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

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

Editors:
ISODA Masami
GAN Teck HockTEH Kim HongWith the support of:
Maitree Inprasitha (KKU)
Supattra Pativisan (IPST)
Wahyudi & Sumardyono (SEAQIM)

A Collaboration Project:Published by
CRICED
University of Tsukuba
1 Chrome- 1 - 1 Tennodai, Tsukuba, Ibaraki, 305-8577, Japan
Cover design by Nur Syafiyah Binti Mohd Kassim

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

Copyright © 2023 by MASAMI ISODA
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.
CRICED
University of Tsukuba
1 Chrome- 1 - 1 Tennodai, Tsukuba, Ibaraki, 305-8577, Japan
Phone: +81-298537826
Email: criced@un.tsukuba.ac.jp
Publication of this book is supported by JSPS grant number 19H01662.
ISBN: 978-4-910114-40-8

Foreword |v

FOREWORD
Congratulations to CRICED, University of Tsukuba and SEAMEO RECSAM
for another collaboration project in the publication of three series ofguidebooks entitled Mathematics Challenges for Classroom Practices basedon the SEAMEO Basic Education Standards: Common Core Regional
Learning Standards (SEA-BES CCRLS) in Mathematics. Two series are forthe lower and upper primary level while the third is for the lower secondary
level. Generally, curriculum standards of subjects are not widely scrutinisedby classroom practitioners and teacher educators compared to curriculum
specialists. The publication of these three new guidebooks anchors well andconsolidates 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 SEABES CCRLS to the newly published book series will create 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
three aspects, namely highlighting the misconceptions, developing new ideas from the previously
learned knowledge and explanation of new mathematical concepts. The task-based approach will
surely help readers to enhance their mathematical understanding and ultimately provide bettersupport for classroom teaching and learning.
I sincerely hope that the Minister of Education of SEAMEO members would support and promote
this guidebook series among educators and teachers in their respective countries. This effort andspirit of cooperation among SEAMEO members and associate members can be realised to bring
benefits to classroom practices, which will eventually benefit children of our future.
My sincere appreciation and congratulations to CRICED, the University of Tsukuba as a project
proponent and provided financial support, SEAMEO RECSAM as the main collaborator, othercollaborating partner institutions and individual educators and specialists for their expertise,commitment and contributions in this endeavour.

Dr ETHEL AGNES PASCUA 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 appreciation
to the Centre for Research on International Cooperation in Educational Development(CRICED), the University of Tsukuba for inviting the Centre as the main collaborator in
the publication of the guidebook series, titled “Mathematics Challenges for Classroom
Practices” for the i) Lower Primary Level, ii) Upper Primary Level and iii) Lower
Secondary Level. Besides the involvement of SEAMEO RECSAM, many educatorsand specialists of other collaborating partners such as Khon Kaen University,
Thailand; the Institute for the Promotion of Teaching Science and 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 their writings
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 gratefulto the outstanding writing team who made this possible. I shall start by acknowledging the contribution ofProfessor Dr Masami Isoda, who initiated the idea and the project and 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 team and other mathematics specialists who
were also invited to contribute their writings. Despite facing time constraints and changes in staff members,
the writing team stayed intact with their contributions and commitment until this guidebook series is published.
Besides the writing team, I also like to thank the panel reviewers of RECSAM who provided their constructive
suggestions to improve the content of this guidebook series.
The guidebook series covers the mathematics content across Grades 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 latter to the newly published guidebook series will create awareness among classroom
teachers and teacher educators regarding the importance and relevance of curriculum standards in theplanning of teaching and learning. The presentation of this guidebook series emphasised highlighting
misconceptions, and contradictions, and developing new ideas from previously learned knowledge toenhance learning and develop mathematical thinking. Such an approach to contradiction will foster deeper
thinking among readers, thus enhancing mathematical understanding, and translating into better support forclassroom teaching and learning.
Without a doubt, much commitment and hard work had been invested to produce these guidebooks. I hopethat 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 their support
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, to the promotion and development of mathematics education in this region. This would
not have been possible without CRICED, the University of Tsukuba’s content expertise and 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
and teachers to develop 21st-century skills and OECD competency (2005)
in education. For SEA-BES-M, I collaborated with Pedro Montecillo Jr., Kim
Hong Teh, and the late Mohd Sazali bin Hj. Khalid of RECSAM, along withthe contribution from curriculum developers in SEAMEO countries, specialists in APEC economies
and several internationally leading researchers. Finally, SEA-BES-M was incorporated into the‘SEAMEO Basic Education Standards (SEA-BES): Common Core Regional Learning Standards(CCRLS) in Mathematics and Science’ which was published by RECSAM in 2017.
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 arecompartmentalised between every content description. Therefore, a major contribution of SEABES-M to the world, particularly in 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 recognised that there are difficult 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 in 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 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 made some minor revisions to the SEA-BES-M to align with the needs ofthe Era of the 4th Industrial Revolution to develop stakeholders and users of Artificial Intelligenceand Big Data in business as well as establish a successful life under the reality of humanity withtechnology. The minor revision was made based on the curriculum reform recommendation(2020) by the APEC InMside Project with the purpose of promoting mathematical capitalism undermathematical-statistical-informational sciences on the demands of the Era. It is hoped that this bookseries is used in teacher education to develop new curriculum content knowledge for teaching inthis Era. I would like to acknowledge the SEAMEO secretariat and centres for their collaborations,especially Shah Jahan Bin Assanarkutty, the Director of RECSAM, who made possible thispublication. 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) who
provided us with innovative ideas.

ISODA MASAMI
Director of CRICED, University of Tsukuba, Japan

Preface | ix

PREFACE
Realising reform in school curricula 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 acquire a
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 their classroom
practices for developing competency and professional development aligning with the trends of the
4th Industrial Revolution.
Mathematics Challenges for Classroom Practices at the Upper Primary Level consistsof mathematical tasks for the following strands:••••

Extension of Numbers and Operations
Measurement and Relations
Plane Figures and Space Figures
Data Handling and Graphs

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). In developing the tasks, the English edition of Japanesemathematics textbooks published by Gakko Tosho had been used as the main reference. These
textbooks provided major guides for learning mathematics through problem solving approach todevelop 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 misconceptionsof ideas, (b) developing ideas from previously learnt knowledge, and (c) using the 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 schoolmathematics were also incorporated into the tasks. It is hoped that these elements will set off a newbreath of mathematics learning in the SEAMEO community, shaping our students to be critical andcreative 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
the tasks can also be solved by students in classrooms as they are also aligned with the schoolmathematics curriculum. In addition, teachers and teacher educators are expected to solve
them without many difficulties. However, studying the tasks carefully will raise awareness of thedepth of SMK and PCK required to complete the tasks. This will trigger off a need to upskill theirunderstanding of mathematical ideas and function effectively to develop students’ mathematicalcompetency. Thus, solving tasks in this book will provide readers with opportunities to relearn themathematics content for teaching. Furthermore, it will also help them to identify (a) the objectivesof 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 embedded in a task, it is recommended that readers
should solve 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 perspectiveof 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 in various contexts. Firstly, as all thetasks were designed based on the 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 creating
similar mathematical tasks that incorporate mathematical thinking. Secondly, when the book is usedin 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 order to preparea unit of a lesson plan based on the standard chosen. This may help to improve the effectivenessof lesson planning and anticipate the responses of students to the tasks. Thirdly, in the context of
pre-service teacher education, the tasks in this book can serve as a means to acquire MKT which
may be required for any teacher employment examination or entrance examination for an education
graduate programme. Fourthly, in the context of mathematics education research, this book can beused 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
do not exist in their current curriculum and textbooks.
Mathematics Challenges for Classroom Practices at the Upper 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’potential in facing the challenging and demanding era ahead.

ISODA Masami
GAN Teck HockTEH Kim Hong

Acknowledgement | xi

ACKNOWLEDGEMENTS
The authors/editors would like to thank and express our gratitude to the following contributors,
institutions and organisations for making this project and publication possible:
Dr Ethel Agnes Pascua-Valenzuela, Director of SEAMEO Secretariat, Bangkok, Thailand forproviding CRICED with 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 in 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,
and 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 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 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
Director of SEAMOLEC, 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 thetasks and submitted the writings for considerations;
Pedro Lucis Montecillo Jr, mathematics specialist of RECSAM (until May 2018) for assisting in thecoordination at the early stage of the project;
All staff of SEAMEO RECSAM supported the production of these books by providing ideas andcoordinating and managing 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 and 19K21743.

Publication of this book is supported by JSPS grant number 19H01662.

Table of Contents | xiii

TABLE OF CONTENTS
Forewordv

Prefaceix

Acknowledgement

xi

Table of contents

xiii

Authors, Contributing Members & Reviewers

xv

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

1

Chapter 2: Extension of Numbers and Operations•
Extending Numbers with Base Ten up to Billion and also to Thousandths with aThree-Digit Numeral System Gradually•
Making decisions of Operations on Situations with Several Steps and Integratingthem into One Expression and Thinking About the Order of Calculations andProducing 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 Finding
the Appropriate Place Value such 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

3
3

17
21
24

Chapter 3: Measurement and Relations•
Introducing Angle and Measuring it•
Exploring and Utilising Constant Relation•
Extending Measurement of Area in Relation to the Perimeter•
Extending Measurement of Volume in Relation to Surface•
Approximating Quantities•
Extending Proportional Reasoning to Ratio and Proportion•
Producing New Quantities Using Measurement Per Unit•
Investigating the Area of a Circle•
Exchanging Local Currency with Currency in the ASEAN Community•
Extending the Relation of Time and Use of Calendars•
Converting Quantities in Various Systems of Units•
Showing Relationships Using a Venn Diagram

80
80
86
92
105
114
118
140
149
159
165
170
178

Chapter 4: Plane Figures and Space Figures•
Exploring Figures with their Components in the Plane•
Exploring Space Figures with their Components in Relation to the Plane•
Exploring Figures with Congruence, Symmetry and Enlargement

185

Chapter 5: Data Handling and Graphs•
Arranging Tables for Data Representations•
Drawing and Reading Graphs for Analysing Data•
Using Graphs in the PPDAC Cycle (Employing the Problem – Plan – Data Analysis– Conclusion)•
Applying Data Handling for Sustainable Living

239
239

11

35
50
64
71

185
205
217

241
257
261

xiv | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Bibliography

265

Appendix A – Framework for CCRLS in Mathematics

267

Appendix B – Terminologies for Mathematical Thinking and Processes

273

Appendix C – Mathematical Values

287

Appendix D – Strand on Mathematical Processes-Humanity (Key Stage 2)

290

Appendix E – Learning Standards for Key Stage 1

293

Appendix F – Learning Standards for Key Stage 3

305

Authors and Contributing Members & Reviewers | xv

AUTHORS, CONTRIBUTING MEMBERS & REVIEWERS
Lead Authors:
Masami Isoda (CRICED, Japan)
Gan Teck Hock (ASMEP, Malaysia)
Teh Kim Hong (ASMEP, 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)
ASMEP (Association for Science and Mathematics Education Penang)xvi | Mathematics Challenges for Classroom Practices at the Upper 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, Sngapore)
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 community through mutual
understanding in the era of artificial intelligence and data science. 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. A comprehensive
illustration of the framework is attached in Appendix A. The revised framework in Figure 1 shows
the interconnection of the three components.
Mathematical Values, Attitudes and Habits for Human Character

Mathema�cal A�tude

Mathema�cal ValuesA�emp�ng to  See and think mathema�cally Pose ques�ons and develop
explana�ons
 Generalise and extend
 Appreciate others’ ideas andchange representa�ons formeaningful elabora�ons

Seeking  Generality andexpandability
 Reasonableness andharmony Usefulness andeﬃciency
 Simpler and easier Beau�fulness

Mathema�cal Habits of Mind

For living  Reasonably and cri�cally while
respec�ng and apprecia�ng others
 Autonomously and socially Crea�vely, innova�vely andharmoniously to develop ci�zenship Judiciously in using various tools
 With empowerment in predic�ng
the future through lifelong learning

Mathematical Thinking and Processes

Mathema�cal Ideas of:Set, Unit, Comparison,
Opera�on, Algorithm,
Fundamental Principles,
Permanence of Form,
Various Representa�ons
and Transla�on,Pa�ern, Recursion andInvariant.

Mathema�cal Ways of Thinking:









Generalisa�on and Specialisa�on
Extension and Integra�on
Induc�ve, Analogical and Deduc�ve
Reasoning
Abstrac�ng, Concre�sing and
Embodiment
Objec�fying by representa�on andsymbolizing
Rela�onal and func�onal thinking
Thinking Forward and Backward

Mathema�cal Ac�vi�es:

Problem Solving
Explora�on and Enquiry
Mathema�cal Modeling,
Mathema�sa�on and
Programming Conjecturing, Jus�fying andProving Conceptualisa�on andProceduralisa�on Representa�on and SharingContents
Key Stage 1
 Numbers & Opera�ons
 Quan�ty & Measurement
 Shapes, Figures & Solids
 Pa�ern & Data
Representa�ons

Key Stage 2
 Extension of Numbers &Opera�ons
 Measurement & Rela�ons
 Plane Figures & Space Figures
 Data Handling & Graphs

Key Stage 3
 Numbers & Algebra Rela�ons & Func�ons
 Space & Geometry
 Sta�s�cs & Probability

Figure 1. Revised CCRLS Framework in Mathematics
This book is written to guide readers in acquiring a better understanding of this framework,
particularly on mathematical thinking and processes which are embedded in all the tasks. A detailed
explanation of mathematical ideas, mathematical ways of thinking and mathematical activities canbe found in Appendix B. Mathematical ideas and mathematical values are elaborated in Appendix

2| Mathematics Challenges for Classroom Practices at the Upper Primary Level

C. The standards for the strand on Mathematical Processes-Humanity are attached in Appendix D
to provide readers with challenging activities to promote metacognitive thinking at different levels ofarguments to make sense of mathematics. In order to understand the development and progressionof learning from the primary level to the secondary level (Key Stage 3), the learning standards of
Key Stage 1 and Key Stage 3 can be referred to in Appendix E and Appendix F, 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, the 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
for learners to recognise their cognitive skills derived from 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 the human character are
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 mathematical thinking and processes, which are
the key learning objectives. Completing the tasks will 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 anddiscussing with others should always be done habitually in order to gain a deeper understanding ofmathematical processes. This may enable the readers to discover any hidden mathematical ideas
and structures in the tasks with appreciation. The tasks are specifically designed to cater for thispurpose. Context 1 is real world problem solving and context 2 is task sequence in mathematics.
In a nutshell, an important target of solving the tasks in this book is to enable you to acquire abetter insight into the learning standards. This insight will help you understand and appreciate yournational mathematics curriculum from the perspective of SEABES CCRLS. Furthermore, since thelearning standards are developed based on the framework which emphasised the components ofcontents, thinking and processes, and mathematics values, ultimately, you will be able to acquire
mathematical teaching knowledge with appreciation.

Chapter 2: Extension of Numbers and Operations |3

CHAPTER 2
Extension of Numbers and Operations
Topic 1: Extending Numbers with Base Ten up to Billion and also to Thousandths withThree-Digit Numeral System GraduallyStandard 1.1:
Extending numbers using base-ten system up to billion with three-digit numeral systemi.
Adopt the three-digit numeral system, extend numbers up to billion with the idea of relative
size of numbers
ii. Compare numbers such as larger, smaller with base-ten system of place values throughvisualisation of relative size of numbers using cube, plane (flat), bar (long) and unit
Sample Tasks for Understanding the Standards
Task 1: How Large is 1 Billion?

Diagram 1i.

Diagram 1 shows four base-ten blocks, a cube, a plane, a bar and a unit, normally used
to represent 1000s, 100s, 10s and 1s, respectively.
•

How many sets of 1s are there in 1 thousand?

•How many sets of 10s are there in 1 thousand?

•How many sets of 100s are there in 1 thousand?

4| Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 2ii. These base-10 blocks can be used to represent different values as shown by the two
examples in Diagram 2.
•

In the first exampes, if the unit represents 1000, the bar will represent 10 000.
 State the values represented by the plane and the cube.
 How many thousands are there in a million?

•

In the second example, if the cube represents 1 billion, then the plane will represent
100 million.
 State the values represeted by the bar and the unit.
 How many times is 1 billion compared to 1 million?
 How many thousands are there in 1 billion?

iii. According to the latest United Nations Population Division estimates, China and India are
the only two countries in the world with a population of more than 1 billion. Table 1 shows
the population of these two countries as of 1 July 2020.
Table 1
Population of China and India as of 2020

Retrieved from WorldOmeter website
https://www.worldometers.info/world-population/population-by-country/•

Read and write the population of China and India in words, respectively.

Chapter 2: Extension of Numbers and Operations |iv. Diagram 3 shows the three-digit numeration system extended to billion.

Diagram 3•v.

How can this chart be used to help your students read the population of China andIndia, respectively?

As of 1 July 2020, the world population is estimated to be 7 794 798 739.
•

What is the value of the underlined digit 9 in each of the following cases?

		(a) 7 794 798 739
		(b) 7 794 798 739•

What is the value of the underlined digit 7 in each of the following cases?

		(a) 7 794 798 739
		(b) 7 794 798 739
vi. Consider the number 8 820 406 200.
•

Read and write the number in words.

•

How many times is the 8 on the left compared to the 8 on the right?

•

How many times is the 2 on the left compared to the 2 on the right?

5

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

Diagram 4vii. Diagram 4 shows two different number lines.
•

Find the number indicated by each ↑ on the number lines.

Diagram 5
viii. Diagram 5 shows 12 number cards. Use all the 12 cards to make various numbers.
•

What number is the largest?

•

What number is the smallest?

ix. About how many years is 1 billion seconds? (Take 1 year as 365 days.)Chapter 2: Extension of Numbers and Operations |7

Standards 1.2:
Extending decimal numbers to hundredths, and to thousandthsi.
Use the idea of quantity and fractions, extend decimal numbers from tenths to hundredths
ii. Compare decimal numbers such as larger, smaller with base ten system of place valueiii. Adopt the ways of extension up to thousandths and so on, and compare the relative sizes
Sample Tasks for Understanding the Standards
Task 1: Reading Decimal Numbers

Diagram 1i.

A ribbon was measured using a metre rule and the length was recorded as 0.342 m. Two
students, John and Jason, read the length in different ways as shown in Diagram 1.
•

Explain the two different ways of reading decimal numbers.

•

Which way will you encourage your students to learn? Justify your choice.

ii. Write and read the following decimal numbers in words.
•

2.6

•

2.06

•

2.46

•

0.308

•

0.038

| Mathematics Challenges for Classroom Practices at the Upper Primary Level

8

Task 2: Visualising Decimal Numbers① Decimal Number System

5.267 is read as “five and two hundred sixty-seven thousandths.

5267 is read as “five thousand two hundred and sixty-seven”.

Diagram 2i.

Diagram 2 shows the decimal number and the whole number systems. Compare the twosystems.
•ii.

Explain the similarities and differences between the two systems.

Fill in the blanks in the following statements.
•

734 is made up from ____ sets of hundreds, ____ sets of tens and ____ sets of ones.

•

7.34 is made up from ____ sets of ones, ____ sets of tenths and ____ sets ofhundredths.

•

0.734 is made up from ____ sets of tenths, ____ sets of hundredths and ____ sets ofthousandths.

Chapter 2: Extension of Numbers and Operations |Diagram 3

Diagram 4iii.

Diagram 3 shows a place-value chart for whole numbers whereas Diagram 4 shows theextension of the chart to include decimal numbers.
•

What rules are there for the whole number place-value system?

•

Do the same rules still apply for the extended decimal number place-value system?

•

Delete the wrong word in the ().

“For both the whole and decimal numbers, a digit is shifted to the next (higher/lower) placewhen there are 10 in a place, and a number is shifted to the next (higher/lower) place when it isdivided into 10.”iv.

How are decimal numbers related to fractions? Illustrate your answers with appropriate
examples.

9

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

Task 3: Comparing Decimal Numbers

Diagram 5i.

Diagram 5 shows a pair of decimal numbers represented by base-ten blocks.
•

Compare the pair of decimal numbers.

•

Write the result of comparison using the symbols > or < .

Diagram 6ii.

Diagram 6 shows the comparison between a pair of decimal numbers, 7.82 and 7.634,
using a place-value chart.
•

Write the result of comparison using the symbols > or < .

Diagram 7iii.

Diagram 7 shows part of a number line.
•iv.

Compare the numbers 1.4 and 1.27 using the number line.

How would you sequence the above three comparison tasks in order to develop yourstudents’ understanding on comparison of decimal numbers by themselves.

Chapter 2: Extension of Numbers and Operations | 11

Topic 2: Making Decisions of Operations on Situations with Several Steps and Integrate Themin One Expression and Think about the Order of Calculations and Produce the Rule (PEMDAS)Standard 2.1:
Finding easier ways of calculations using the idea of various rules of calculations such as theassociative, commutative and distributive rules
i. Find the easier ways of addition and subtraction and use it, if necessary, such as answer is thesame if add the same number to the subtrahend and minuend
ii. Find the easier ways of multiplication and division and use them in convenient ways such as 10times of multiplicand produce the product 10 timesiii. Use associative, commutative and distributive rules of addition and multiplication for easierways of calculation, however, commutative property does not work in subtraction and division
iv. Appreciate the use of simplifying rules of calculations
Sample Tasks for Understanding the Standards
Task 1: Addition and Subtraction Made Easier

Diagram 1i.

ii.

Diagram 1 shows two rules used to make addition and subtraction easier.
•Explain each rule.

•Why does each rule work?

Use the rules to make the following calculations easier.
•126 + 78•

214 – 59•

$27.50 + $6.85•

125 m – 34 m

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

Task 2: Multiplication Made Easier

Diagram 2i.

Diagram 2 shows an algorithm and the vertical form multiplication for 28 x 53.
•

What multiplication rule is used in the algorithm? Explain your answer.

•

Explain how the algorithm can be used to explain the vertical form of multiplication.

Diagram 3
ii.

iii.

Diagram 3 shows two multiplications made easier using a rule.
•

Complete the sentences by filling in the missing number in each•

What rule is used to make the multiplications easier?

•Why does the rule work?

Use the rule to make the following multiplications easier.
•628 x 5•244 x 125.

Chapter 2: Extension of Numbers and Operations | 13

Task 3: Mixed Operations Made Easieri.

John and Jason used different methods
to calculate 95 + 68 + 32 as shown in
Diagram 4.
Compare the two methods.
•

Which method is easier for yourstudents? Why?

•

What rule of addition was used tomake the calculation easier?

Diagram 4ii.

Diagram 5 shows John’s method to
calculate 25 x 63 x 4. He used two rulesof multiplication to make the calculationeasier.
•What are the two rules?

•

Explain how the rules were used.

Diagram 5iii. When trying to calculate 48 – 34 + 12,
John and Jason interpreted the PEMDAS
rule differently as shown in Diagram 6.
•

Explain the two interpretations.

•

Which of the interpretations iscorrect? Why?

iv. Explain why each of the following
sentences is true.

Diagram 6•

60 – 20 – 15 = 60 – 15 – 20•

60 – 20 – 15 = 60 – (20 + 15)

14 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelv.

John and Jason used the distributive rule
to simplify 23 x 48 as shown in Diagram 7.
•

Compare the two methods.


Did John and Jason get a sameanswer?



Why or why not?

184•

Which method will you recommend to
your students? Justify your choice.

•Do you think it is worth for students
to see more than one method of
doing computational problem? Whyor why not?

Diagram 7

vi. John and Jason used different methods tosimplify 276 ÷ 12 as shown in Diagram 8.
•

Diagram 8

Compare the two methods.
What rule was used to simplify
the division in each method?

Which method will you
recommend to your students?
Justify your choice.

Chapter 2: Extension of Numbers and Operations | 15

Standards 2.2 :Thinking about the order of calculations in situations and produce rules and order of operationsi.
Integrate several steps of calculation into one mathematical sentenceii. Produce the rule of PEMDAS and apply it to the multi-step situationiii. Think about the easier order of calculation and acquiring fluency of PEMDAS and rules withappreciation
Sample Tasks for Understanding the Standards
Task 1:

Mixed Operation in Daily Situationsi.

A multi-level parking lot has 10 levels and there
is a total of 1 799 parking lots. There are 152
parking lots on the first level. The other parking
lots are distributed equally on
the other 9 levels.

Diagram 1 shows a daily-life problem.
•ii.

How many parking lots are
there in the last two levels?
Diagram 1

Write the calculation in one
mathematical sentence.

Build a daily-life story problem based oneach of the following mixed-operation
sentences.
•

12 x (6 – 3)•

12 x 6 – 3•

12 – 6 ÷ 3•(12 + 6) ÷ 3

iii. Diagram 2 shows two mathematicalsentences involving 7 + 2 x 5.
•

Draw a diagram to show the difference
between (7 + 2) x 5 and 7 + (2 x 5).

•

Determine whether each of thesentences is true or false.

Diagram 2

iv. Calculate 8 – 2 + 3. Is your answer 9 or 3? Why?
v.

Calculate 12 ÷ 3 x 2. Is your answer 8 or 2? Why?

vi. Calculate each of the following mixed operations and explain each case using a real-lifesituation.
•

12 ÷ (3 x 2)•

12 ÷ 2 x 3•

12 ÷ 3 x 2

16 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelvii. The acronym PEMDAS is sometimes used by teachers to help students remember the rule
involving order of calculation.
•

Explain the rule.

•

Why is the rule important?

viii. Calculate•

7 + 3 x 42•

7 + (3 x 4)2

ix. Build a daily-life story problem for each of the following mixed operations.
•

12 x (6 – 3)•

12 x 6 – 3•

12 + 6 ÷ 3•(12 + 6) ÷ 3

Chapter 2: Extension of Numbers and Operations | 17

Topic 3: Producing the Standard Algorithm Using Vertical Form Division with Whole
Numbers
Standard 3.1:
Knowing the properties of division and use it for an easier way of calculationi.
Find the easier ways of division and use it, if necessary, such as answer is the same if
multiplying the same number by the dividend and divisor
ii. For confirmation of the answer of division, use the relationship among divisor, quotient andremainder and appreciate the relationshipSample Tasks for Understanding the Standards
Task 1: Division Made Easier

Diagram 1i.

