2021
will2021
be byThen, the students were asked to tell which day of the week their birthdays in 2021 will be byplacingtheirsmiley cardsona squaregridgrid
drawn
on a bigpiece
ofpiece
cardboardas shown inas shown
Diagram5.
by placing
theirsmileycards
onasquare
drawn
on
a
big
of
cardboardplacing their smiley cards on a square grid drawn on a big piece of cardboard as shown in
Diagramin Diagram
3. 5.5.
Diagram

Diagram 2

Birthday on Days of the Week
BirthdayBirthdayononDays
Daysofofthe
theWeek
Week

Diagram55
Diagram
Diagram
Diagram 3
5
Whatdoesdoeseacheach•   What
does
each
What
What does
each

represent?
represent?
represent?
represent?

How
many
more
students
have birthday
on Tuesday
than Friday?
 How
many
more
students
birthday
How
many
more
studentshave
birthdayon
onTuesday
Tuesday than
than Friday?
Friday?
• Howmany
more
studentshave
havebirthdayon Tuesday
than Friday?
Isitnecessary
to
order
the
categories
of
the
data
when
drawinga pictogram?
your  Is itIsnecessary
toto
order
pictogram?Explain
Explainyourit necessaryorderthethecategories
categoriesofofthe
thedatadata when
when drawing
drawing aa pictogram?
Explainyouranswer.
answer.
answer.
•Is it necessary to order the categories of the data when drawing a pictogram? Explain

iii.

iii.
Determinewhether each of the following statements about pictogram is true or false.
answer.
iii. iii. yourDeterminewhether
pictogram isis true
trueororfalse.
false.
Determinewhethereacheachofofthe
thefollowing
followingstatements
statements about
about pictogram__________ (a)A pictogram may not have a title.
__________
pictogrammay
maynotnothavehave aa title.
title.
__________(a)(a) A Apictogram
Determinewhetherof thecan
followingabout
pictogram is true or false.
__________(b) each
A picture
only be statements
used to represent
1 unit.
__________(b)A
picture
can
only
be
used
to
represent 1 unit.
__________ (b)A picture can only be used to represent
unit.
__________ (c)The pictures used must be of same size.
__________(c) TheThepictures
picturesusedmustabebeof samesame size.
size.
__________(a) A(c)pictogram
may nothave
title.
__________
used
must
of
__________ (d)The pictures used must be equally spaced.
__________
Thepictures
picturesused
usedmust
mustbebe equallyequally spaced.
spaced.
__________(d)(d) The__________
(e)
A key
must
bebe
providedthe1valueof each picture.
__________(b)A
picturecan
onlyused to
totodetermine
represent__________
(e)
A
key
must
be
provided
determine
the unit.
value__________ (e)A key must be provided to determine thevalue ofof eacheachpicture.
picture.
__________ (f)A pictogram can be presented both horizontally and vertically.
__________(f)A
pictogramcanbepresented
bothhorizontallyandvertically.
__________ (f)A pictogram can be presented both horizontally and vertically.

__________ (c) The pictures used must be of same size.

__________ (d) The pictures used must be equally spaced.
__________ (e) A key must be provided to determine the value of each picture.
__________ (f) A pictogram can be presented both horizontally and vertically.

Chapter 5: Patterns and Data Representations | 185

iv.
iv.

The
Thepictogram
pictogramin
inDiagram
Diagram66shows
showsthethenumbernumberofofchildren
childrenusing
usingdifferent
differentmeans
meansofofiv. transportation
The pictogramin Diagram 4 shows the number of children using different means oftotransportation
toschool.
school.
transportation to school.
Means
Means ofof Transportation
Transportation toto School
SchoolCar
Car

Motorcycle
Motorcycle

BicycleBicycleBusBusWalking
Walking

represents 10 children
represents
represents1010children
children
Diagram4
Diagram
Diagram66
Lookingthepictogram,
astudent
student
said
there
are
9children
childrenwho
walkedschool.
 • Looking
atthepictogram,
aastudent
said
there
are
99children
who
walkedtoschool.
Lookingatat
thepictogram,
said
there
are
who
walkedtoto
school.

isisthemisconceptionofstudent?
WhatWhat
themisconception
ofthe
the
student?
What
is
the
misconception
ofthestudent?

How
will
you
help
the
student
overcome
 How will you help the student overcomethe
themisconception?
misconception?
 How will you help the student overcome the misconception?

186 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Standards 3.3:
Representing a data structure by using bar graph to predict the future of communities
Standards
3.3: Representing
a datastructureusingusingbar graph
to predict the
futurei. Understandhow to draw
bar graphfrombytabledata categories
andsort ofthecommunities
graph fori. showing
Understandhow to draw bar graph from table using data categories and sort the graph for
its structure
showingits
structure
ii. Appreciate
ways
of presenting
databysuch
pictogramsbar graphs withStandards
3.3: Representing
a data structureusingasbarusing
graphtables,
to predict
the future ofand
communities
ii.i. sorting
Appreciate
wayspresentingdata
as using
pictograms
andgraphbar graphswith sortingfor predicting
theirfutureUnderstand
how
to of
draw
bar
graphcommunities
from suchtable using
datatables,
categories
and sort theforfor predicting
their future
showing
itsthestructureiii. Appreciate
usingof datacommunities
for making decisionii.
Appreciatepresenting
as using
tables, pictograms and bar graphs with sortingiii.
Appreciateways
the ofusingof datadata
for suchmakingdecisionfor predicting their future communities
SampleTasks for Understanding the Standardsiii.
Appreciateusing of data forthemakingdecision
SampleTasks forthe
UnderstandingStandards

Task 1:Tasks
Bar Graphs
Samplefor Understanding the Standards
Task
1: Bar Graphs
Task 1: Bar Graphs

Table
1 shows
thethenumberof ofchildrenin eachgrade
whovisitedthetheschoollibrary
duringrecessTable
1 shows
numberchildrenin eachgrade
whovisitedschoollibrary
duringrecess
time
for
a
week.
Due
to
unforeseen
circumstances,
the
data
for
Grade
6
is
missing.
time for a week. Due to unforeseen circumstances, the data for Grade 6 is missing.
Table 1 shows the number of children in each grade who visited the school library during recesstime for a week. Due to unforeseen circumstances, the data for Grade 6 is missing.

Table
Table 11

NumberTable
1 of
Numberof ChildrenChildren Who
Who Visited
Visited thethe School
School Library
Library forfor Grade
Grade 1
1 to
to Grade
Grade 6
6
Number of Children Who Visited the School Library for Grade 1 to Grade 6
Grade

Grade

1

1

2

2

3

3

4

NumberofofChildrenChildren 6 6 10 1015 1531
Number

4

5

6

31
36

36
?

?

5

NumberVisitedSchoolLibrary
NumberofChildrenChildrenVisitedSchoolLibrary i.
NumberofofChildren
VisitedtheSchoolLibrary

i.

ii.

ii.

iii.

iii.

Diagram
Diagram71
Diagram 7

6

i. aDraw
a bar ongraph
on Diagram
7 to
Draw
bar graph
Diagram
7 torepresent
the datathein Table
1. Table 1.
represent
data in

Draw a bar graph on Diagram 1 to
 What
the
scale
used
on usedthe on the
represent
the
data
Table
1.
 is
What
is inthescaleverticalverticalaxis? axis?
•
What is the scale used on the How do you draw the bars forvertical How
Grades3axis?
anddo
4? you draw the bars for
Grades 3 and 4?
•BasedHow
do baryougraph,draw predict
the barson thethe forii. Grades
Basedpredict themissing
numberofthechildrenfor Grade3onand4? bar graph,6.
missing number of children for Grade
Based6.on the bar graph, predict the Explain the reasons for yourmissing
number of children for Grade 6.
prediction.
 Explain the reasons for yourprediction.
• How
Explainthe arereasons
confident
you withfor
your yourprediction. Explain your answer.
prediction?
 How confident are you with yourprediction?
Explainyouranswer.
can bar aregraphsto your•In what
Howwayconfident
youhelp
withdevelopappreciation
towards
prediction?
Explain youranswer.
mathematics
amongiii. In what
way yourcan students?
bar graphs help todevelopappreciation
towards
In what way can bar graphs help
mathematics among your students?
to develop appreciation towards
mathematics among your students?

Chapter 5: Patterns and Data Representations | 187

Number ofof Children
theSchool
School Library
Diagram
2 shows
a baragraph
drawn
by a
NumberChildrenVisited
VisitedLibrary iv. iv.
Diagram
8 shows
bar graphdrawn
student.
aretwo major
indrawntheNumber of Children Visited School LibraryaTherestudent.
There
two
major
iv. byDiagram
8 shows
aare
barerrors
graph
bar graph.
by a instudent.
errorsthe barTheregraph.are two major
errors in the bar graph.
•
Identify
and andexplainthe major
errors.
 Identify
explainthe major
 errors.
Identify and explain the major•
How will
you help your students toerrors.
avoid
these
 How
willerrors?
you help your students to avoid
Howthesewill youhelp your students toerrors?
avoid these errors?

Diagram 28
Diagram
Diagram 8v. Diagram 9 shows a bar graph representing the number of teachers for each favourite colour.
Diagram
3 shows
a bar
graph
representing
the numberof teachers
forfavourite
each favourite
v. v. Diagram
9 shows
a bar
graph
representing
the numberof teachersfor eachcolour.
colour.

Diagram 9
Diagram 9
Diagram 3
There are two major errors in the scale used in this bar graph.
There are two major errors in the scale used in this bar graph.
There are two major errors in the scale used in this bar graph.
 Identify and explain the major errors.
 Identify and explain the major errors.
•
Identify and explain the major errors.
 How will you help your students to avoid these error?
 How will you help your students to avoid these error?
•
How will you help your students to avoid these error?

188 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Bibliography | 189

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Grades_1_-_9.pdf
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Singapore Mathematics Curriculum: Secondary One to Four
https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/2020-ntmaths_syllabus.pdf
https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/2020express_na-maths_syllabuses.pdf
Vietnamese Mathematics Curriculum: Years 1 to 12
https://www.futureschool.com/vietnam-curriculum/Appendix A – Framework for CCRLS in Mathematics | 191

APPENDIX A
Framework for CCRLS in Mathematics: Edited Edition
Nature of Mathematics
Mathematics has been recognized as a necessary literacy for citizenship and not only living economically
but also to establish a society with fruitful arguments and creations for better living. It has been taught as abasic language for all academic subjects using visual and logical-symbolic representations. In this informationsociety, mathematics has increased its role to establish 21st century skills through reviewing mathematics as
the science of patterns for future prediction and designing with big data which produces innovation not only fortechnology advancement but also for business model.
Mathematics is an essential subject to establish common reasoning for sustainable development of society
through viable argument in understanding each other and develop critical reasoning as the habits of mind.
Mathematics should be learned as basis for all subjects. For clarifying the framework in CCRLS on mathematics
and by knowing the role of mathematics education, the humanistic and philosophical natures of mathematics
are confirmed as follows:
The Humanistic nature of mathematics is explained by the attitudes of competitiveness and understanding ofothers by challenging mathematicians such as Blaise Pascal, Rene Descartes, Isaac Newton and Gottfried
Wilhelm Leibniz. For example, if you read the letter from Pascal to Pierre de Fermat, you recognise thecompetitive attitude of Blaise Pascal to Fermat’s intelligence and seek the way to be understood on the
excellence of his finding on Pascal’s Triangles. If we read Pascal’s Pensées, you recognise how Pascal denied
Descartes geometry using algebra from the aspect of ancient Greek geometry. On the other hand, Descartes
tried to overcome the difficulties of ancient geometry by algebra. If you read the letter from Descartes to
Elisabeth, you recognise how Descartes appreciated and felt happy the Royal Highness Elisabeth used his
ideas of algebra in geometry. Despite being a princess, Elisabeth had been continuously learning mathematics
in her life.
There were discussions on who developed calculus between Britain and Continent. On that context, Johann
Bernoulli, a continental mathematician, posed a question on the journal about the Brachistochrone problem,
locus of the point on circumference of the circle when it rotates on the line. There was no response andBernoulli extended the deadline of the answer and asked Newton to reply. Newton answered it within a day.
Finally, six contributions of the appropriate answer including Newton and other Continental mathematicians
were accepted. All those stories show that mathematics embraces the humanistic nature of proficiency forcompetitiveness and understand others for sharing ideas.
The Philosophical nature of mathematics can be explained from ontological and epistemological perspectives.
From the ontological perspective, mathematics can be seen as a subject for universal understanding andcommon scientific language. Plato and Aristotle are usually compared from this perspective. Plato believes
that the existence of the world of “idea” and mathematics existed in the world of “idea” on Platonism. In thiscontext, mathematical creation is usually explained by the word “discover” which means taking out the coverfrom which it has already existed. At the moment of discovery, reasonable, harmony and beautifulness ofmathematical system is usually felt. Aristotle tried to explain about reaching an idea from the “material” tothe “form”. This explains that abstract mathematics can be understood with concrete materials using terms
such as “modelling”, “instruments”, “embodiment”, “metaphor” and “change representation”. On this context,"invention" is also used for development of mathematical instruments, representation. and so on. From both
ontological perspectives, mathematics can be understood and acquired by anyone and if acquired, it serves
as a common scientific language that is used to express in any subject. Once representing the ideas using theshared common language, the world can be perceived in the same view autonomously.
From the epistemological perspective, mathematics can be developed through processes that are necessary
to acquire mathematical values and ways of thinking. From this perspective, idealism and materialism arecompared. In the context of Hegel, a member of German idealism, Imre Lakatos explained the development ofmathematics through proof and refutation by using counter example. In another words, beyond the contradictionis the nature of mathematical activity and it provides the opportunity to think mathematically for overcoming. On
this context, mathematics is not fixed but an expandable system that can be restructured through a process192 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelof dialectic in constructing viable arguments. Plato also used dialectic for reaching ideas with examples ofmathematics. The origin of dialectic is known as the origin of indirect proof. In education today, dialectic is apart of critical thinking for creation. Parallel perspectives for mathematical developments are given by George
Polya and Hans Freudenthal. For the discovery of mathematics, Polya explained mathematical problem-solving
processes with mathematical ideas and mathematical ways of thinking in general. Freudenthal enhanced theactivity to reorganize mathematics by the term mathematisation on the principle of reinvention.
Genetic epistemologist Jean Piaget established his theory for operations based on various theories, including
the discussion of Freudenthal and explained mathematical development of operations by the term reflective
abstraction. Reflection is also a necessary activity for mathematisation by Freudenthal. On materialism,
under the Vygotskyian perspective, intermediate tools such as language become the basis for reasoning
in the mind. Under his theory, high-quality mathematical thinking can be developed depending on the highquality communication in mathematics classrooms. Dialectical-critical discussion should be enhanced in themathematics class. From both the epistemological perspectives, mathematics can be developed throughthe processes of communication, problem solving and mathematisation which include reorganization ofmathematics. Those processes are necessary to acquire mathematical values and ways of thinking throughreflection.
Aims of Mathematics in CCRLS
epistemological perspectives, mathematics can be developed through the processes of communication, problem

and mathematisation
which forincludereorganizationbasicof mathematics.
processescreative
are necessary
toThe aims ofsolving
mathematics
in CCRLS
developing
human Thosecharacters,
humancapital, and wellacquire mathematical values and ways of thinking through reflection.
qualified citizens in ASEAN for a harmonious society are as follows:
Aims of Mathematics
in CCRLS•
Developmathematicalvalues, attitudes and habits of mind for human character,•
Developmathematicalthinking
and ableto humanengagein appropriate
processes,
The aims of mathematics in CCRLS
for developing
basic
characters,
creative humancapital, and wellqualified citizens
in ASEANin
formathematics
a harmonious society
are as follows:•
Acquireproficiency
contents
and apply mathematics in appropriate situations.
•
Develop mathematical values, attitudes and habits of mind for human character,•
Develop mathematical thinking and able to engage in appropriate processes,•
Acquirein mathematics
contentsin
andFigureapply mathematics
in appropriate
situations.
Framework for CCRLS
in proficiency
Mathematics
as shown
3 is developed
based
on the three components

with discussionsof fortheCCRLS
humanistic
and asphilosophicalof mathematics.
framework
also depicts the
Frameworkin Mathematics
shown in Figure 3nature
is developed
based on the threeThis
components
withdiscussionsof the humanisticlearningand philosophicalof mathematics.
concrete ideas
of mathematicsof thenature
above
aims. This framework also depicts the concrete
ideas of mathematics learning of the above aims.