Diagram 1 shows a division made easier using a rule.
•

Complete the division sentence by filling in the missing number in each•

What rule is used to make the divisions easier?

•Why does the rule work?

Diagram 2
ii.

Diagram 2 shows another two cases of division made easier using some rules.
•

Fill in the missing number in each•

For each case, what rule is used to make the divisions easier?

•Why does the rule work?

.

.

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

Diagram 3iii.

A student used an easy method to solve a division problem as shown in Diagram 3. In
this method, the student simply “cancelled off” the zero in both the dividend and divisor.
•

What division rules can be used to justify this method? Explain your answer.

•

Discuss the significance of writing down the remainder of zero in the vertical form
division algorithm.

•

How will this help your students learn other mathematical ideas such as the Remainder
Theorem in higher grades?

Diagram 4iv.

v.

Two students, John and Jason simplify the division 396 ÷ 99 as shown in Diagram 4.
•

Compare and justify the two methods.

•

Which method will you recommend to your students? Justify your decision.

Use the division rules to make the following divisions easier.
•

220 ÷ 20•1300 ÷ 25•

255 ÷ 15•348 ÷ 50•

13800 ÷ 2300

vi. How can the division rules be used to make the following division involving fractions easier?

Chapter 2: Extension of Numbers and Operations | 19

Standards 3.2 :Knowing the Algorithm of Division in Vertical Form and Acquiring Fluencyi.
Know the division algorithm with tentative quotient and confirm the algorithm by the relationshipamong divisor, quotient and remainderii. Interpret meaning of quotient and remainder in situations
iii. Acquire fluency for division algorithm in the case of up to 3-digit whole number divided by 2-digit
iv. Think about the situations with or without remainder in relation to situations for quotative andpartitive division
Sample Tasks for Understanding the Standards
Task 1:Division Algorithm in Vertical Form

Diagram 1i.

ii.

Diagram 1 shows the definition of the division algorithm for whole numbers.
•

Why is the phrase “andincluded in the definition?

•

In this definition, a, b, q and r are called the dividend, divisor, quotient, andremainder, respectively.
Explain the relationship among dividend, divisor, quotient, and remainder.

Explain how the relationship can be used to check the answer for 38 ÷ 4.

Two conceptual models that are commonly used by teachers to develop division algorithm
are the repeated-subtraction model, and the missing-factor model.
•

Using appropriate examples, explain the meaning of division based on each of thesemodels.

Diagram 2

20 | Mathematics Challenges for Classroom Practices at the Upper Primary Leveliii.

Diagram 2 shows the steps in calculating 278 ÷ 13 using a division algorithm in verticalform. In each of the steps, a teacher asked the following questions to guide students:“What number when multiplied by 13 will give you 27 or less?”“What number when multiplied by 13 will give you 18 or less?”•

In using this vertical form of division algorithm, what model of division did the teacherbase on? Explain your answer.

Diagram 3iv.

v.

Two students, Jason and John used another algorithm to calculate 517 ÷ 16.
Diagram 3 shows their calculation steps and explanations, respectively.
•

In this algorithm, what model of division did Jason and John base on?

•

Which student’s thinking is more efficient? Why?

In using any division algorithm, how do you know when you are done with a long divisioncalculation?

vi. What values in the SEA-BES CCRLS Framework in Mathematics (refer page 1) could be
nurtured through this task? How would you do it?

Chapter 2: Extension of Numbers and Operations | 21

Topic 4: Extending the Vertical Form Addition and Subtraction with Decimals to Hundredths
Standard 4.1:
Extending the vertical form addition and subtraction in decimals to hundredthsi.
Extend the vertical form addition and subtraction to hundredths place and explain it with models
ii. Appreciate the use of addition and subtraction of decimals in their life
Sample Tasks for Understanding the Standards
Task 1: Relationships Among 1, 0.1 and 0.01

Diagram 1i.

A teacher used the base-10 blocks to represent numbers 1, 0.1 and 0.01 as shown in
Diagram 1. The teacher then drew the picture in Diagram 1 to show the relationships
among the three decimal numbers.
•

Fill in the missing number in each .

•

What are the relationships among 1, 0.1 and 0.01?

Diagram 2
ii.

Another teacher, used a metre rule to represent decimal numbers up to hundredths asshown in Diagram 2.
•

What number is represented by location ⓐ and ⓑ, respectively?

•

Express the following measurements in metres.


329 cm

1 m 30 cm



5 m 23 cm



16 m 5 cm

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

iii.

iv.

Compare the block model and the metre-rule model.
•

Which model do you think will be easier for your students to see the relationships
among 1, 0.1 and 0.01?

•

Justify your answer.

How will you use the two models to shows the procedure in doing the following calculations?
•

0.4 + 0.35•

0.36 + 0.8•

0.78 – 0.2•

1.62 – 0.5

Diagram 3v.

Diagram 3 shows two tasks to help students understand the decimal place-value system.
•

How will you sequence the order of the two tasks?

•

Justify your choice.

Chapter 2: Extension of Numbers and Operations | 23

vi. Diagram 4 shows the work of John involving
addition of decimal numbers in vertical form.
•

What is the error of John’s calculation?

•

Suggest a possible cause for the error.

vii. When asked to correct John’s work,
Jason explained his thinking as shown in
Diagram 5.
•

Why must the decimal points belined up when adding two decimal
numbers?

Diagram 5
viii. Diagram 6 shows an extended placevalue chart involving decimal numbers up
to hundredths.
•

What are the strengths and weaknessof using this chart to help students
add and subtract decimal numbers up
to hundredths?

•

How will you use this chart to help
your students to solve the following
problem?

Diagram 6

Nadia did a long jump with a distance of
3.48 m and her friend Maria only cleared
2.5 m. How long did Nadia jump more thanMaria in metres?

Diagram 7ix. Many students have difficulties comparing measurements in different units. Diagram 7
shows two students’ errors in comparing lengths.
•

What are the possible reasons for each of them to think so?

•

How will you help each of them to overcome the difficulty?

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

Topic 5: Extending the Vertical Form Multiplication and Division with Decimals and Find theAppropriate Place Value such as Product, Quotient and RemainderStandards 5.1:
Extending the multiplication from the whole number to decimal numbers
i. 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 themwith appreciation in situations
ii. Extend the vertical forms multiplication of decimals up to 3 digits by 2 digits with considerationof the decimal places step by stepiii. Obtain fluency using multiplication of decimals with sensible use of calculators in learners’ life
iv. Develop number sense in multiplication of decimals such as comparing sizes of products
before multiplying
Sample Tasks for Understanding the Standards
Task 1: Multiplication Involving Decimal Numbers

Diagram 1i.

Diagram 1 shows a problem used by a teacher to teach multiplication involving decimal
numbers.
•
What are the possible difficulties a student may have in solving this problem?

Diagram 2
ii.

Another teacher modified Problem ① to become Problem ② as shown in Diagram 2.
•
What are the strengths of Problem ② as compared to Problem ① in helping students
understand multiplication involving decimal numbers?

Diagram 3
iii.

John filled thewith 2 kg and 3 m and drew a proportional number line and construct atable of Rule of Three as shown in Diagram 3.
••

Explain how the number line and the table can help John to understand the situationin this problem.
Find the missing number in ? .

Chapter 2: Extension of Numbers and Operations | 25

Diagram 4
iv.

Jason filled thewith 6.3 kg and 5 m and his solution is shown in Diagram 4.

•

Explain the method used by Jason to solve the problem.

•

Construct a table of Rule of Three to check Jason’s solution.

•

Find the missing number in ? .

Diagram 5v.

Carol filled thewith 3 kg and 2.7 m and her solution is shown in Diagram 5.

•

Write an expression for ? .

•Explain Carol’s solution.

•

Find the missing number in ? .

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

Diagram 6
vi. After studying Carol’s solution, Cindy drew another proportional number line and a tableof Rule of Three to explain her thinking as shown in Diagram 6.
•

Write an expression for ? .

•Compare Cindy’s method with Carol’s method.
What is the same and what is different about the two students’ methods?

Which method will you recommend to your students? Why?

vii. Catherine filled the•with 4.8 kg and 3.5 m.

Draw a proportional number line and a Rule of Three table to help Catherine solve
the problem.

Diagram 7
viii. Diagram 7 shows John’s idea to simplify the multiplication 32.8 x 4.2 in vertical form bychanging both the multiplicands and the multipliers to whole numbers.
•

Explain the rule used by John to simplify the multiplication.

•

Find the missing number in ? .

Chapter 2: Extension of Numbers and Operations | 27

Diagram 8ix. Diagram 8 shows ideas of Jason and John to simplify the multiplication 4.7 x 50.

x.

•Compare the rules used by each of them.

•Find the missing number in each ? .

Simplify and calculate the answer for each multiplication in vertical form.
•

8.4 x 320•

12.64 x 70•

7.35 x 26•

72.4 x 3.65

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

Diagram 9
xi. Diagram 9 shows Cindy’s reasoning to get the answer for 1.87 x 1.3.
•

Explain Cindy’s reasoning.

xii. Without actual calculation, put the missing decimal point on the product for each of thecalculations.
•8.24 x 3.5 = 2 8 8 4•12.45 x 7.8 = 9 7 1 1•

30.2 x 19.5 = 5 8 8 9Check your answers with a calculator.

Chapter 2: Extension of Numbers and Operations | 29xiii. What is the area, in m², of a rectangular mat that is 1.3 m wide and 2.4 m long?
•

Write an expression for the area of the rectangular mat.

•Approximately, what is the area in m2 ?

Diagram 10•

Diagram 10 shows how Jeff solved the problem by drawing an area diagram.
 Fill in the missing number in each.

 Explain Jeff’s method.
 Calculate (1.3 x 2.4) m² to check the answer in Diagram 10.
 What are the strengths of this problem in helping your students understandmultiplication involving decimal numbers?

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

Standards 5.2:
Extending the division from the whole number to decimal numbers
i. Understand how to represent division situations using diagrams such as number lines, andextend the diagram of decimal numbers for explaining division by decimal numbers
ii. Extend the division algorithm in vertical form of decimal numbers and interpret the meaning ofdecimal places of quotient and remainder with situations
iii. Acquire fluency in division algorithm of decimals up to 3 digits by 2 digits with consideration ofdecimal places step by stepiv. Obtain fluency using division of decimals with sensible use of calculators in life
v. Develop number sense of division in decimals such as comparing sizes of quotient before
dividing
vi. Distinguish the situations with decimal numbers of multiplication and division
Sample Tasks for Understanding the Standards
Task 1: Division Involving Decimal Numbers

Diagram 1i.

A supermarket in Manila sells orange juice in 2-litre and 1.8-litre cartons at 90 pesos and72 pesos respectively as shown in Diagram 1.
•

Compare the two packaging.
What calculations will you do to determine which packaging is the better buy?

Explain your answer.

Diagram 2

Chapter 2: Extension of Numbers and Operations | 31•

Diagram 2 shows a proportional number line and Rule of Three table to determine thecalculation for the price of 1l of orange juice for the 2- l packaging.


Write an expression for ? .

Calculate the price per litre for the packaging.

Diagram 3•

Diagram 3 shows a proportional number line and Rule of Three table to determine thecalculation for the price of 1 l of orange juice for the 1.8 l packaging.
Fill in the missing number in each ? .

Write an expression for ? .

Calculate the price per litre for the packaging.

l
l

l

l

l

l

l

l

Diagram 4ll

Diagram 5

32 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelii.

Diagram 4 shows two students’ ideas to calculate 72 ÷ 1.8. The same ideas are also
represented in tables shown in Diagram 5.
•

Discuss what the two ideas have in common.

•

Which idea corresponds to each of the two tables?

Diagram 6iii.

Diagram 6 shows a rope-cutting problem and Jackson’s solution to the problem.
•

Explain the rule used by Jackson to simplify the division?

•

Is Jackson’s solution correct?

•

How many metre of rope was left over?

Chapter 2: Extension of Numbers and Operations | 33

Diagram 7
iv.

Diagram 7 shows the steps to do the division 5.76 ÷ 3.2 in vertical form.
•

Explain each step.

•

In Step 5, what do you think is the purpose of writing down the remainder 0?

Diagram 8v.

Diagram 8 shows the procedure used by Jackson to do the division 12.8 ÷ 2.5 in verticalform.
•

Discuss what Jackson could do with the remainder.

•

Complete the division procedure for Jackson on Diagram 8.

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

Diagram 9vi. Diagram 9 shows three problems involving different contexts of division.
•

Discuss the possibility of using a proportional number line and Rule of Three table tosolve each of the problems.

vii. Diagram 10 shows a painter painting the interior wallof a house. He used 3.8 of paint to cover 45.6 m2
of wall.
How many litres of paint will he need to cover 98.6
m² of wall?
Diagram 10

Chapter 2: Extension of Numbers and Operations | 35

Topic 6: Using Multiples and Divisors for Convenience
Standards 6.1:
Using multiples and divisors for convenience with appreciation to enrich number sensei. Understand set of numbers by using multiple and divisor
ii. Find common multiple and appreciate the use in situations, and enrich number sense with
figural representations such as arrangement of rectangles to produce a squareiii. Find common divisor and appreciate its use in situations, and enrich number sense with figuralrepresentations such as dividing a rectangle into pieces of square
iv. Understand numbers as composite of multiplication of numbers as factors
v. Appreciate ideas of prime, even and odd numbers in situations using multiples and divisors
vi. Acquire number sense to see its multiple for convenience
Sample Tasks for Understanding the Standards
Task 1: Multiples and Divisors

Diagram 1i.

Diagram 1 shows a figural representation of 3 x 4.
•

Explain the relationships between multiples and divisors.

•

Find all other divisors of 12.

•

Is 1 a divisor of 12? Why or why not?

•Is 12 a divisor of 12? Why or why not?

[Note. In this topic, a divisor of a whole number is a whole number which evenly divides thenumber and leaves no remainder. A divisor is also known as a factor.]

Diagram 2
ii.

Diagram 2 is a number line used to represent skip counting by 4.
•

How is skip counting related to multiples?

•

Diagram 2 shows that 4, 8, 12 and 16 are multiples of 4.
Is zero a multiple of 4? Why or why not?

36 | Mathematics Challenges for Classroom Practices at the Upper Primary Level•

Find the first five multiples of 13.
What is the least non-zero multiple of 13?



Can the greatest multiple of 13 be found? Why or why not?
iii.

Diagram 3 shows a problem involving
stacking boxes of chocolate.
•

How does the task promote theunderstanding of multiples?

Diagram 3
iv.

Diagram 4 shows a hundred chart. Circle allmultiples of 2 between 1 and 100.
•

Describe the pattern of multiples of 2.

Diagram 4v.

Diagram 5

Explore the patterns of multiples of othernumbers between 2 and 10 using thehundred chart in Diagram 5.
•

Which of these patterns do you find theminteresting? Why?

•

How does this exploration of patterns
promote your students’ understandingof multiples?

Chapter 2: Extension of Numbers and Operations | 37

vi. Krit’s father intends to train him in managing
his pocket money. So, instead of everyday,
Krit’s father decided to give him a bigger sumof pocket money every four days.
Diagram 6 shows the calendar for January ofa year. The dates for the first two allowances
are marked with .
•What will be the date for Krit’s third
allowance?

•

What are the remaining allowance dates
in January for Krit?

•

How can this problem inculcate yourstudents’ appreciation towards multiples?

Diagram 6vii. Diagram 7 shows a big basket of 28
mangoes. José wants to pack all the
mangoes into smaller baskets with equal
number of mangoes, and with no left over.

Diagram 7•

What are the different ways for José topack the mangoes?

•

How many smaller baskets of mangoes
can he pack?

•How does this problem inculcate yourstudents’ appreciation towards divisors?

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

Task 2: Common Multiples
i.

Mark with on all multiples of 3, andmultiples of 5 in Diagram 8.

on all•What numbers are marked with both theand the ?

•These numbers the common multiples of 3
and 5.


What is common multiple?

What is least common multiple (LCM)?



Find the LCM of 3 and 5.

Diagram 8ii.

Diagram 9iii.

Diagram 10

Diagram 9 shows a 4 x 6 rectangular card.
•

Cut out the rectangular cards in Material
Sheet 1 at the end of this topic.

•

Arrange some of these cards to construct asquare.

•Construct another square of different sizes.

•

Explain how the squares are related to thecommon multiples of 4 and 6.

•

Find the LCM of 4 and 6 using thesesquares.

Diagram 10 shows a 15 x 12 rectangular card.
•

Cut out the rectangular cards in Material
Sheet 2 at the end of this topic.

•

Explain how these cards can be used to findthe LCM of 15 and 12.

•

Find the LCM of 15 and 12 using thesecards.

Chapter 2: Extension of Numbers and Operations | 39

Diagram 11
iv.

Diagram 11 shows two student’s methods to find the common multiples of 8 and 12.
•

Describe each method and explain the student’s ideas.

•

What are the strengths and weaknesses of each method?

•

What is the LCM of 8 and 12?

•

Find the LCM of 18 and 27.
v.

8 bunsa pack6 sausagesa packA supermarket sells buns that come in a packof 8 and sausages in a pack of 6 as shown in
Diagram 12. Tata plans to prepare hot dogs withthese buns and sausages for a party. She tries
to match the number of sausages with the bunsin order not to leave with surplus of either one.
•

How many packs of each will you
recommend for Tata to buy? Show morethan one way of solving the problem.

•

What is the least number of packs for buns
and sausages that Tata can buy?

•

What concept of multiples is enhanced insolving this problem?

Diagram 12

40 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Common Divisors

Diagram 13i.

Diagram 13 shows the methods used by two students to solve the following problem:
12 apples and 18 oranges are to be distributed equally amongst some children.
What is the greatest number of children who can get the fruits ?
•Explain each method.

•

What are the strengths and weaknesses of each method?

•

What is the greatest common divisor (GCD) of 12 and 18?

•

Find the GCD of 12 and 28.

•

Find the greatest common divisor of 18 and 27.

Chapter 2: Extension of Numbers and Operations | 41

Diagram 14
ii.

iii.

Diagram 14 shows John’s method of finding the GCD of 68 and 48 using the Euclidean
algorithm.
•

What is the GCD of 68 and 48?

•

Explain John’s method.

•

Find the GCD of 210 and 45 using John’s method.

Both the numbers 12 and 8 have 6 divisors each.

Diagram 15•

What pattern do you see when the divisors are grouped as shown in Diagram 15?

•

Make a conjecture about the number of divisors of a number.
Is your conjecture sometimes or always true? Justify your decision.

42 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 16
iv.

Diagram 16 shows a problem to divide a piece of land.

Diagram 17
Cindy figured out a solution to the Land Problem as shown in Diagram 17.
•

Explain Cindy’s ideas and solution.

•

Solve the Land Problem using any other method.

•

Compare your method with Cindy’s method.
In what ways is your method different from Cindy’s method?

Is there any similarity between your method and Cindy’s method?

Chapter 2: Extension of Numbers and Operations | 43

Task 4: Prime Numbers and Composite Numbers

Diagram 18i.

Diagram 18 shows four tasks involving arrangement of square tiles and the solutions given
by Cindy as well as her understanding on the meaning of prime and composite numbers.
•

Cut out the square tiles in Material Sheet 3 at the end of this topic.

•

How many different ways to arrange 15 square tiles into a rectangle?

•



How is this subtask related to the factors of 15?



Is 15 a prime or composite number? Justify your decision.

How many different ways to arrange 19 square tiles into a rectangle?


How is this subtask related to the factors of 19?



Is 19 a prime or composite number? Justify your decision.

44 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelii.

Diagram 19

Diagram 19 shows an ancient method of finding
all prime numbers within 100 which is known as
the Sieve of Eratosthenes.
•

How many prime numbers less than 100
do you get from the Sieve of Eratosthenes?

•

What is a prime number?

•

Why is 1 not considered as a prime number?

Chapter 2: Extension of Numbers and Operations | 45

Task 5: Even Numbers and Odd Numbers

Diagram 20i.

Diagram 20 shows a representation of the smallest even number 2. Using this as a unit,
a student represented five other even numbers as shown in Diagram 21.

Diagram 21
Based on these representation of even numbers, a student makes a conjecture that “evennumbers are multiples of 2”.
•

Is the conjecture sometimes or always true? Justify your decision.

•

How are odd numbers related to multiples of 2?

Diagram 22ii.

Two students make a conjecture respectively on the relationship between odd numbers
and multiples of 2 as shown in Diagram 22.
•

Which conjecture is true? Justify your decision.

8 + 4 = 󠄻

7 + 5 = 󠄻

8 + 5 = 󠄻Diagram 23
iii.

Diagram 23 shows three sums involving even and odd numbers.
•

Fill in the missing sum in each 󠄀.

•

Based on these sums, make a conjecture for each of the following case:•

Sum of two even numbersSum of two odd numbersSum of an even number and an odd number

Is each of your conjectures sometimes or always true? Justify your decision.

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

8 – 4 = 󠄻 7 – 5 = 󠄰 8 – 5 = 󠄻 Diagram 24
iv.

v.

Diagram 24 shows three cases of subtraction involving odd and even numbers.
•

Fill in the missing number in each 󠄀.

•

Make a conjecture involving subtraction of odd and even numbers for each case.

•Is each of your conjectures sometimes or always true? Justify your decision.

Decide whether each of the following statements is true or false. Justify your decision.
•

All prime numbers are odd.

•

An even number cannot be a prime number.

•

Zero is an even number.

vi. In Malaysia, every citizen is issued an identification card with a 12-digit number
such as xxxxxx-xx-5207 and xxxxxx-xx-2318. The last digit indicates the genderof the card holder, even digit for female and odd digit for male. So, the holderof xxxxxx-xx-5208 is a female and the holder of xxxxxx-xx-2317 is a male.
•Why do odd and even numbers work for Malaysian identification system?

•

What other application of odd and even numbers can you find in your own country?
Explain the application clearly.

Chapter 2: Extension of Numbers and Operations | 47

Material Sheet 1: Rectangular Card (4 x 6)

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

Material Sheet 2: Rectangular Card (15 x 12)

Chapter 2: Extension of Numbers and Operations | 49

Material Sheet 3: Square Tile50 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 7: Introducing Improper and Mixed Fractions and Extending to Addition and
Subtraction of Fractions to Dissimilar Fractions
Standards 7.1: Extending fractions to improper, mixed and equivalent fractions
i. Extend fractions to improper and mixed fractions using number line of more than one bymeasuring with unit fraction
ii. Find ways of determine equivalent fractions with number lines and with the idea of multiple ofnumerator and denominatoriii. Compare fractions using number line and the idea of multiple
Sample Tasks for Understanding the Standards
Task 1: Fraction More than 1

Diagram 1i.

A student measured the length of a tape using a metre rule as shown in Diagram 1. The
tape was found to be 1 m and a remaining part Ⓐ. The student also found that 3 partsⒶ were equal to 1 m.
•

How many metre is one remaining part Ⓐ?

•

How many metre is the length of the entire tape?


Express your answer as an improper fraction and also a mixed fraction.

Diagram 2
ii.

The student measured another tape and found that it was again 1 m and a remainding
part, Ⓑ. He cut out part Ⓑ and found that two parts of Ⓑ is a little less than 1 m. He thencontinued the process and found that 2 parts of Ⓒ is the same as 1 part Ⓑ as shown in
Diagram 2.
•

How many part Ⓒ is the same as 1 m?

•

How many metre is one part Ⓒ?

•

How many metre is one part Ⓑ?

•

How many metre is the entire tape?

  

Express your answer as an improper fraction and also a mixed fraction.

Chapter 2: Extension of Numbers and Operations | 51

Diagram 3
iii.

Both the procedures used in Diagram 1 and 2 involve the Euclidean algorithm to find theremaining part as a fraction are similar to the procedure of finding the GCD of 68 and 48
as illustrated in Diagram 14, Task 3 of Topic 6.
Diagram 3 shows the same procedure for finding the entire length of tape Ⓗ.
•

Explain the procedure.

•

Find the entire length of tape Ⓗ, in m.

Diagram 4iv.

v.

Diagram 4 shows 3 litres of water divided equally into 2 jugs.
•

Shade the portion for 1 jug on Diagram 4.

•

Express in litre, the amount of water in 1 jug.
Give your answer as an improper fraction and also a mixed fraction.

Fill in eachwith an appropriate number for the following statement to be true.

1•

In what way can the unit fraction 5– help your students understand the idea ofimproper fraction?

•

Discuss how the idea of unit fraction can be used to compare 5– m, 5 m and 1 5– m.

8

12

2

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

Diagram 5vi. Diagram 5 shows a fraction number line.
•

What is the unit fraction Ⓐ?

•

Fill in the missing mixed fraction in each  to complete the number lines.

Diagram 6vii. Diagram 6 shows a fraction number line drawn by a student.
•

What is wrong with the number line?

Diagram 7
viii. Diagram 7 shows three tasks involving mixed fractions and improper fractions.
•

How will you order the sequence of the three tasks? Justify your decision.

Chapter 2: Extension of Numbers and Operations | 53

Diagram 8ix. Diagram 8 shows a tape divided into 4 equal parts. A student said that the length of the2tape is 3– of 2 m.
•

Is the student’s claim correct?

•

What is the length of the tape in m?

Diagram 9x.

3

Diagram 9 shows three students’ reasoning process when changing 2 5– to an improper
fraction.
••

Compare the three students’ reasoning.


Why is John’s method possible?

How is John’s method related to Cindy’s method?

How is Cindy’s method related to Jason’s method?

Which methods will you promote to your students, Cindy’s, John’s, Jason’s or all thethree?


Justify your decision.

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

Task 2: Equivalent Fractions

Diagram 10i.

Diagram 10 shows a problem involving a ratio of lucky draw prizes to a number ofplayers. Cindy drew a proportional number line and constructed a table of Rule of Three
to find the solution as shown in the diagram.
•

Find the missing number in ? and ? .

•The ratio of prize to 5 players is 5 . What is the ratio of prize to 10 players?

•2

If there are 45 players participating in the game, draw a proportional number line andconstruct a table of Rule of Three to find the number of prizes.

•

What is the ratio of prize to player for the 45-player case?

•

What can you conclude on the ratios of these three cases?

•

How are these ratios related to equivalent fractions?

Diagram 11ii.

Diagram 11 shows three equivalent fractions 23 , 69 and 12
18 .
•Find the missing number in each ? .

•Draw a proportional number line to show the relationship between 3 and 18 .

2•

Make a conjecture on methods of finding equivalent fractions.

•

Test your conjecture with other fractions.

•

Make a conclusion about your conjecture.

12Chapter 2: Extension of Numbers and Operations | 55

Diagram 12iii.

Diagram 12 shows nine fraction number lines for unit fractions
respectively.

and•Fill in the missing fraction in each .

•

Use the number lines to compare the values of the following pairs of fractions.

•

Explain how the number lines can be used to find equivalent fractions of•

Use the number lines to find equivalent fractions of.

56 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Comparing Fractionsi.

ii.

Pick any two fractions with the same numerator, such as 35 and 103 from the number lines in
Diagram 12.
•

Compare the values of the two fractions.

•

Make a conjecture about the values of fractions with the same numerator.

•

Test your conjecture.

•

Make a conclusion about your conjecture.

•

Justify your conclusion.

Pick any two fractions with the same denominator, such as 37 and 67 .
•Compare the values of the two fractions.

•

After investigating the values of some pairs of fractions with the same denominator,
a student concluded that “when the denominators are the same, a fraction becomeslarger as the numerator increases”.
Justify the student’s conclusion.

Diagram 13iii.

Diagram 13 shows some fractions with similar numerator 3.
•

Arrange the fractions in order, from least to greatest.

Diagram 14
iv.

Diagram 14 shows some fraction with similar denominator 9.
•

Arrange the fractions in order, from greatest to least.

Diagram 15v.

Diagram 15 shows three fractions with different numerators and denominators.
•

Using common multiples, compare the values of the three fractions in two different
ways.

Chapter 2: Extension of Numbers and Operations | 57

Standards 7.2:
Extending addition and subtraction of similar fractions to improper and mixed fractions, anddissimilar fractions
i. Extend addition and subtraction of similar fractions to proper and mixed fractions with
explanations using models and diagrams
ii. Extend addition and subtraction into dissimilar fractions with explanations using diagrams andcommon divisorsiii. Acquire fluency of addition and subtraction of fractions with appreciation of idea to produce
the same denominators
Sample Tasks for Understanding the Standards
Task 1: Addition and Subtraction Involving Mixed Fractions

Diagram 1i.

Diagram 1 shows two containers with•andof water, respectively.

Shade on Diagram 1 to show the sum ofWrite the sum as an improper fraction.

Convert the sum to a mixed fraction.

58 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 2
ii.

iii.

Diagram 2 shows Cindy’s method to add two mixed fractions.
•

Explain Cindy’s method.

•

Use Cindy’s method to add the following mixed fractions.

With the help of a diagram, explain how you will extend Cindy’s thinking to do subtraction
involving mixed fractions such as

Chapter 2: Extension of Numbers and Operations | 59

Diagram 3
iv.

Diagram 3 shows two tapes, Ⓐ and Ⓑ. Use the number line to answer the following
questions:•

What is the total length of the two tapes in metres?

•

What is the difference in metres between tape Ⓐ and tape Ⓑ?

Diagram 4v.

Diagram 4 shows John’s and Jason’s methods to subtract mixed fractions using a
number line.
•

What is the difference between the two methods?

•

Which method will you promote to your students? John’s, Jason’s or both?

•

Justify your decision.

vi. Explain how you use the strategy of “carrying” and “borrowing” to do the following
calculations.

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

Task 2: Addition and Subtraction of Dissimilar Fractionsi.

Diagram 5 shows how a student calculated
12
the sum of 5 and 5 . The student also drewa diagram as shown to explain his method.
•

Referring to the diagram, explain why thedenominator of the sum remain unchanged.

Diagram 5ii.

Diagram 6 shows contradicting ideas of John
and Jason regarding the solution of the OrangeJuice problem.
•

Diagram 6

How will you use the contradicting ideas topromote a dialectic discussion among yourstudents?

Chapter 2: Extension of Numbers and Operations | 61iii.

John drew a diagram as shown in Diagram 7 toargue for his case.
•

Conceptually, what is wrong with John
diagram?

Diagram 7

iv.

Jason tried to draw a diagram to support hisargument that
However, he got
stuck with his drawing as shown in Diagram 8.
•

Help Jason to complete the drawing anduse it to explain that

Diagram 8v.

Diagram 9

Diagram 9 shows a common error of a student
when subtracting fractions.
•

What is the misconception of the student?

•

With help of a diagram, explain how you will
guide the student to correct the error.

62 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelvi. Diagram 10 shows two tasks, ① and ②, todevelop fluency of addition and subtraction
with similar fractions, respectively.
•

Explain the respective rule for adding andsubtracting similar fractions.

•

Explain how the rules can be extended
to addition and subtraction of dissimilar
fractions such as
and.

Diagram 10vii. Diagram 11 shows another two tasks, ③ and ④, to
help students understand the respective processof adding and subtracting dissimilar fractions.
•

Explain each process.

•How can these two tasks inculcate
appreciation towards the idea of common
multiples?

•

In Task ④, another student used
24 instead of 12 as the common
denominator.


Diagram 11

Will this student get the sameanswer? Why or why not?

Chapter 2: Extension of Numbers and Operations | 63

viii. Diagram 12 shows two students’ methods to
add 123 and 168 .

Diagram 12•

Compare the two methods.

•

Which method is correct? John’s, Jason’sor both?

•Justify your decision.

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

Topic 8: Extending Fractions as Numbers and IntegrateStandards 8.1: Seeing fractions as decimals and seeing decimals as fractions
i.
ii.
iii.

See fractions as decimals using division and define quotient with divisible which includes
repetition of the remainder
See decimals as fractions such as hundredths are per hundred
Compare decimals and fractions and order them on the number line

Sample Tasks for Understanding the Standards
Task 1: Fraction as a Quotient

Diagram 1i.

Diagram 1 shows 3 pieces of 1-metre paper strips, folded into 3, 4 and 5 parts respectively.
•

Shade on the diagram and find the quotient of (1 m ÷ 3), (1 m ÷ 4) and (1 m ÷ 5) as afraction, respectively.

•

What is the relationship between fraction and division?

2

Diagram 2
ii.

Diagram 2 shows John’s reasoning when comparing the answers of 5 ÷ 2 and 7 ÷ 3. He
concluded that 5 ÷ 2 = 7 ÷ 3 since both answers seem to be the same.
•

Why is John’s reasoning not correct?

Chapter 2: Extension of Numbers and Operations | 65

Diagram 3
iii.

Another student, Jason argued differently based on his own calculations shown in
Diagram 3.
•

Explain how Jason’s argument can be used to help John correct his misconception.

•

Why is decimal important in division of whole numbers?

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

Task 2: Expressing Fractions as Decimals

Diagram 4i.

Diagram 4 shows a ribbon problem and a student’s solution.
•

Fill in the missing fraction in each .

•

Convert each of the fractions to a decimal number.

Diagram 5
ii.

1

Diagram 5 shows a number line with unit fraction 5 .

•
•
iii.

4

Using the number line, draw a diagram to show that 5 is the quotient of 4 ÷ 5.
Convert the quotient to a decimal number.
4

Draw and use a number line to show that 3 is the quotient of 4 ÷ 3.
•

Convert the quotient to a decimal number.

Chapter 2: Extension of Numbers and Operations | 67

Method ①

Method ②

Method ③

Diagram 6
iv.

4

Diagram 6 shows three students’ methods to convert 5 to a decimal number.
•Compare the three methods.
Which method will you promote to your students?

Justify your decision.

Diagram 7v.

Diagram 7 shows four fractions.
•

Express each fraction as a decimal or whole number.

•

Sort the fractions into three groups.
Ⓐ Whole numberⒷ Accurate decimal numberⒸ Repeating decimal number

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

Task 3: Expressing Decimals as Fractionsi.

How is a decimal related to a fraction?

ii.

Diagram 8 shows three squares each
representing 1. The shaded region ofeach square represents a fraction of thewhole square.
•
•
•
•

1

How many sets of 10 are there in 1?
1

How many sets of 100 are there in 1?

Fill in the missing number in each 
to show the fractions.
Match each fraction with its
corresponding decimal number.

Diagram 8iii.

Jason tried to explain why 2.1 is not
the same as 0.21 to Cindy as shown in
Diagram 9.
•

Explain Jason’s way of thinking.

•

How can Jason’s way of thinking beused to compare the values of 2.1
and 0.21?

Diagram 9iv.

v.
Diagram 10

Diagram 10 shows Jason’s way of
converting 2.1 and 0.21 to fractions.
•Fill in the missing fraction in .

•Will you recommend Jason’s way of
thinking to your students? Why or
why not?

Express the decimal numbers as
fractions.
•

2.8

•

2.08

•

2.80

Chapter 2: Extension of Numbers and Operations | 69

Task 3: Ordering Fractions and Decimals

3

Which is longer, 5 m or 0.5m?

Diagram 11i.

Diagram 11 shows a problem about comparing a fraction with a decimal number and three
students’ ideas to find the answer.
•

Which idea is easier to do the comparison?

•

Justify your choice.

Diagram 12ii.

Diagram 12 shows a fraction number line and a decimal number line.
•

Fill in the missing number in each .

•

Use the number lines to compare the values of

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

Diagram 13iii.

Mark with a ↓ on the number line in Diagram 13 to show each of the following numbers.

iv.

Arrange the following numbers in order from least to largest.

Chapter 2: Extension of Numbers and Operations | 71

Topic 9: Extending Multiplication and Division to Fractions
Standards 9.1: Extending multiplication to fractions
i.
ii.
iii.

Extend the multiplication to fractions with situations using diagrams such as number lines, stepby step, and find the simple algorithm for the multiplication of fractions
Acquire fluency with the multiplication of fractions
Develop number sense of multiplication of fractions such as comparing sizes of products before
multiplying

Sample Tasks for Understanding the Standards
Task 1: Multiplication of Fractions
A 1-metre stick weighs 3 kg.
1

How many kg does a 5-metre stick weigh?

Diagram 1

Diagram 2i.

Diagram 1 shows a problem involving the mass and length of a stick. Diagram 2 shows
John’s and Jason’s methods to solve the problem.
•

Compare the two methods.
What is the crucial learned knowledge required to use each of the methods?



Find the missing number in .

72 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 3
ii.

Another student, Catherine, found the answer using a table of Rule of Three as shown in
Diagram 3.
•

iii.

Explain Catherine’s method.
2

Use each of the three methods to calculate 3× 5 .

Diagram 4
iv.

3

•

3

1

3

Using the diagram, explain why 4 × 2 = 8 .

Diagram 5v.

1

Diagram 4 shows John’s method using two proportional number lines to calculate 4 × 2 .

1

2

Diagram 5 shows John’s method to calculate 3 × 5 .
•Explain how the diagram can be used to find the missing number in .

Chapter 2: Extension of Numbers and Operations | 73

Iron Rod Problem
The mass of an iron rod is 23 kg per metre.
How many kg will a 5 m long iron rod weigh?
4

Diagram 6vi. Diagram 6 shows a problem involving the mass of an iron rod and Cindy’s solution to the
problem.
•Find the missing number in .

•

Compare the proportional number lines drawn by Cindy to that drawn by John in
Diagram 2.
What are the similarities and differences between the two ways of using the
proportional number lines?

What are the strengths and weaknesses of each way?

Diagram 7vii. Diagram 7 shows Carol’s solution to the Iron Rod Problem.
•

Fill in the missing number in .

•

Explain Carol’s method of finding the solution.

•

Between Cindy’s and Carol’s methods, which will you promote to your students?
Justify your decision.

74 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Printable Area Problem
2

A can of paint will cover 5 m2 of a wall.
1

What area in m2 will 3 can of this paint cover?

Diagram 8viii. Diagram 8 shows a problem involving a paintable area on a wall and Jackson’s solution
for the problem.
•

Explain Jackson’s method of solving the problem.

•

Write a mathematical expression for the problem.

•

Compare Jackson’s method of multiplying fractions with John’s and Cindy’s methods.
What is the main mathematical idea supporting each method?

Which method will you promote to your students? Why?

[Note. It is important to ensure the two fractions are represented by two sides of asquare instead of a rectangle. Otherwise, the shaded regions of the area diagram
may be a major source of misconceptions if students recognise fractions only by thepart-whole relationship.]Diagram 9

Chapter 2: Extension of Numbers and Operations | 75

12

ix. Diagram 9 shows Connie’s method of calculating 3 × 5 .
•Compare Connie’s method with Jackson’s method.
What are the similarities and differences between the two methods?

Which method will you recommended to your students? Why?

Diagram 10x.

Diagram 10 shows three open-ended tasks for multiplication of fractions.
•Find as many solutions as possible for the three tasks.

•

How do these tasks help your students acquire fluency with multiplication of fractions?

xi. Calculate.
•
•
•

2

5× 3
2
5

3

×4

3

1

25 ×7

Task 2: Comparing Products of Fractionsi.

Justify each statement without calculate the products.
•
•

ii.

Which multiplication has a product that is less than 8?
••

Justify each answer without calculate the product.

•Calculate each product to verify your answer.

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

Standards 9.2: Extending division to fraction
i.
ii.
iii.

Extend the division to fraction with situations using diagrams such as number lines, step bystepAcquire fluency with the division of fractions
Develop number sense of division of fractions such as comparing sizes of quotients before
dividing

Sample Tasks for Understanding the Standards
Task 1: Modeling Division of Fraction

Diagram 1i.

Diagram 1 shows a problem and a Rule-of-Three table constructed by John to solve theproblem.
••

32

Explain why the solution to the problem is 5 ÷ 3 (m²).
Find the missing number in.

Diagram 2

ii.

3

2

Diagram 2 shows Cindy’s method to calculate 5 ÷ 3 .

•Find the missing number in•

What are the strengths of Cindy’ method?

•

Use Cindy’s method to calculate 7 ÷ 5 .

using Cindy’s method.

2

3

Chapter 2: Extension of Numbers and Operations | 77

Diagram 3
iii.

Diagram 3 shows Jason’s method to calculate 35 ÷ 23 .

•Explain Jason’s method.

•

Find the missing number in•

Use Jason’s method to calculate 4 ÷ 3 .

using Jason’s method.
3

7

Diagram 4
iv.

3

2

Diagram 4 shows Jeff’s method to calculate 5 ÷ 3 . The procedure is based on therepeated-subtraction model of division.
•

Explain the division procedure.

•

Find the missing number in .

•

Use the same method to find 4 ÷ 2 .

31

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

Task 2: Fluency with Division of Fractionsi.

Diagram 5 shows 18 digit-cards and aproblem involving the multiplication of
two fractions to get a product of 1.
•

Pick and arrange any four cards tofind as many solutions as possible
for the problem.

•

What rule is there between
multiplicand and multiplier to makethe product 1?

Diagram 5ii.

Catherine made a claim about the divisionof fractions as shown in Diagram 6.
•

What is the inverse number of afraction?

•

Use Catherine’s method to calculate
3
4



1
5


iii.

l
l

2

÷3

Diagram 7 shows a problem on bottlingof orange squash and Jason’s method insolving the problem.
•

Explain Jason’s method.

•

Find the missing number in .

•

What rule of division did Jason use
84
to calculate 3 ÷ 9 ?

••Diagram 7

2

÷5

1

2

Find 5 ÷ 3 using Jason’s method.
What are the strengths andweaknesses of Jason’s method?

Chapter 2: Extension of Numbers and Operations | 79

iv.

Diagram 8v.

Diagram 9

Diagram 8 shows a student’s idea in
dividing two fractions.
•Find the missing number in .

•

Make a conjecture base on Jeff’s
idea.

•

Is your conjecture always true,
sometime true, or never true?

•

Make a conclusion on yourconjecture.

•

Find 15 ÷ 23 using Jeff’s method.

Diagram 9 shows another student’s idea incalculating the same division of fractions.
•Find the missing number in .

•

Explain Carol’s method.

•

Find 5 ÷ 3 using Carol’s method.

12

vi. How will you use Catherine’s, Jason’s, Jeff’s or Carol’s ideas to help your students acquire
fluency with division of fractions?
vii. Calculate•••

viii. Which value is larger?
•••

Complete the following statements:


If the divisor is smaller than 1, then the quotient is __________ than the dividend.



If the divisor is larger than 1, then the quotient is __________ than the dividend.”
3

2

ix. A bottle contains 1 5 of mango juice. If you pour 3 of juice into a cup, how many cup of
juice can you get?
x.

1

The area of a rectangular board is 2 45 m² and its length is 1 5 m. Find its width inmetres.

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

CHAPTER 3
Measurement and Relations
Topic 1: Introducing Angle and Measuring it
Standards 1.1:
Introducing angle by rotation, enabling measure and acquire fluency using the protractor
i. Compare the extent of rotation and introduce degree as a unit for measuring an angle
ii. Recognise right angle is 90 degrees, and adjacent angle of two right angles is 180 degrees,and 4 right angles are 360 degrees
iii. Acquire fluency in measuring angles using the protractor
iv. Draw equivalent angles with addition and subtraction using multiples of 90 degrees
v. Appreciate measurement of angles in geometrical shapes and situations in life
Sample Tasks for Understanding the Standards
Task 1: Introducing Angles and Unit for Measuring Angle

Diagram 1i.

Diagram 1 shows some animals that can open their mouths very wide.
•

Which animal can open its mouth the widest?

•

How do you compare how wide the mouths are opened?
Is the wideness same as the angle?

•

Arrange the size of angles Ⓐ,Ⓑ,Ⓒ,and Ⓓ in ascending order.

•

How do you make the comparison without any measuring tool?

•

Explain the meaning of angle by rotation.

Chapter 3: Measurement and Relations | 81

Diagram 2
ii.

Diagram 2(a) shows a 2-arm plastic stick joined at a vertex. One arm rotates in anticlockwise direction to form angle ⓐ as shown by the circular arrow in Diagram 2(b).

Diagram 3
Diagram 3 shows five other angles formed by rotating the arm in anti-clockwise direction.
•

Draw a circular arrow to show each of the angles on Diagram 3.

•

If ⓒ forms a right angle, how many right angles are there in, ⓓ, ⓔ, and ⓕ?

•

For angle ⓕ, the arm rotates and making one complete revolution back to the original
position.
•

What is the difference in the size of the angles formed in ⓕ and the original
position?

Which of the angles ⓐ to ⓔ show the angle of a half revolution?

82 | Mathematics Challenges for Classroom Practices at the Upper Primary Leveliii.

Angle of one revolution is divided into 360 equal parts. The size of one part is one degree
and is a unit to measure the size of angles.
••

State the following angles in degree.


1 right angle



2 right angles



3 right angles



4 right angles

Historically, why is one revolution divided into 360 instead of 100 or 10?

Diagram 4
iv.

Diagram 4 shows a protractor for measuring angles. Students A, B and C read angle P as
35°, 145° and 155°, respectively.
•

Discuss the three readings of the students.

•

Use a circular arrow to mark 145° on Diagram 4.

•

What is the relationship between 145° and angle P ?

Chapter 3: Measurement and Relations | 83

Task 2: Measuring Angle

Diagram 5i.

Diagram 5 shows two sides of an angle extended from point O. A, B, C and D are 4 points
on the two sides.
Angle ∠AOB is formed by line segments OA and OB, angle ∠COD is formed by line
segments OC and OD and ∠AOC is formed by line segment OA and OC.
•

A student claims that both ∠AOB and ∠AOC are smaller than ∠COD because theylook smaller.


Suggest and explain how you will help the student to overcome the 		
misconception.

Diagram 6ii.

Diagram 6 shows an angle Ⓐ and a protractor. When a student, Wijaya was asked tomeasure the angle using the protractor, he was confused and claimed that the angle wastoo small to be measured.
•

Explain how your suggestion in question (i) is applied to help Wijaya overcome his
confusion.

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

Diagram 7iii.

Diagram 7 shows two angles, Ⓑ and Ⓒ which are larger than 180°.
•

What difficulties do you anticipate from your students if they are asked to measurethese angles using the protractor in Diagram 6.

•

Explain how the protractor is used to measure these two angles.

Diagram 8iv.

Diagram 8 shows a task involving angles formed by two intersecting straight lines.
•

What is the learned knowledge required to do this task?

•

How can this task be used to enhance your students’ mathematical reasoning?

•

What is the next task related to angles that you will give to your students?

What mathematical contents do you expect the students to acquire from the task?



Justify your answer.

Chapter 3: Measurement and Relations | 85

Task 3: Angles in Daily-life Situations

Diagram 9i.

Diagram 9 shows a camping hut which has a unique geometrical shape with two slanting
roofs and a floor base.
Angles of large structure such as the angle between a slanting roof and the floor base ofthis camping hut are usually difficult to measure directly.

Diagram 10
Diagram 10 shows a protractor, a cardboard and a pendulum.
•

Discuss how you will use these materials to design a tool that can measure the angleof large structure in an easier way.

•

Discuss how this task can incalcate your students’ appreciation of the usefulness ofmathematics.

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

Topic 2: Exploring and Utilising Constant Relation
Standards 2.1:
Exploring equal constant relation with utilisation of letters to represent placeholders
i. Explore two possible unknown numbers such that their sum (or difference/ product/quotient)
is constant, for example □+Δ=12 (□ and Δ are placeholders)ii. Use letters instead of placeholder (empty box) to derive equivalent relationiii. Understand the laws for operations (e.g. associative, commutative and distributive, etc.) to
explain the simpler way of calculation
iv. Appreciate the use of diagrams such as number lines and area to represent relation whenfinding solutions
Sample Tasks for Understanding the Standards
Task1: Sorting Out the Possible Numbers

Diagram 1i.

Diagram 1 shows cupcakes with two different flavours. Erika plans to buy some cupcakes.
Let □ represents the number of chocolate cupcakes and ○ represents the number of vanillacupcakes bought by Erika.
•

Explain the meaning of the expression □ + ○ = 8.

•

List all the possible values of □ and ○ using a suitable strategy.

Diagram 2
ii.

iii.

Let ◊ represents the number of cupcakes and each piece of the cupcake costs 9 baht.
•

Write an expression to represent the cost of all the cupcakes.

•

Diagram 2 shows another box of cupcakes bought by Erika. Find the cost of this box
of cupcakes.

Instead of using symbols such as □, ○, and ◊, letters can also be used to represent thesituation.
•

Suggest a letter to replace □, ○, ◊ each.

•

Use the letters to show the relationship between □, ○ and ◊.

Chapter 3: Measurement and Relations | 87

Diagram 3

Diagram 4iv.

On New Year day, a bakery put on a sale for the cupcakes. Customers can pick any numberof cupcakes. Diagram 3 shows the boxes of cupcake bought by Anong and Somchai andthe price of each box.
Diagram 4 shows a student’s reasoning process in finding the price of 1 chocolate and 1
vanilla cupcakes.
•

Explain the student’s reasoning.

•

To what extend will this reasoning process help your students learn solving equations?

88 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 5v.

A bakery in Hatyai sells cupcakes in two boxes as shown in Diagram 5.
•

If □ and ○ represents the price of 1 chocolate cupcake and 1 vanilla cupcake,respectively, write an equation base on each of the boxes.

•

Explain how the idea of substitute values for the symbols □ and ○ can be used to findthe price of

1 chocolate cupcake.



1 vanilla cupcake.

Chapter 3: Measurement and Relations | 89

Task 2: Laws of Operationsi.

Diagram 6 shows a box of cupcakes,
P, with vanilla and chocolate flavour.
Let letter c represents the number of
chocolate cupcakes and letter v thenumber of vanilla cupcakes.
•

Explain how box P can be used to
explain commutative law for c + v.

•

Diagram 7 shows another box of
cupcakes, Q. Based on this box, astudent claims that c – v = v – c.

Diagram 6

What is the possible cause of
the student’s misconception?

What can you do to help
thestudent
correcthis
misconception?

Diagram 7ii.

Diagram 8 shows 3 boxes of vanillacupcakes repackage into 4 boxes.
Let letter b represents the number ofboxes and letter v the number of vanillacupcakes.
•

Diagram 8

Explain how the situation can beused to explain commutative law forb × v.

90 | Mathematics Challenges for Classroom Practices at the Upper Primary Leveliii.

Diagram 9 shows two situations involving
packaging chocolate cupcakes intoboxes. Let letter b represents the number
boxes and letter c the number of chocolate
cupcakes.
•

Explain how you will use the situations
to help your students learn that thecommutative law does not apply forc ÷ b.

Diagram 9

Diagram 10
iv.

Diagram 10 shows an 8-piece box and a 12-piece box that contain only chocolate cupcakes.
A bakery is offering a “Buy-one-free-one” promotion with buying a 12-piece box and get an
8-piece box for free.
•

Mr Krit buys p 12-piece boxes during the promotion. How many cupcakes will he getin total?

•

Ms Nisakorn buy some boxes of cupcakes. If she get a total of 60 cupcakes, what isthe value of p?

•

Explain how the distributive law can be used to simplify the calculation.

Chapter 3: Measurement and Relations | 91

Task 3: Problem Solvingi.

Diagram 11 shows a problem involving a
gift box filled with chocolates and another5 pieces outside the box.
•

Solve the Chocolate Problem.

•

Diagram 12 shows Anong’s andSomchai’s methods to solve theproblem.


Explain each of the method.

Is your method of solving thesame as or different from the two
students’ methods?

Which of the methods will you
promote to your students?
Justify your choice.

Diagram 11

x
x
x

x

Diagram 12

92 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 3: Extending Measurement of Area in Relation to PerimeterStandards 3.1:
Introducing area and produce formula for the area of rectangle
i. Compare extent of area and introduce its unit, and distinguish it from perimeterii. Introduce one square centimetre (cm2) as unit for area and its operation using addition andsubtractioniii. Investigate area of rectangles and squares and produce the formula of area
iv. Extend square centimetre to square metre and to square kilometre for measure of large areas
v. Convert units and use appropriate units of area with fluency
vi. Draw the equivalent size of a rectangular area based on a given area with the factors of awhole number
vii. Appreciate the use of areas in daily life such as comparing land sizes
Sample Tasks for Understanding the Standards
Task1: Area and Perimeter

Diagram 1i.