Mathematical Values, Attitudes and Habits for Human Character
Mathematical Values:
Seeking ➢ Generality andexpandability➢ Reasonableness andharmony➢ Usefulness andefficiency➢ Simpler and easier➢ Beautifulness

Mathematical Attitude
Attempting to ➢ See and think mathematically➢ Pose questions and develop
explanations➢ Generalize and extend➢ Appreciate others’ ideas andchange representations formeaningful elaborations

Mathematical Habits of Mind for Citizen
To live ➢ Reasonably and critically while
respecting and appreciating others➢ Autonomously and socially➢ Creatively, innovatively andharmoniously➢ Judiciously in using various tools➢ With empowerment in predicting thefuture through lifelong learning

Mathematical Thinking and Processes
Mathematical Ideas:
Set, Unit, Comparison,
Operation, Algorithm,
Fundamental principles,
Permanence of form,
various representations
and Translations.

Mathematical Thinking:➢ Generalisation and specialisation➢ Extension and integration➢ Inductive, analogical and deductive
reasoning➢ Abstracting, concretising andembodiment➢ Objectifying by representation andsymbolising➢ Relational and functional thinking➢ Thinking forward and backward

Mathematical Activities:➢ Problem solving➢ Exploration and inquiry➢ Mathematical modeling➢ Conjecturing, justifying
and proving➢ Conceptualisation andproceduralisation➢ Representation andsharing

Contents
Key Stage 1➢ Numbers & Operations➢ Quantity &Measurement➢ Shapes, Figures andSolids➢ Pattern & Data
Representations

Key Stage 2➢ Extension of Numbers andOperations➢ Measurement & Relations➢ Plane Figures & Space Solids➢ Data Handling & Graphs

Key Stage 3➢ Numbers & Algebra➢ Relations & Functions➢ Space & Geometry➢ Statistics & Probability

Figure 3. CCRLS Framework for Mathematics and Aims of Mathematics Learning: Old edition
Figure 3. CCRLS Framework for Mathematics and Aims of Mathematics Learning.

Appendix A – Framework for CCRLS in Mathematics | 193

Mathematical Values, Attitude and Habits for Human Character
For cultivating basic human characters, values, attitudes and habits of mind are essentials to be developed
through mathematics. Values are basis for setting objectives and making decisions for future directions.
Attitudes are mindsets for attempting to pursue undertakings. Habits of mind are necessary for soft skills tolive harmoniously in the society. Mathematical values, mathematical attitudes and mathematical habits of mind
are simultaneously developed and inculcated through the learning of the content knowledge.
Essential examples on values, attitudes and habits of mind are given in Figure 3. On mathematical values,
generalisable and expandable ideas are usually recognised as strong ideas. Explaining why a proving isnecessary in mathematics is a way of seeking reasonableness. Harmony and beautifulness are described
not only in relation to mathematical arts, but also in the science of patterns and system of mathematics.
Usefulness and simplicity are used in selection of mathematical ideas and procedures.
On mathematical attitudes, “seeing and thinking mathematically” means attempting to use the mathematics
learned for seeing and thinking about the objects. Posing questions and providing explanation such as the“why” and the “when” are ordinary sequence for thinking mathematically. Changing representation to otherways such as modelling can overcome the running out of ideas in problem solving. The mindset for trying tounderstand others is the basis to explain one’s own ideas that is understandable by the rest with appreciation.
Producing a concept with definition operationally is a manner of mathematics.
On mathematical habits of mind for citizens to live, mathematical attitudes and values are necessary forreasoning critically and reasonably. Appreciating and respecting other ideas is also necessary. Mathematics isdeveloped independently for those who appreciate life creatively, innovatively and harmoniously. Seeking theeasier and effective manner of selecting appropriate tools is necessary. Mathematics is a subject to challenge
and experience competitiveness, appreciation with others, develop the mindset for lifelong learning, personal
development and social mobility.
Mathematical Thinking and Processes
For developing creative human capital, mathematical ideas, mathematical ways of thinking and mathematicalactivities are essential. Mathematical ideas are process skills involving mathematical concepts. Mathematicalthinking is mathematical way of reasoning in general which does not depend on specific concepts. Mathematicalactivities are various types of activities such as problem solving, exploration and inquiry. Mathematicalprocesses which include these components are necessary skills to use mathematics in our life, such as
innovation in this society (e.g. Internet of Things (IOT)). In the context of education, competency referring tomathematical processes is the basis for STEM and STEAM1 education as well as basis for social science andeconomy education.
Mathematical ideas serve as the basis of content knowledge related to promoting and developing mathematicalthinking. Some key ideas of mathematics are used as special process. The fundamental ideas of set and unit
lead to a more hierarchical and simple structural relationship. The ability to compare, operate, and perform
algorithm of related functions enables efficient ways of learning mathematics and solving problem in learners’
life with mathematics.
In the case of set, set is a mathematical ideas is related with conditions and elements. It is related with activity
in grouping and distinguishing with other groups by conditions. Example, 3 red flowers and 4 white flowers
become 7 flowers, if we change the condition of the set by not considering the colours. “A” and non A” is a
simple manner to distinguish sets with logical reasoning. For categorizing, we use intervals such as x > 0, x <0, x = 0. This situation can be seen in the hyperbolic graph where y=1/x.
In the case of unit, it is a mathematical idea that is related with the process to produce and apply the unit with
operations. On some cases, trying to find the common denominator is the way to find the unit of two given
quantity. Tentative unit such as arbitrary units can be set and applied locally whereas standard units are used
globally. In the combination of different quantities, it produces new measurement quantity such as distancewith respect to time produce speed. Square unit such as square centimetres is a unit for area.
STEM refers to Science, Technology, Engineering and Mathematics. STEAM refers to Science, Technology, Engineering, Arts
and Mathematics or Applied Mathematics.
1

194 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Mathematical thinking is well discussed by George Polya. Inductive, analogical and deductive reasoning
are major logical reasoning at school. However, deductive reasoning is enhanced in relation to formal logic
and inductive and analogical reasoning are not well recognized. Polya enlightens the importance of thosereasoning in mathematics. On the process of mathematisation by Hans Freudenthal, objectifying of the method
is necessary. David Toll mentioned it by the term thinkable concept on the process of conceptual development.
Polya mentioned thinking forward and backward in relation to ancient Greek term analysis.
Mathematical activities are ways to represent mathematical process. Problem solving process was analysedby Polya. He influenced problem solving with various strategies. Technology enhances the activities
of conjecturing and visualizing for inquiries. Conceptualization is done based on procedures such as theprocedure 3+3+3+3=12, become the basis for 4x3. The proceduralisation of multiplication is done throughdeveloping the multiplication table, idea of distribution and memorizing.
Content
For cultivating well qualified citizens, content knowledge of mathematics is essential. Content of mathematics
is usually divided by the set of mathematics. However, for developing human characters and creative humancapitals, it should be developed through the mathematical processes. Values, attitudes and habits of mind are
driving force for engagement in mathematical processes. Thus, without involving human character formations
with mathematical process skills, content knowledge of mathematics cannot be realised. The content is divided
into three stages in CCRLS and every stage has four strands. Between the stages, the names of the strands
are directly connected and those on the standard level are well connected too. The names of strands for every
key stage are as follows:
Key Stages

		Strands

Key Stage 1 		Numbers and Operations
			Quantity and Measurement
			Shapes, Figures and Solids
			
Pattern and Data Representations
Key Stage 2 		Extension of Numbers and Operations
			Measurement and Relations
			Plane Figures & Space Figures
			
Data Handling and Graphs
Key Stage 3 		Numbers and Algebra			
Relations and Functions
			Space and Geometry
			Statistics and Probability
In every stage, four content strands are mutually related2. Between the key stages, all strands in different
key stages are mutually related. The same content strand names are not used to indicate development andreorganisation beyond each stage. For example, “Numbers and Operations” in Key Stage 1, “Extension of
Numbers and Operation” in Key Stage 2, and “Numbers and Algebra” in Key Stage 3 are well connected.
These names of the content strands show the extension and integration of contents. For example, even andodd numbers can be taught at any stage with the different definition. At Key Stage 1 even numbers can be
introduced as “counting by two” which does not include zero. In Key Stage 2, it can be re-defined by a number
divisible by two. Finally, in Key Stage 3 it can be re-defined as a multiple of two in integers which includes zero.
Although we use the same name as even number, they are conceptually different. The definition in Key Stage
1 is based on counting, Key Stage 2 is based on division while Key Stage 3 is based on algebraic notation3.
Strands used to explain mutual relation of content (Jeremy Kilpatrick, Jane Swafford, Bradford Findell. “Adding it up”,
National Academies Press. 2001). The term domain is sometimes used for compartmentalization through categorisation ofcontent.
3
In algebraic notation of numbers, addition and multiplication are major operations. Subtraction can be represented by additionof negative numbers and division can be represented by reciprocal or multiplicative inverse property.
2Appendix A – Framework for CCRLS in Mathematics | 195

Expressing such theoretical differences requires name of strands for content be distinguished. In the case ofmeasurement, there is no strand name of measurement in Key Stage 3. Key Stage 1 relates with quantity andsetting the units. In Key Stage 2, it extends to non-additive quantity beyond dimension. In Key Stage 3 theidea of unit and measurement is embedded in every strand. For example, square root in Numbers and Algebra
strand is an irrational number which means unmeasurable, Pythagorean Theorem in Space and Geometry
strand is used for measuring, proportional function in Relations and Functions strand is used for counting thenumber of nails by weight, and in Statistics and Probability strand, new measurement units are expressed suchas quartile for boxplot.
Context to Link the Three Components
Three components in Figure 3 should be embedded in every key stage as standards for the content ofteaching. “Mathematical values, attitudes, habits for human character” component and “Mathematical thinking
and processes” component cannot exist without “Content” component. The first two components can betaught through teaching with the content. For teaching those three components at the same time, context is
introduced as shown in Figure 4. For example, context 1 is considered as problem solving situation in the realworld situation and context 2 is considered as mathematical situation under the mathematical task sequence
to develop further mathematics itself.
Mathematical Values, Attitudes andHabits for Human Character

Mathematical Thinking
and Processes

Content

Figure 4. Interlinking of the three components with the context

Figure 4. Interlinking of the three components with the context

On a given context, three components are well connected. For this reason, classroom activities for developing
competencies should be designed to link all of them. The following contexts are samples:
On a given context,
three
components
well connected.
Forwell mathematicalconnecting,classroom activities for•
Explorea problemwith curiosityare
in a situation
and attempting toformulate
problems•
Apply the mathematics learned, listen to other’s ideas and appreciate the usefulness, power and beauty ofdeveloping competencies
should be designed to link all of them. The following contexts are samples:mathematics•
Exploreproblemwithcommunications
curiosity inona mathematicalsituation andto formulatemathematicalproblems• aEnjoyclassroom
ideasattempting
in solving problems
with patienceand develop•
Apply theperseverance
mathematics learned, listen to other’s ideas and appreciate the usefulness, power and•Feel the excitement of “Eureka” with enthusiasm for the solutions and explanation of unknown problems
beauty• of Thinkmathematics
about ways of explanation using understandable representations such as language, symbols,
diagrams andnotation of mathematics•
Enjoy classroom
communications
on mathematical ideas in solving problems with patience and•Discuss the differences in seeing situations before and after learning mathematics
developperseverance•
Explain, understand others and conclude mathematical ideas•
Exploreideas throughinductive anddeductive
reasoning whenproblems to andfosterexplanation
mathematical of unknown•
Feel theexcitementof “Eureka”
withenthusiasm
for solving
the solutions
curiosity
problems•
Explore ideas with examples and counter examples•
Think about
ofinexplanation
using
understandable
representations
as language,
symbols,•
Feelwaysconfident
using mathematics
to analyseand solve contextual
problems both in such
school andin realsituations
diagrams life
andnotation of mathematics•
Promote knowledge, skills and attitudes necessary to pursue further learning in mathematics•
Discuss
differences
in seeing
and after learning mathematics• theEnhancecommunication
skills withsituations
the languagebeforeof mathematics• understandPromote abstract,
logical,and
criticalconclude
and metacognitive
thinking to assess
one’s own and other’s work•
Explain,othersmathematicalideas•
Foster critical reasoning for appreciating other’s perspectives•
Exploreideas
through
inductive
anddeductive
reasoningwhen solving
problems to foster•
Promote critical appreciation on the use of information and communication technology
in mathematics•
Appreciate
the universality of mathematics and its multicultural and historical perspectives
mathematicalcuriosity

•
•
•
•
•
•

Exploreexamplesand counterexamplesThose ideas
contextswith
are chosen
for illustrating
the interwoven
links of the two components with contents. It looks likeof teaching,
however
all the three components
are theandsubjectsof teachingon the contexts.
Feelmethods
confident
in using
mathematics
to analyse
solvecontextual
problems both in school andin real-life situations
Extracted
and edited from:
Promoteknowledge,
skills and attitudes necessary to pursue further learning in mathematics
Enhancecommunication
skills with the language of mathematics
SEAMEO Basic Education Standards (SEA-BES): Common Core Regional Learning Standards (CCRLS) in
Promoteabstract,
logical,criticaland metacognitive
top.assess
one’s own and other’s work
Mathematics
and Science.
SEAMEORECSAM.
(Mangao, Ahmad, &thinking
Isoda, 2017,
2-11)
Foster critical reasoning for appreciating other’s perspectives

196 | Mathematics Challenges for Classroom Practices at the Lower Primary Level••

Promote critical appreciation on the use of information and communication technology in mathematics
Appreciate the universality of mathematics and its multicultural and historical perspectives

Those contexts are chosen for illustrating the interwoven links of the two components with contents. It looks
like methods of teaching, however all the three components are the subjects of teaching on the contexts.
Extracted and edited from:
SEAMEO Basic Education Standards (SEA-BES): Common Core Regional Learning Standards (CCRLS) in
Mathematics and Science. SEAMEO RECSAM. (Mangao, Ahmad, & Isoda, 2017, p. 2-11)

Appendix B – Terminologies for Mathematical Thinking and Processes | 197

APPENDIX B
Terminologies Explained
Mathematical Thinking and Processes
Higher order thinking is the terminology for curriculum but it is not specified in mathematics. Here, it is explained
generally as acceptable terms from the perspective of mathematics in education. On the Mathematical thinking
and processes, the following terms are the sample which can be seen on SEA-BES: CCRLS for making
clear descriptions of the objective of teaching. If you can use these terminologies for writing the objectivesof teaching, you would be able to consider how you teach them in the process. Mathematical Thinking canbe explained through Mathematical Ideas, Mathematical Ways of Thinking, and Mathematical Attitude.
Mathematical attitude is a component of the Value, Attitude and Human Character Formation in Appendix A.
Mathematical Ideas, Mathematical Ways of Thinking can be developed through the reflection of the process
and Value and Attitude can be developed through appreciation.
Mathematical Ideas
Even through every mathematics content embedded some ideas, there are essential mathematical ideas
which are used in various occasions. Mathematical ideas are not exclusive but functions as complementary.
The followings are samples of essential mathematical ideas.
Terminology

ExplanationSet

A set is a collection of elements based on certain conditions. When the conditionof the set changes, result of reasoning related to the set may change too. Sets
are compared by one-to-one correspondence. Basically, the idea of set is reflected
through activities that require us to think about membership (elements andconditions) of a set. In addition, activities involving subsets, cardinality and power
are extended ideas of set. Number of elements called cardinal or cardinal number(or set number). Ordinal number does not imply number of elements. Other ideas
include operations of sets such as union, intersection, complement, the ordered
pair/combination of elements such as Cartesian products and dimension mapping.
A number system is a set with structures that has the structures of equality, order(greater, less than), and operations, which is developed and extended throughout
the curriculum from natural number to complex number: the set of complex
numbers does not have the structure of order. In Mathematics, a set is the bases
to set algebraic structure and the bases for logic such as universal proposition andexistential proposition.