Ms Apinya was making rectangular enclosures for ladybugs to move around using 1 cmlength plastic blocks. Diagram 1 shows two enclosures, ① and ②, each made using 20
pieces of the plastic blocks. Ms Apinya then cut out the two enclosed spaces. When she
tried to compare the two enclosed spaces, she found two extra regions, one longer andanother shorter. She then repeated the comparison process with the two extra regions
until she finally got a square, ③, that fitted nicely onto the shorter extra region. The spaceof square ③ is 1 square centimetre.
••

How many square centimetres is

the shorter extra region?

the longer extra region?



enclosure ①?

enclosure ②?

Which enclosure is larger, ① or ②?

Chapter 3: Measurement and Relations | 93

Diagram 2ii.

Ms Apinya gave 30 pieces of 1 cm-length plastic blocks for each of her students to make arectangular enclosure. Two students made the shapes, Ⓐ and Ⓑ, as shown in Diagram 2.
•

The two students agreed that the enclosed space in enclosures Ⓐ and Ⓑ are thesame because both rectangles were made from the same number of plastic blocks.
iii.

Explain how direct comparison can be done to help the students rectify their
misconception.

The space 1 square cm, commonly written as 1 cm² is a unit used to measure area.
•

Explain the meaning of 1 cm².

•

Explain how you will guide your students to find the area in cm², of rectangles Ⓐ andⒷ, respectively•

Find the area, in cm², of rectangles Ⓐ and Ⓑ, respectively.

iv.

What is the difference between area and perimeter?

v.

Investigate other possible rectangles formed by the 30 pieces of 1 cm-length plastic blocks.
•

What dimensions of rectangle will give the minimum area?

•

What dimensions of rectangle will give the maximum area?

•

Make a conjecture on the relationship between the dimensions of a rectangle to its
area.

•

Verify your conjecture with other perimeters such as 38 cm.
•What if you use a 38-cm string instead of the plastic blocks?

Make a conclusion on your conjecture.

94 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelvi. Diagram 3 shows a rectangle with square
grids.
•One square grid is shaded.
•How many square grids are there
altogether?


How is this number of square
grids related to the area of therectangle?

What mathematical operation
will enable you to find thenumber of square grids 		
directly? Explain your answer.

Diagram 3

Diagram 4

What is the area of this squaregrid in cm²?

•

Make a conjecture on a formula tofind the area of a rectangle.

•

Verify your conjecture and make aconclusion.

•

Does your conclusion also work for asquare such as Diagram 4. Why or
why not?

Chapter 3: Measurement and Relations | 95

Task 2: Adding and Subtracting Areas

Diagram 5i.

Diagram 5 shows a composite shape. Four students, John, Jason, Cindy and Catherine
find the area of the shape with different methods as shown in Diagram 6.

Diagram 6

96 | Mathematics Challenges for Classroom Practices at the Upper Primary Level•

The main idea used in Jason’s and John’s methods is the same. What is the main
idea?

•

What is the main idea used in Cindy’s method?

•

Catherine explained that “when I rearrange the shapes, the total area is still the same.”

Explain what does Catherine mean.

Diagram 7ii.

Explain how the main ideas in John’s, Jason’s, Cindy’s and Catherine’s methods can beused to develop formula of finding the area of triangle Ⓐ, trapezium Ⓑ, and parallelogramⒸ in Diagram 7.
•

Find the formula for finding the area of triangle Ⓐ, trapezium Ⓑ, and parallelogram Ⓒ.

Diagram 8iii.

Diagram 8 shows a piece of rectangular land with two crossed paths. Each path is 4 m
wide.
•

Calculate the area of the shaded region.

•

One student claimed that the area can be calculated using (46 m x 76 m). Explain how
this calculation is possible.

Chapter 3: Measurement and Relations | 97

Task 3: Larger Units of Areai.

Diagram 9 shows a 1 m x 1 m large square
divided into 100 medium squares. One
medium square at the bottom left corneris further divided into another 100 small
squares. The area of the large square is
1 square metre.
•

A medium square is a 10 cm x 10 cm
square. What is the area of a medium
square, in cm²?

•

What is the area of one small square,in cm²?

•

What is the relationship between 1 m²
and 1 cm²?

Diagram 9ii.

When asked to convert 2.5 m² to cm², two
students, John and Jason, showed their
methods of conversion as in Diagram 10.
•

Compare the two methods.
Which method will you promote toyour students?



Why?

Diagram 10

iii.

Diagram 11 shows a square with an area of
1 km².
•

Diagram 11

Explain how you will use the diagram
to help your students derive therelationship between km² and m².

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

Task 4: Equivalent Size of Rectangular Area

Diagram 12i.

Diagram 12 shows a rectangle P with an area of 36 m² drawn on a square grid.
•State the dimensions of rectangle P.

•

Draw other rectangles with the same area as P on Diagram 12.
•State the dimensions of each rectangle.

Explain how you can find the dimensions of each rectangle without drawing.

Chapter 3: Measurement and Relations | 99

Standards 3.2:
Extending area of a rectangle to other figures to derive formulae
i. Explore and derived a formula for the area of a parallelogram by changing its shape to arectangle without changing its area
ii. Explore and derived a formula for the area of a triangle by bisecting a rectangle into two
triangles without changing its areaiii. Appreciate the idea of changing or dividing shapes of a rectangle, parallelogram, or/and
triangle for deriving the area of other figures
iv. Use formulae to calculate areas in daily life
Sample Tasks for Understanding the Standards
Task1: Finding Area of Parallelogram and Triangle

Diagram 1i.

Diagram 1 shows a rectangle Ⓟ drawn on a 1 cm² grid.
•

What is the perimeter of rectangle Ⓟ?

•

Draw others rectangles on Diagram 1, each with the same perimeter with rectangle Ⓟ.
Which rectangle has the maximum area?



What can you conclude on the rectangle with maximum area?

100 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 2
ii.

Diagram 2 shows a rectangle Ⓟ and two parallelograms, Ⓠ and Ⓡ.
•

How could you show that Ⓟ, Ⓠ and Ⓡ have the same area, without any calculation,

Based on your answer, explain how you could derive a formula to find thearea of a parallelogram.

Given that each square grid is 1 cm². Use your formula to find the area ofparallelogram Ⓡ, in cm².

Diagram 3
iii.

Diagram 3 shows three triangles, Ⓐ, Ⓑ,and Ⓒ, each with a base b, and a height h.
••1

Prove that the area of each triangle is 2 × b × h.
How will you sequence the order of the three tasks in order to facilitate your students’reasoning process? Justify your decision.

Chapter 3: Measurement and Relations | 101

Diagram 4iv.

Mr. Ram gave triangle Ⓕ in Diagram 4 as a task for his students to think of different
methods to find the area. He picked four samples among the students to share their ideas
as shown in the diagram.
•

Which student’s idea is the same as yours?

•

What are the similarities and differences among the ideas of the 4 students?

•

Student ① changed the triangle into a rectangle.
•Student ② changed the triangle into a parallelogram.


•

•

v.

Write a mathematical expression to find the area of triangle Ⓕ.

Write a mathematical expression to find the area of triangle Ⓕ.

Compare students ② and ③ methods.
What is the same and what is different about the two methods?



Explain how the formula to find the area of triangle Ⓕ can be derived fromstudent ③ method.

Compare students ① and ④ methods.
What is the same and what is different about the two methods?



Explain how the formula to find the area of triangle Ⓕ can be derived fromstudent ④ method.

You intend to use the four students’ ideas in a classroom discussion.
•

Which students’ ideas will you discuss first? Justify your decision.

•

How will you sequence the order of discussion for the other three ideas? Justify yourdecision.

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

Diagram 5vi. Diagram 5 shows a geoboard. Three pieces of rubber bands were used to construct
triangles Ⓙ, Ⓚ, and Ⓛ with a common base AB.
•

Without any calculation, explain why the areas of the three trianges are the same.

•If the distance between any two dots on the geoboard is 2 cm, find the area of thetriangle in cm².

Diagram 6vii. Diagram 6 shows three triangles Ⓐ, Ⓑ and Ⓒ, with a common base RS constructed on a
geoboard.
•

Without any calculation, explain why the area of triangle Ⓒ is half the area of triangleⒷ.

•

Without any calculation, find the ratio of the area of triangle Ⓐ to the area of triangleⒷ.

Chapter 3: Measurement and Relations | 103

Task 2: Area of Other Figures

Diagram 7i.

Diagram 7 shows a trapezium PQRS. John found the area of the trapezium by cutting thetrapezium as shown in the diagram.
•

Find the area of the trapezium, in cm².

•

Explore at least 3 other ways of drawing to find the area of the trapezium.

•

What are the similarities and differences found in all the ways?

•

In what way will this task inculcate the appreciation of knowing the formulae of figures
such as triangle and rectangle?

Diagram 8ii.

Diagram 8 shows the trapezium ABCD and Jason’s calculation to find the area of the
trapezium.
•

Explain how Jason did the calculation.

•

Is Jason’s solution correct?

•

Justify Jason’s solution.

104 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Problem Solving Involving Land Areas

Diagram 7
Diagram 9i.

Diagram 9 shows two plots of land, Ⓐ and Ⓑ owned by Uncle Nathan. He called his twosons to choose one plot each. Both the sons wanted plot Ⓐ because it appears larger toboth.
•What would you do to find out if the choice was correct?

•

Which piece of the land is actually larger? Justify your answer with calculations.

Diagram 10
ii.

Diagram 10 shows a regular pentagon and a regular hexagon, with centres A and B
respectively.
•

Use a ruler to make some measurements needed for the calculation of the area of thehexagon and pentagon, respectively.
•

Use the measurements to calculate the area of each shape.

What is the minimum number of measurements needed for calculating the area of
each shape?
Calculate the area of each shape using the minimum number of 			measurements.

Chapter 3: Measurement and Relations | 105

Topic 4: Extending Measurement of Volume in Relation to Surface
Standards 4.1:
Introducing volume from area and derive formula for cuboid
i. Compare the extent of volume and introduce its unit, and distinguish it from surface
ii. Introduce one cubic centimetre as the unit for volume and its addition and subtractioniii. Investigate the volume of a cuboid and cube and produce the formulae
iv. Extend cubic centimetre to cubic metre to measure large volume
v. Convert units and use appropriate units of volume with fluency
vi. Appreciate the use of volume in life such as comparison of capacity of containers
Sample Tasks for Understanding the Standards
Task1: Relating Area to Volume

Diagram 1i.

Diagram 1 shows two boxes, ① and ②. Box ① is 3 cm x 3cm x 1cm and box ② is 4cm x
2cm x 1cm. A student compared the sizes of the two boxes and found that there was an“extra part” from each box, ③ and ④. He continued the process of comparison and finally
got a 1 cm cubic box as shown in the diagram. The volume of the 1 cm cubic box is 1 cubic
cm, written as 1 cm³.
••

What is the volume of the following, in cm³?
Extra part ④Extra part ③



Box ②



Box ①

What previously learned knowledge is required for your student to learn the process ofderiving 1 cm³ by Euclidean algorithm?

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

Diagram 2
ii.

Diagram 2 shows two nets used to make two boxes, Ⓐ and Ⓑ, Catherine tried to compare
the volume of the two boxes by puting them side by side. However, she seem to havesome difficulties doing it.
•

Explain how 1 cm cubic blocks can be used to help Catherine determine which box islarger.

•Which box is larger and how much larger?

Diagram 3
iii.

Diagram 3 shows a box, Ⓡ, filled with a layer of 1 cm cubic blocks.
•

How many 1 cm cubic blocks are there in this layer?

•

How many layers of 1 cm cubic blocks can fill the whole box?

•

How many 1 cm cubic blocks altogether can fill the whole box?

•

What is the volume of box Ⓡ, in cm³?

•

Explain the relationship between the base area of a rectangular box and its volume.

•

Base on the relationship, state the formula of the volume of a rectangular box.

Chapter 3: Measurement and Relations | 107

Diagram 4
iv.

Diagram 4 show a rectangular box Ⓢ. The area of one of its vertical faces is 24 cm².
•

Find the volume of box Ⓢ, in cm³.

•

How can this task help your students learn other mathematical ideas such as volume
of a prism in future grade?

108 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Larger Units for Volume

Diagram 5i.

Diagram 5 shows a 1 cm cubic block, Ⓐ, and another 1 m cubic block, Ⓑ.
•

How many blocks Ⓐ are there in one block Ⓑ?

•

What is the relationship between cm³ and m³?

•

What is the volume of block Ⓑ, in m³?

•

How can this task help your students understand the size of 1 m³?

Diagram 6•

Diagram 6 shows a rectangular block Ⓣ which is made up of several blocks Ⓑ.
How many blocks Ⓑ are there in block Ⓣ?

What is the volume of block Ⓣ in m³?

Chapter 3: Measurement and Relations | 109

ii.

Table 1 shows a list of dialy items.
•

Choose from cm³ or m³, as appropriate unit to measure the volume of each item.

•

Justify each of your choices.

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

Standards 4.2:
Extending volume of a cuboid to other solid figures to derive formulai. Extend the formula for the volume of a cuboid as base area x height for exploring solid figures
such as prism and cylinder
ii. Extend the formula for the volume of a prism and a cylinder to explore and derive the volume
formula of a pyramid and coneiii. Use formulae to calculate volume in daily life
Sample Tasks for Understanding the Standards
Task1: Volume of Prisms and Cylinder

Diagram 1i.

Diagram 1 shows two prisms, Ⓐ and Ⓑ, where the base area of Ⓑ is ½ of Ⓐ.
•

Name the two prisms.

•

The two prisms are to be filled with 1 cm cubic blocks.
How many blocks will be needed to fill one layer of Ⓐ and Ⓑ, respectively?

How many layers is needed to fill each prism?

What is the volumn of each prism, in cubic cm?

•

Explain the relationship between the base area of a prism and its volume.

•

Base on the relationship, state the formula of the volume of a prism.

Chapter 3: Measurement and Relations | 111

Diagram 2
ii.

Diagram 2 shows a prism with two parallel hexagonal faces. The area of one face is
61 square cm and the perpendicular distance between the two faces is 10 cm.
•

Find the volume of the prism, in cubic cm.

Diagram 3
iii.

Diagram 3 shows a prism with two parallel pentagonal faces. The area of one face is
52 square cm.
•

What missing information is required for you to calculate the volume of the prism?

•

Provide a value for the missing information and find the volume of the prism.

Diagram 4
iv.

Diagram 4 shows a circular card with an area of 12.5 sqrare cm. The cards are stacked up
to 4 cm high to form a cylinder.
•

In what way can the knowledge of finding volume of a prism helps your students findthe formula for the volume of a cylinder?

•

Find the volume of the cylinder, in cubic cm.

112 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Volume of Various Space Solids

Diagram 5i.

Diagram 1 shows a cube ABCDEFGH. The cube is trisected into three identical pyramids,Ⓐ, Ⓑ and Ⓒ.
•

Given the side AB = a cm.
•Explain why the volume of the pyramid is V= 13 × a × a × a.

If a = 6 cm, find the volume of the pyramid, in cm³.

Diagram 6ii.

Diagram 6 shows two containers, Ⓡ and Ⓢ. Container Ⓡ is a rectangular prism witha base of 4 cm x 3 cm, and a height of 6 cm. Container Ⓢ is a pyramid with the same
rectangular base and height.
•

John filled container Ⓢ with sand fully. Then, he poured the sand into container Ⓡ.
Mark the level of sand in container Ⓡ on Diagram 7.

Chapter 3: Measurement and Relations | 113

Diagram 7•

What is the relationship between the volume of Ⓡ and Ⓢ.

•

Verify the relationship.

•

Base on the relationship, state the formula for the volume of a pyramid.
What if the base of the pyramid is a square?

Diagram 8
iii.

iv.

Diagram 8 shows a cone and a cylinder with the same base diameter and height.
•

What is the relationship between the volume of the cone, V1, and the Cylinder, V2.

•

Verify the relationship.

•

Base on the relationship, state the formula for the volume of a cone.

Digram 9 shows a L-shaped solid Ⓖ and a pentagonal prism Ⓗ.
•

Calculate in different ways, the volume of each solid in cm³.

114 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 5: Approximating Quantities
Standards 5.1:
Approximating numbers with quantities depending on the necessity of contexts
i. Understand the ways of rounding such as round up and round down
ii. Use rounding as an approximation for making a decision on the quantity with related context
iii. Critique approximation beyond the context with a sense of quantity such as based on therelative size of units
Sample Tasks for Understanding the Standards
Task1: Different Ways of Rounding

Diagram 1i.

ii.

Diagram 1 shows a problem and two students’ ideas on the solution.
•

Explain the different ways of rounding used by each of the students.

•

Which way of rounding is inappropriate for this situation? Justify your decision.

There are 1 685 sheets of greeting cards to be packed into boxes for sale. Each box musthave 100 sheets of cards.
•

How many boxes of greeting cards can be packed for sale?

•

Which way of rounding is appropriate in this situation? Justify your decision.

Chapter 3: Measurement and Relations | 115

Diagram 2
iii.

iv.

Diagram 2 shows part of a number line.
•

Explain how the number line can be used to help your students round 2 134 to thenearest hundred.

•

What way of rounding is used in this case? Explain your answer.

•

Explain briefly, the rule used in this way of rounding.

A number ⓐ is rounded to the nearest ten, and the answer is 60.
•

What are the possible numbers for ⓐ?

•Show your answers on a number line.
v.

Table 1 shows the exact number of newcases of COVID-19 reported on specificdates in a country. A student roundedthese numbers to the nearest thousand.
•Which set of numbers should a healthofficer refer to for more efficientmonitoring of the situation? Justify
your decision.

•

Which set of numbers is sufficient
to provide a real situation to acommoner? Justify your decision.

vi. The following are some numbers.
56 476,57 501,58 573,58 231,57 499,58 500,57 634,57 256•

Which numbers become 58 000 when rounded to the nearest thousand?

•Which numbers become 57 000 when rounded down to the nearest thousand?

•Which numbers become 57000 when rounded up to the nearest thousand?

116 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Making Decisions in Context

Diagram 3i.

Diagram 3 shows a list of items on Anada’s shopping list. In one month, Anada had saved
1 000 Bahts.
•

Using appropriate estimation, what is the most number of items he can buy?

•

Estimate the amount of money required for him to buy all the items.


Explain how you do the estimation.

How do you ensure that your estimation can guarantee you enough money tobuy all the items?

Diagram 4ii.

Diagram 4 shows how John and Jason did their estimation of visitors to a shopping complex
in their trip report.
•

Whose way of rounding is more sensible? Verify your decision.

iii.

Why is it necessary to use estimation in our daily life?

iv.

A primary school has 2 856 students. The headmaster informs the board of directors that
there are about 3 000 students, whereas the assistant headmaster informs the parentteacher association that the school has about 2 900 students.
•v.

Which approximation is more appropriate? Justify your decision.

In the recent mathematics test, Lina’s teacher announced that 92% of the students passed
the test. When reaching home, Lina told her mother that about 100% of her class students
passed the test.
•

Is it appropriate for Lina to tell the mother an estimated passing percentage? Justify
your decision.

Chapter 3: Measurement and Relations | 117

vi. Table 2 shows the information collected from 121 factory workers on the transportation
they used to go to work. A student calculated the ratio of each group of workers to the total
number of workers for each type of transportation. The ratios were then rounded to thenearest percentage as shown in the table.
Table 2
Transportation to work•

Explain how each of the ratios was rounded to the nearest percentage?

•

Why the total percentage is more than 100%?

118 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 6: Extending Proportional Reasoning to Ratio and Proportion
Topic
6: Extending
Proportional
Reasoningto Ratioand Proportion
Topic
6: Extending
Proportional
Reasoningto Ratioand Proportion
Standards 6.1:
Standards
Standards
6.1: 6.1:
Extending proportional reasoning to ratio and percent for comparison
proportional
to ratioand percent
forcomparison
Extending
proportional
to between
ratio
and two
percent
forquantities
comparison
i.Extending
Understandratio asreasoning
thereasoning
relationshipsame
or between two differenti.
Understand
ratio
as
the
relationshipbetweentwosame
quantities
or between
two differenti.
Understand
ratio
as
the
relationshipbetweentwosame
quantities
or between
two different
quantities (the latter idea is rate)quantities(thelatter
idea
is
rate)
quantities(thelatter
idea
is
rate)
ii. Express the value of a ratio by quotient such as the rate of two different quantities
ii. UnderstandExpress
the percent
value
ofas
a ratioby quotient
such
asthe
the
rate
of
two
quantities
ii. Express
the value
of
ratio
by
such
assamethe rateof different
twodifferent
quantitiesiii.
thea valueof quotienta ratiowithquantity
andthenecessityof roundingiii.
Understandpercent
as
the
value
of
a
ratio
with
the
same
quantity
andthenecessityof roundingiii.
Understandpercent
as
the
value
of
a
ratio
with
the
same
quantity
andthenecessityof rounding
iv. Understand proportional reasoning for ratio as part-whole and part-part relationshipsiv.
Understandproportional
reasoning
for
ratio
as
part-wholeandpart-part
relationships
iv. Understandproportional
for ratio as part-whole and part-part relationships
v. Apply
the rule of three
in usingreasoning
ratio
v. Apply
the rulethree
in using
ratio ratio
v. Apply
the ofruleof threein using
Sample Tasks for Understanding the Standards
SampleTasks
for Understanding
the Standards
SampleTasks
for Understanding
the Standards
Task 1: Proportional Reasoning in a Context
Task 1: Proportional Reasoning in a Context
Task 1: Proportional Reasoning in a Context

1 month later
1 month later

1 month later
1 month later

1010 g1010 g(2 months old)
(2 months old)

1460 g1460 g(3 months old)
(3 months old)
Diagram 1
Diagram
Diagram
1 11910 g1910 g(4 months old)
(4 months old)

Diagram
Diagram11 1shows
showsaa akitten’skitten’sbody
bodyweightweightfor
forthreethreeconsecutive
consecutivemonths.
months.
Diagramshows
kitten’sbodyweightfor
threeconsecutive
months.
i. i. What
is is
your
prediction
onon
thetheweightof ofthethekittenwhen
it is
5 months
old?
What
your
prediction
weightkittenwhen
it is
5 monthsold?
i. Whatis yourprediction
on theweightof thekittenwhen
it is
5 monthsold?
 Explain the reason of your prediction.
 • Explain
thethereasonof yourprediction.
Explainreasonof yourprediction.
 Discuss the accuracy of your prediction.
 Discuss the accuracy of your prediction.
•  What
Discuss
the accuracyof yourprediction.
assumption
do youmakefor your prediction?
 What assumption do you make for your prediction?
 How
can youjustify it? do you make for your prediction?
assumption 
HowWhatcan youjustify it?
 How do you determine if the situation is proportional to help you make a decision?
 How
you determine
if theit?
situation is proportional to help you make a decision?
 doHow
can you justify
ii. What is your guess on the weight of the kitten at birth?
ii. What is your guess on the weight of the kitten at birth?
• manyHow minimum
do you determine
if theis toproportional
to help you
make prediction?
a decision?
iii. Howsets of data
dosituation
you needmake a reasonably
reliableiii. How many minimum sets of data do you need to make a reasonably reliable prediction?

ii. What is your guess on the weight of the kitten at birth?
Note. For this case, we are able to find any possible function that fit three known ordered pairs,
Note. mathematically.
For this case, we are able to find any possible function that fit three known ordered pairs,iii. mathematically.
How many minimum sets of data do you need to make a reasonably reliable prediction?
Note. For this case, we are able to find any possible function that fit three known ordered pairs,
mathematically.

Chapter 3: Measurement and Relations | 119

Task 2: Comparing Quantities by Ratiosi.

Table 1 shows the number of passengers
and the number of seats available inthree buses. A student compared thecrowdedness between the buses asfollows:
Comparison ①: Bus Ⓐ and Bus Ⓑ
Comparison ②: Bus Ⓐ and Bus Ⓒ
Comparison ③: Bus Ⓑ and Bus Ⓒ

ii.

iii.

Diagram 2•

Why is comparison ③ more difficult todo than comparisons ① and ②?

•

Explain how the idea of ratio can beused to make comparison ③.

Ambrose constructed two tables as shownin Diagram 2 to compare the degree of thecrowdedness of Bus Ⓑ and Bus Ⓒ.
•

Explain Ambrose’s method.

•

Find the missing number in each•

Which bus is more crowded ?

.

There are two ways to look at thecrowdedness of the bus: (a) the number ofpassengers per seat, and (b) the numberof seats per passenger.
•

Explain how the two ideas are used
to determine the degree of thecrowdedness of the bus.

•

Which idea is used in Ambrose’s
method?

•Use the other idea to determine thedegree of the crowdedness of the bus.

120 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

iv.

v.

Amrin drew two graphs as shown in
Diagram 3 to compare the degree of thecrowdedness of Bus Ⓑ and Bus Ⓒ.
•

Explain Amrin’s method.

•

Find the missing number in each.

Compare Ambrose’s method with Amrin’s
method.
•What is the same and what isdifferent between the two methods?

vi. Compare the degree of the crowdednessof Bus Ⓐ, Bus Ⓑ and Bus Ⓒ.

Diagram 3•

Which bus is the most crowded?

•

Which bus is the least crowded?

vii. Diagram 4 shows a class of 16 girls and 20
boys.
•

Find the ratio of

the number of girls to the numberof boys.
(Note. The base value forcomparison is the number ofboys.)

the number of boys to the numberof girls.



the number of girls to the total
number of students.

the number of boys to the total
number of students.

•

Why are the values of the four ratios
not the same?

•

What is the purpose of using a
different base value for comparison?

Diagram 4

Explain your answer with a reallife example.