Unit

Unit is necessary for counting, measurement, number line, operations andtransformation. It is represented as “denomination’ for discrete quantity, such as 1
“apple’ for situations involving counting, or continuous quantity, such as 1 gram forsituations involving measurement. Mathematically, unit is used to indicate a number
by mapping it with the quantity in a situation.
In a situation, it can be fixed based on the context of comparison, which can either
be direct or indirect comparison. In this context, a remainder or a difference from acomparison can be used for fixing a new arbitrary unit for measurement which is afraction of the original unit. This process of determining a new unit is the applicationof Euclidean algorithm for finding the greatest common divisor.
For the base-10 place value number system, every column is defined by the units
such as ones, tens, hundreds and so on. However, in other place value number
system such as the binary system, every column is defined by the units such as

198 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelones, twos, fours and so on. Therefore, in a place value number system, the unit is
not always a multiple of the power of ten.
In addition, various other number systems are made up of different units. For thecalendar system, the lunar calendar is based on 30 (29.5) days, while the solar
calendar is based on 365 (365.25) days. On the other hand, the imperial and U.S.
customary measurements include units in the base-12 and base-16 systems. In
the ancient Chinese and Japanese systems, there were units in the base-4, thus
also included the base-16. On the other hand, the units used in different currency
systems are dependent on differing culture and countries. However, many countries
had lost the unit of 1/100 on their currency systems, which originated from “percentos” that means percent. Even though the based-10 place value system is used
to represent the value of money, many currency systems are using the units for 2,
5, and 25 in their denominations instead.
Unit for a new quantity can be derived from ratio of different quantities. For example,
the unit for speed (km/h) is the ratio of distance (km) over time (h) which cannot be
added directly. A car moves at 30km/h and then moves at 20km/h does not mean
the car moves at 50km/h.
Identity element for multiplication is one but additive identity is zero. Identity for
multiplication is the base for multiplicative and proportional reasoning. Inverse
element for multiplication is defined by using one.
Comparison

Comparison of concrete objects can be done directly or indirectly without
measurement unit.  As mentioned at the Unit, direct comparison can be used to fix
a new unit of measurement, whereas indirect comparison can be used to promotelogic for transitivity which includes syllogism.
Comparison of multiple denominated numbers with different unit quantities on thesame magnitude such as 5.2 m and 5 m 12 cm can be done if they are represented
by a single denominated number by the unified unit quantity, such as 520 cm and512 cm.  Furthermore, comparison of expressions which has the same answers
on the same operation such as 2+4, 3+3 and 4+2 can be used to find rules andpatterns.  For example, 2+4 = 4+2 can show commutative rule for addition whereas
for 2+4 = 3+3 can be used to show pattern when 1 is added to 2 and subtracted
from 4, the sum is still the same.
Comparison of fractions is an activity to find the unit fraction. For comparison of
fractions such ad
and

.

and

, we have to find the unit fractionis the common denominator forandwhich can measure. The algorithm to find the unit

fraction as the common denominator is called ‘reduction of fraction’.
On numbers, the relationship of two numbers can be equal, greater or less. The
number set up to real number is a total/linear order set, thus up to real number set,
two numbers can be compared. However, complex number as an extension of real
number cannot be compared directly because it has two dimensions.
On the number line as real number, the size of number (distance) is defined by thedifference, 1 = 2-1 = 3-2 = 4-3 =.... Here the difference is the value of subtraction
as binary operation and can be seen as the equivalence class. On the idea of
equivalence class, the value of operations can be compared. On the plane suchas complex plane, even though the number is not simply ordered, the size ofAppendix B – Terminologies for Mathematical Thinking and Processes | 199

number (distance) is defined by Pythagorean theorem. By using this definition,
I1I= l()( + i)l = liI = I()( - i)I=... the theorem produce the distance on the plane

and the distance can be compared.
Explained at the Unit, on measurement, the magnitude is given by defining theunit of magnitude. One of the ways to produce the unit magnitude is a directcomparison which provides the difference and Euclidean Algorithm produce the unitof measurement as the greatest common devisor. Another way to produce the unit
is by using the ratio and so on. Such a newly produced magnitude usually lost thelinearity. In Physics, ‘db’ is the size of volume which is produced by the common
logarithm of sound pressure. ‘db’ is fitting well for human impression of the sizeof sounds on its linearity. It is known as Weber-Fechner’s law that human senses
are proportional to the logarithm of stimulus. On science, logarithmic scale is used
for Semi-log graph and Log-log graph for demonstrating the linearity even it is anexponential phenomenon. Logarithm produce the scale to illustrate multiplicative
phenomena as an additive phenomenon.
In any mathematical investigation, particularly in the mathematics classroom,
problem solving approach, comparisons of various ideas, representations andsolutions are key activities for discussion and appreciation. This comparison is anature of mathematical activity.
Operations

Addition, Subtraction, multiplication and division are four basic arithmetic
operations. Mathematically, these are binary operations involving any two numberswith symbols of operations, +, –, × and ÷. Polynomial expressions can be seen
as a combination of binary operations. Mental arithmetic may be used in column
method with the base-10 place value system. An operation is not just a rule but
can be demonstrated by using various representations. For example, an operation
can be represented by the manipulation of concrete objects as well as expressions.
However, the manipulating process is not the same as the operation process
because it cannot be recorded without using a diagram.
From Key Stage 3 up to the field theory at the university levels, arithmetic operations
are expressed as addition and multiplication. On the axiom of vector space, negative
vector is represented by using the minus symbol. Mathematically, operations andset are used for to establish mathematical structure.

Algorithm

Algorithm is a set of sequential activity at the special situation to produce a particularsolution to a task. The column method is based on the base-10 place value system.
When using this method, 200 + 300 is done by just 2 + 3 on the hundreds placevalue. This algorithm is adapted for mental calculation. Representation of column
method is not universal like an expression as algebraic representation. Algorithm
for the column method is fixed as a formal form based on culture, however it can becreated from the manipulation of the base ten blocks.
A formula also functions like an algorithm. It can be applied without understanding
its meaning. However, it cannot be created without understanding and recognising
its underlying structure or meaning. If we understand the structure or meaningsuch as ratio and proportionality, it is not necessary to memorise it. In Informatics,
algorithm is usually defined by programing language through analysing procedureand setting several functions by the program are necessary.

200 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Fundamental
Principles

Fundamental principles are the rules which is related with mathematical structuresand forms in general.
Commutativity, Associativity and Distributivity are three fundamental principles forarithmetic operations. Commutativity does not work on subtraction and division.
On the discussion of Distributivity, if a, b, and c, are positive numbers, then theexpressions a(b + c), (b + c)a, a(b – c), and (b – c)a are different. However, if a, b,
c is both positive and negative numbers, then the four expressions can be seen as
the same.
There are also other fundamental principles for arithmetic operations at theelementary level such as the followings:
1 + 9 = 10↓+1 ↓–1
2 + 8 = 10

8.1 ÷ 9 = 0.9
↓×10
↑÷10
81 ÷ 9 = 9

2 × 3 = 6↓×10↓×10
20 × 3 = 60

8.1 ÷ 9 = 0.9
↓×10
↓÷10
8.1 ÷ 90 = 0.09

Principles can be identified through comparison. They are necessary for the
explanation of algorithms and thinking about how to calculate by using models andother representations. On the extension of numbers and operations, principles areused for the discussion of the permanence of form (see the permanence of form).
Not only algebraic participles, there are other forms that can be extended. In
geometry, the extendable nature of a line changes its functions in curriculum. For
example, shape is extended to figure; edge, which may include the inner part of ashape, is extended to side, which may not include the inner part of a figure. Then,the side is extended to a line which enable the discussion on the possibility of
escribed circles. In addition, parallel lines are necessary to derive the area formulafor triangles with various heights.
Permanence of
Form

Permanence of the equivalence of form, Hankel’s Principle, is known as
Commutativity, Associativity, and Distributivity for algebra for the field theory.
Permanence of form had appeared in history of mathematics in 16th century andfunctioned to shift from arithmetic algebra to symbolic algebra. On Peacock’s
Permanence of Form, it is not only the limited three rules like Hankel’s, but it isapplied to algebraic symbolic form for overcoming arithmetic algebra which had
limited the positive number and 0.
In Education, the form is not a limited expression but includes the patterns and thepermanence of form can be used in various occasions. Especially, it is used forthe extension of numbers and operations from elementary level to secondary level
education like the followings:(+3) + (+2) = +5↓–1(+3) + (+1) = +4↓–1(+3) + 0 = +3↓(+3) + (–1) = ?

(+3) – (+2) = (+1)↓+1(+3) – (+1) = (+2)↓+1(+3) – 0 = (+3)↓(+3) – (–1) = ?

Appendix B – Terminologies for Mathematical Thinking and Processes | 201

The ‘?’ are unknown, not yet learned. However, people could imagine the ‘?’ byanalogical reasoning with the idea of the permanence of the patterns. Here, thepermanence of patterns is used as hypothesis and it makes possible to apply it tothe unknown cases.
Permanence of form is used for initiation of number in Key Stage 1 and later
throughout all other key stages. For examples, it is used to explain the necessity ofzero (0) and the sum of any number with zero (0) as at Key Stage 1.
Various
Representations
and Translations

Every specified representation provides some meanings on its essential nature of
representation which can be produced by specified symbols and operations. Different
representations have the different nature, use different symbols and operations.
Every representation of specific mathematical idea has the limitation of interpretation
on its nature, thus, thinking by using only one specified representation provides thelimitation of reasoning and understanding. If one type of representation is translated
to another type of representation, then the representation of idea can be interpreted
in other ways. If the idea of specified representation with certain embedded nature istranslated into various representations, a rich and comprehensive meaning and use
will be produced. However, for making the translation meaningful, it is necessary
to know the way of translations which consists with the correspondences between
symbols and operations on different representations.
For example, proportional number lines only functions for teachers and students
who know well how to represent the proportionality on the tape diagrams andnumber lines, and so on. If they know what it is, they can use it for explanation to
produce the expression.
Comprehensive learning of mathematics by using various representations andtranslations are necessary however students have to learn how to represent it inother representations and translate, at first, as well as between expressions andsituations.

Mathematical Ways of Thinking
Mathematical ways of thinking support the student’s thinking process by and for themselves.
Generalisationand Specialisation

Generalisation is to consider the general under the given conditions of situations
through extending set or reducing conditions. Any task in mathematics textbooks
is usually explained by using some examples which are known as special cases,
however, it is usually preferred to discuss the general ideas. Given conditions areusually unclear in textbooks, however if students are asked to consider other casessuch as by saying ‘for example’, the conditions become clear because we may needto consider the condition and set for producing other example. In the upper grades,
variable, domain, range, parameter, discreet, continuous, zero, finite and infinite
become the terms of set to consider the conditions for general.
Specialisation is to consider the thinkable or more simple example on situations
and necessary to find the hidden conditions. Considering general with thinkable
example is called generalisable-special example. In mathematics, general theory isusually stronger than local theory. It is one of the objectives for generalisation andspecialisation.
In mathematical inquiry, the process is usually on going from special cases to a
general case to establish a stronger theory. Thus, so many cases, task sequence ofevery unit in textbooks is progressed from special cases to general for generating202 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

simple procedures, exceptionally the tasks for exercise and training the procedures
which usually progress from general to special because students already learned
the general.
For students engaging in generalisation and specialisation by and for themselves,
students have to produce examples. Thus, developing students who say ‘forexample’ by and for themselves is a minimum requirement for teaching mathematics.
Extension andIntegration

Extension means extend the structure beyond the known set on the process of theenlargement of set. On this meaning, extension can be seen as a generalization.
The product of multiplication is usually increased if both the factors are natural
numbers. However, if we extend it into fractions and decimals, there are cases
where the products decease.
On the extension of structure beyond set, learned knowledge produce themisconception which is explained by the over generalisation of learned knowledge.
In mathematics curriculum, we cannot initiate numbers from fractions and decimals
instead of natural numbers. Thus, producing misconception is an inevitable natureof mathematics curriculum and its learning. Even through it is a source of difficulty
for students to learn mathematics, it is the most necessary opportunity to thinkmathematically and to justify the permanent ideas to be extended. After experiencing
the extension with misconceptions, if students learn what ideas can be extended, it
is a moment of integration.
Extension and Integration is a mathematical process for mathematisation toreorganise mathematics. It implicates that the school mathematics curriculum is akind of nets which connect local theory of mathematics as knot even though theirstrings/paths include various inconsistencies with contradictions. It is a long-term
principle for task sequence to establish relearning on the spiral curriculum
In order to develop the way of thinking for extension and integration, teachers needefforts to clarify the repetition of both, the same patterns for extensions and thereflections after every extension in classroom. For example, there are repetitions forthe extensions of numbers from Key Stage 1 to 3. On every extension of numbers,
there are discussions of the existence of number with quantity, the comparison
with equality, greater and lesser, and constructing operations of numbers with thepermanence of form. Through the reflection of every repetition, students are able tolearn what should be done on the extension of numbers.

Inductive,
Analogicaland Deductive
Reasoning

These three reasoning are general ways of reasoning as for the components oflogical reasoning in any subject in our life, however mathematics is the best subjects
in schools to teach them.
Inductive reasoning is the reasoning to generalise the limited number of cases tothe whole set or situations. Three considerable cases or more will be the minimum
to be considered inductive reasoning. To promote inductive reasoning, teachers
usually provide the table for finding the patterns, however it is not the way to develop
the inductive reasoning. To promote inductive reasoning, students need to consider
various possible parameters in a situation and choose two or further parameters by
fixing other parameters. Then, consider the relationship in the situation for knowing
the cause and effect, and subsequently, set the ways to check the cases by wellordering of the cause and get the effect, follow by recording the data in a table.
To promote inductive reasoning, teachers usually provide the table first for finding
the patterns, however it is not the way to develop the inductive reasoning. It isthe concern of students because to develop inductive reasoning, students have toAppendix B – Terminologies for Mathematical Thinking and Processes | 203

learn the ordering of natural numbers is vital because it is the cause of effect andbeautifulness of patters that is originated from the order of natural number. Tableis a tool for finding pattern but students cannot produce their inductive reasoningby and for themselves as long as it is given by teachers. Students have to learn
the way of ordering for finding the pattern, inductively because it is the manner ofexperiment in Science and Engineering.
Analogical reasoning is the reasoning to apply known ideas to the unknown set
or situations when we recognise the similarity with the known set or situations.
Depending on the context it is called abduction. Most of the reasoning to findways of solution for unknown-problem solving by using what we already knew isanalogical reasoning. In analogical reasoning, it is to recognising similarity between
the unknown-problem and the known problems.
Even though the rule of translations between different representations are not wellestablished, analogical reasoning may function as a metaphor for understanding.
Many teachers explain operations by using diagrams. It appears meaningful toprovide the hint for solving but most of students cannot use the hint by and forthemselves because students do not recognise the similarity as in their own analogy.
To develop analogical reasoning, the most necessary way is to develop the habit to
use what students learned before by and for themselves. Providing assisting tasks
before posing the unknown problem is also used as a strategy to find similarity.
Deductive reasoning is the reasoning with components of already approved
notions and given by using ‘if… then’ and logics for propositions such as transitivity
rule. ‘If not’ also functions for proof by contradiction as well as counter examples.
In cases where the rules of translations are well established, the translations ofvarious representations still function under their limitations too. Various methods forproving such as a complete induction is also done by deductive reasoning.
For finding the ways of explaining and proving, inductive and analogical reasoning
are necessary, and analytical reasoning, thinking backward from conclusion to
the given, is also used but these reasoning do not allow to write at the formal
proof by deductive reasoning. Arithmetic and algebraic operation can be seen as
automatised deductive reasoning. Most of students do it just by recognising thestructure of expression intuitively without explaining why. For clarifying the reason,teachers are necessary to ask why?
For developing the three reasoning, knowing objectives of reasoning are necessary.
Inductive reasoning is applied to find general hypothesises. Analogical reasoning isapplied to challenge unknown problem solving. Deductive reasoning is applied to
explain or proof in general on local system.
Abstracting,Concretising and
Embodiment

Abstracting and concretising are changing perspectives relatively by changing
representations such as expressions. Abstracting is usually done for making
clear a structure. Concretising is usually done for making ideas meaningful by
concrete objects. For numerical expressions, manipulative and diagrams function
as concrete. For algebraic expression, numerical expressions function as concrete
objects. For both examples, abstract representations and concrete representations
do not correspond one-to-one on translation because concrete representation
usually have some limitations but concrete representations function as metaphorsof abstract ideas.
Embodiment functions in both abstracting and concretising. When abstract ideas
can be concretised, it implicates those abstract ideas are embedded with somespecified concrete ideas. When concrete ideas can be abstracted, it implicatesconcrete ideas are embedded into abstract ideas. Both the embodiments function

204 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelfor understanding ideas using metaphors but their translations are usually limited
only for corresponding contents. In the case abstract and concrete objects are welltranslated on both directions, both of them function as mathematical representation.
Otherwise, embodiment is a useful word for explaining the metaphor on the given
context in process to establish mathematical representations in general.
Objectifying
operations forsymbolising andestablishing new
operations onmathematisation

A mathematical representation can be characterised by its symbols and operations
with specified purpose and context. In the process of mathematisation, lower-level
operational matters are usually objectified for new symbolising and its operations.
Until Key Stage 2, numbers do not mean the positive and the negative number. The
number in red on financial matter is large if the number is ‘large’. The number inblack on financial matter is large if the number is ‘large’. Here, the meaning of ‘large’are defined at the opposite number rays, thus cannot be compared the numbers in
red and black easily. At Key Stage 3, as for integration, we have to alternate new
symbols and operations. We represent the red number by the negative symbol ‘-’and the black number by the positive symbol ‘+’ and integrate the one direction forcomparison into one dimensional number line.
Here, on Key Stage 3, ‘larger’ for comparison (operational matter on numberrays) on lower level become the object of higher level to produce the comparison(operational matter on number line) for new number symbols with positive andnegative.
It is the process of mathematisation by objectifying the operational matter toestablish new symbols and operations. The process of abstracting from concrete
can be seen as the process of mathematisation.

Relational andFunctional
Thinking

Relational and functional thinking are ways of thinking that can be represented byrelation and function if we need describing them by mathematical notation. It has
been used as major terminologies to explain mathematical thinking at the Klein
Movement, 100 years ago. Relation in pure mathematics is a basic axiom. In
relation to functional thinking, it is known as the ordered pairs between two sets.
Mapping is a relationship and ordering is also a relationship and it supports keeping
the structure or not. Equivalence class is a relationship in which axiom is defined byreflective, symmetric and transitive properties.
Function is a binary relation between two sets that associates every element of thefirst set to exactly one element to the second set. The second set is usually numbers
in mathematics and both sets are numbers in education. Relational thinking is thethinking which can be represented by various representations such as graphs. It is
the activity for students who do not know such representations, and teachers haveto teach such representations on the necessity of students to engage in their activity:
It is a welcome for students to create their necessary informal representation suchas graph and diagram by themselves.
Historically, characteristics of functional thinking was described by Hamley (1934).
He defined functional thinking by four elements: Class, Order, Variable andCorrespondence, Class is a set which has possibility to include equivalence inoperations such as 5 can be seen as the value of 1+4, 2+3, 3+2, 4+1. Order isdiscussed within a set and between sets. Variables are domain and range which are
not the same as the origin set and the destination set. Correspondence is discussed
between domain and range.

Appendix B – Terminologies for Mathematical Thinking and Processes | 205

Functional thinking is useful to predict and control a situation. In elementary level,
proportionality and operations are used for functional thinking on this objective. For
example, proportional number line is a representation of functional thinking. For
knowing the properties of each function, tables and graphs are useful. Teachers
usually provide the tables and the graph papers from the beginning of experimentfor finding patterns and properties of function. However, this is not the way todevelop functional thinking because teachers take over the opportunities fromstudents to consider and fix the sets, orders, variables, correspondences and soon by themselves.
Rate of change is used to check if a function has linearity. In the case if it is constant,
it is a liner function. If not, otherwise. The limit of the rate of change for finding thetangent is the definition of differentiation. To show pattern beyond order, semi-log
graph and logarithm graph are in Physics and Engineering.
Correlation in statistics is a relation but not necessary functions as a causal relation.
Thinking Forward
and Backward

Thinking forward and backward are the terminologies of Polya. In Pappus of
Alexandria (4th Century A. D.), thinking forward corresponds to synthesis andthinking backward corresponds to analysis. Synthesis is the deductive reasoning(proving) from the given and known whereas analysis is the reasoning from theconclusion for finding the possible ways of reasoning from the given and known.
From Ancient Greece, analysis is a method of heuristics which was the ways tofind adjoin lines on construction problem, valance for area and volume problem.
Descartes used unknown x for algebraic problem. Leibniz used unknown limit x of
the function for calculus. These are hypothetic-heuristic reasoning beginning fromthe conclusion, such as if the construction is achieved, if the valance is kept, if
the unknown x is given, and if the limit of x existed. Since analysis began fromhypothesis, without proving from the given, people used to believe that analysis
produced tautology which is not allowed to write in the system. On the other hands,
modern mathematics system itself begins from the axiom as presupposition. Thus,
if the ways of analytic reasoning become a part of presupposition, it is allowed in a
written form. On this reason, the unknown x is able to be written in Algebra and thelimit of x is able to be written in Calculus. However, in school mathematics, before
such reformations of mathematics, analysis, a method of heuristics, do not allow to
be written in a part of proof. It is a reason why some mathematics textbooks lookvery difficult to understand because they have to be written in the form of deductive
reasoning for the construction of the system from axiom which does not include
heuristics and ways of findings. On this consequence, there are old fashioned
textbooks which just oriented to exercise the procedures as rules without explaining
why.
Current school textbooks, which orients to write the problem solving process withvarious solutions as well as misconception by using what already learned, include
the heuristics such as thinking backward and so on. In the standards of Key Stage
1 and 2, addition and subtraction are inverse operation, and multiplication anddivision are inverse operation as for verification of answers on operations. Suchideas are reformulated at the algebra in Key Stage 3.
In the case where ‘If your saying (conclusion) is true, it produces contradiction which
we already knew’ is known as dialectic in communication and it was formalised as
the proof by contradiction in mathematics. It is also the way of analysis for thinking
backward but accepted as the way of proving by counter example and Reductio ad
absurdum. In mathematical communication, thinking backward is a part. Without

206 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

the preparation of lesson plan which includes thinking backward, teachers cannot
realise the classroom communication including misconception because the counterexample is the component for the proof by contradiction. Though the communicationof both objective and way of reasoning such as thinking backwards by using what
students already learned, we are able to develop students’ mathematical thinking.

Mathematical Activities
Mathematical Activities usually explain the teaching and learning process but also embedded the mathematicalideas, thinking, values and so on. As for style of teaching approach, they are usually enhanced however thestyle of teaching itself does not make clear them.
Problem Solving

Pure mathematicians inquire problems that are never solved yet and they develop
new theorems for solving their problems. It produces a part of system. Such authentic
activity is the model of problem solving in education because it usually embedded
rich ideas, ways of thinking and values in mathematics. As mathematicians usuallypose problems for themselves, the activity includes problem posing and reflection
which is necessary for establishing new theories.
In education, there are two major approaches for embedding them in learning.
The first one is setting the time, unit or project for the problem solving. Here, solving
problem itself is an objective for students. It focused on heuristics: it is usuallyobserved unexpectedly and to plan it is inevitably not easy. On this difficulty, theproblem solving tasks are usually provided in two types. The first type focuses onmathematical modelling from the real world. The second type focuses on open
ended tasks for students because it provides the opportunity for various solutions.
The second one is called the problem solving approach, tries to teach content
through problem solving in classes. This case is only possible if teachers prepare
the task sequence that enabling students to challenge the unknown-task by using
what students already learned (ZPD). In problem solving approach, the tasks given
by teachers are planned for students where they are able to learn the content,
mathematical ideas and ways of thinking. For teachers, solving the tasks itself is
not the objective of their classes but students reveal the objectives of teaching byteachers through recognising problematic as an unknown and finding the way ofsolutions. For students to be able to learn by and for themselves, it is necessary
to plan the class with the preparation of future learning as well as by using learned
knowledge. Textbooks such as the Japanese textbooks equipped task sequence
for this purpose. In such textbooks, heuristics is not an accidental matter but apurposeful matter because every task in the textbook will be solved by utilising thealready learned representation and so on. It is called a ‘guided discovery’ under
ZPD because it expects well-learned students on the learning trajectory and never
expects the genius students on producing unknown ideas.
For observers, the way of teaching by the problem solving approach in a classcannot be distinguished with the open-ended approach. The open approach is
not necessary to prepare the task sequence because it is characterised by anindependent open-ended task. If teachers set the open-ended task independently,
it is the first approach. If teachers set the task sequence of open ended tasks forlearning mathematics itself, it is the problem solving approach, the second one.
Both of the approaches were known as Japanese innovation in textbooks since
1934 for elementary level and since 1943 for secondary level which was doneby Prof. Shimada Shigeru and others. Prof. Nobuhiko Nohoda was known as hetheorized the open approach with open-ended task and task sequence. Currently,

Appendix B – Terminologies for Mathematical Thinking and Processes | 207

Japanese textbooks until the middle school level equipped the task sequence forproblem solving approach.
Even though problem solving in education resemble activities of authentic
mathematicians, it is actually not the same because problems of mathematicians
are usually unsolvable beyond years and decades, while tasks in classroom
can be explained by teachers who posed the problems. When teachers referproblem solving in education, it includes various objectives such as developing
mathematical ideas, thinking, values and attitudes. These terminologies are usedin education to develop students who learn mathematics by and for themselves
while mathematicians only use some of them. On teacher education, if teachers
only learned the content of mathematics, they may lack the opportunity to learn
the necessary terminology. If teachers do not know it, the higher order thinking
becomes a black box which cannot be explained.
Exploration andInquiry

Exploration has been enhanced in finding hypothesis by using technology such as
Dynamic Geometry Software and Graphing Software. These software provide theenvironment for students to explore invariant in diversity, easily. Exploration on theenvironment support to produce hypothesis.
Inquiry which includes exploration orients to the justification and proving throughreflections.
Both exploration and inquiry enhance questioning by students. Thus, the process
and finding will be depending on students’ questioning sequence, not likely on
problem solving by the task which teachers are able to design before the class.

Mathematical
Modelling,
Programming andMathematisation

Mathematical modelling is a necessary way to solve real world problems in theworld. Mathematical model is hypothetically set by using mathematics to represent
the situation of the problem. Mathematical answers based on the model are
confirmed in the situation through interpretation. Modelling enhanced various
possibility to apply various representations in mathematics.
In education, modelling has been enhanced for the problem solving after the newmath movement which recognised school mathematics with set and structure.
In this era of Artificial Intelligence (AI) and Big Data, modelling is done throughprogramming with algorithms and it is also a part of mathematics.
Before mathematical modelling, mathematics had functioned as the metaphor,
theory and language for the nature. In Ancient Greece, music and astronomy were
exactly theorised under geometric representations. Today, mathematics itself has
various theories with algebraic representations, thus, modelling means hypothetical
approach for the real world by using universal mathematical language.
Mathematisation has two usages: The first usage is for science and engineering in
the establishment of mathematical model and produce new mathematics theories
based on the model. In physics, mathematical problem solving of nature has
produced the various theorisation in mathematics for solving problems in general.
The second usage is used in education as the principle of mathematics curriculum
sequence which enhances reorganisation of mathematical experiences. It includes
the process of extension and integration by using the prior learned knowledge.
From the perspective of mathematical problem solving, second usage aimed todevelop or reinvent mathematics by students through the solving mathematics
problems with support of teachers, textbook and curriculum. For mathematisation,
setting appropriate task sequence has the key role to develop or revise known
mathematical ideas, concepts and procedure.

208 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Conjecturing,
Justifying andProving

Conjecturing, justifying and proving have been on the context of proof andrefutation. In education, students conjecture hypothesis with reasoning on
exemplar. Conjecture is conceived through generalisation and justifying withappropriate conditions. Proving includes not only the formal proof in a local systembut also the various ways of explanations. Counter example is a way of refutation.
Counter example is meaningful because it sets off the reasoning from which if theconclusion is true. Proving is the activity to reach proof. On this meaning, probing
is not the proof itself. Collecting and explaining with examples can be also included.

Conceptualisation
andProceduralisation

Conceptual knowledge is the knowledge to explain the meaning and why, and used
for the conscious reasoning, and procedural knowledge is the skilful knowledge
used for unconscious-automatised reasoning. A unit of mathematics textbook
usually begins from the initiations of new conceptual knowledge by using learned
procedural-conceptual knowledge which is called conceptualisation. After the
initiations, new conceptual knowledge formulates the new procedural knowledge
for convenience and the exercises produce proficiency of procedure which is called
proceduralisation.
In mathematics, the proposition format ‘if…, then…’ is the basic format to represent
knowledge, however in school mathematics, it is impossible to make clear the
proposition from the beginning because ‘if’ part can be clarified later. For example,‘number’ changes the meaning several times in school curriculum. In multiplication,
products become large if it is […] number. Until numbers are extended to decimal,[…] part cannot be learned. On this problematic, it is normal that students meet thedifficulty in their learning and challenge to produce appropriate knowledge if they dohave a chance to over-generalise knowledge for knowing […] part Misconception(overgeneralized conception) is usually appear in the process of learning. It is anature of mathematics curriculum.
On this reason for producing exact knowledge in mathematics, conceptualisationand proceduralisation is a journey that continues recursively in mathematics
learning to change the view of mathematics. It is the opportunity to learn
necessary mathematical ideas and ways of thinking, and to develop value andattitude. The process of mathematisation also can be seen from the perspectiveof conceptualisation and proceduralisation on the context of number and algebra.