Chapter 3: Measurement and Relations | 121

Task 3: Ratio and Rate
Table 2
Distance Travelled with TimeA man went for a brisk walk. Table 2 shows the record of time taken and distance travelled by
the man.
i.

ii.

iii.

iv.

Distance and time are two different quantities with different units of measure. The ratio of
distance to time is also known as speed, which is an example of rate.
•

What is the difference between ratio and rate?

•

What is the unit of measure for the ratio of distance to time?

The ratio of distance to time for the interval 0 – 15 mins is shown in Table 2.
•

Find the ratio of distance to time for every other interval.

•

What can you say about the speed of walking of the man?

The man walked for 240 minutes.
•

Calculate the total distance travelled, in km.

•

What assumption do you make in the calculation?

The man walked 10 km.
•

Calculate the time taken, in minutes.

•

What assumption do you make in the calculation?

122 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 4: Percent as the Value of Ratio

Diagram 5i.

Diagram 5 shows the conversation between three students, Amrin, Ambrose and Asiah,
discussing the solution for the bus crowdedness problem.
•

Compare the ideas of Amrin and Ambrose.
•What is the same and what is different between the two ideas?

How does Asiah obtain the degree of crowdedness as 1?
What does it mean by crowdedness of 1?

Diagram 6ii.

Ambrose explained his ideas further by drawing a graph as shown in Diagram 6.
•

Explain Ambrose’s ideas.

•

Find the missing number in•Express the ratio 0.03 and 0.3 as a percentage, respectively.

.

Chapter 3: Measurement and Relations | 123

•
•

9

3

Express the ratio 100 and 4 as a percentage, respectively.
On another day, taking the 50 seats in a bus as the base value, the degree of thecrowdedness of the bus is found to be 90%.
How many passengers are there on the bus?

Table 3
Types of Vehicles Passed by a School Gate

iii.

iv.

Table 3 shows the number of various vehicles on the road in front of a school gate for
1 hour.
•Use Amborose’s method to find the ratio of each type of vehicle to the total number ofvehicles.

•Fill in the blank in Table 3.

•

What is the total of all the percentages?

A bus with 50 seats is filled with 55 passengers.
•

Calculate the degree of crowdedness as a percentage.

•

What does it mean when the percentage is larger than 100%?

124 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 4: Equivalent Ratios

Diagram 7
Diagram 7 shows an orange concentrate with the value of ratio 1 : 9 to make the drink with adilution of water.
i.

What does the ratio 1 : 9 mean?

ii.

If you need to make a glass of 300 mℓ orange drink, how much of concentrate and water isrequired?

iii.

There are 25 people in a meeting and each is served a glass of orange drink.
•iv.

How would you prepare the drink?

Christ prepares his own drink by adding 20 mℓ concentrate to 180 mℓ water. Hardy preparesthe drink for a few friends. He adds 120 mℓ concentrate into 1 ℓ 80 mℓ of water.
•

Will both drinks taste the same? Explain why or why not.

•What is equivalent ratio?

•

If Hardy wants to use only 80 mℓ of concentrate, how much water should he use?

Chapter 3: Measurement and Relations | 125

Task 5: Applying Ratio and Rate to Solve Problems

Diagram 8i.

A supermarket sells apples and oranges in three different packages, small, medium andlarge. As shown in Diagram 8, a small packet has 1 apple and 2 oranges, a medium packet
has 3 apples, and a large packet has 8 oranges. Given that the number of apples and thenumber of oranges in all the packets are in the same proportion.
•

What is the ratio of the number of apples to the number of oranges in the small packet?

•

What does it mean by “the number of apples and the number of oranges is in the same
proportion”?

•

Find the number of oranges in the medium packet.

•

Find the number of apples in the large packet.

Diagram 9ii.

Diagram 9 shows a tree and a 2-metre wooden pole on a sunny day. The shadow of thewooden pole and the tree is 3 m and 18 m, respectively.
•

Explain how the idea of proportion can be used to find the height of the tree.

•

Using the rule of three, calculate the height of the tree.

126 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 10
iii.

iv.

The student-teacher ratio for a school in a country is a maximum of 24 students to
1 teacher. Diagram 10 shows Catherine’s method to find the number of teachers for aschool with 432 students.
•

Explain Cindy’s method.

•

Find the missing number•

Find another way to get the answer. [Hint: Read the table horizontally.]•

Another school has 25 teachers. Use the rule of three to guess the number of students.

.

Mr Bảo used 8 litres of emulsion paint to paint 2 rooms.
•

How many litres of emulsion paint is needed to paint 6 similar rooms?

•

How many similar rooms can he paint using 28 litres of emulsion paint?

Diagram 11v.

aA ratio of two quantities, a and b, can be written either in the form of a : b or b .
Diagram 11 shows a recipe to prepare a salad dressing using olive oil and lemon juice.
•

Let a represent the volume of olive oil and b represents the volume of lemon juice.
Write the ratio a : b in its simplest form.



Rewrite the ratio in the form of b .

How much lemon juice should be used with 40 mℓ of olive oil to prepare the 		salad dressing?



What is the unit of measure for the ratio of olive oil to lemon juice?

aChapter 3: Measurement and Relations | 127•

Let a represent the volume of olive oil and c represents the number of teaspoons of
salt.
a



Write the ratio c in its simplest form.



Rewrite the ratio in the form of a : c.

How much salt should be used with 40 mℓ of olive oil to prepare the salad 		dressing?

What is the unit of measure for the ratio c ?
avi. What are the major difference between the ratios b and c ?
aavii. The recipe for salad dressing is the quantity for 6 people.
•

What should be the quantity used in the recipe for 18 people?
viii. Diagram 12 shows a problem involving
pancakes and two students’ solutions to
the problem.

•

Why did Jason divide the ratio by 2?

•

How did Cindy get 30 g of pancake
mix and 25 g of milk for each cake?

•

Comparesolutions.


Diagram 12

Jason’sandCindy’s

Which is easier for your students
to understand the meaning of
proportion? Justify your decision.

128 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

ix. Find the missing numbers in the (
•

4:6=(

•

3 : 7 = 63 : (•120 : (•(

) for the following proportions.

) :150)) = 10 : 18

) : 240 = 3 : 8

Diagram 13

x.

Diagram 13 shows two squares. The proportion of the length of the sides is 3 : 4.
•

When the length of a side of the smaller square is 18 cm, what is the length of thecorresponding side of the larger square?

•

If the length of a side of the larger square is 64 cm, how many cm is the length of theside of the smaller square?

•

Is it possible to make the square becomes a rectangle with the same proportion?


Explain your answer with an example.

Chapter 3: Measurement and Relations | 129

Standards 6.2:
Extending proportional reasoning to proportioni. Extend proportional reasoning on multiplication tables as equal ratios and understandproportions
ii. Understand proportion by multiple and a constant quotient, not changing the value of the ratioiii. Demonstrate simple inverse proportion by constant productiv. Express proportion in a mathematical sentence by letters and graphv. Use properties of proportionality to predict and explain phenomena in daily life
Sample Tasks for Understanding the Standards
Task1: Exploring Ratios in Multiplication Tablesi.

Diagram 1 shows the multiplication table
for the number 6.
•

Diagram 1

ii.

Diagram 2iii.

Diagram 3

Study the first five pairs of numbers aand b.


What pattern do you observe ineach successive pair of a and b?



Does the pattern also hold for the
next four successive pairs of aand b?

Why do you think such a pattern
exists?

Pick any pairs of a and b such as theexample shown in Diagram 2.
•

Find the missing number in each•

What pattern do you observe?

•

Does the pattern also hold for any
other pairs of a and b?

•

Why do you think such a pattern
exists?

.

The pairs of a and b are extended beyond
the multiplication table as shown in
Diagram 3.
•

Find the missing number in each•

Does the pattern in Diagram 2 still
exist? Why or why not?

.

130 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

iv.

v.

Diagram 4b

The quotient a is the ratio of b to a for eachpair of a and b.
•

What can you conclude about theseratios?

•What about the ratio of a to b?

Diagram 4 shows two pairs of numbers
from another different multiplication table.
•

Find the missing number•

Explain your reasoning for finding themissing number.

.

Diagram 5vi. Diagram 5 shows some marbles that will be kept in fancy boxes. Each box can keep 18
marbles. Table 1 shows the number of marbles for 1 to 5 boxes.
Table 1
Number of Marbles in Boxes•

The numbers 18, 36, 54, 72 and 90 are multiples of 18.


Explain the additive property in these multiples.
Why is zero significant in the additive property?

Explain the multiplicative property in these multiples.
Why is zero problematic in the multiplicative property?

Chapter 3: Measurement and Relations | 131•

The number of marbles is said to be proportional to the number of boxes.
Find the ratio of the number of marbles to the number of boxes for each pair of
numbers in Table 1 except the ‘zero’ pair.

What can you conclude about the ratios?



If there are 15 boxes, how many marbles are there?

How many boxes are needed to keep 240 marbles?

Diagram 6•

Diagram 6 shows a proportional number line drawn to determine the number ofmarbles kept in 13 boxes.
•Using the Rule of Three, find the missing number in.

Draw a proportional number line and use the Rule of Three to determine the numberof boxes needed to keep 414 marbles?

132 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Proportionsi.

A group of students collected some data to study the relationship between the number ofmarbles and mass. Table 2 shows the data collected.
•

What pattern do you observe between x and y in Table 2?

Table 2
Number and Mass of Marbles

Diagram 7•

Ambrose and Amrin reported an interesting observation as shown in Diagram 7.
•What does it mean by the mass of marbles is proportional to the number ofmarbles?

For each pair of x and y, calculate the quotientyx

and fill in the blank in Table 2.



Write a mathematical sentence involving x and y to represent the proportional
relationship.

Use the mathematical sentence to find(a) the mass of 15 marbles.
(b) the number of marbles when the mass is 360 g.

Chapter 3: Measurement and Relations | 133

Diagram 8

ii.

•

Miss Sarsi asked her students to find the mass of 12 marbles. Diagram 8 shows threesolutions from her students.

•

Whose solution is the easiest to show the proportion? Why?

Tables 3 and 4 show two sets of data.
Table 3Distance Travelled with Volume of Petrol

Table 4Temperature Change with Timey•Calculate the quotient, x , and fill in the blank in Table 3.

•

Calculate the quotient, Tt , and fill in the blank in Table 4.

•

Which table shows two quantities that are in direct proportion? Justify your decision.

134 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 4: Inverse Proportion

Diagram 9i.

Diagram 9 shows the duration of time in months, required by different numbers of workers
to build a house.
•

Find the missing number in each•If the value of x changes 2 times, 3 times and so on, how does the value of y change?

.

Table 5
Number of Workers and Duration of Time to Build a House••For each pair of x and y, calculate the product x × y and fill in the blank in Table 5.
Fill in the blank in Table 5.

What pattern do you observe between the products of x and y?

In this case, y is said to be inversely proportional to x.
Write a mathematical sentence involving x and y to represent the relationship.

Use the mathematical sentence to solve the following problems.
(a) If 10 workers are available to build the house, how many months will it taketo complete the house?
(b) If we want to complete the house in 5 months time, how many workers do weneed?

Chapter 3: Measurement and Relations | 135

Task 5: Graphs of Proportion
Table 6Mass of Apples and Total Costi.

A supermarket in Chengmai is having
a “Crazy Sale” to sell apples bykilogrammes. 1 kg of apple is sold for
20 Thai Baht. Table 5 shows the massof apple x, and the total cost y.
•

Find each missing value of y and fillin each blank in Table 6.

•

What is the relationship between
the x and y?

•

Write a mathematical sentence torepresent the relationship between
x and y.

•Plot points that represents all pairsof values, the value of x and its
corresponding value of y, on thegraph in Diagram 10.

•

Connect the points with a line.

•

Diagram 10

What kind of line do you get?



Use the graph to find the costof 3 kg and 5 kg of apples,
respectively.

Explain how the graph can be used
to find the cost of 8 kg of apples.

136 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 11ii.

Diagram 11 shows graph Ⓖ and graph Ⓡ, representing the relationship between the massof apples, x in kg and the cost, y in Thai Baht for green apples and red apples, respectively.
•

Which type of apple is more expensive? Explain how you find it from the graph.

•

Aki has 100 Baht only.

•How many kg of red apples can he buy?

How many kg of green apples can he buy?

Sui has 140 Baht only. She wants to spend all the money to buy apples.
Explain how the graphs can be used to find different combinations of green
apples and red apples that she can buy.

Chapter 3: Measurement and Relations | 137

Task 6: Graph of Inverse Proportion

Diagram 12i.

Diagram 12 shows a rectangle with length, x cm and width, y cm. The rectangle has a fixedarea of 24 cm². Table 7 shows some combinations of different values of x and y.
Table 7
Length and Width of a Rectangle with a Fixed Area of 24 cm²•

Find and fill in each blank with the missing value of y in Table 7.

•

What relationship do you observe between x and y?

•

Plot a point that represents each pair of values, the value of x and its corresponding
value of y, on the graph in Diagram 13. The first three points have been plotted foryou.
Connect the points with straight line.

Diagram 13•

Compare the graph with the graph of direct proportion in Diagram 10.


Describe the differences between the two graphs.

138 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 14ii

You leave Kuala Lumpur and use the PLUS highway to drive to Melaka, the historical city of
Malaysia. The distance between Kuala Lumpur and Melaka is 150 km. Diagram 14 shows
the graph of speed, y (km/h) and travelling time, x (hour).
•What is the relationship between the speed, y and the travelling time, x?
•Write a mathematical sentence to represent the relationship.

Use the graph to solve the following problems.


If you want to arrive in Melaka in 1 hour 30 minutes, how fast in km per hour doyou need to drive?



If you drive at 75 km/h, how long will the journey take?

Check your solutions using the mathematical sentence that represent therelationship between y and x.

Chapter 3: Measurement and Relations | 139

Task 7: Making Prediction with Properties of Proportionality

Diagram 15
Global warming is causing global sea level to rise mainly due to the melting of mountain glaciersand ice in the North and South poles. According to Climate Change Knowledge Portal, thecurrent rise is approximately 3 mm per year.
[Source: https://climateknowledgeportal.worldbank.org/country/malaysia/impacts-sea-level-rise ]i.

It is predicted that the sea level will rise at least 3 cm in next 10 years. Based on thisprediction, draw a graph of rise of sea level, y cm and time, x years later on Diagram 15.

ii.

Use the graph to make the following prediction.
•

After how many years, will the land that is 24 cm above the sea level now be coveredby the sea completely?

•

After 100 years, how many cm will the sea level rise?

140 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 7: Producing New Quantities Using Measurement Per Unit
Standards 7.1:
Producing new quantities using measurement per unit
i. Introduce average as units for distribution and comparison of different sets of values
ii. Introduce population density with the idea of average and appreciate it for comparison
iii. Introduce speed with the idea of average and appreciate it for comparison
iv. Appreciate using diagrams such as number lines and tables to decide the operations on thesituations of measurement per unit quantity
v. Compare the context of different quantities with the idea of average as a rate
vi. Apply the idea of measurement per unit quantity in a different context
Sample Tasks for Understanding the Standards
Task1: Understanding Average

Table 1Students Saving from Pocket Moneyi.

Table 1 shows the pocket money savedby three students over five days. Min was
absent for one of the days.
•

Compare the saving of the three
students.
•Diagram 1

Who had saved the most pocket
money?

John and Jason had different opinions
on who had saved the most pocket
money. Their argument is shown in
Diagram 1.
What is the difference between
John’s and Jason’s opinions?



Whose argument do you support?
Why?

Chapter 3: Measurement and Relations | 141

Diagram 2
ii.

iii.

Diagram 2 shows Cindy’s method to find the average daily saving of Sui by drawing a bar
graph.
•

Explain Cindy’s method.

•

Find the average saving of Sui from the diagram.

Use Cindy’s method to draw a separate bar graph on Diagram 3 to find the average dailysaving of Ann and Min, respectively.

Diagram 3

142 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 4
iv.

There is some orange juice in five containers as shown in Diagram 4.
•

What difficulties may your students face if they are to use Jason’s method to find theaverage amount of orange juice?

Diagram 5•

Diagram 5 shows Jason’s idea for finding the average amount of orange juice.
What formula for finding the average can you derive from Jason’s method?

What are the strengths and weaknesses of Jason’s method compared to Cindy’s
method?

Chapter 3: Measurement and Relations | 143v.

The same number which is averaged from some numbers is also called the mean of theoriginal numbers. Table 2 shows the number of pages of the storybook read by 6 students
in a week.

		
		

•

•

Table 2
Number of Pages of Story Book Read

When asked to calculate the mean number of pages read by the six students, Jason
and John answered 68 and 56.7 pages, respectively.
Why are there different answers given as the mean value?

What is the correct value of the mean for the 6 students?		

How many pages of storybook must student ② read so that the mean will be 70 pages?

144 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Average as Population Density

Diagram 6i.

ii.

Diagram 6 shows some chickens in enclosures Ⓟ, Ⓠ and Ⓡ. Enclosure Ⓟ, Ⓠ and Ⓡ has
an area of 4 m², 9 m² and 9 m², respectively.
•

Between Ⓟ and Ⓠ, which one is more crowded? Explain your reasoning.

•

Between Ⓠ and Ⓡ, which one is more crowded? Explain your reasoning.

•

Between Ⓟ and Ⓡ, which one is more crowded? Explain your reasoning.

Crowdedness of the chicken is affected by two measures, the number of chickens and thearea of the enclosure.
•

How many chickens are there per 1 m² for each enclosure? Show your calculations.

•

The number of chickens per 1 m² expresses the mean of crowdedness. Fill in theblank for the following two statements.

•
iii.



If the value of the mean of crowdedness is higher, the level of crowdedness is
____.



If the value of the level of crowdedness is lower, the mean of crowdedness is____.

Based on your calculations, which enclosure is the most crowded?

Table 3 shows the population and area of 3 capital cities of ASEAN countries.

				

Table 3Population and Land Area of Three ASEAN Capital Cities

1 982 112Chapter 3: Measurement and Relations | 145•

Population density is the number of people per square kilometre.


iv.

Find the population density of the three cities, respectively.

•

How is population density related to the level of the crowdedness of a city?

•

Based on the population densities, which city is the most crowded?

Explain how average is related to population density as a measurement per unit quantity.

146 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Average as Speed
Three boys, Joe, Tim and Sim were running in a field. Table 4 shows the distance covered in
metres (m) and the time taken in seconds (s) by the three boys.
Table 4
Distance Run and Time Takeni.

Compare the distance covered and the time taken by the three boys.
•

Who ran faster between Tim and Joe? Explain your reason.

•

Who ran faster between Joe and Sim? Explain your reason.

•

Why is it more difficult to compare who ran faster between Tim and Sim?

Diagram 7
ii.

iii.

Cindy calculated the average distance per second for Tim and Sim to make a comparisonas shown in Diagram 7.
•

Explain how proportional reasoning is used by Cindy.

•

Find the missing value•

Who ran faster between Tim and Sim?

.

Speed is an example of measurement per unit.
•

How is speed expressed in this case?

•

What is the measurement unit for speed in this case?

•

Give some other examples of measurement units for speed?

Chapter 3: Measurement and Relations | 147

Task 4: Applying Measurement Per Unit Quantity
Finding Average Speed

Diagram 8i.

ii.

Diagram 8 shows the speed of a car that changes with situations of traffic on the road. The
number line shows the speed of the car travelling on a road for 2 hours.
•

What is the average speed of the car? Show how you calculate.

•

Why is the average speed necessary to be used in telling the speed of a travelling car?

Town P is 150 km away from town Q. The highway connecting the two towns has a speed
limit of 110 km/h.
•

What is the shortest possible time for a car driver to travel from P to Q?

•

Do you think the driver can reach Town Q as shown by your calculation? Why or why
not?

Rate and Averagei.

Ricky, Andy and Danny cycle to school every day. Table 5 shows the distance from home,time taken and cycling speed of the three students on a schooling day.
Table 5Distance, Time and Cycling Speed of Three Students•
ii.

Rate can be expressed as a quantity measured per unit of another quantity.
•iii.

Find the missing value for each question mark ‘?’.

Explain the meaning of each of the following examples of rate.
A speed of 90 km/h.

Water flows from a pipe at 2.5 litres per second.



A tree grows at 58 cm per year.

How is the rate related to the average? Illustrate your answer with a suitable example.

148 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Problem Solving with Measurement Per Unit Quantity

Diagram 9i.

ii.

Ms Kana gave a problem to her students as shown in Diagram 9.
•

Explain how the idea of a number of words per unit of time can be used to solve theproblem.

•Find the rate of typing for Mr Ken and Ms Siva, respectively.

Aida is preparing food and drink for her birthday party. Consider the following problems
she needs to solve in the preparation.
Problem ①: Each friend is allocated 8 cherries. How many cherries are needed for
12 friends?
Problem ②: She prepares 12 pieces of doughnuts for 4 special friends coming to theparty. How many doughnuts will each special friend get?
Problem ③: The cost of 8 cupcakes is 160 Philippines pesos. How much will she needto buy 35 cupcakes?

Diagram 10•

These are problems related to measurement per unit quantity. Aida works out thesolutions for the problems using the Rule of Three as shown in Diagram 10.
•Find the missing value for each.

Sequence the tasks to show the development of ideas on measurement per unit 		quantity.

Chapter 3: Measurement and Relations | 149

Topic 8: Investigating the Area of a Circle
Topic 8: Investigating the Area of a Circle
Standards 8.1:
Standards 8.1:
Topic 8: Investigating the Area of a Circle
Areas of a circle are discussed through the relationship between the radius and the circumference
Investigate
the relationshiprelationship between
between thethe diameter
diameter ofof aa circle
circle andand its
its circumference
circumference using
using thethe ideai.i. Investigate
Standards
8.1: theofproportionideaof
proportion
Areas of a circle are discussed through the relationship between the radius and the circumferenceii. Investigate
Investigatethe
therelationshiparea ofof aa circle
circleby transforming
transforming
intoa triangle
triangleorparallelogram
and
find
the
Investigate
the
area
by
into
find
the
i. ii.
betweenthe diameteritit ofaacircleand oritsparallelogram
circumferenceand
using
the
idea
formulaofthecircleof the circle
offormula
proportioniii. Investigate
Estimate the
the
area
of
inscribed
andcircumscribedshapesbased
on aa known
known formula
formulaof areaarea
Estimateandcircumscribedshapesbasedofii.iii.
thearea
areaofofinscribeda circle bytransforming
it intoa triangle
or onparallelogramand find
the
iv.
Enjoy
estimating
the
area
of
irregular
shapeswithfluencyinlifeof the circle
iv. formula
Enjoy estimating
the area of irregular shapes with fluency in lifeiii. Estimate the area of inscribed and circumscribed shapes based on a known formula of area
SampleTasks
for Understanding
the Standards
iv.Sample
EnjoyTasks
estimating
the area of irregularshapes with fluency in life
for Understanding
the Standards
Task 1: Relationship
Between Diameterand Circumference of Circular Objects
Samplefor Understanding
the Standards
Task1: Tasks
Relationship
Between Diameterand Circumference of Circular Objects
Task 1: Relationship Between Diameter and Circumference of Circular Objects

Diagram 1
Diagram 1
i. Jason drew three circles, ⓐ,ⓑ, ⓒ on cardboards with diameters 10 cm, 20 cm and 30 cm
as threeshown
in Diagram
he rolled
the
one complete
i.respectively
Jason drew
circles,ⓐ,ⓑ,1.ⓒThen,on cardboards
withcircles
diameters
10 cm, 20rotation
cm andand
30 cm
Diagram1
investigate
how
far
each
of
them
advanced
as
shown
by
tapes
①,
②,
③
in
the
diagram.
respectively as shown in Diagram 1. Then, he rolled the circles one complete
rotation andinvestigate
how fareachⓐ,ⓑ,of themas shown
by tapes ①,
in cm
the anddiagram.
i. Jasondrew threecircles,ⓒ advanced
on cardboards
with diameters
10②,
cm,③2030 cm• respectively
What do you
think thedistance 1.
rolledis related
to?the circles one complete rotation and
as
shown
in
DiagramThen,herolled What do you think the distance rolled is related to?
investigate
how
far
each
of themadvanced
as shown
by cm,
tapes①, ②, ③the diagram.
Jason
cutoutoutanother
anothercirclewithaadiameter
diameter40
estimatetheindistancedistancerolled for•  If IfJason
cut
circlewith
ofof40cm, estimate
therolledthiscircle.
 for
What
you think the distance rolled is related to?
thisdocircle.
cardboard
to draw
andwithcut aout
the four
Roll each
onerolled
complete  If Use
Jason
cut out anothercirclediameterof circles.
40 cm, estimate
thecircle
distanceforrotation
to
find
the
distance
advancedandcompleteTable
1.
this circle.
•
Use
cardboard to draw and cut out the four circles. Roll each circle one complete rotation
Use cardboard
to distancedraw andadvanced
cut out thecircles.
Roll
to find the
andfourcompleteTable
1. each circle one complete
Table 1to find the distance advanced and complete Table 1.
rotation
Distance Advanced for One Complete Rotation
Table 1
Table 1 Advanced for One Complete
Distance
RotationⓐⓑⓒDistance Advanced for One Complete Rotation
Diameter (cm)Circumference
Diameter(cm) (cm)

ⓐ10

ⓑ20

ⓒ30

10

20

30

40
40

(cm)
 Circumference
How does the circumferenceof a circle change when its diameter increases 2 times,
3 times and 4 times?
How does
does thethe circumference
circumference ofofaacircle
circlechangechangewhen
whenitsitsdiameter
diameterincreases
increases2 2times,
times,•  Howtimes andand 44 times?
times?
33 times

150 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 2
ii.

Jason tried to investigate further the relationship between the circumference and diameterof a circle by taking some measurements of some circular objects as shown in Diagram 2.
He measured the circumference and diameter of each object using the methods shown indiagrams 3 and 4. The measurements are shown in Table 2.

Diagram 3

Measure the circumference.
Roll the round object one complete rotation and measure how far it moves.

Diagram 4
Table 2
Circumference and Diameter of Round Objects•

Calculate the ratio

Circumference

Diameterfor each circular object.