Representations
and Sharing

A mathematical representation consists of symbols, operations, and objective(context). Solving equations in algebra can be seen as specific ordered elements ofevery equation which has the same answer. Order of equations shows the context
that it is the process of solving. Operation of equations between an equation toanother equation can be explained by the property of equality. Each equation is asymbolic sentence.
If operations of representation are missing, it is not a mathematical representationeven through it has some artistic images such as diagrams. When students
draw diagrams, it is sharable if the rule of the drawing (operation) are shared.
In a classroom, students produce their own images in diagrams. It is helpful toencourage for their explanation by themselves but every explanation is independent
until knowing the hidden ideas as in the comparison. Other students cannot represent it until the ways of drawing (operation) is shared. Thus, comparison ofvarious representations is a part of the process to recognise ideas for producing
symbol and operations, and translations as in mathematical representations.
In mathematics, different representations use different symbols and operations. If
there is a rule for correspondence between symbols and operations on different

Appendix B – Terminologies for Mathematical Thinking and Processes | 209

representations, it can be translated and produce rich meanings. It is the way to
produce a mathematics system.
In mathematics, representation system can be defined universally, which is the
product of convention by mathematicians. For teachers, it looks far for students’
activity in the classroom, however it is the opportunity for students to reinvent the
representation and its system through considering the why and how. For example,
to produce metre as the measurement quantity includes such activities: There
are historical episodes on why and how ‘m’ was defined by Condorcet and othersin the middle of the French Revolution. On the area of Engineering, Informaticand Science, applied mathematicians usually try to produce new measurements
based on their necessity of research and development to conceptualise the idea
mathematically and operationally. Setting the measurable quantity is a part ofmathematical modelling for real-world problem solving because if it is measurable,
we can apply known mathematics.

210 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

APPENDIX C
Strand: Mathematical Process – Humanity for Key Stage 1
Enjoyable mathematical activities are designed to bridge the standards in different strands. Exploration ofvarious number sequences, skip counting, addition and subtraction operations help to develop number sensethat is essential to support explanations of contextual scenarios and mathematical ideas. Mathematical waysof posing questions in daily life are also necessary to learn at this stage. Ability to select simple, general andreasonable ideas enables effective future learning. Application of number sense provides facility for preparing
sustainable life. The use of ICT tools and other technological tools provide convenience in daily life. At the initialstage, concrete model manipulation is enjoyable, however, drawing diagram is most necessary for explaining
complicated situations by using simple representation.
Standards:
Enjoying problem solving through various questioning for four operations in situations
Enjoying measuring through setting and using the units in various situations
Using blocks as models and its diagram for performing operations in base ten
Enjoying tiling with various shapes and colours
Explaining ideas using various appropriate representations
Selecting simple, general and reasonable ideas which can apply for future learning
Preparing sustainable life with number sense
Utilising ICT tools such as calculators as well as other tools such as notebooks and other instruments suchas clocks
Promoting creative and global citizenship for sustainable development of neighbourhood using mathematics
Enjoying problem solving through various questioning for four operations in situations1

i.

ii.
iii.
iv.
v.
vi.
vii.

In addition situations, pose questions for ‘altogether’ and ‘increase’In subtraction situations, pose questions for ‘remainder’ and ‘difference’In multiplication situations, pose questions for ‘number of groups’
In division situations, pose questions for ‘partitive’ and ‘quotative’
Enjoy questioning in various situations by using combination of operations
In operations, pose questions to find the easier ways of calculation
Use posing questions for four operations on measurements in daily life

Enjoying measuring through setting and using the units on various situations2

i.

ii.
iii.
iv.
v.

Compare directly and indirectly
Set different tentative units for measuring
Give appropriate names (denominations) for counting units
Use measurement for communication in daily life
Use tables and diagrams for showing the data of measures

Using blocks as models and its diagram for performing operations in base ten3

i.

ii.
iii.
iv.

Show increasing and decreasing patterns using blocks
Show base ten system using blocks, a unit cube is 1, a bar stick is 10 and a flat block represents 100
Explain addition and subtraction algorithm in vertical form using base ten block model
Explain multiplication table with number of grouped blocks

It is related to Numbers and Operations and Quantity and Measurement both in Key Stage 1.
It is related to Quantity and Measurement and Pattern and Data Representations both in Key Stage 1.
3
It is related to Patterns and Data Representations and Numbers and Operations both in Key Stage 1.
1
2

Appendix C – Strand on Mathematical Processes-Humanity (Key Stage 1) | 211v.
vi.

Explain division using equal distribution of blocks and repeated subtraction of blocks
Use the number of blocks for measurement in daily life

Enjoying tiling with various shapes and colours4
i.
ii.
iii.

Appreciate to produce beautiful designs with various shapes and find the pattern to explain it
Reflect, rotate and translate to produce patterns
Cut and paste various shapes and colours to form a box and ball such as develop the globe frommap

Explaining ideas using various appropriate representations5
i.
ii.
iii.
iv.
v.
vi.

Explain four operations using pictures, diagrams, blocks and expressions for developing ideas
Explain measurement using measuring tools, tape diagrams, container and paper folding for sharing
ideas
Make decision on how to explain figures and solids by using manipulative objects or diagrams or
only verbal explanation
Explain patterns using diagrams, numbers, tables and expressions with blank box
Ask questions using terms such as why, how, what, if and if not, and reply using examples and ‘forexample’ in discussion
Change the representation and translate it appropriately in daily life

Selecting simple, general and reasonable ideas which can apply for future learning6
i.
ii.
iii.
iv.

Discuss the argument for easier ways for addition and subtraction algorithm in vertical form
Extend the algorithm to large numbers for convenience and fluency
Use the pattern of increase in multiplication table for convenience
Use multiplication tables for finding the answers of division

Applying number sense7 acquired in Key Stage 1 for preparing sustainable life8
i.
ii.
iii.
iv.

Use mathematics in situations with minimum and sequential use of resources
Estimate for efficient use of resources in situations
Maximise the use of resources through appropriate arrangement in space
Understand equally likely of resources in situations

Utilising ICT tools such as calculators as well as other tools such as notebooks and other instruments suchas clocks9
i.
ii.
iii.
iv.

Use calculators for multi addition in situations
Use mental calculations for estimations
Use a balance scale to produce equality and inequality
Use cups, tapes, stop watches, and weighing scales for measuring distances and mass

It is related to Shapes, Figures and Solids and Pattern and Data Representations both in Key Stage 1.
It is related to all strands in Key Stage 1.
6
It is related to Numbers and Operations and Pattern and Data Representations both in Key Stage 1.
7
It is related to Numbers and Operations, Pattern and Data Representations and Quantity andMeasurement all included in Key Stage 1.
8
Sustainable development goals were crafted at the 70th Session of the United Nations General Assembly and indicated themas universal value in education.
9
STEM education is enhanced. Mathematics is the major and base subject for STEM Education in Key Stage 1 hence,technological contents are included in Mathematics.
4
5

212 | Mathematics Challenges for Classroom Practices at the Lower Primary Levelv.

Use calculators to explain the process of calculation by solving backward and understand therelationship of addition and subtraction, and multiplication and division.
vi. Enjoy using notebooks to exchange learning from each other such as mathematics journal writingvii. Enjoy presentations with board writingviii. Use various tools for conjecturing and justifying
Promoting creative and global citizenship for sustainable development of neighbourhood using mathematics
i.
ii.
iii.
iv.
v.

Utilise notebooks and journal books to record and find good ideas and share with others
Prepare and present ideas using posters to promote good practices in neighbourhood
Listen to other’s ideas and asking questions for better creativity
Utilise information, properties and models as basis for reasoning
Utilise practical arts and outdoor studies to investigate local issues for improving welfare of life

Appendix D – Learning Standards for Key Stage 2 | 213

APPENDIX D
The four strands on content learning standards and the strand on Mathematical Process-Humanity of Key
Stage Two
KEY STAGE 2
Key Stage 2 (KS2) can be learned based on the Key Stage 1. This stage provides the extension of numbers
and operations, measurement and relations, plane figures and space figures as well as data handling andgraphs. This stage enables the extension of the four operations to daily use of numbers such as decimal andfraction and allows the use of mathematical terminologies, performing investigations and establish the groundfor analysing, evaluating and creating in learners’ life. Appreciating the beauty of the structure of mathematics
will enable them to enjoy and sustain their learning which provides basis for Key Stage 3.
Strand: Extension of Numbers and Operations
Numbers are extended to multi-digits, fractions and decimals. Multiplication and division algorithms arecompleted with fluency. Fraction becomes numbers through the redefinition as a quotient instead of part-whole
relationship. Multiplication and division of decimals and fractions are also explored to produce procedures forcalculation. Various representations are used to elaborate and produce meaning for the calculation. Number
sense such as approximating numbers, relative size of numbers and values are enhanced for practicalreasoning in the appropriate context of life.
Topics:Extending Numbers with Base-ten Up to Billion and also to Thousandths with Three Digit Numeral System
Gradually
Making Decision of Operations on Situations with Several Steps and Integrate them in One Expression andThink About the Order of Calculations and Produce the Rule (PEMDAS)
Producing the Standard Algorithm for Vertical Form Division with Whole Numbers
Extending the Vertical Form Addition and Subtraction with Decimals to Hundredths
Extending the Vertical Form Multiplication and Division with Decimals and Find the Appropriate Place Valuesuch as Product, Quotient and Remainder
Using Multiples and Divisors for Convenience
Introducing Improper and Mixed Fractions and Extending to Addition and Subtraction of Fractions to Dissimilar
Fractions
Extending Fractions as Numbers and Integrate
Extending Multiplication and Division to Fractions
Extending Numbers with Base-ten Up to Billion and also to Thousandths with Three Digit Numeral
System Gradually
Extending numbers using base-ten system up to billion1 with three-digit numeral system2i.
ii.

Adopt the three-digit numeral system, extend numbers up to billion with the idea of relative size of
numbers
Compare numbers such as larger, smaller with base3-ten system of place values through visualisationof relative size of numbers using cube, plane(flat), bar(long) and unit

Billion is too large for counting and it is introduced in the three-digit system under relative size of number.
In British system, it is referred as short scale.
3
Metric system names of units are discussed under Measurement and Relations.
12

214 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Extending decimal numbers to hundredths, and to thousandths4
i.
ii.
iii.

Use the idea of quantity and fractions, extend decimal numbers from tenths to hundredths
Compare decimal numbers such as larger, smaller with base-ten system of place value
Adopt the ways of extension up to thousandths and so on, and compare the relative sizes

Making Decision of Operations on Situations with Several Steps and Integrate them in One Expression
and Think about the Order of Calculations and Produce the Rule (PEMDAS)
Finding the easier ways of calculations using the idea of various rules of calculations5 such as the associative,
commutative and distributive rule
i.
ii.
iii.
iv.

Find the easier ways of addition and subtraction and use it, if necessary, such as answer is the sameif add same number to the subtrahend and minuend
Find the easier ways of multiplication and division and use them in convenient ways such as 10times of multiplicand produce the product 10 times
Use associative, commutative and distributive rules of addition and multiplication for easier ways of
calculation, however, commutative property does not work in subtraction and division
Appreciate the use of simplifying rules of calculations

Thinking about the order of calculations in situations and produce rules and order of operations
i.
ii.
iii.

Integrate several steps of calculation into one mathematical sentence
Produce the rule of PEMDAS and apply it to the several steps situation
Think about the easier order of calculation and acquiring fluency of PEMDAS and rules withappreciation

Producing the Standard Algorithm Using Vertical Form Division with Whole Numbers
Knowing the properties of division and use it for easier way of calculationi.
ii.

Find the easier ways of division and use it, if necessary, such as answer is the same if multiplying
the same number to the dividend and divisor
For confirmation of answer of division, use the relationship among divisor, quotient and remainderand appreciate the relationshipKnowing the algorithm of division in vertical form and acquiring fluency
i.
ii.
iii.
iv.

Know the division algorithm with tentative quotient and confirm the algorithm by the relationshipamong divisor, quotient and remainderInterpret meaning of quotient and remainder in situations
Acquire fluency for division algorithm up to 3-digit whole number divided by 2-digit
Think about situations with or without remainder in relation to situations for quotative and partitive
division

Under the three-digit system, if we teach until thousandths, we can extend by three-digit.
In Measurement and Relations, use of constant sum, difference, product and quotient are described. (e.g.25 -21 = 4, 26–22 =4, 27– 23 = 4)45

Appendix D – Learning Standards for Key Stage 2 | 215

Extending the Vertical Form Addition and Subtraction with Decimals to Hundredths
Extending the vertical form addition and subtraction in decimals to hundredthsi.
ii.

Extend the vertical form addition and subtraction to hundredths6 place and explain it with models
Appreciate the use of addition and subtraction of decimals in life

Extending the Vertical Form Multiplication and Division with Decimals and Find the Appropriate Place
Value Such as Product, Quotient and Remainder
Extending the multiplication from the whole number to decimal numbers
i.
ii.
iii.
iv.

Extend the meaning of multiplication with the idea of measurement by the number of unit length
for multiplication of decimal numbers and use diagrams such as number lines to explain them withappreciation in situations
Extend the vertical form multiplication of decimals up to 3 digits by 2 digits with consideration of thedecimal places step by stepObtain fluency using multiplication of decimals with sensible use of calculators in life
Develop number sense in multiplication of decimals7 such as comparing sizes of products before
multiplying

Extending the division from the whole number to decimal numbers
i.
ii.
iii.
iv.
v.
vi.

Understand how to represent division situations using diagrams such as number lines, and extend
the diagram of decimal numbers for explaining division by decimal numbers
Extend the division algorithm in vertical form of decimal numbers and interpret the meaning ofdecimal places of quotient and remainder with situations
Acquire fluency in division algorithm of decimals up to 3 digits by 2 digits with consideration ofdecimal places step by stepObtain fluency using division of decimals with sensible use of calculators in life
Develop number sense of division in decimals such as comparing sizes of quotient before multiplying
Distinguish situations with decimal numbers of multiplication and division

Using Multiples and Divisors for Convenience
Using multiples and divisors for convenience with appreciation to enrich number sense
i.
ii.
iii.
iv.
v.
vi.

Understand set of numbers by using multiples and divisors
Find common multiples and appreciate the use in situations, and enrich number sense with figuralrepresentations such as arrangement of rectangles to produce a square
Find common divisor and appreciate its use in situations, and enrich number sense with figuralrepresentations such as dividing a rectangle into pieces of square
Understand numbers as composite of multiplication of numbers as factors8
Appreciate ideas of prime, even and odd numbers in situations using multiples and divisors
Acquire number sense to see the convenient use of multiples and divisors

Discussion of decimals to hundredths is related to the use of money. It is a minimum requirement. If teaching to hundredths,
further extension of place value can be understood.
7
Applying the idea of multiplication into ratio, percent and proportion is discussed in Measurement and Relations.
8This idea is related to the strand on Measurement and Relations in Key Stage 2 at the area of rectangle.
6216 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Introducing Improper and Mixed Fractions and Extending to Addition and Subtraction of Fractions to
Dissimilar Fractions
Extending fractions to improper, mixed and equivalent fractions
i.
ii.
iii.

Extend fractions to improper and mixed fractions using number line of more than one by measuringwith unit fraction9
Find ways to determine equivalent fractions with number lines and with the idea of multiple ofnumerator and denominator
Compare fractions using number line and the idea of multiple

Extending addition and subtraction of similar fractions to improper and mixed fractions, and dissimilar fractions
i.
ii.
iii.