(Give your answer to the nearest hundredth by rounding the thousandth.)•

Approximately, how many times is the diameter to the circumference?

•

Based on your calculation, what conclusion can you make on the relationship between
the circumference and diameter of a circle?

Chapter 3: Measurement and Relations | 151

Circumference

Note: The ratio Diameter is the same value for all circles regardless of the circles’ sizes.
This ratio for a circle is named pi, represented by the symbol π. It is a number that
continues infinitely like 3.14159 … and we usually take 3.14 as its approximate value in thecalculation.
iii.

Jason concluded the following formula for any circle from his investigation.

•

Based on this formula, explain what it means by “the circumference of a circle isdirectly proportional to its diameter”.

Diagram 5
iv.

Diagram 5 shows another two circular objects, a dart board and a bicycle rim.
•

Calculate the circumference of the dart board.

•

Calculate the radius of the bicycle rim.

152 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Area of a Circle
Formula for the Area of a CircleNguyênDiagram 6i.

Diagram 6 shows a circle divided into 16 equal parts. A student, Nguyên cut out andrearrange the parts into a triangle as shown in the diagram.
•If the radius of the circle is r cm, approximately how many cm willThe height of the triangle be?

The base of the triangle be?
(Express your answer in terms of r.)•

  Do you agree with Nguyên’s formula for finding the area of the triangle?
Justify your decision.

How is the formula for finding the area of the circle related to this formula?



If the radius of the circle is 10 cm, find its area in cm².

Chapter 3: Measurement and Relations | 153

Diagram 7
ii.

Another student, Liên, cut out and tried to rearrange the 16 parts into a rectangle. However,the new shape looks more like a parallelogram than a rectangle as shown in Diagram 7.
•

What does Liên mean by the parallelogram will approach a rectangle?

•

Do you agree with her idea? Justify your decision.

Diagram 8iii.

Liên tried to test her idea by cutting another two circles into 32 and 64 parts respectively.
as shown in Diagram 8.
•

Find the missing value for each•

Do you agree with Liên’s conclusion the area of the rectangle will approach the areaof the circle eventually? Why or why not?

in Diagram 8.

154 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

iv.

Based on Liên’s conclusion, find the formula for the area of a circle with a radius of r cm.
•

Find the area of the circle if its radius is 10 cm.

Diagram 9v.

Diagram 9 shows two circles. The diameter of the big circle is double the diameter of thesmall circle.
•

What will happen to the circumference and the area of a circle when its diameter isdouble? Justify your answer with an appropriate calculation.
What if the diameter is tripled?

Diagram 10
vi. A restaurant in Kuala Lumpur sells pizza in three different sizes as shown in Diagram 10.
Two brothers, Jackson and Jeff are given RM20 to spend in the restaurant for their lunch.
Upon seeing the menu, the brothers have contradicting ideas on what pizza to order asshown in the diagram.
•

Between the brothers, who has a misconception about the size of the pizza?
What is the misconception?

•

What should the brothers order? Why?

•

How would you help the brother correct the misconception?

Chapter 3: Measurement and Relations | 155

Task 3: Finding Areas of Inscribed and Circumscribed Shapes

Diagram 11i.

Diagram 11 shows two quarter-circles overlapping to form a coloured region in a square.
•     Find the area of the coloured region.
•

Find the area of the uncoloured region.

•

What are the subtasks that the students need to learn to solve the problem?

•

How would you sequence the subtasks to facilitate your students’ effort in solving the
problem by themselves?

Diagram 12ii.

Diagram 12 shows two semi-circles overlap with a quarter circle to form a coloured region.
•

Find the area of the coloured region.

•

What are the subtasks that students need to learn to solve the problem?

•

Why is task sequence important in developing students’ ability to solve the problem?

156 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

		Diagram 13
iii.

Find the area of the figure Ⓟ and Ⓠ in Diagram 13.

Chapter 3: Measurement and Relations | 157

Task 4: Estimating Areas of Irregular ShapesNguyên’s MethodNguyênDiagram 14i.

Diagram 14 shows two students’ methods to estimate the area of a leaf. Each square is
1 cm x 1 cm.
•     Use Nguyên’s method to estimate the area of the leaf.
•

ii.

Use Liên’s method to estimate the area of the leaf.

Which method do you think is easier for your students? Justify your decision.

158 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 15
iii.

Diagram 15 shows the map of Penang Island. The measurement of each square is
2 km x 2 km.
•      What is the area of a square, in square km.
•

Estimate the area of Penang Island using Nguyên’s and Liên’s methods, respectively.

•

Check your answers with information on Penang Island from the Internet.
Which method gives you a closer estimate of the actual area?

Note:
The area is a 2-dimensional measure. Notice that the area of a trapezium is (a + b)h x ½; thearea of a triangle is ah x ½; and the area of a circle is pi x r ². Thus, each of these area formulas
can be seen as a representation of proportionality with a constant.

Chapter 3: Measurement and Relations | 159

Topic 9: Exchanging Local Currency with Currency in the ASEAN Community
Standards 9.1:
Exchanging local currency in the ASEAN community with the idea of rate
i. Extend the use of ratio for currency exchange (rate of exchange)
ii. Apply the four operations for money in appropriate notation in lifeiii. Appreciate the value of money
Sample Tasks for Understanding the Standards
Task1: Ratio for Currency Exchange in ASEAN Community
The exchange rate is the price of one country’s currency in terms of another currency. Table
1 shows the exchange rates between the Malaysian Ringgit (MYR) and currencies in the
ASEAN community and U.S. Dollar (USD).
Table 1The Exchange Rates of ASEAN Countries and the USA

Based on OANDA’s Currency Converteri.

The rate for Brunei Dollar (BND) is 0.315.
•

What does it mean?

•

How many MYR can exchange for 1 BND?

160 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelii.

The rate for Singapore Dollar (SGD) is also 0.315.

Diagram 1
Diagram 1 shows a pair of proportional number lines used by a student to exchange
50 SGD for MYR.
•

Find the missing number in each•

How many MYR can 50 SGD change for?

.

Diagram 2iii.

A store in Đồng Xuân market, Hanoi is selling crispy lotus seeds at 360 000 VND per kg.
The rate for Vietnamese Dong (VND) is 5274.150. Diagram 2 shows a Rule of Threetables to change 360 000 VND to MYR.
•

Find the missing number in each•

How many MYR will 1 kg of crispy lotus seeds cost?

.

Diagram 3
iv.

Diagram 3 shows three proportional number lines representing currency exchange rates
for three currencies.
•

Explain how you can use diagram 3 to exchange 100 000 Indonesian Rupiah for
Combodia Riel?

Chapter 3: Measurement and Relations | 161

Diagram 4v.

Diagram 4 shows six proportional number lines representing currency exchange rates for
six countries.
••

Use the number lines to exchange 500 THB for

Philippines Peso, PHP

Lao Kip, LAK

Indonesian Rupiah, IDR

Use the number lines to exchange 20 USD for

Myanmar Kyat, MMKThai Baht, THB

Malaysian Ringgit, MYR

vi. US dollar is a common based currency for exchange.
•

How many MYR can be exchanged for 1 USD?

•

Explain how the exchange rate is calculated.

•

Calculate the exchange rates of other currencies to 1 USD and complete Table 1.

vii. Mr Joseph from the Philippines accompanied 10 grade-6 students to visit Borobudur in
Jogjakarta. The entrance ticket for each foreign visitor is 25 USD.
•

Based on the rates in Table 1, how much is the price of each ticket in Philippines Peso,
PHP?

•

What is the total amount for 11 tickets in PHP?

•

How much is the total amount of 11 tickets when converted to Indonesian Rupiah,
IDR?

162 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelviii. Based on the rates in Table 1, arrange the following currencies from the highest value tothe lowest.
•

100 Indonesian rupiah, 100 Viatnamese Dong, 100 Thai Bhat, and 100 Peso•

Explain how you make the comparisons.

Task 2: Four Operations for Money
Task 2: Four Operations for Money

Diagram 5
Diagram 5

i.

Madam
Aishah took her three children to Jollibee for a meal while in Cebu, Philippines. She
i. Madam Aishah took her three children to Jollibee for a meal while in Cebu, Philippines. She
ordered
3 setsof meal
C1C1
and
1 set
shownininDiagram
Diagram
5. Thechildrenordered
3 setsof mealand
1 setofofmeal
meal C3C3 as
as shown
5. Thechildrenalso also
requested
individual
Mini
Sundaerequested
individual
Mini
SundaeTwirl.
Twirl.
 What is the total amount she has to pay in the Philippines Peso? Show the procedure of•     Whatcalculation.
is the total amount she has to pay in the Philippines Peso? Show the procedureof calculation.
 Aishah was not sure if the charges for the food were expensive. What would you suggest
she do in order to find out if the food prices were reasonable?

 The following day, Madam Aishah ordered 2 sets of C4 with Chocolate Sunday Twirl for
her two children, 1 set of C5 with Cookies’n Cream Sundae Twirl for another child, 1 setof C6 with a Vanilla Cone Twirl for herself.

Chapter 3: Measurement and Relations | 163•

Aishah was not sure if the charges for the food were expensive. What would you
suggest she do in order to find out if the food prices were reasonable?

•

The following day, Madam Aishah ordered 2 sets of C4 with Chocolate Sunday Twirlfor her two children, 1 set of C5 with Cookies’n Cream Sundae Twirl for another child,
1 set of C6 with a Vanilla Cone Twirl for herself.
What is the average cost of food for each person, in the Philippines Peso?

Diagram 6ii.

Diagram 6 shows the exchange rates of the Malaysian Ringgit to six other ASEAN countries.
It also shows the price of seven items in the respective country.
•     What item can you buy in other countries with one Malaysian Ringgit?
•

How much Malaysian Ringgit do you need to buy a bottle of Coca-Cola in Thailand?

•

Which is the most expensive item listed in the diagram?

•

How much does an apple cost in your country?

•

Is it cheaper than the apple in Cambodia?

Why do you think apples cost so much in Cambodia?

An ice-cream cone costs S$ 2 in Singapore.
How much does that cost when the value is exchanged to your local currency?

If you are at home, will you pay such a price for a piece of such ice cream? 		
Why or why not?



What if you are travelling in Singapore?

164 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 7
iii.

iv.

Diagram 7 shows the prices of two best-seller sets of KFC fried chicken in Malaysia.
•

Find out the prices of similar sets of KFC fried chicken in your country.

•

Find out the current exchange rate of your country’s currency to the Malaysian Ringgit.


Is KFC selling the same sets at the same prices in your country?

Do you think it is justifiable for the price difference if any?

Suggest Two ways to make students aware of the values of money in different currencies.

Chapter 3: Measurement and Relations | 165

Topic 10: Extending the Relation of Time and Use of Calendar in Life
Standards 10.1:
Extending the relation of time and use of calendar in life
i. Convert time in 12-hour system with abbreviation a.m. and p.m. to 24-hour system and viceversa
ii. Investigate the numbers in calendar to relate days, weeks, months and year using the idea of
number patternsiii. Appreciate the significance of various calendars in life
Sample Tasks for Understanding the Standards
Task1: Telling Time in 24-hour Systemi.

Diagram 1

Diagram 1 shows the departure and arrival
time of a flight from Penang to Kuala
Lumpur.
•When is the flight, in the morning or in
the afternoon?

•

What is the departure time? Write thetime with abbreviation a.m. or p.m.

•

What is the arrival time? Write thetime with abbreviation a.m. or p.m.

Diagram 2
ii.

Diagram 2 shows a chart connecting the 24-hour system and the 12-hour system in telling
time.
•

Using the chart as a guide, convert the following time to the 12-hour system. Write thetime using the abbreviation a.m. or p.m.


14:55 hours



09:10 hours

18:30 hours

166 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

•

iii.

Using the chart as a guide, convert the following time to the 24-hour system.


5:00 a.m.



8:15 a.m.



4:35 p.m.

What are the advantages of using the 24-hour system?

Chapter 3: Measurement and Relations | 167

Task 2: Exploring Calendar Numbersi.

Diagram 3 shows the calendar for
December 2022. There are 7 columnsof numbers in the calender, namely the
Sunday, Monday, Tuesday, Wednesday,Thursday, Friday and Saturday columns.
•

What pattern do you observe between
the numbers in each column?

•

Why does the pattern exist?

Diagram 3ii.

Diagram 4iii.

Diagram 5

Diagram 4 shows a left-to-right diagonal of
numbers in the same calendar.
•

What pattern do you observe between
these numbers?

•

Does the same pattern exist for otherleft-to right diagonals too?

•Why does the pattern exist?

Diagram 5 shows a right-to-left column of
numbers in the same calendar.
•

What pattern do you observe between
these numbers?

•

Does the same pattern exist for otherright-to-left diagorals too?

•Why does the pattern exist?

168 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Relationship of Numbers in Calendars

i.

Diagram 6

Diagram 6 shows three consecutive
numbers from a column in the same
calendar.
•What relationship do you observe
between these numbers?

•

Is the relationship always true forany other three consecutive column
numbers such as 14, 21 and 28?

•

Let the expressions (a – 7), a, and(a + 7) represent any three consecutive
column numbers.
ii.

Prove the relationship using these
expressions.

Diagram 7 shows 1 Dec on the 2022
calendar.
•Diagram 7•

1 Dec is a Thursday.
What day is 7 days later?

What day is 10 days later?

What day is 14 days later?

What day is 20 days later?

What pattern do you observe?

What calculation can you do tosolve similar problems?

Solve the following problems without
refering to the calendar.


4 Dec is a Sunday. What day is
21 days later?



20 Dec is a Tuesday. What day is
25 days later?



26 Dec is a Monday. What day is
10 January 2023?



Check all your answer with acalendar.

Chapter 3: Measurement and Relations | 169

iii.

Diagram 8 shows a ‘2-by-2 square’ of four
numbers in the same calendar.
•

What relationship do you observe
between these numbers?

•

Is the relationship true for other ‘2-by2 squares’?

•Why does the relationship exist?

•

Clare claims that the sum of these four
numbers is (4 x 5) + 16 as shown inthe diagram.

Diagram 8

iv.

v.



Verify Clare’s claim.

Is the claim true for any other‘2-by-2 square’ of numbers?



Why is the claim true?

Diagram 9 shows a ‘3-by-3 square’ of nine
numbers in the same calendar and a claimon the sum of these numbers made by Jeff.
•

What does Jeff mean?

•

Verify Jeff’s claim.

•

Is the claim also true for any other‘3-by-3 squares’ of numbers in thecalendar?

•Why is the claim true?

Diagram 9
The Gregorian calendar that is universally used now is a solar calendar.
•

What does it mean by solar calendar?

•

Name other calendars that you know.
Compare the calendars. Explain the similarities and differences among thecalendars.

vi. How will the tasks in this section inculcate your students’ appreciation towards the valuesof mathematics?
vii. A calendar is used to track passage of time. How will our life be affected if there is no
calendar?

170 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 11: Converting Quantities in Various System of Units
Standards 11.1:
Converting measurement quantities on international and non-international system with idea ofbase-10i. Convert measurement system of metre and kilogram with prefixes deci-, centi-, and mili-, andwith deca-, hecto-, and kiloii. Convert measurement system of litre with cubic centimetreiii. Convert measurement system of area using are (a) and hectare (ha) with square metre
iv. Convert measurement of local quantities with standard quantities
v. Understand the unit system with power, such as metre, square metre and cubic metre
Sample Tasks for Understanding the Standards
Task1: Quantity and Unit of Measurementi.

Any quantity can be measured using abasic unit of measurement such as metre(m), gram (g), and litre (ℓ). Centi, kilo, andmili are common prefixes used to modify
the magnitude of a unit. Table 1 showssome examples of prefixes.
•

Write the measurements in therespective basic unit.
1 kilometre = ________ m
1 kilogram = ________ g
1 hectometre = ________ m
1 decametre = ________ m
1 decigram = ________ g
1 centimetre = ________m
1 centilitre = ________ ℓ
1 mililitre = ________ ℓ,ii.

Arrange each of the following sets ofmeasurements from least to the largest.
•

3 dm, 563 cm, 750 mm•

2.7 kg, 42.5 hg, 499.6 cg

Chapter 3: Measurement and Relations | 171iii.

Kilo, hecto, and deca are examples ofprefixes used to make the units larger,
whereas deci, centi, and mili are prefixes
used to make the units smaller. Tables 2
and 3 are more examples of prefixes.
•

Fill in the missing factors in thetables.

172 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Conversion Between Units of Lengthi.

Diagram 1 shows the relationships among
units of length.
•

Diagram 1ii.



10 x 1 mm = ________ cm

100 x 1 cm = ________ m1000 x 1 m = ________ km____ x 1 mm = 1 m

____ x 1 mm = 1 km____ x 1 cm = 1 km

Diagram 2 shows a chart used to convert
between units of length based on therelationships shown in Diagram 1. Two
examples of conversions as shown inthe chart are 5 km = 500 000 cm and12 m = 12 000 mm.
••
Diagram 2

With the help of Diagram 1, complete
the following conversion of units forlength.

Use the chart to convert


2 km to mm



235 m to cm



37 cm to mm



5.2 m to cm

Use the chart to convert
7900 cm to m

860 mm to m

2375 cm to km

Chapter 3: Measurement and Relations | 173

iii.

Diagram 3 shows a problem involving
comparing the length of two ropes.
Jackson make a mistake as shown in the
diagram.

Diagram 3•

What are the possible sources of themistake?

•

How to help Jackson correct the
mistake?

174 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Conversion Between Units of Massi.

Diagram 4

Ton (t), kilogram (kg), gram (g) andmilligram (mg) are units of mass and theirrelationships are shown in Diagram 4.
•

Fill in the missing factor in each

?

.

•

With the help of Diagram 4, complete
the following conversion of units formass.
		 3 000 000 g = ________ t

•

ii.

2 000 000 mg = ________ kg

What next conversion tasks will you
give to your students? Explain yourreasons.

Diagram 5 shows an incomplete chartsimilar to Diagram 2 for conversion of units
involving mass.
•

Complete the chart.

•

Use the chart to convert

2.8 kg to mg



6.37 t to kg



692 g to mg

Diagram 5

•

iii.



4500 kg to t

43860 mg to g



120 g to kg

Diagram 6 shows four fruit baskets. The
mass of each basket is written on a card
next to it.
•

Diagram 6

Use the chart to convert

Arrange the fruit baskets from lightest
to the heaviest.

Chapter 3: Measurement and Relations | 175

Task 4: Conversion Between Units of Area

Diagram 7i.

Hectare and 'are' are two units in Metric System commonly used to measure land area.
The size of 1 are (a) and 1 hectare (ha) is an area equal to a square that is 10 m and 100
m on each side respectively, as shown in Diagram 7.
•

How many m² is 1 are?

•

How many m² is 1 hectare?

•

How many are is 1 hectare?

•

How many km² is 1 hectare?

ii.

1 ha 9 a and 1.9 ha, which is larger?

iii.

Arrange the following land areas from least to the largest.

Diagram 8iv.

Diagram 8 shows some measurements of area.
•

Match each area with its appropriate measurement.

176 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 5: Units of Volume

Diagram 9i.

Diagram 9 shows one large and one small cube, respectively. The small cubes can fill
1 cubic centimetre (cm³), also known as 1 mililitre (mℓ) of water and the large cube can fill
1 litre (ℓ) of water.
•

How many small cubes are there in 1 big cube?

•

How many cm³ are there in 1 ℓ ?

Diagram 10
ii.

Diagram 10 shows the relationships among units for measuring volume of liquid.
•

Fill in the missing factor in each•With the help of Diagram 10, complete the following conversion of units for volume ofliquid.

?

.



1 mℓ

= __________ x 1 cm³



1 dℓ

= __________ x 1 cm³

1ℓ= __________ x 1 dℓ



1 kℓ

= __________ x 1 cm³

Chapter 3: Measurement and Relations | 177

Diagram 11iii.

Diagram 11 shows some measurements of water capacity.
•

Match each water capacity with its appropriate measurement of volume.

Diagram 12
iv.

Diagram 12 shows a chart used to convert between units of volume. Two examples of
conversions as shown in the chart are 3 kℓ = 3000 ℓ and 53 ℓ = 530 dℓ.
•v.

Use the chart to convert

2 kℓ to cm³



18 dℓ to mℓ

6.5 kℓ to dℓ



2.6 ℓ to mℓ

3 kℓ 600 dℓ and 3.6 kℓ, which is larger?

vi. Arrange the following volume of liquid from least to the largest.

178 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Topic 12: Showing Relationships Using a Venn Diagram
Standards 12.1:
Using a Venn diagram to show relationships of numbers and figures for making a clear logicaldeductioni. Sort objects by their defining characteristics
ii. Show relationships of squares, rectangles, rhombuses, parallelograms, trapezium andquadrilaterals by using a Venn diagramiii. Show the relationship between numbers
iv. Critique ambiguous reasoning by using a Venn diagram for making clear a definition
Sample Tasks for Understanding the Standards
Task1: Who am I?
i.

Diagram 1 shows a number telling about its
characteristics.
•

Can the number be 8? Why or why not?

•Can the number be 17? Why or why
not?

•

What can the number be?

Diagram 1

ii.

Diagram 2

Diagram 3

iii.

iv.
Diagram 4

Diagram 2 shows another number.
•

Can the number be 2? Why or why not?

•Can the number be 28? Why or whynot?

•

Can the number be 29? Why or whynot?

•

Can the number be 115? Why or why
not?

•

What can the number be?

Diagram 3 shows a space figure telling about
its characteristics.
•

Can the figure be a triangle? Why or
why not?

•

What can the figure be?

Diagram 4 shows another space figure.
•

What can the figure be?

Chapter 3: Measurement and Relations | 179

Task 2: Relationships among Quadrilateralsi.

Quadrilaterals can be classified into (a) trapezium, (b) parallelogram, (c) rhombus,(d) rectangle, and (e) square.
•State the properties of each of the quadrilaterals.

Diagram 5ii.

Diagram 5 is a Venn diagram showing the relationships among quadrilaterals, rectangles
and squares.
••

Decide whether each of the following statements is true or false.
All squares are quadrilaterals.

All rectangles are square.

Some quadrilaterals are squares.



Some squares are quadrilaterals.



A square is also a rectangle.



A quadrilateral is also a rectangle.

Justify each of your decisions.

Diagram 6iii.

Diagram 6 shows a Venn diagram illustrating the relationships among quadrilaterals.
•

A rectangle is also a trapezium. Why?

•

A square is also a rhombus. Why?

180 | Mathematics Challenges for Classroom Practices at the Upper Primary Leveliv.

Diagram 7 shows two quadrilaterals, Q1
and Q2. In Q1, the lengths of all the sides
are equal; whereas two of the sides of Q2
are parallel.
•

Decide whether each of the statements
about the quadrilaterals is true or false.

		

Diagram 7•v.

Diagram 8



Q1 is a parallelogram.



Q1 is a trapezium.



Q2 is a trapezium.



Q2 is a parallelogram.

Justify each of your decisions.

Diagram 8 shows three tasks to help
students understand the relationships
between a rectangle, a square and aparallelogram.
•

How will you order the sequence ofthe tasks?

•

Justify your decision.

•

What responses will you anticipate
getting from your students for each of
the tasks?

•

Based on the anticipated		
responses, what additional 		specific questions will you 		
ask to support your students’ 		
mathematical thinking in each task?

•

What next task will you give to yourstudents? Explain your reasons.

Chapter 3: Measurement and Relations | 181

vi. Diagram 9 shows a square, a rectangle, arhombus and a parallelogram.
•Complete eachstatements.

of

the

followingA square is also a rhombus, but
a rhombus may not be a square.
So, a rhombus is a square if__________.

A rectangle is also a parallelogram,but a parallelogram may not be arectangle. So, a parallelogram is arectangle if _________________.

Diagram 9vii. Diagram 10 shows a plane figure F1 withtwo pairs of equal adjacent sides. Jason
said F1 is a kite.
•

Diagram 10

Is F1 a kite? Why or why not?

182 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Relationships among Numbers

Diagram 11i.

Diagram 11 shows a number describing itself and a Venn diagram.
•Can the number be 188? Why or why not?

•Shade on the Venn diagram, the region that represents the number.

Diagram 12ii.

Diagram 12 shows another number describing its characteristics, and a Venn diagram
represents whole numbers less than 100.
•

Draw on the Venn diagram to express the relationship among multiples of 2, multipleof 3, and multiples of 6.

•

Shade on the Venn diagram, the region that represents whole numbers less than 100
which are multiples of 6.

•

Find a number that is a multiple of 2, but not a multiple of 6.
•

Shade on the Venn diagram, the region for the number.

Can you find a number that is a multiple of 6, but not a multiple of 3? Explain yourreasoning.

Chapter 3: Measurement and Relations | 183

Venn Diagram ①

Venn Diagram ②

Venn Diagram ③

Diagram 13

iii.

o

The relationship of even numbers to multiples of 4.

o

The relationship of multiples of 4 to multiples of 5.

o

The relationship of odd numbers to multiple of 6.

Diagram 13 shows three Venn diagrams and three relationships of numbers. Set E has
whole numbers between 100 and 200.
•

Match the Venn diagrams with the relationships.

•

Explain each of the relationships using specific examples.

Diagram 14
iv.

The Venn diagram in Diagram 14 shows the relationship of prime numbers to evennumbers.
•

Give an example of a number that is in the shaded region.

•

Shade the region of odd numbers that are also prime numbers.

•

Shade the region of even numbers that are not prime numbers.

Diagram 15v.

The Venn diagram in Diagram 15 shows the relationship of multiples of 2 to multiples of 3
which are less than 100.
•

How many numbers are multiples of 2?

184 | Mathematics Challenges for Classroom Practices at the Upper Primary Level•

How many numbers are multiples of 3?

•

How many numbers are multiples of 2 and multiples of 3?

•

•

•



Shade this region on the Venn diagram.



What is the largest common multiple of 2 and 3?

How many numbers are multiple of 2 or multiples of 3?
Shade this region on the Venn diagram.

Give an example of such a number.

How many numbers are multiple of 2, but not multiples of 3?
Shade this region on the Venn diagram.

Give an example of such a number.