Extend addition and subtraction of similar fractions to proper and mixed fractions with explanations
using models and diagrams
Extend addition and subtraction into dissimilar fractions with explanations using diagrams andcommon divisors
Acquire fluency of addition and subtraction of fractions with appreciation of idea to produce the samedenominators

Extending Fractions as Numbers and Integrate10
Seeing fractions11 as decimals and seeing decimals as fractions
i.
ii.
iii.

See fractions as decimals using division and define quotient from the case of indivisible
See decimals as fractions such as hundredths is per hundred
Compare decimals and fractions and ordering them on a number line

Extending Multiplication and Division to Fractions
Extending multiplication to fraction
i.
ii.
iii.

Extend multiplication to fraction with situations using diagrams such as number lines, and find thesimple algorithm for the multiplication of fractions, step by step
Acquire fluency with multiplication of fractions
Develop number sense12 of multiplication of fractions such as comparing sizes of products before
multiplying

Extending division to fraction
i.
ii.
iii.

Extend division to fraction with situations using diagrams such as number lines, step by step
Acquire fluency with division of fractions
Develop number sense of division of fractions such as comparing sizes of quotients before dividing
Strand: Measurement and Relations

Additive quantity such as angles, areas and volume and relational quantities such as population density, andspeed are introduced. Additive quantity can be introduced by establishment of the standard unit which is thesame way as the Quantity and Measurement of Key Stage 1. Relations of quantities in situations are discussed
9

Extension of fraction more than one is done by using the fraction with quantity for situation such as m

10

Selecting the appropriate denomination of quantities and units for fraction in the context ( m is two of

1m, howeverm and the whole is

m is 3x( )m; the structure is the same as tens is ten of units and discussed under Measurement and Relations.

Fraction as ratio is introduced in Measurement and Relations.
12
Applying the idea of the multiplication of fraction into ratio, proportion, percentage and base is discussed in Measurement andRelations.
11Appendix D – Learning Standards for Key Stage 2 | 217with patterns such as sum is constant, difference is constant, product is constant and quotient is constant using
tables and represented by mathematical sentences and letters. Proportion and ratio are introduced withrepresentations of diagrams, graphs and tables for multiplication, and connected with decimals and fractions.
Percent is introduced with diagrams in relation to ratio and proportion. Relational quantity is produced bydifferent quantities with understanding of ratio. Area of a circle is discussed through a proportional relationshipbetween radius and the circumference. Ideas of ratio and proportion are fluently applied for real world problem
solving.
Topics:Introducing Angle and Measuring it
Exploring and Utilising Constant Relation
Extending Measurement of Area in Relation to Perimeter
Extending Measurement of Volume in Relation to Surface
Approximating with Quantities
Extending Proportional Reasoning to Ratio and Proportion
Producing New Quantities Per Unit
Investigating the Area of a Circle
Exchanging Local Currencies in ASEAN Community
Extending the Relation of Time and Use of Calendar
Converting Quantities in Various System of Units
Showing Relationship Using Venn Diagram
Introducing Angle and Measuring It13
Introducing angle by rotation, enabling measure and acquire fluency using the protractor
i.
ii.
iii.
iv.
v.

Compare the extent of rotation and introduce degree as a unit for measuring angle
Recognise right angle is 90 degrees, and adjacent angle of two right angles is 180 degrees, and 4
right angles is 360 degrees
Acquire fluency in measuring angles using the protractor
Draw equivalent angles with addition and subtraction using multiples of 90 degrees
Appreciate measurement of angles in geometrical shapes and situations in life14

Exploring and Utilising Constant Relation
Exploring equal constant relation with utilisation of letters to represent placeholders15
i.
ii.
iii.
iv.

Explore two possible unknown numbers such that their sum (or difference /product/quotient) isconstant,16 for example □+∆=12 (□ and ∆ are placeholders).
Use letters instead of placeholder17 (empty box) to derive equivalent relation
Understand the laws for operations (e.g. associative, commutative and distributive etc.) to explain
the simpler way of calculation
Appreciate the use of diagrams such as number lines and area to represent relation when finding
solutions

Right angle is learned at Key Stage 1 in Shapes, Figures and Solids for explaining the properties of figures.
Conservation of angles will be re-learnt in triangle under Key Stage 2 Plane Figures and Space Figures.
15
The idea for the use of Numbers and Operations (Key Stage 2) in finding the easier ways of calculations with the idea of rulesof calculations.
16
Constants of multiplication and division, corresponds proportionality in multiplication table at Key Stage 1 under number andoperations. Constant of addition and subtraction are treated in Key Stage 1 under Pattern and Data Representations.
17
Place holder is introduced in Key Stage 1.
13
14

218 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Extending Measurement of Area in Relation to Perimeter
Introducing area and produce formula for the area of a rectangle
i.
ii.
iii.
iv.
v.
vi.
vii.

Compare extent of area and introduce its unit, and distinguish it from perimeter
Introduce one square centimetre as a unit for area and its operation using addition and subtraction
Investigate area of rectangles and squares and produce the formula of area18
Extend square centimetre to square metre and to square kilometre for measure of large areas
Convert units and use appropriate units of area with fluency
Draw the equivalent size of rectangular area based on a given area with the composite numbers19
Appreciate the use of areas in daily life such as comparing of land sizes.

Extending area of a rectangle to other figures to derive formulae
i.
ii.
iii.
iv.

Explore and derived formula for the area of parallelogram by changing its shape to rectangle withoutchanging its area
Explore and derived formula for the area of a triangle by bisecting a rectangle into two triangles
without changing its area
Appreciate the idea of changing or dividing shapes of rectangle, parallelogram, or/and triangle forderiving the area of other figures
Use formulae to calculate areas in daily life

Extending Measurement of Volume in Relation to Surface
Introducing volume from area and derive formula for cuboid
i.
ii.
iii.
iv.
v.
vi.

Compare the extent of volume and introduce its unit, and distinguish it from surface
Introduce one cubic centimetre as unit for volume and its addition and subtraction
Investigate volume of cuboid and cube and produce the formulae
Extend cubic centimetre to cubic metre to measure large volume
Convert units and use appropriate units of volume with fluency
Appreciate the use of volume in life such as comparison of capacity of containers

Extending volume of cuboid to other space figures to derive formula
i.
ii.
iii.

Extend the formula for the volume of a cuboid as base area x height for exploring space figures suchas prism and cylinder
Extend the formula for the volume of a prism and a cylinder to explore and derive the volume formulaof pyramid and cone
Use the formulae to calculate volume in daily life

Approximating with Quantities
Approximating numbers with quantities depending on the necessity of contexts
i.
ii.
iii.

Understand the ways of rounding such as round off, round up and round down
Use rounding as approximation for making decision on the quantity with related context
Critique over approximation beyond the context with a sense of quantity such as based on relative
size of a unit

Multiplications were studied in Key Stage 1 Number and Operations.
The idea of composite numbers such as 2 times 10 equals 5 times 4 is related to factors in extending the numbers and operations
at the same Key Stage 2.
18
19

Appendix D – Learning Standards for Key Stage 2 | 219

Extending Proportional Reasoning to Ratio20 and Proportion
Extending proportional reasoning to ratio and percent
i.
ii.
iii.
iv.
v.

Understand ratio as a relationship between two same quantities or between two different quantities(the later idea is rate)21
Express the value of ratio by quotient such as the rate of two different quantities22
Understand percent as the value of ratio with the same quantities23
Understand proportion with ratio
Apply the rule of three24 in using ratio

Extending proportional reasoning to proportion
i.
ii.
iii.
iv.
v.

Extend proportional reasoning to multiplication tables as an equal ratio and understand proportions
Understand proportion using multiple and constant quotient, not changing the value of ratio25
Demonstrate simple inverse proportion by constant product26
Express proportion in mathematical sentence using letters and graph27
Use properties of proportionality to predict and explain phenomenon in daily life

Producing New Quantities by Per Unit
Producing new quantities by per unit with the idea of average such as population density and speed
i.
ii.
iii.
iv.
v.
vi.

Introduce average as a unit for distribution and comparison of different sets of values
Introduce population density with the idea of average and appreciate it for comparison
Introduce speed with the idea of average and appreciate it for comparison
Appreciate using diagrams such as number lines and tables to decide the operations situations ofmeasurement per unit quantity
Comparing on the context of different quantities with the idea of average as rate28
Apply the idea of measurement per unit quantity in different context29

Investigating the Area of Circle
Areas of a circle are discussed through the relationship between the radius and circumferencei.
ii.

Investigate relationship between the diameter of a circle and its circumference using the idea of
proportion
Investigate area of a circle by transforming into a triangle or parallelogram and find the formula of thecircleBand graph and pie chart for representing ratio are discussed under Key Stage 2 under the strand Data Handling and Graphs.
Ratio of different quantities is a rate. Ratio of the same quantities has a narrow meaning of ratio.
22
The value of fraction as a ratio is not necessary a part of a whole in situations. Fraction as a ratio is usually used in the context
of multiplication situation, where denominator is the base or a unit for comparison.
23
Percent is used in Data Handling and Graphs.
24
Rule of three is the method on the table to find one unknown term from the three known terms using proportional reasoning
such as
		a
c
		?
d
20
21

Enlargement is discussed in Key Stage 2 under Plane Figures and Space Figures. The graph is treated at Key Stage 2 under
the strand Data Handling and Graphs.
26
Proportion and Inverse proportion are necessary in Key Stage 3 in science.
27
This will be discussed in detail in same Key Stage 2 under Data Handling and Graphs.
28
On number and operations Key Stage 2, rate is the value of division as quotient.
29
Using measurement per unit quantity with fluency to make logical judgment in daily life, refer to Key Stage 2 Data Handlingand Graphs.
25

220 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

iii.
iv.

Estimate the area of a inscribed and circumscribed shapes based on known formula of areas30
Enjoy to estimate the area of irregular shapes with fluency in life

Exchanging Local Currency with Currency in ASEAN Community
Exchanging local currency in ASEAN community with the idea of rate
i.
ii.
iii.

Extend the use of ratio for currency exchange (rate of exchange)
Apply the four operations for money in appropriate notation in life
Appreciate the value of money

Extending the Relation of Time and Use of Calendar in Life
Extending the relation of time and use of calendar in life
i.
ii.
iii.

Convert time in 12-hour system with abbreviation a.m. and p.m. to 24-hour system and vice versa
Investigate the numbers in calendar to relate days, weeks, months and year using the idea of number
patterns
Appreciate the significance of various calendars in life

Converting Quantities in Various System of Units
Converting measurement quantities based on international and non-international system with the idea ofbase-10
i.
ii.
iii.
iv.
v.

Convert measurement system of metre and kilogram with prefixes deci-, centi-, and milli-, and withdeca, hecto-, and kiloConvert measurement system of litre with cubic centimetre
Convert measurement system of area using are (a) and hectare (ha) with square meter
Convert measurement of local quantities with standard quantities
Understand the unit system with power, such as metre, square metre and cubic metre

Showing Relationship Using Venn Diagram
Using Venn diagram to show relationships of numbers and figuresi.
ii.

Show relationship of squares, rectangles, rhombus, parallelogram, trapezoid and quadrilateral byusing a Venn diagram
Show relationship of numbers
Strand: Plane Figures and Space Figures

Through tessellation, figures can be extended through plane figures. Parallelogram and perpendicular lines
are tools to explain properties of triangles and quadrilaterals as plane figures. They are also needed for
identifying and recognising symmetry and congruence parallelogram and perpendicular lines. Plane figures
are used to produce solids in space and vice versa. Opening faces of solids would produce plane figures which
are referred as nets. Activities related to building solids from plane figures are emphasised and encouraged
to facilitate finding the area of a circle through numerous sectors of the circle to construct a rectangle. Circles
are used for explaining the nets of cylinders.

30

Relationship on polygons and circles are discussed in Key Stage 2 under Plane Figures and Space Figures.

Appendix D – Learning Standards for Key Stage 2 | 221

Topics:Exploring Figures with their Components in the Plane
Exploring Solids with their Components in Relation to the Plane
Exploring Figures with Congruence, Symmetry and Enlargement
Exploring Figures with their Components in the Plane
Exploring figures with their components in the plane and use their properties
i.
ii.
iii.
iv.

Examine parallel lines and perpendicular lines by drawing with instruments
Examine quadrilaterals using parallel and perpendicular lines, and identify parallelogram, rhombus,
trapezium through discussion
Find properties of figures through tessellations such as a triangle where the sum of the angles is 180
degrees, a straight angle
Extend figures to polygons and expand it to circles with knowing and using its properties

Exploring Solids with their Components in Relation to the Plane
Exploring rectangular prisms and cubes with their components
i.
ii.
iii.
iv.

Identify relationship between faces, edges and vertices for drawing of sketch
Explore nets of rectangular prism and find the corresponding position between the components
Explore the perpendicularity and parallelism between faces of a rectangular prism
Explain positions in rectangular prisms with the idea of 3-dimensions

Extending rectangular prism to other solids such as prisms and cylinders
i.
ii.
iii.

Extend the number of relationships between faces, edges and vertices for drawing of sketch
Explore nets of prisms and cylinders, and find the corresponding position between components
Distinguish prism and cylinder by the relationship of their faces

Exploring Figures with Congruence, Symmetry and Enlargement
Exploring the properties of congruence
i.
ii.
iii.

Explore properties of figures which fit by overlapping and identify conditions of congruency withcorresponding points and sides
Draw congruent figures using minimum conditions and confirm by measuring angles and sides
Appreciate the power of congruent figures by tessellation

Exploring the properties of symmetry
i.
ii.
iii.

Explore the properties of figures which reflect and identify conditions of symmetry with line and its
correspondence
Draw symmetrical figures using conditions in appropriate location
Appreciate the power of symmetry in designs

Exploring the properties of enlargement31
i.
ii.
iii.
31
32

Explore properties of figures in finding the centre of enlargement in simple case such as a rectangle
Draw enlargement of rectangle using ratio (multiplication of the value of ratio)32
Appreciate the power of enlargement in interpretation of map

General case will be discussed in Key Stage 3, Space and Geometry.
Ratio and rate are discussed in Key Stage 2 under Measurement and Relations.

222 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Strand: Data Handling and Graphs
The process of simple data handling is introduced through data representation such as using table, bar graph,line graph, bar chart and pie chart. Graphs are utilised depending on the qualitative and quantitative data
used such as bar graph is for distinguishing and counting in every category. The discussion of producing theline graph includes taking data at specific intervals, suitable scale used and slope. Histogram is necessary
for interpreting the data representation of social study and science, and is also used as a special type of bar
graph. Average is introduced based on the idea of ratio for making the dispersion of bar chart even, and used
for summarising and comparing data on a table. Problem-Plan–Data- Analysis–Conclusion (PPDAC) cycles
are experienced through the process of data handling by using those data representation skills. Those skills
are necessary for learning of sustainable development.
TopicsArranging Tables for Data Representations
Drawing and Reading Graphs for Analysing Data
Using Graphs in PPDAC Cycle (Employing the Problem - Plan - Data - Analysis - Conclusion) Appropriately
Applying Data Handling for Sustainable Development
Arranging Tables for Data Representations
Exploring and arranging tables for data representations
i.
ii.
iii.

Explore how to collect multi category data based on a situation
Explore how to arrange and read multi category data on appropriate tables.
Appreciate the use of multi category tables in situations

Drawing and Reading Graphs for Analysing Data
Drawing and reading line graphs for knowing the visualised pattern as basis for tendency of change
i.
ii.
iii.
iv.