How many numbers are neither multiple of 2 nor multiple of 3?
Shade this region on the Venn diagram.

Give an example of such a number.

Diagram 16
vi. The Venn diagram in Diagram 16 shows the number of students who like apples and/or
mangoes in Mr Wahid’s class. Two students, Carol and Connie have different opinions on
the number of students who like mangoes.
•

How will you use this contradicting opinions in helping your students learn Venn
diagram?

•

How many students are there altogether in Mr Wahid’s class?

Chapter 4: Plane Figures and Space Figures | 185

CHAPTER 4
Plane Figures and Space Figures
Topic 1: Exploring Figures with their Components in the Plane
Standards 1.1:
Exploring figures with their components in the plane and using their properties
i. Examine parallel lines and perpendicular lines by drawing with instruments
ii. Examine quadrilaterals using parallel and perpendicular lines, and identify parallelogram,rhombus, and trapezium by discussioniii. Find properties of figures through tessellations such as a triangle where the sum of the angles
is 180 degrees, a straight angle
iv. Extend figures to polygons, and expand them to circles by knowing and using their properties
Sample Tasks for Understanding the Standards
Task 1: Perpendicular Linesi.

Diagram 1 shows a quadrilateral formed byfour lines drawn on a dot paper.

• Line① and Line ③ are perpendicular.
		 What instrument can be used tomeasure an angle?
•

Diagram 1

ii.

Measure angles ⓐ,ⓑ,ⓒ, and ⓓrespectively.

Is line ② perpendicular to line ③?
Explain your way of finding theanswer.



Confirm your answer by paper
folding.



Why is paper folding an error-freemethod to confirm your answer?

•

Which other line is also perpendicular
to line ①?

•

Draw another line on Diagram 1 that isperpendicular to line ③.

What are perpendicular lines?

186 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 2iii.

Diagram 2 shows two pairs of lines.
•

Which pairs of lines are perpendicular?

•

Explain your way to find the answer.
•

Explain how the properties of a square can be used to confirm your answer.

Think of other ways to find the answer.

Diagram 3
iv.

Diagram 3 shows another two pairs of lines.
•

Which pairs of lines are perpendicular?

•

Explain your ways of finding the answers.


Explain how the properties of a square and parallel lines can be used to confirm
your answers.

Chapter 4: Plane Figures and Space Figures | 187

Diagram 4v.

Diagram 4 shows a clock geoboard.
•

Design a task to help your students learn perpendicular lines using a clock geoboard
with rubber bands.

Diagram 5vi. Diagram 5 shows line ① and two points, A and B. Use a ruler and a protractor or a set
square to help you with the following subtasks.
•

Draw a line such that it passes through point A and is perpendicular to line ①.

•

Draw another line such that it passes through point B and is perpendicular to line ①.

vii. Get a piece of A4 paper. Fold the paper to make two perpendicular lines.
•

How many folds do you need to make?

•Must each fold always divide the paper into two equal parts? Explain your answer.

•

What are the possible different reasons that your students may suggest to explain why
the two folded lines are perpendicular?

188 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Parallel Linesi.

Diagram 6 shows a quadrilateral formed byfour lines drawn on a dot paper.
•Line ③ is perpendicular to both lines① and ②.
		
 What is the value of the angle ofintersection between line③ andlines ① and ②, respectively?
•

Diagram 6•Line ④ is another line crosses bothlines ① and ②.


Measure the values of angles ⓐand ⓑ.



What do you find out about these
two angles?

Lines ① and ② are said to be a pair ofparallel lines.


ii.

Diagram 7

Based on what you have foundso far, explain the meaning ofparallel lines.

Diagram 7 shows two parallel lines, ① and②. Five other line segments, AP, AQ, AR,
AS and MN intersect with lines ① and ②.
•

Arrange the distance, AP, AQ, AR,
and AS in ascending order.

•

How do you determine the order?

•

Compare the distance between AQ
and MN.
•

What can you conclude from yourcomparison?

Will lines ① and ② meet if they areextended at both ends? Why or why
not? 		

Chapter 4: Plane Figures and Space Figures | 189iii.

Two students, Jose and Pedro were asked to draw a parallel line to line ①. They useddifferent methods as shown in Diagram 8.

Diagram 8•

Explain the two students’ methods and the reason why both methods are appropriate.

Diagram 9iv.

Diagram 9 shows line ① and point A.
•

Draw a line which passes through point A and is parallel to line ①.

•

Draw another two lines that are parallel to line ① and 3 cm apart.

190 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 10v.

Diagram 10 shows a sheet of rectangular paper with side AB.
•

Using a scaled ruler only, draw a parallel line 2 cm away from side AB on Diagram 10.

Diagram 11

vi. Diagram 11 shows another sheet of A4 paper, PQRS. Three line segments, L① , L② , andL③ are drawn as shown in the diagram.
•

Are the three line segments, L① , L② , and L③ parallel lines? Why or why not?

•Using a scaled ruler only, draw another parallel line to L① , L② , and L③.

Chapter 4: Plane Figures and Space Figures | 191

Diagram 12vii. Diagram 12 shows a third sheet of A4 paper, WXYZ. Taking the diagonal ZX as a reference,
two line segments, L④ and L⑤, are drawn as shown in the diagram.
•

Are the two line segments, L④ and L⑤ parallel to diagonal ZX? Why or why not?

•Using a scaled ruler only, explain how you could fold the A4 paper to construct aparallel line segment that is 5 cm away from diagonal ZX.

192 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Perpendicular and Parallel Lines Surrounding Us

Diagram 13i.

Diagram 13 shows some objects surrounding us.
•

Identify pairs of perpendicular lines and pairs of parallel lines in each of these objects.

Diagram 14ii.

Diagram 14 shows the drawing of a pair of parallel railway lines.
•

Supposedly, parallel lines never intersect but why do the two railway lines draw tomeet at one endpoint?

Chapter 4: Plane Figures and Space Figures | 193

Diagram 15
iii.

Diagram 15 shows a clock face with line ① joining 3 to 12.
••

Using a ruler and pencil only, drawa line joining any two numbers on the clock face that is perpendicular to line ①.

A line joining any two numbers on the clock face that is parallel to line ①.

Other than measuring the angles of intersection, explain how you know that the lines
are perpendicular or parallel.

194 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 4: Different Types of Quadrilaterals

Diagram 16i.

A quadrilateral is a closed figure with 4 straight sides. Diagram 16 shows three types ofquadrilaterals formed by joining the dots with four lines.
•

Name each of the quadrilaterals.

•

How many pairs of parallel lines can you find in each of the quadrilaterals?

•

Draw a Venn diagram to show the relationships that exist among these quadrilaterals.

•

Explain each relationship.

Diagram 17ii.

Diagram 17 shows another three types of quadrilaterals.
•

Explain why each of these quadrilaterals is also a parallelogram.

•

Explain how these three quadrilaterals fit into your Venn diagram in subtask (i).

Diagram 18
iii.

Diagram 18 shows some trapezium found in our surroundings.
•

Look for more examples of the trapezium in your home surrounding.

•

Look for examples of parallelograms and rhombuses in your home surrounding.

Chapter 4: Plane Figures and Space Figures | 195

Diagram 19
iv.

Diagram 19 shows a point, A. What figure will A become after making the movet?

Diagram 20v.

Diagram 20 shows another point, B. What figure will B become after making the move?

Diagram 21
vi. Diagram 21 shows a figure telling about its properties. What can the figure be?

Diagram 22vii. Diagram 22 shows another figure.
•Can the figure be a square? Why or why not?

•What can the figure be?

Diagram 23
viii. Diagram 23 shows a third figure.
•Can the figure be a rectangle? Why or why not?

•Can the figure be a square? Why or why not?

•What else can the figure be?

196 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 5: Sum of Interior Angles of Polygons

Diagram 24i.

Diagram 24 shows a tessellation pattern formed by a rhombus Ⓐ.
•

Explain what is a tessellation pattern.

Diagram 25
ii.

Diagram 25 shows a tessellation pattern formed by a triangle Ⓑ, with interior angles
X, O, and ∡. Cut out and use triangle Ⓑ in Material Sheet 1 at the end of this topic tomake the tessellation pattern.
•

Based on the tessellation pattern, explain why the sum of the three interior angles of
the triangle Ⓑ is 180°.

•

Other than using the tessellation pattern, how will you prove that the sum of threeinterior angles of any triangle is 180°?

Diagram 26
iii.

Diagram 26 shows quadrilateral Ⓒ.
•

Cut out and use quadrilateral Ⓒ in Material Sheet 2 at the end of this topic to create atessellation pattern.

•

Use the tessellation pattern to find the sum of all interior angles of a quadrilateral Ⓒ.

•

Explain your reasoning for finding the sum of the interior angles.

Chapter 4: Plane Figures and Space Figures | 197

iv.

Diagram 27 shows a quadrilateral Ⓓ.
Without forming a tessellation pattern, two
students, Aroon and Boon-Mee, used two
different ideas to find the sum of all interior
angles of the quadrilateral Ⓓ as shown in
Diagram 28.
•

Explain the two ideas.

Diagram 27

Divide the quadrilateral into two
triangles by drawing a diagonal.

Divide the quadrilateral into four
triangles by drawing two diagonals.

So, the sum of the four interior anglesof the quadrilateral is 2 times 180°.

So, the sum of the four interior anglesof the quadrilateral is 4 times 180°,
subtract the extra 360°.

Diagram 28v.

Diagram 29

Diagram 29 shows a regular pentagon Ⓔwith five interior angles. Aroon and BoonMee used two different methods to find thesum of all interior angles of the pentagonⒺ as shown in Diagram 30.
•

Explain the reasonings for the two
methods.

•

Explain why the pentagon Ⓔ cannot
form a tesselation pattern.

•

Explain how this task can nurture
your students’ appreciation towards
the beautifulness of mathematicalideas.

198 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Divide the pentagon into three triangles
by drawing two diagonals.

Divide the pentagon into one triangle
and one quadrilateral by drawing one
diagonal.

So, the sum of the five interior anglesof the pentagon is 3 times 180°.

So, the sum of the five interior anglesof the pentagon is 180° adds up with360°.

Diagram 30

vi. Diagram 31 shows a regular hexagon Ⓕ,
with 6 interior angles.

Diagram 31•

Find the size of each interior angle.

•

Explain your reasoning for finding theinterior angle.

•

Can the hexagon Ⓕ be used to forma tessellation pattern? Explain yourreasons.

Chapter 4: Plane Figures and Space Figures | 199

vii. Diagram 32 shows four tasks involving
interior angles of polygons.

Task ①:
Prove that the sum of interior
angles of any heptagon is 900°.

•

Which task will you choose to be thefirst task for your students? Explain
your reasons.

•Explain how the knowledge learnt
in Task ① can help to simplify the way
to solve Task ④?

•

How will you sequence the four tasks
for your students?

•

Explain the reasons for your sequenceof tasks.

•
		How can sequencing the tasks
encourages your students to learn
mathematics by and for themselves?

Task ②:
Prove that the sum of interior
angles of any triangle is 180°.
Task ③:
Prove that the sum of interior
angles of any quadrilateral is360°.
Task ④:
Prove that the sum of interior
angles of any octagon is 1080°.
Diagram 32

viii. Diagram 33 shows a concave quadrilateral
CDEF.
•

Find the sum of the interior angles.

Diagram 33
ix. Diagram 34 shows a concave pentagon
PQRST.
•

Diagram 34

Find the sum of the interior angles.

200 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 6: Circle

Diagram 35i.

Diagram 35 shows a line JK.
•

Explain how a compass together with a ruler can be used to measure the length of JK.

Diagram 36
ii.

Diagram 36 shows a zigzag road connecting five towns A, B, C, D, and E on a map and anumber line.
•

Explain how a compass can be used to mark the total distance from A to E, going
through B, C and D, on the number line.

•

With the help of a compass, mark the total distance AE on the number line.

Diagram 37
iii.

A triangle ABC is constructed by drawing two circles, with centres at A and B on line L①
as shown in Diagram 37.
•

Explain why triangle ABC is an equilateral triangle.

•

Using a compass and a ruler only, reproduce triangle ABC, with AB = 5 cm.

Chapter 4: Plane Figures and Space Figures | 201

Diagram 38
iv.

Diagram 38 shows a circle with centre O and radius OA. Six other identical circles are thenconstructed with centres on the circumference of the circle as shown in the diagram. A
regular hexagon is constructed by joining the points A, B, C, D, E and F.
•

Explain why the six sides of the hexagon are of the same length.

•

Using a compass and ruler only, reproduce hexagon ABCDEF, with AB = 5 cm.

Diagram 39v.

Diagram 39 shows a design created by drawing 13 identical circles along a straight line.
•

Using a compass and ruler only, reproduce the design.

•

What property of a circle is used in creating the design? Explain your answer.

202 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 40
vi. Diagram 40 shows a circle with centre O and eight other points, A, B, C, D, E, F, G and H,
on its circumference. The design is created by constructing eight other identical circles
with each of the eight points as the centre for each circle.
•

Using a compass and ruler only, reproduce the design.

•

What mathematical ideas are embedded in this task?

•

In what ways can this task help to develop an appreciation towards circles?

vii. With the help of a compass, create your artistic design using circles.
viii. What mathematical ideas in future grades can be extended from Task 6? Explain youranswer with examples.

Chapter 4: Plane Figures and Space Figures | 203

Material Sheet 1: Tessellation of Triangles

204 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Material Sheet 2: Tessellation of Quadrilaterals

Chapter 4: Plane Figures and Space Figures | 205

Topic 2: Exploring Space Figures with Their Components in Relation to the Plane
Standards 2.1:
Exploring rectangular prisms and cubes with their components
i. Identify the relationship among faces, edges and vertices for drawing sketch
ii. Explore the nets of a rectangular prism and find the corresponding position between
components
iii. Explore the perpendicularity and parallelism between the faces of a rectangular prism
iv. Explain positions in rectangular prisms with the idea of 3 dimensions
Sample Tasks for Understanding the Standards
Task 1: Rectangular Prism and Cubei.

Diagram 1 shows a rectangular prism
ABCDEFGH, with a shaded base EFGH.
•

Determine all pairs of parallel faces.

•

Which faces are perpendicular to•
Diagram 1face EFGH?

face ABFE?

Which edges are perpendicular toface EFGH?

face ADHE?

•

Which edges are perpendicular to edge
BC?

•

Which edges are parallel to edge BC?

206 | Mathematics Challenges for Classroom Practices at the Upper Primary Leveli.

Diagram 2 shows a cube and a rectangularprism each with two shaded parallel faces.
•

What is the same and what is different
between these two solid figures?

•

Find the number of vertices (V), edges(E), and faces (F) for each solid figure.

•For each of the solid figures, let 󠅘  be  the number of edges of each of the shadedbase shapes.
Find the relationshipbetween

Diagram 2

•
ii.



V and 󠅘

E and 󠅘F and 󠅘Find the relationship between V, E, and F.

Diagram 3 shows a net with six faces of arectangular prism.
•

What shape is the base of this prism?
Explain your answer.

•

Determine all pairs of parallel faces ofthis prism.

•

Determine all faces that areperpendicular to the face ⓕ.

Diagram 3

Chapter 4: Plane Figures and Space Figures | 207

Diagram 4iii. Diagram 4 shows a net and an incomplete sketch of a rectangular prism.
•

Complete the sketch of the prism.

•

Determine all pairs of opposite faces of the prism.

Diagram 5
iv.

Diagram 5 shows an arrangement of six congruent rectangles.
•

Explain why this arrangement will not form a rectangular prism.

•

Explain how you will modify the arrangement to make a net for a rectangular prism.

208 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 6v.

Diagram 6 shows four arrangements of six faces to make a rectangular prism.
•

Determine, which of these arrangements are not nets of the rectangular prism.

•

Explain any clue that you use to identify nets of a rectangular prism.

Diagram 7vi. Diagram 7 shows ten arrangements of squares.
•

Determine which of these arrangements are nets of a cube.

•

Explain any clue that you use to determine the nets of a cube.

Chapter 4: Plane Figures and Space Figures | 209

Diagram 8vii. Diagram 8 shows a giant rectangular prism with a length 5 m, width 3m, and height 4 m.
A beetle is resting at one of its vertices as shown and this beetle can crawl around on thefaces of the prism. However, it only knows three directions of movement, which are leftright, inward-outward, and upward-downward as shown in the diagram.
•

The beetle crawls 5 m to the right, turns and crawls 3 m inward, and finally turns again
to crawl 4 m upward.
•At what location is the beetle now?

Describe the movement of the beetle from its original position tolocation ②.

location ③.

location ④.

•

Describe the movement of the beetle to crawl from location ④ to location ⑤.

•

Sometime, the beetle will fly to move beyond the prism faces. From location ②, thebeetle flies 2 m upward, then 2 m to the right and stop in the air.
•

From the location in the air, describe the movement of the beetle to location ③.

From location ③, the beetle flies 1 m to the right, then 2 m outward, and finally 2 m
downward.


At what location is the beetle now?

210 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 9

Diagram 10

viii. Diagram 9 shows a rectangular prism with a beetle resting at vertex A, whereas Diagram
10 shows a net of the rectangular prism when it is opened up.
••Mark and label on Diagram 10, vertices C, D, E, F, G, and H.

•

Explain why these vertices appear twice in the net.

•

Explain why vertices A and B appear once only in the net.

•

The beetle is crawling from vertex A to vertex G. A student claimed that the shortest
route for the beetle to crawl from vertex A to vertex G is to crawl along edge AE, thenalong EG in a straight line.
Is the student’s claim valid?

Explain how you will use the net in Diagram 10 to help the student visualise theshortest route.

Chapter 4: Plane Figures and Space Figures | 211

Standards 2.2:
Extending rectangular prism to other solids such as prisms and cylinders
i. Extend the number of relationships among faces, edges and vertices for drawing sketch
ii. Explore nets of prisms and cylinders, and find the corresponding position between
components
iii. Distinguish prism and cylinder by the relationship of their faces
Sample Tasks for Understanding the Standards
Task 1: Prisms and Cylinders
What is a prism?

Diagram 1i.

Diagram 1 shows some examples and non-examples of prisms.
•

What is a prism?
ii.

Diagram 2 shows a cylinder.
•

Diagram 2

Why is a cylinder not a prism?

212 | Mathematics Challenges for Classroom Practices at the Upper Primary Leveliii.

Diagram 3 shows a file holder.
•

Is the file holder a prism? Why or why
not?

Diagram 3

Diagram 4
iv.

Diagram 4 shows a structure from Egypt.
•

Is the structure a prism? Why or why not?

Diagram 5v.

Diagram 5 shows a traditional house in Malaysia. Study the structure of the house.
•

Identify and sketch any prism in the structure of the house.

Chapter 4: Plane Figures and Space Figures | 213

Relationship Among Faces, Edges and Vertices of a Prism

Diagram 6i.

ii.

Diagram 6 shows four prisms, each with two shaded parallel faces, known as the bases ofthe prism.
•

What can you say about the shape and sizes of the two parallel faces for each prism?

•

What is the shape of all the other faces that are not shaded?

•

Which faces are perpendicular to the shaded bases?

For each prism, determine and fill in Table 1, its number of vertices (V), edges (E) and faces(F). Then, determine the relationship among V, E and F.
Table 1
Number of Vertices, Edges and Faces•

Determine and write a mathematics sentence to explain the relationship betweenV and nE and nF and n•Determine and write a mathematics sentence to explain the relationship between V, E
and F.

•

If your students have difficulty in dealing with the use of algebraic symbols such as n,
V, E and F, how will you present this task to them?

214 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Nets of Prism and Cylinder
Net of Prism

Diagram 7i.

Diagram 8

Diagram 7 shows a prism with a right-angled triangle base. The prism is cut out to form a
net as shown in Diagram 8.
•Mark and label the vertices D, E, and F on the net in Diagram 8.

•

Identify and label the edge AD, BE, and CF on the net in Diagram 8.

•

Sketch another net for this triangular prism.

Diagram 9ii.

Diagram 9 shows a net of a prism.
•

Based on the net, sketch the prism and name the prism.

•

How many pairs of parallel faces are there in the prism? Shade the parallel pairs of
faces with different colours on the net in Diagram 9.

•

How many faces are perpendicular to face ABCDE? Shade the perpendicular faces
with another colour on the net in Diagram 9.

Chapter 4: Plane Figures and Space Figures | 215

Net of Cylinder

Diagram 10i.

A face of a cylinder is cut out and flattened to be a rectangle ABCD as shown in Diagram
10.
•

Find the length of


AB



AC

Diagram 11ii.

Diagram 11 shows two arrangements of circles and rectangles, Ⓐ and Ⓑ.
•

Which of these arrangements is or is not a net for a cylinder? Why or why not?

Diagram 12iii.

Diagram 12 shows the net of a solid with two semi-circular faces.
•

Sketch the solid.

216 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Who am I?

Diagram 13i.

Diagram 13 shows a solid figure talking about how it looks from different directions.
•

Can the solid figure be a prism? Why or why not?

•What can the solid figure be?

Diagram 14ii.

Diagram 14 shows another solid figure talking about how it looks from different directions.
•

Can the solid figure be a prism? Why or why not?

•What can the solid figure be?

Diagram 15
iii.

Diagram 15 shows a third solid figure talking about how it looks from different directions.
•

Can the solid figure be a prism? Why or why not?

•What can the solid figure be?

Diagram 16
iv.

Diagram 16 shows a fourth solid figure talking about its look from different directions.
•

Can the solid figure be a prism? Why or why not?

•What can the solid figure be?

Chapter 4: Plane Figures and Space Figures | 217

Topic 3: Exploring Figures with Congruence, Symmetry and Enlargement
Standards 3.1:
Exploring the properties of congruencei. Explore properties of figures which fit when overlapped and identify conditions of congruency
with corresponding points and sides
ii. Draw congruent figures using minimum conditions and confirm by measuring angles andsidesiii. Appreciate the usefulness of congruent figures by tessellation
Sample Tasks for Understanding the Standards
Task 1: Congruent Figures
Congruent Figures

Diagram 1i.

Diagram 1 shows a tessellation of dark and white bear heads. The bear heads arecongruent with one another. Examine the shape and size of each bear’s head.
•

Explain the meaning of congruent figures.

•

Explain how the tessellation of figures can be used to develop the concept ofcongruency.

•

Explain how the tessellation of congruent figures can be used to nurture appreciation
towards the beautifulness of mathematics.

218 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 2
ii.

Diagram 2 shows a tessellation of congruent quadrilaterals.
•

What is the relationship between the corresponding angles of the quadrilaterals?

•

What is the relationship between the corresponding sides of the quadrilaterals?

Chapter 4: Plane Figures and Space Figures | 219

Congruent Triangles
i.

Diagram 3

Jason drew a triangle and he told his friend
about his triangle. Diagram 3 shows theinformation he gave to his friends. From theinformation, his friend was trying to draw atriangle.
•

Based on the information given, is it
possible for Jason’s friend to be certain ofdrawing another triangle exactly the sameas triangle ABC? Why or why not?

•What is the minimum information needed
for Jason’s friend to be certain of drawing
the exact triangle ABC?

Diagram 4
ii.

iii.

Diagram 4 shows a triangle ABC and eight other triangles.
•

Determine which of the eight triangles fit by overlapping with triangle ABC.

•

What can you conclude about the congruency of these triangles with triangle ABC?

A student claimed that: “Congruent triangles are triangles that have the same size andshape.”•

iv.

Is the student’s claim valid? Why or why not?

Determine if each of the following statements is true or false. Explain your reasoning.
•

Two triangles are congruent if they have the same three sides.

•

Two triangles are congruent if they have the same three angles.

220 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelv.

Two sides and a corresponding angle oftriangle ABC are given as shown in Diagram 5.
Jason used a protractor and compass to draw
the triangle as shown in Diagram 6.
•

How many different triangles can be drawn
to fulfil the given condition? Explain youranswer.

Diagram 5

Diagram 6
vi. Two sides and the included angle of triangle
DEF are given as shown in Diagram 7.
Diagram 8 shows part of the process ofdrawing the triangle.
•

How many different triangles can be
drawn? Explain your answer.

Diagram 7

Diagram 8

Chapter 4: Plane Figures and Space Figures | 221vii. Two angles and the included side of triangle
PQR are given as shown in Diagram 9.
•

Use a ruler, a protractor and a 		compass to draw the triangle.

How many different triangles can be
drawn? Explain your answer.
		•

Diagram 9viii. Two angles and a corresponding sideof triangle STU are given as shown in
Diagram 10.

Diagram 10•

Use a ruler, a protractor and a 		compass to draw the triangle.

•

How many different triangles can be
drawn? Explain your answer.

222 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Standards 3.2:
Exploring the properties of symmetry.
i. Explore the properties of figures which reflect and identify conditions of symmetry with line
and its correspondence
ii. Draw symmetrical figures using conditions in an appropriate locationiii. Appreciate the usefulness of symmetry in designs
Sample Tasks for Understanding the Standards
Task 1: Line Symmetrical Designs

Diagram 1i.

Diagram 1 shows three ethnic designs from ASEAN countries. Duplicate and cut out eachof the designs. Fold each of the cut-out designs into two congruent parts that fit on top ofeach other. The folded line is a line of symmetry.
•

What is a line of symmetry?

•

Which design has more than one line of symmetry? Explain your answer.

Chapter 4: Plane Figures and Space Figures | 223

Diagram 2
ii.

Line symmetry is a common motif in many designs. Diagram 2 shows nine logos of some
automobile companies.
•

Determine if each of the logos has any line of symmetry. If yes, draw all the lines ofsymmetries. If not, redesign the logo so that it will have at least one line of symmetry.
iii.

Diagram 3 shows an incomplete drawing
of a butterfly. Given that AB is the line ofsymmetry of the butterfly.
•

iv.

Diagram 3

Complete the drawing.

Explain how the tasks in this section
will nurture your students’ appreciation
towards the beautifulness and usefulnessof line symmetry.

224 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 2: Line Symmetry and Paper Cutting

Diagram 4i.