Introduce line graphs based on appropriate situations such as rainfall, temperature and others
Distinguish line graph from bar graph for observation such as increase, decrease, and no-change
Introduce the graph of proportion using the idea of line graph and read the gradient by constantratio33
Appreciate the line graph in various situations

Drawing and reading band graph and pie chart for representing ratio in a whole34
i.
ii.
iii.

Explore how to scale the band or circle for representing ratio or percent
Use the band graph and pie chart for comparison of different groups
Appreciate the band graph and pie chart in a situation

Reading histogram35 for analysing frequency distribution
i.
ii.
iii.

Draw a simple histogram36 from frequency table on situations
Read various histograms for analysing data distribution
Use mean37 to compare different groups in the same situation with histograms

Proportions are learnt in the Key Stage 2 under Measurement and Relations.
Ratio is learnt under Key Stage 2 Measurement and Relations.
35
How to draw histogram is discussed in Key Stage 3 under Statistics and Probability. Reading histograms is necessary in social
studies and science.
36
Using ICT for drawing graph will be mentioned in mathematical activities.
37
Mean is introduced as average in Key Stage 2 under Measurement and Relations.
33
34

Appendix D – Learning Standards for Key Stage 2 | 223

Using Graphs in PPDAC38 Cycle Appropriately
Identifying appropriate graphs for data handling in PPDAC cycle
i.
ii.
iii.
iv.
v.

Analyse a situation and discuss the expected outcomes before collecting data to clarify the purposeof a survey
Plan the survey for the intended purpose
Collect the data based on the purpose of the survey
Use appropriate graphical representation which is most suitable for the purpose
Appreciate the use of graphs before making the conclusion

Applying Data Handling for Sustainable Living
Applying data handling for sustainable development39 and appreciate the power of data handling for predicting
the future.
i.

ii.
iii.

Read data related to sustainable development such as emergency preparedness and resiliency,
food and energy security, world weather warming, inclusion and human connectivity in society, andlifelong learning in the changing society such as TVET (Technical and Vocational Education andTraining) and adopting positive views for changing the society
Understand the idea of probability as ratio and percentage in reading the data for situations related
to sustainable development such as weather report and risk analysis
Experience implementing a project of reasonable size in data handling for the purposes of achieving
sustainable development and appreciate the power of data handling
Strand: Mathematical Processes – Humanity

As a follow up of Key Stage 1, activities are designed to enable an appreciation of knowledge and skills learned
and the ways of learning such as applying knowledge of number sense to solve daily problems. Mathematicalprocesses such as communication and reasoning are used to provide explanation for mathematical problems
and modelling. The ability to connect and reason mathematical ideas would trigger an excitement among
learners. Discussions of misconceptions are usually enjoyable and challenging. Mathematics learning usually
begins from situations at Key Stage 1. In Key Stage 2, the development of mathematics is possible throughdiscussions for the extension of the forms. Appreciation of ideas and representations learned become partof the enjoyable activities. Through the consistent use of representations such as diagrams, application oflearning becomes meaningful.
Standards:
Enjoying problem solving through various questioning and conjecturing for extension of operations into
decimals and fractions with proportionality and new quantities such as area and volume
Enjoying measuring through settings and using the area and volume in situations
Using ratio and rate in situations
Using number lines, tables, and area diagrams for representing operations and relationships in situations
Establishing the idea of proportion to integrate various relations with consistency of representations
Enjoying tiling with various figures and blocks
Producing valuable explanation based on established knowledge, shareable representations and examples
Performing activities of grouping and enjoy representing with Venn diagram
Experiencing PPDAC (Problem-Plan-Data-Analysis-Conclusion) cycle and modelling cycle in simple projects
in life
Preparing sustainable life with number sense and mathematical representations
Utilising ICT tools as well as notebooks and other technological tools
Promoting creative and global citizenship for sustainable development of community using mathematics

38
39

PPDAC itself will be described in the mathematical activities later.
This standard which is related to SDG as inter subject content between social studies and science.

224 | Mathematics Challenges for Classroom Practices at the Lower Primary Level

Enjoying problem solving through various questioning and conjecturing for extension of operations into
decimals and fractions with proportionality and new quantities such as area and volume40i.
ii.

Pose questions to develop division algorithm in vertical form using multiplication and subtraction
Pose questions to develop multiplication and division of decimal numbers using the idea of
proportionality with tables and number linesiii. Pose questions to develop multiplication and division of fractions using the idea of proportionality
with tables, area diagrams and number lines
iv. Pose questions to extend multiplication and division algorithm in vertical form to decimal numbers
and discuss about decimal pointsv.
Pose questions to use decimals and fractions in situations
vi. Pose questions to use area and volume in life
vii. Pose questions to use ratio and rate in life
viii. Pose conjectures based on ideas learned such as when multiplying, the answer becomes larger
Enjoying measuring through settings and using the area and volume in situations
i.
ii.
iii.
iv.

Compare directly and indirectly areas and volumes
Set tentative units from difference for measuring area and volume41
Give the formula for the area and volume for counting units
Use measurement for communication in daily life

Using ratio and rate in situations42
i.
ii.
iii.

Understand division as partitive (between different quantities) and quotative (between same quantity)
in situations
Develop the idea of ratio and rate utilising the idea of average and per unit with tables and number
lines
Communicate using the idea of population density and velocity in life

Using number lines, tables, and area diagrams for representing operations and relations in situations43
i.
ii.

Represent proportionality on number lines with the idea of multiplication tables
Use number lines, tables, and area diagrams for explaining operations and relations of proportionality
in situations

Establishing the idea of proportion to integrate various relations with consistency of representations44
i.
ii.
iii.
iv.
v.
vi.

Use the idea of proportion as the relation of various quantities in life
Identify through the idea of proportion using tables, letters, and graphs
Adopt the idea of proportion to angles, arcs and area of circles
Adopt the idea of proportion to area and volume
Adopt the idea of proportion to enlargement
Use ratio for data handling such as percent and understand the difficulties to extend it to proportion

Enjoying tiling with various figures and blocks45
i.
ii.
iii.

Appreciate to produce parallel lines with tessellation of figures
Explain the properties of figures in tessellations by reflections, rotations and translations
Develop nets from solids and explain the properties of solids by each of the component figures

This is connected to the three strands, Extension of Numbers and Operations, Measurement and Relations, and Plane Figures
and Space Figures.
41
Euclidean algorithm is a method of finding the largest common divisor of two numbers.
42
Ratio and proportion bridge multiplication and division in situation of two quantities with reference to Extension of Numbers
and Operations and Measurement and Relations.
43
This is a bridge to the Extension of Numbers and Operations and Measurement and Relations.
44
Bridge to the three strands, Measurement and Relations, Plane Figures and Space Figures and Data Handling and Graphs.
45
Connected to the two strands, Measurement and Relations, and Plane Figures and Space Figures.
40

Appendix D – Learning Standards for Key Stage 2 | 225iv.

Use the idea of tiling for calculating the area and volume

Producing valuable explanations based on established knowledge, shareable representations and examples
i.
ii.
iii.
iv.

Establish the habit of explanation by referring to prior learning and ask questions using terms suchas why, how, what, if and if not, and reply using examples and ‘for example’ in discussion
Assessing the appropriateness of explanations using representations such as generality, simplicity
and clarity
Use other’s ideas to produce better understanding
Use inductive reasoning for extending formulae

Performing activities of grouping and enjoy representing with Venn diagrami.
ii.

Use the idea of Venn diagram for social study
Understand classifications based on characteristics and represent by using Venn diagrams

Experiencing PPDAC (Problem-Plan-Data-Analysis-Conclusion) cycle and modelling cycle in simple projects
in life
i.
ii.
iii.
iv.

Understand the problem of context
Plan appropriate strategies to solve the problem
Gather data and analyse using appropriate methods and tools
Draw conclusion with justification based on data analysis

Preparing sustainable life with number sense and mathematical representations46
i.
ii.
iii.
iv.
v.

Use minimum and sequential use of resources in situations
Use data with number sense such as order of quantity and percentage for the discussion of mattersrelated to sustainable development
Estimate the efficient use of resources in situations
Maximise the use of resources through appropriate arrangement in a space such as a room
Understand “equally likely” of resources in situations

Utilising ICT tools as well as notebooks and other technological tools
i.
ii.
iii.
iv.
v.
vi.

Use internet data for the discussion of matters related to sustainable development
Distinguish appropriate or inappropriate qualitative and quantitative data for using ICT
Use calculators for organising data such as averageUse calculators for operations in necessary context
Use projectors for sharing ideas as well as board writing
Enjoy using of notebooks to exchange learning experiences between each other such as inmathematics journal writingvii. Use protractors, triangular compasses, straight edges, clinometers for drawing and measuringviii. Use the idea of proportionality to use mechanism such as rotate once and move twice (wheels,
gears)
ix. Use various tools for conjecturing and justifying
Promoting creative and global citizenship for sustainable development of community using mathematics
i.
ii.
iii.
iv.
v.

Utilise notebooks, journal books and appropriate ICT tools to record and find good ideas and sharewith others
Prepare and present ideas using posters and projectors to promote good practices in the community
Listen to other’s ideas and ask questions for better designs
Utilise information, properties, models and visible representations as the basis for reasoning
Utilise practical arts, home economics and outdoor studies to investigate local issues for improving
welfare of life

It is related to Numbers and Operations, Pattern and Data Representations and Quantity and Measurement all under Key
Stage 1.
46

226 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

APPENDIX E
The four strands of content learning standards and the strand on Mathematical Process-Humanity of Key
Stage ThreeKEY STAGE 3
Key Stage 3 (KS3) can be developed based on Key Stage 2. This stage focus on numbers and algebra,
relations and functions, space and geometry, and statistics and probability. Symbolic representations allow thedealing of abstract ideas and concepts that enhance critical and creative thinking through the application ofknowledge. Understanding and using mathematical concepts and principles in this stage through discussions,dialogue, and arguments enable learners to participate in contemporary societal, economic, technological,
political, environmental and mathematical issues. This stage is the basis for the creation of a better future withpredictions. It bridges further mathematics learning in various job demands.
Strand: Numbers and Algebra
Numbers are extended to positive and negative numbers and square roots. Algebraic expressions are alreadyintroduced by the mathematical sentences and symbols at Key Stage 2. At Key Stage 3, algebra is operated
by expressions and equations until the second degree. On the extension from numbers to symbolic algebra,
various possible ways of calculations are explored until their appropriateness is established. Like and unlike
terms are introduced in an algebraic sentence and in simplifying expressions. Properties of equations are
introduced for finding simple equivalent and solving equations with fluency. Substitution, addition and subtractionof equations enable further operations of simultaneous equations. Expansion and factorisation enable further
operations of the polynomials. Finally, quadratic equations can be solved using various operations.
Topics:Extending Numbers to Positive and Negative Numbers
Utilising Letters for Algebraic Expressions and Equations
Algebraic Expressions, Monomials and Polynomials and Simultaneous Equations
Expansion and Factorisation of Polynomials
Extending Numbers with Square Roots
Solving Quadratic Equations
Extending Numbers to Positive and Negative Numbers
Extending numbers to positive and negative numbers and integrate four operations into addition andmultiplication
i.
ii.
iii.
iv.

Understand the necessity and significance of extending numbers to positive and negative numbers
in relation to directed numbers with quantity
Compare numbers which is greater or less than on the extended number line and use absolute value
for distance from zero
Extend operations to positive and negative numbers and explain the reason
Get efficiency on calculation in relation to algebraic sum

Utilising Letters for Algebraic Expressions and Equations
Extending the utilisation of letters for general representation of situations and find ways to simplify algebraic
expressionsi.

Appreciate the utilisation of letter for general representation of situations to see the expression asprocess and value

Appendix E – Learning Standards for Key Stage 3 | 227

ii.
iii.

Find ways to simplify expressions using distributive law and figural explanations, establish thecalculation with like and unlike letter
Acquire fluency of simplifying an expression and appreciate it for representing the pattern of situation

Thinking about set of numbers in algebraic expression with letters as variables and represent them withequality and inequality
i.
ii.
iii.
iv.

Recognise numbers as positive and negative numbers, and explain integers as a part of numbers
Represent a set of numbers using variables with equality and inequality
Translate given sets of numbers on the number lines using interval and inequality notations
Appreciate redefining of even and odd numbers using letters to represent different sets of variables

Thinking about how to solve simple linear equation
i.
ii.
iii.
iv.

Review the answers of equations from the set of numbers and thinking backwards
Know the properties of equations which keep the set of answers of equation
Appreciate the efficient use of the properties of equations to solve linear equation
Use equations based on life situations to develop fluency, to solve equation, and interpret the solution

Algebraic Expressions, Monomials and Polynomials and Simultaneous Equations
Thinking about the calculations of monomial and polynomials, simple case1
i.
ii.
iii.

Introduce terms, monomials and polynomials
Introduce a number raised to the power of two as square, and a number raised to the third power as
cube
Get fluency for the calculation of polynomials such as combining like terms and the use of the four
operations in simple cases

Thinking about how to solve simultaneous equations in the case of linear equations
i.
ii.
iii.
iv.

Understand the meaning of solution of linear equations and simultaneous linear equations as a pairof numbers
Know the substitution and elimination methods of solving simultaneous linear equations
Get fluency for selecting the methods from the form of the simultaneous linear equations
Appreciate simultaneous linear equations in situations

Expansion and Factorisation of Polynomials
Acquisition to see the polynomials in the second degree with expansion and factorisation and use it
i.
ii.
iii.
iv.

Use the distributive law to explain the formulae for expansion and explain them on diagrams
Acquire proficiency for selecting and using the appropriate formulae
Use the expansion formulae to factorise the second degree expression and recognise both formulaewith inverse operation
Solve simple second-degree equation using the factorisation and apply it in life situations

Extending Numbers with Square Roots
Extending numbers with square roots and calculate the square roots algebraically
i.
ii.

1

Define square root and discuss ways to estimate the nearest value of a square root by Sandwich
Theorem
Understand that some square roots cannot be represented as fractions

Simple case may vary from and depending on the countries based on the mapping of curriculum.

228 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

iii.
iv.
v.

Compare square roots using number line and understand that the order does not change but thedifferences between two consecutive square roots varied
Think about multiplication and addition of square root and understand the algebraic way of calculation
which is similar to polynomial
Appreciate square roots in applying to situations in life2

Solving Quadratic Equations
Solving simple second-degree equation using the factorisation and apply on the situation
i.
ii.
iii.

Find the answers of simple second-degree quadratic equations by substitution and explore bycompleting the square, quadratic formula, and factorisation3
Get fluency to select the appropriate ways for solving quadratic equations
Apply quadratic equations in life situations
Strand: Relations and Functions

Relationships are represented by equations and system of equations. Functional relations are treated amongst
situation, table, and equation of function are introduced based on patterns and relations with algebraic
representation on Key Stage 2 and Key Stage 3. Solution of simple equation is done by equivalence deduction
based on algebra learnt earlier. Two variables in simultaneous equations as simple system of equations are
solved by substitution and additive-subtractive methods. Three representations, table, equation and graph, areused as methods to analyse the properties of every function. Proportion and inverse proportions are redefinedwith those representations mentioned. Proportional function is extended to line functions. The comparisons of
inverse proportion and line functions are made clear by the property of linearity with ‘constant ratio of change’.
The concept of proportion is extended to function of y = ax2. Ways of translations between table and equation,
equation and graph, graph and table are specific skills for every function with fluency.
Topics:Extending Proportion and Inverse Proportion to Functions with Variables
Exploring Linear Function in Relation to Proportions
Exploring Simple Quadratic Function
Generalising Functions
Extending Proportion and Inverse Proportion to Functions with Variables
Extending proportion and inverse proportion to functions with variables on positive and negative numbers
i.
ii.
iii.
iv.
v.
vi.

Extend proportions to positive and negative numbers, using tables and equations on situations
Plot set of points as graph for proportions defined in ordered pairs (x, y) in the coordinate plane using
appropriate scales precisely4
Introduce inverse proportion using tables, equations and graphs
Introduce function as correspondences of two variables in situations
Explore the property of proportional function with comparison of inverse proportional function
Appreciate proportion and inverse proportion functions in life

Pythagorean Theorem is discussed in Space and Geometry strand.
The graph of quadratic equation will be treated in Relations and Functions strand.
4
Utilising ICT is recommended in mathematical activities.
2
3

Appendix E – Learning Standards for Key Stage 3 | 229

Exploring Linear Function in Relation to Proportions
Exploring linear function in relation to proportion and inverse proportions
i.
ii.
iii.
iv.
v.
vi.

Identify linear functions based on situations represented by tables and compare it with proportional
functions
Explore properties of linear function represented by tables, equations and graphs and compare itwith direct and inverse proportional functions
Acquire fluency to translate the rate of change of a linear function represented in table, as coefficientin an expression and gradient in a graph5
Acquire fluency to translate y values of x = 0 in a table, constant in an expression, and y intercept ingraph
Apply the graphs of linear functions to solve simultaneous equations
Apply the linear function for data representation on situations to determine best fit line

Exploring Simple Quadratic Function
Exploring quadratic function y = ax2 in relation to linear function
i.
ii.
iii.

Identify the quadratic function on situations using tables and comparing it with linear function
Explore properties of quadratic function using tables, equations as well as graphs and comparing itwith linear function
Apply the quadratic function on situations in daily life and appreciate it

Generalising Functions
Generalising functions with various representations6 on situations
i.
ii.
iii.
iv.

Distinguish domain, range and intervals and is appropriately used for explaining function
Use various situation for generalising ideas of functions such as moving point A and moving point B
with time
Compose a graph as a function of two or more graphs with different domains in a situation
Introduce situations of step-functions7 with graph for generalisation the idea of function which cannot
be represented by equation
Strand: Space Geometry

Space and Geometry provide the ways of reasoning for exploring properties in geometry and produce theways of argument to explain justifications of visual reasoning. The calculations of angles are not just simple
calculation but also the ways of using the geometric propositions to justify answers through explaining why
it is correct based on basic properties. By explaining the relationship of figures using transformation, theproperties of congruency and describing similarity are identified and described. Finding the value of angles
and building arguments for proving are means for developing the habit of reasoning in the properties of plane
figures. The conditions of congruence and similarities, properties of circles, are also used to explain andprove appropriateness of geometric conjectures in relation to triangles, quadrilaterals, and circles. Dynamic
geometric software as well as simple compass and ruler are used for conjecturing. It shows general ideas fromconsistency in variations. Counter example is also found as special case from variations.

Family of linear functions are recommended to use ICT tool under mathematical activities strand.
Utilising functions as model in daily life which is necessary for STEM education.
7
Intervals are taught in the Key Stage 3 on Numbers and Algebra.
56230 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

TopicsExploring Angles, Construction and Designs in Geometry
Exploring Space with its Components
Exploring Ways of Argument for Proving and Its Application in Geometry
Exploring Angles, Construction and Designs in Geometry
Exploring angles to explain simple properties on the plane geometry and do the simple geometrical Construction8
i.
ii.
iii.

Explain how to determine the value of angles using the geometrical properties of parallel lines,
intersecting lines, and properties of figures
Use ruler and compass to construct a simple figure such as perpendicular lines and bisectors
Appreciate the process of reasoning that utilises the properties of angles and their congruency insimple geometrical constructions

Exploring the relationship of figures using congruency and enlargement for designs
i.
ii.
iii.

Explore the congruence of figures through reflection, rotation and translation and explain the
congruency using line of symmetry, point of symmetry and parallel lines
Explore similarity of figures with enlargement using points, ratio, and correspondences
Enjoy using transformations in creating designs

Exploring Space with its Components
Exploring space by using the properties of planes, lines and their combinations to form solids
i.
ii.
iii.

Explore the properties produced by planes, lines and their combinations, such as parallel lines
produced by intersection of parallel planes with another plane
Produce solids by combining planes such as nets and motion such as rotation, reflection andtranslation
Recognise the space of an object based on its properties and projection

Exploring Ways of Argument for Proving and its Application in Geometry9
Exploring properties of congruency and similarity on plane geometry
i.
ii.
iii.
iv.

Explore ways of arguments using the congruence of two triangles and appreciate the logic ofargument in simple proving
Explore ways of arguments using the similarity of two triangles based on ratio and angles andappreciate the logic of arguments in simple proving
Explore the proof of the properties of circles such as inscribed angles, and intercepted arcs
Appreciate proving through making the order of proven propositions to find new propositions

Exploring Pythagorean theorem in solving problems in plane geometry and spaces
i.
ii.
iii.
iv.

Explore the proving of Pythagorean theorem using diagram and use it in solving problems involving
plane figures
Apply Pythagorean theorem on prism by viewing the figures through faces.
Explore the situations for simple trigonometry using special angles in relation to the Pythagorean
theoremAppreciate the use of Pythagorean theorem in life10

Simple geometric construction is discussed by the ruler and compass with reasoning. Dynamic Geometric software usuallydraws entire circles. For knowing invariant dynamic geometric software is useful.
9
Dynamic Geometry software is useful to find the invariant properties which is discussed in Mathematical Processes andHumanity strand in Key Stage 3.
10
Pythagorean theorem is used for re-understanding the topic on square root under Key Stage 3 Numbers and Algebra.
8Appendix E – Learning Standards for Key Stage 3 | 231

Strand: Statistics and Probability
Data handling are extended to explore the dispersion of histogram with mean, median, mode and range.
Exploratory data analysis (EDA) attempting to represent and visualise the structure from the given datausing Information Communication and Technology (ICT) is enhanced. Histogram shows different dispersion
if we change the class. Probability is introduced as ratio with the law of large numbers. Sample space withassumption of equal probability becomes the point of discussion. Logical analysis to understand whole possible
cases such as a tree diagram is introduced for knowing the ways to represent logical reasoning. Histogram canbe seen relatively and produce frequency distribution polygon. Difference between sample and population is
discussed. Boxplots with quartile is an extension of median and range is used for comparisons of distributions.
Using skills of statistics and probability make problem solving in situations possible. Analysing and identifying
the trends in situations for making decisions are necessary such as issues for sustainable living.
Topics:Exploring Distribution with the Understanding of Variability
Exploring Probability with Law of Large Numbers and Sample Space
Exploring Statistics with Sampling
Exploring Distribution with the Understanding of Variability
Exploring distribution with histograms, central tendency and variability
i.
ii.
iii.
iv.
v.

Use histogram with different class intervals to show different distribution of the same set of data
Identify alternative ways to show distribution such as dot plots, box-plot and frequency polygon
Investigate central tendencies such as mean, median, mode11 and their relationships in a distribution
Investigate dispersion such as range and inter-quartile range in a distribution
Appreciate the analysis of variability through the finding of the hidden structure of distribution on
situations using the measure of central tendency and dispersion

Exploring Probability with Law of Large Numbers and Sample Space
Exploring probability with descriptive statistics, law of large numbers and sample space
i.
ii.
iii.
iv.
v.

Experiment with tossing coins and dice to explore the distribution of the relative frequency andunderstand the law of large numbers
Use the idea of equally likely outcomes to infer the value of probability12
Analyse sample space of situations represented by a table to determine the probability and use it forpredicting occurrenceUse various representations such as table, tree diagram, histogram and frequency polygon forfinding probability
Analyse data related to issues on sustainable development and use probability to infer and predict
future events

Mean, median and quartile are fixed depending on the data. However, mode changes depending on the class.
The probability here is called equiprobability when all possible cases are equally likely. In the upper grade level, probability will
be redefined based on distributions.
1112

232 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Exploring Statistics with Sampling
Exploring sampling with the understanding of randomness
i.
ii.
iii.
iv.

Discuss the hidden hypothesis behind sample and population
Use randomness to explain sampling
Analyse the data exploratory such as dividing the original into two for knowing better data
representations and discuss appropriateness such as regrouping
Appreciate data sampling in a situation with sustainable development
Strand: Mathematical Processes – Humanity

Critical argument in mathematics is enhanced through communication with others beyond Key Stage 2. This
proposed challenging activities will promote metacognitive thinking at different level of arguments to makesense of mathematics. Translating real life activities into mathematical models and solving problems using
appropriate strategies are emphasised in functional situations. The process of doing mathematical activities
involves patience that develops perseverance in learners and takes responsibility of one’s own learning. At thisstage, the habitual practice of self-learning will eventually develop confidence, thus, opportunity for challenges
to extend mathematics and the ability to plan sequence of future learning are also enhanced.
Standards:
Enjoying problem solving through various questioning and conjecturing for extension of operations into algebra,
space and geometry, relationship and functions, and statistics and probability
Enjoying measuring space using calculations with various formulae
Producing proof in geometry and algebra
Utilising tables, graphs and expressions in situations
Using diagrams for exploring possible and various cases logically
Exploring graphs of functions by rotation, by symmetry and by translation of proportional function
Understanding ways for extension of numbers
Designing sustainable life with mathematics
Utilising ICT tools as well as other technological tools
Promoting creative and global citizenship for sustainable development of society in mathematics
Enjoying problem solving through various questioning and conjecturing for extension of operations into algebra,
space and geometry, relationship and functions, and statistics and probability13
i.

Pose questions to extend numbers and operations into positive and negative numbers, algebraic
operations, and further extension into polynomial operations, and numbers with square roots
ii. Pose questions to solve linear equations, simultaneous equations and simple quadratic equations
iii. Pose assumptions in geometry as objects of argument and proof
iv. Pose questions to transform three dimensional objects into two dimensional shapes and vice versav.
Pose questions in relations and functions for knowing properties of different types of function
vi. Pose questions to exploratory of data handling for knowing structure of distribution
vii. Pose questions that apply PPDAC in relation to statistical problem solving
viii. Pose questions in relation to “equally likely” events
ix. Pose assumptions to discuss hypothesis based on sample and population
x. Pose conjecturing such as if x increases and y decreases then it is inverse proportion
Enjoying measuring space using calculations with various formulae14
i.
ii.
iii.
13
14

Extend the number line to positive and negative numbers and compare the size of numbers with theidea of absolute value
Derive the square root using unit squared paper through the idea of area
Explain the expansion of polynomials using area diagrams

Connected to the three strands, namely Numbers and Algebra, Relations and Functions, and Space and Geometry.
Connected to the three strands, namely Numbers and Algebra, Relations and Functions, and Space and Geometry.

Appendix E – Learning Standards for Key Stage 3 | 233

iv.
v.
vi.

Use the projection of space figures to plane figures using the Pythagorean theorem
Apply similarity and simple trigonometry for measurement
Use common factors to explain factorisation of the area of a rectangle based on the area of a square

Producing proof in geometry and algebra
i.
ii.
iii.
iv.

Have an assumption through exploration and produce propositions
Justify the proposition using examples and counter examples for understanding
Rewrite propositions from sentences to mathematical expressions by using letters and diagrams
Search the ways of proving by thinking backward from the conclusion and thinking forward from thegivenv.
Show the proof and critique for the shareable and reasonablenessvi. Deduce other propositions in the process of proving and after proving using what if and what if notvii. Adapt ways of proving to other similar propositions of proof
viii. Explain the written proof in geometry and algebra by the known
ix. Revise others’ explanation meaningfully
Utilising tables, graphs and expressions in situations15
i.
ii.

Explore the properties of functions by using tables, graphs and expressions and establish the fluency
of connections among them for interpreting functions in context
Analyse distribution of raw data by using tables, graphs and expressions in daily life

Using diagrams for exploring possible and various cases logically16
i.
ii.
iii.
iv.

Use number line with inequality to identify range and set
Use circle to identify relationship between the circumference and the central angle (acute, obtuse
and right)
Use rectangle and rotating a point on the side rectangle to draw the graph of the area
Use tree diagram for thinking about all possible cases sequentially

Exploring graph of functions by rotation, by symmetry and by translation of proportional function17
i.
ii.
iii.
iv.

Use the slope of a graph for the proportional function to rotate the graph or to determine the point ofintersection
Explore to know the nature of two simultaneous equations by using translation
Use the y-axis, x-axis and y=x as the line of symmetry to explore proportional function
Explain the graph of linear function by translation of proportional function.

Understanding ways for extension of numbers18
i.
ii.
iii.

Extend the numbers based on the necessity of solving equations such as x + 5 = 3 and , and show
examples for demonstrating the existence such as on the number lines, and understand it as set
Compare the size of number and identify how to explain the order of numbers and its equivalence
Extend operations for keeping the form19 beyond the limitations of meaning20

Connected to the strand on Relations and Functions.
Connected to the two strands on Relations and Functions and Space and Geometry.
17
Connected to the two strands on Numbers and Algebra and Relations and Functions.
18
Connected to the strand on Numbers and Algebra.
19
There are three meanings of form: (1) Permanence of form means “keep the pattern of operation” such as (-3)x(+2)=-6,(-3)x(+1)=-3, (-3)x0=0, and (-3)x(-1)=+3, and (-3)x(-2)=+6. Here, the product of the pattern increases by 3; (2) The form
means “Principle of the permanence of equivalence form” which means to keep the law of commutativity, associativity anddistributivity; and (3) The form means the axiom of field in Algebra. Normally, in education, we only treat (1) and( 2).
20
For the extension of numbers to positive and negative numbers, beyond the limitations of meaning such as subtract smaller
number from larger number. For the extension of numbers to irrational number, beyond the limitation of meaning such as
rational number is quotient (value of division).
15
16

234 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Designing models for sustainability using mathematics21
i.
ii.
iii.
iv.

Discuss and utilise probabilities in life such as weather forecasting for planning
Design cost saving lifestyle models through comparison of data such as cost of electricity, water
consumption, and survey
Plan emergency evacuation such as heavy raining and landslide where the calculations on theamount of water in barrel per minute exceeds the maximum standards
Forecast the future with mathematics

Utilising ICT tools as well as other technological tools
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.

Use dynamic geometry software for assumption, specialisation and generalisation
Use graphing tool for comparison of the graph and knowing properties of function
Use data to analyse statistics with software
Use internet data for the discussion of sustainable development
Use calculators for operations at necessary context
Use projector for sharing ideas such as project survey, reporting and presentation
Use the idea of function to control mechanism
Use ICT tools for conjecturing and justifying to produce the object of proving.

Promoting creative and global citizenship for sustainable development of society using mathematics
i.
ii.
iii.
iv.
v.
vi.

21

Utilise notebooks, journal books and appropriate ICT tools to wisely record and produce good ideas
for sharing with others
Prepare and present ideas using posters, projectors, pamphlets and social media to promote goodpractices in society
Promote the beautifulness, reasonableness and simplicity of mathematics through contextual
situations in the society
Listen to other’s ideas and ask questions for better designs, craftsmanship and innovations
Utilise information, properties, models and visible representations as the basis for making intelligent
decisionsUtilise practical arts, home economics, financial mathematics and outdoor studies to investigate
local issues for improving welfare of life

Connected to the three strands, Numbers and Algebra, Relations and Functions and Space and Geometry.