Diagram 4 shows a piece of A4 paper being folded into halves. Then, several parts were
cut off from the folded paper as shown.
••

Sketch on Diagram 4, the outcome when the paper is unfolded.
How many lines of symmetry are there?

Mark the line of symmetry.

Reproduce the outcome by folding and cutting a piece of A4 paper to check youranswer.

Diagram 5ii.

Diagram 5 shows another piece of A4 paper being folded into quarters. Similarly, several
parts were cut off as shown.
••

Sketch on Diagram 5, the outcome when the paper is unfolded.
How many lines of symmetries are there?

Mark the lines of symmetries.

Reproduce the outcome by folding and cutting to check your answer.

Chapter 4: Plane Figures and Space Figures | 225

Diagram 6iii.

Diagram 6 shows a “double hearts” design created by folding and cutting a piece of square
paper.
•

Explain the folding and cutting process to produce the “double hearts”.

•

How many lines of symmetry does the “double hearts” have?

Diagram 7
iv.

Diagram 7 shows an artistic design created from a piece of square paper.
•

Explain how the square paper can be folded twice and cut to produce the design.

•

How many lines of symmetry does the design have?

•

Reproduce the design from a piece of square paper.

226 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 8v.

Diagram 8 shows an artistic design created from another piece of square paper.
•

Explain how the square paper can be folded three times and cut to produce the design.

•

How many lines of symmetry does the design have?

•

Reproduce the design from a piece of square paper.

Diagram 9vi. Diagram 9 shows a piece of rectangular paper, folded three times, and cut to create a chainof dolls.
•

Explain how the chain of dolls can be created.

•

How many lines of symmetry does the chain of dolls have?

•

Reproduce the chain of dolls.

vii. Using a piece of paper, create your artistic design by folding and cutting the paper.
•

Explain how you fold and cut the paper in creating the design.

Diagram 10
viii. After making some symmetrical designs by folding and cutting paper, Catherine, as in
Diagram 10, makes a conjecture about the relationship between the number of lines ofsymmetry and the number of folding.
•

Is Catherine’s conjecture always, sometimes, or never true?

•

Explain your answer.

Chapter 4: Plane Figures and Space Figures | 227

Task 3: Properties of Figures with Line Symmetry

Diagram 11i.

In Diagram 11, when the shape is folded along the line of symmetry AH, the two parts ofthe shape fit exactly on top of each other.
•

B and N is a pair of corresponding points. Draw a line segment from B to N.
••

Identify and state all other pairs of corresponding points. Repeat drawing a line
segment joining each pair of corresponding points.


What is the angle of intersection between each line segment and AH?

What conclusion can you make about these angles of intersection?

What is the relationship between the perpendicular distance from each point of acorresponding pair to AH?

BA and AN are a pair of corresponding sides.
•What is the relationship between the lengths of each pair of sides?

∠ABC and ∠AMN are a pair of corresponding angles.
•Compare the lengths of BA and AN. What can you conclude about these twocorresponding sides?

Identify and state all other pairs of corresponding sides.
•What is the angle of intersection between the line segment and AH?

Compare the sizes of the two corresponding angles. What can you conclude
about these two corresponding angles?

Identify and state all other pairs of angles.
What is the relationship between the sizes of each pair of angles?

228 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 12ii.

Diagram 12 shows incomplete drawings of three shapes. Given that AB is the line ofsymmetry.
•

Complete the drawings.

•

What different ways can be used by your students to complete these tasks?

•

Many students may have difficulties completing these tasks. What different ways will
you suggest to support these students’ learning?

Chapter 4: Plane Figures and Space Figures | 229

Task 3: Polygons and Line Symmetry

Diagram 13i.

Diagram 13 shows four quadrilaterals.
•

Which quadrilaterals have line symmetry?

•

How many lines of symmetry does each have?

•

Duplicate and cut out each quadrilateral. Fold the quadrilaterals to verify your answers.

230 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 14ii.

A regular polygon has all its sides with equal length. Diagram 14 shows six regular polygons
with a different number of sides.
•

A 3-sided regular polygon is known as an equilateral triangle, whereas a 4-sided
polygon with all equal sides as shown is known as a square.
What is the name of the 5-sided regular polygon in Diagram 14?

•

How many lines of symmetry does each of the regular polygons have?

•Complete Table 1.

		
		

•

Table 1
Number of Lines of Symmetry

Make a conjecture about the relationship between the number of sides and thelines of symmetry of a regular polygon.



Verify your conjecture using the 6-sided, 7-sided, and 8-sided polygons in Material
Sheet 3 at the end of this topic.

•How many lines of symmetry does a 10-sided regular polygon have?
•

Cut out and fold the 10-sided polygon in Material Sheet 3 to verify your answer.

What mathematical reasoning can this task develop among your students?
Explain your answer.

Chapter 4: Plane Figures and Space Figures | 231

Standards 3.3:
Exploring the Properties of Enlargement
i. Explore properties of figures in finding the centre of enlargement in simple cases such asrectangle
ii. Draw an enlargement of a rectangle using the ratio (multiplication of the value of the ratio)iii. Appreciate the usefulness of enlargement in the interpretation of a map
Sample Tasks for Understanding the Standards
Task 1: Changing the Size of a ShapeDiagram 1i.

Diagram 1 shows three shapes, ①, ②, and ③. Shape ② is a different shape from ①, butshape ③ is considered the same shape as ①. Table 1 recorded the length of each side of
the three shapes, whereas Table 2 shows the ratio of each side of shape ② and shape ③
to shape ①.
•

Complete Table 1

Table 1
Length of Sides•

Complete Table 2

Table 2
Ratio of Side Compare to Shape ①

232 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

ii.

There are seven interior angles in each of the shapes. These angles are ∠A, ∠B, ∠C,∠D, ∠E, ∠F, and ∠G.
•

Duplicate and cut out shapes ①, ② and ③. Compare the corresponding angles ofshapes ② and ③, with shape ①, then complete Table 3.

		Table 3		Comparison Between Corresponding Angles of Shapes ② and ③ with Shape ①iii.

Why is shape ③ considered same as shape ①, but shape ② is not?

iv.

Shape ③ is an enlarged shape of ①, whereas shape ① is a reduced shape of ③.
•v.

•Conclude the relationships between corresponding sides and corresponding angles
between ① and ③.

What is the ratio of the area of shape ③ to shape ①?
•

Make a conjecture about the ratio of the side and the ratio of the area of an enlarged
or reduced shape.

•

Test your conjecture with shapes ④ and ⑤ in Diagram 2, as well as shapes ⑥ and ⑦
in Diagram 3.

Diagram 2

Diagram 3

Chapter 4: Plane Figures and Space Figures | 233

Task 2: Drawing Enlarged or Reduced Shapes Using Sides and Angles

Diagram 4i.

Diagram 4 shows a quadrilateral ABCD.
•

With the help of a ruler, compass and protractor, draw a 2 times enlarged drawing and
a ½ time reduced drawing of ABCD in the space provided in Diagram 4.

234 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Drawing Enlarged or Reduced Shapes Using a Centre of Enlargement

Diagram 5i.

In Diagram 5, triangle ABC is enlarged 2 times from centre E to become triangle PQR.
•

Study the distance from the centre of enlargement, E, to points A, B, C, P, Q, and R.
•Rectangle GHIJ is also enlarged 2 times from centre E to become rectangle STUV.
•

What can you conclude about the ratios EP : EA, EQ : EB, and ER : EC?

Draw a rectangle STUV on Diagram 5.

Square KLMN is reduced ½ time from centre E to become square WXYZ.
Draw square WXYZ on Diagram 5.

Chapter 4: Plane Figures and Space Figures | 235

Diagram 6ii.

In diagram 6, shape Ⓐ is enlarged to become shape Ⓑ.
•

Locate and mark the centre of enlargement on Diagram 6.

••How many times is shape Ⓑ larger than shape Ⓐ?

Rectangle ABCD is enlarged to
become rectangle PQRS. Determine
the centre of enlargement.

Rectangle ABCD is enlarged to
become rectangle AQRS. Determine
the centre of enlargement.

Rectangle ABCD is enlarged to
become rectangle PQRS. Determine
the centre of enlargement.

Diagram 7iii.

Diagram 7 shows three tasks to find centres of enlargement.
•

How will you order the sequence of tasks for your students?

•

Explain your answer.

236 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Task 3: Map Readingi.

Diagram 8 shows the drawing of a housewith a reduced scale of 1 : 50.
•

What does the reduced scale of 1 : 50
mean?

•

The length of the kitchen is 6 cm on the
reduced drawing. What is the actual
length of the kitchen in m?

•

The actual width of the living roomis 5.2 m. How long is it in cm on the
reduced drawing?

Diagram 8

Diagram 9ii.

Diagram 9 shows two popular street art paintings by the Lithuanian artist, Ernest Zacharevic,
and another popular tourist site on Penang Island. Diagram 10 shows the locations ofthese sites on a map of Georgetown.

Chapter 4: Plane Figures and Space Figures | 237

Diagram 10iii.

A map is a drawing that represents an area of land or sea based on a reduced scale, whichstates the ratio that represents how much is it reduced from the real length. The scale ofthe map in Diagram 10 is 1 : 2500 as indicated in the map.
•

How far is it from site ① to site ② along Beach Street, in the actual distance?

•

A tourist starts walking from site ① to site ② along Beach Street at 3 km per hour.
How many minutes will the walk take?

•

Another tourist wants to go from site ② to site ③.

•

What are the possible routes?

•

Which route is the shortest?

238 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Material Sheet 3: n-sided Polygons

Chapter 5: Data Handling and Graphs | 239

CHAPTER 5
Data Handling and Graphs
Topic 1: Arranging Tables for Data Representations
Standards 1.1:
Collecting and arranging data
i. Explore how to collect multi-category data based on a situation
ii. Explore how to arrange and read multi-category data on appropriate tablesiii. Appreciate the use of multi-category tables in situation
Sample Tasks for Understanding the Standards
Task 1: Collecting Multi-Category Data on a Situation
Table 1 shows a record of injuries in a schoolover a period of one month.

Table 1
Record of Injuries

i.

Grade

Locations4

Playground

Types of Injuries
Cut

5

CorridorBruise5CorridorBruise1Classroom

Scratch3Gymnasium

ScratchTable 2 and Table 3 show two sets of datacollected from this record.
Table 2
Locations of Injuries

3

Playground

Scratch6

Gymnasium

Fracture

5

Classroom

Cut

4

Playground

Scratch5Gymnasium

ScratchClassroom

3

Gymnasium

BruiseGymnasium

Locations

Corridor

6

PlaygroundSprainStairs6Corridor

SprainTotal

2Gymnasium

Scratch1

Classroom

Bruise5Classroom

Cut

5

Gymnasium

Scratch3

StairsBruise4

Gymnasium

Scratch2

Playground

Bruise6Classroom

Scratch4

CorridorBruise4Playground

Scratch2Gymnasium

Sprain2Gymnasium

Sprain3

Playground

Scratch3Gymnasium

SprainComplete Table 2 and Table 3.

Numbers of Students

Playground

Table 3
Types of Injuries
Types
Cut
BruiseScratchFracture
SprainTotal

Numbers of Students

240 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelii.

Based on the tables, answer the following questions:•

Where did injuries happen most frequently?

•

Which type of injury happened most frequently?

•

What can you say about the types of injuries that happened at the gymnasium?

•

What can you say about the locations of Cut?

•
		
iii.

Which of these questions cannot be answered using information from Table 2 andTable 3? Explain your answers.

In order to answer the unanswered questions, a student rearranges the data into Table 4.
Table 4
Locations and Types of Injuries
Type
Location

Cut

BruiseScratchFracture

SprainTotal

Playground
CorridorClassroom
Gymnasium
Stairs
Total

Complete Table 4.
•

In what way is Table 4 different from Tables 2 and 3?

•

Answer the unanswered questions in (ii).

•

What can you conclude about the types and locations of injuries based on informationfrom Table 4?

iv.

In what way is Table 4 more useful than Tables 2 and 3? How will you instil appreciation
towards the usefulness of Table 4 among your students?

v.

Build two other tables to analyse the (a) types of injuries, and (b) locations, related to gradelevels, respectively.
•

How will you analyse the information derived from each table?

•

What relationship can you find between the grade levels and the types of injuries
based on your analysis?

•

What relationship can you find between the grade levels and the locations of injuries?

•

How shall you rearrange the data in Table 4 to see at a glance, any trend in

the type of injury?
the location of injury?

Chapter 5: Data Handling and Graphs | 241

Topic 2: Drawing and Reading Graphs for Analysing Data
Standards 2.1:
Drawing and reading line graphs for knowing the visualised pattern as the basis for tendency ofchangei.
Introduce line graphs based on appropriate situations such as rainfall, temperature and others
ii. Distinguish line graph from bar graph for observation such as increase, decrease, and nochangeiii. Introduce the graph of proportion using the idea of a line graph and read the gradient by aconstant ratio
iv. Appreciate the line graph in various situations
Sample Tasks for Understanding the Standards
Task 1: Line Graphsi.

Table 1 shows the average monthly temperature, measured in ºC, for the periods 19611990 and 1991-2020 in Myanmar.
Table 1
Average Monthly Temperature (ºC) for 1961-1990
and 1991-2020 in MyanmarMonthTemperature (ºC)
1961-1990

1991-2020

1

18.1

18.7

2

19.8

20.4

3

22.8

23.2

4

25.5

25.8

5

25.8

26.0

6

25.2

25.5

7

24.9

25.2

8

24.9

25.2

9

24.9

25.2

10

24.1

24.5

11

21.5

22.0

12

18.7

19.4

Retrieved from:World Bank Group, Climate Change
Knowledge Portal for Development
Practitioners and Policy Makers
https://climateknowledgeportal.worldbank.
org/country/myanmar-burma
andhttps://climateknowledgeportal.worldbank.
org/country/myanmar-burma/climate-datahistorical242 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 1
Diagram 1 shows a line graph plotted for the period 1961-1990.
•

Based on the line graph, describe the changes in temperatures from month to month.

•

In which month is the temperature highest?

•

In which month is the temperature lowest?

•

Which part of the graph shows

Increase in temperature?

Decrease in temperature?

No change in temperature?

•

Between which two consecutive months does the temperature change most?

•

Are the slopes of the graph for Jan-Feb, Feb-Mac, and Mac-Apr the same? Justify
your answer.

•

Are the slopes of the graph for Sept-Oct and Nov-Dec the same? Justify your answer.

•

What can you conclude about the change in temperature from

January to April?



October to December?

Chapter 5: Data Handling and Graphs | 243••

Plot on Diagram 1, another line graph for the period 1991-2020
Compare the changes in temperatures between 1961-1990 and 1991-2020.

What trend in the changes in temperatures do you see?

Look for the average monthly temperature in your country from the World Bank Climate
Change Knowledge Portal. Plot the relevant line graphs to compare the changes intemperatures for different periods of time.
What can you conclude on the issue of global warming in your country?

How will you use line graphs to instil awareness and appreciation towards
sustainable development among your students?

Diagram 2
ii.

Diagram 2 shows a sketch of a line graph of average monthly rainfall (mm) over a periodof 12 months in a village in an ASEAN country.
••

Identify and explain which part of the line graph showsslight increase in monthly rainfall;

significant increase in monthly rainfall;slight decrease in monthly rainfall;

significant decrease in monthly rainfall;

no change in monthly rainfall, for that part of the time.

What can you say about the change in value for partsCD and GH,BC and HI.

244 | Mathematics Challenges for Classroom Practices at the Upper Primary Level•

Look for the average monthly rainfall data for your region.
Plot the relevant line graphs to analyse the data.

What conclusion can you make on the change in rainfall for your region?

Task 2: Graph of Proportion

Diagram 3
Diagram 3 shows eight wires with different lengths, in cm. Each of the wires was weighed,in g, and the result is shown in Table 2.
Table 2
Length of a Wire and Weighti.

Length (cm)

1

2

3

4

5

6

7

8

Weight (g)

3

6

9

12

15

18

21

24

Study Table 2.
•

What pattern do you see in the changes in weight and length?

•

Find the values of the quotient, (weight) ÷ (length) for each pair of the values.

•

Write a mathematical sentence to represent the relationship between the weight andlength of the wire.

Chapter 5: Data Handling and Graphs | 245

Diagram 4
ii.

iii.

iv.

Draw a graph on Diagram 4 to show this set of data.
•

What is the difference between this graph and the line graph in Task 1?

•

Why is there a difference?

Use the graph to find the weight of•a length of 4.5 cm wire,•a length of 9 cm wire.

•

Justify your method.

Another wire weighs 20 g.
•

Use the graph to find the length of the wire.

•

Justify your method.

246 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Standards 2.2:
Drawing and reading band graph and pie chart for representing ratio in a wholei.
Explore how to scale a band or circle for representing the ratio or percent
ii. Use the band graph and pie chart for comparison of different groupsiii. Appreciate the band graph and pie chart in a situation
Sample Tasks for Understanding the Standards
Task 1: Comparing Ratio or Percent with Band Graph
Expressing Ratio as Decimal in a Band Graphi.

Table 1
Number of Passengers

Table 1 shows the number ofpassengers and seats of two buses
on a trip to Kuala Lumpur.
•

Which bus is more crowded?

•

What mathematical ideas do you
use to determine crowdedness?
Explain your answer.

Diagram 1ii.

A student uses ratio to determine the degree of crowdedness. Diagram 1 shows a bandgraph drawn by the student to solve the problem involving Bus ①.
•

What does a ratio of 1 mean in terms of the crowdedness of Bus ①?

•

What is the degree of the crowdedness of Bus ①?

•

Draw another band graph to determine the degree of the crowdedness of Bus ②.

•

Can the degree of crowdedness be more than 1? Explain your answer.
iii.

Table 2
Number of Vehicles
Vehicle

Bus

Van

Car

Number

6

8

12

iv.
Diagram 2

There are 6 buses, 8 vans and12 cars in a parking lot as shown in
Table 2. Diagram 2 shows a bandgraph involving the ratios of thesebuses, vans and cars.
•

What is the ratio of the numberof buses to the number of vans?

•

What is the ratio of the numberof cars to the number of vans?

Draw another band graph to find theratio of the number of vans to thenumber of cars.

Chapter 5: Data Handling and Graphs | 247

Expressing Ratio as Percentage in a Band Graphi.

There are 40 boys in a class of 50 students. Diagram 3 shows a band graph converting theratio of boys to percentages.

Diagram 3•

Find the ratio of the boys to the total number of students in decimal numbers andpercentages.

•

Why do we multiply the ratio with 100 to change it into a percentage?

Diagram 4ii.

Diagram 4 shows a band graph comparing the means of transportation to the school of agroup of students by percentage.
•

What is the percentage of students cycled to school compared to the total number ofstudents?

•

What is the percentage of bus, walking and car compared to the total number ofstudents, respectively?

•

Which means of transportation has the least number of students?

•

There are 150 students in the group. How many students walked to the school?

•

What is the advantage of using a band graph to show the percentages of each meansof transportation?

248 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

iii.

Table 3 and Table 4 show the favourite ice cream flavours of two groups of children.
Table 3Favourite Ice Cream Flavours of First
Graders
FlavourNumber ofChildren

Chocolate

Table 4
Favourite Ice Cream Flavours of Sixth
Graders

Percentage(%)

FlavourNumber ofChildren

12

Chocolate

9

Strawberry

5

Strawberry

10

Vanilla

3

Vanilla

5

Mocha

3

Mocha

2

Others2

Others

6

Total

25

Total

32

Percentage(%)•

Find each ratio to the total and round it to the nearest hundredth.

•

Convert each ratio to a percentage for each flavour and fill up the tables.

•

Draw a band graph for each grade on Diagram 5.

Diagram 5•

Discuss your finding based on the two band graphs.

Chapter 5: Data Handling and Graphs | 249

Diagram 6
iv.

Draw a pie chart for each grade on Diagram 6 to show the favourite ice cream flavours of
the two groups of children.

v.

Band graph or pie chart, which of these graphs is easier to compare the favourite ice creamflavours of the two groups of children? Justify your choice.

Diagram 7
Data retrieved from The World Bank DataBank portal
https://databank.worldbank.org/source/world-development-indicators

vi. The band graphs in Diagram 7 show the percentage of the population using the Internet in
the year 2020 for three SEAMEO countries.
•

A student claimed that more individuals were using the Internet in Thailand than in
Singapore. Is the claim valid? Explain your reasons.

•Another student claimed that more individuals were using the Internet in Singapore
than in Vietnam. Is the claim valid? Explain your reasons.

250 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelvii. Table 5 shows the daily total calorie intake per person in Vietnam and Japan for the year
2011.
Table 5
Daily Calories Intake per Person by Country
Country
Calories Per Person

Vietnam

Japan

2704

2717

Retrieved from What the world eats? National Geography Magazine website.
https://www.nationalgeographic.com/what-the-world-eats/

The band graphs in Diagram 8 show the percentage of daily calorie intake from different typesof food for the countries.

Diagram 8
Ariff wrote down the following conclusions from the graphs.
① The Vietnamese get the least percentage of calories from produce.
② The Vietnamese get more calories from grains than the Japanese.
③ The Vietnamese and the Japanese get the same amount of calories from dairy and
eggs.
•

Decide if each of the conclusions is true or false.

•

Justify your decision.

Chapter 5: Data Handling and Graphs | 251

viii. Table 6 shows the daily total calorie intake per person of Vietnamese 50 years apart.
Table 6
Vietnamese Daily Food Intake for 1961 and 2011
Year

1961

2011

Daily Food Intake(Calories Per Person)

1907

2704

Retrieved from What the world eats? National Geography Magazine website.
https://www.nationalgeographic.com/what-the-world-eats/

Diagram 9
The band graphs in Diagram 9 show the percentage of calories intake per person fromdifferent food types.
•

Based on the band graphs, discuss the change in the Vietnamese diet pattern after
50 years.

•

How can this task be used to promote a healthy lifestyle among your students?

252 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Standards 2.3:
Reading histogram for analysing frequency distribution
i. Draw a simple histogram from the frequency table on situations
ii. Read various histograms for analysing data distribution
iii. Use averages to compare different groups in the same situation with histograms
Sample Tasks for Understanding the Standards
Task 1: Average Scores

187, 199, 182, 184, 191, 204,
194, 196, 187, 201, 185, 196 (cm)Diagram 1
The numbers in Diagram 1 show the height (in centimetres) of 12 members of a basketballteam. Two students, Baskoro and Satria, calculated the average height of this team using two
different methods as shown in Diagram 2.

Baskoro’s Method(187 + 199 + 182 + 184 + 191 + 204 + 194 + 196 + 187 + 201 + 185 + 196) ÷ 12
= 192.2 cm

Satria’s Method
180 + (7 + 19 + 2 + 4 + 11 + 24 + 14 + 16 + 7 + 21 + 5 + 16) ÷ 12
= 180 + 12.2
= 192.2 cm
Diagram 2i.

Why do both methods give the same answer?

ii.

Which method do you think will be easier for your students? Explain your reasons.

Chapter 5: Data Handling and Graphs | 253

Task 2: Analysing Data Distributioni.

The data in Table 3 shows the heights, arranged from shortest to the tallest, of a group of
12-year-old children.
Table 3Height of 12-Year-Old Children (cm)Child

Height (cm)Child

Height (cm)Child

Height (cm)

C1

110

C17

122

C33

127

C2
C3

C20

112

C5

C21

112

C6

C22

112

C7

112

C10

117

C8

115

C11

C13

C29

119

C14

C30

121

C15

C31

121

C16

C32

121

C37

125

C38

125

126

C28

119

C36

125

C26

C27

118

C12

C35

123

125

C25

C34

122

C23
C24

113

C9

ii.

C19

111

C4

•

C18

111

C45

C46

127

C47

127
127

128

133

C44

126

128

C42

C43

126
126

128

129

C41

126

127

C39
C40

126

127

129
132
134
135
135
136
141

Find the average height of these children.

Table 4 shows the data separated into class intervals of 5 cm.
Table 4
Distribution of Heights
Class
①

②

Height (cm)Greater or Equal

Number of Children

Less Than

110 – 1158

115 – 1205③

120 – 125⑤

130 – 135⑦

140 – 145④

125 – 130⑥

135 – 140•

In which class interval does the height of 120 cm belong?

•

Fill in the number of children in each blank space in Table 4.

•

Draw a histogram to show the distribution of children’s heights in Diagram 3.

254 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 3•

Based on the histogram, answer the following questions.
ⓐ Which class interval has the most number of children?
ⓑ Which class interval does the average height that you calculated in (i) belong to?
ⓒ What is the height in the centre (24th place) and which class interval does thisheight belong to?
Based on your answers in ⓐ, ⓑ, and ⓒ, which one explains the data well? Justify
your choice.

•Task 3: Comparing Two Groups of Datai.

Table 1 shows a record of mathematics test scores of two groups of students, 20 boys and18 girls.
Table 1Mathematics Test Score of Two Groups
Group
Boy
Girl

Score
62

82

64

85

71

66

82

65

69

71

71

63

75

63

81

56

58

91

68

73

80

66

69

73

78

77

68

82

75

77

74

70

69

46

64

70

74

77

-

-

•

What are the highest and lowest scores for each group?

•

What is the average score for each group?

Chapter 5: Data Handling and Graphs | 255

ii.

Baskoro and Satria, compare the test scores of the two groups using different measuresand came to different conclusions as follows:
Baskoro’s Reasoning“Both the highest and lowest scores of the boys are larger than the highest and lowest
scores of the girls, respectively. Therefore, the boys performed better than the girls in
the test.”
Satria’s Reasoning“The average score of the girls is larger than the average score of the boys. Therefore,
the girls performed better than the boys on the test.
•iii.

Whose reasoning do you support? Explain your reasons.

The test scores in Table 1 are separated by class intervals of 5 marks as shown in Table 2.
Table 2Distribution of Test Scores
Class
①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

Score

Number of Students
Boy Group

Greater or Equal

Girl GroupLess Than

45 – 50-

50 – 55-

55 – 603

60 – 655

65 – 703

70 – 752

75 – 80-

80 – 855

85 – 901

90 – 95

1•

Fill in the number of students for the girl group in Table 2.

•Compare the test scores of the boys and girls.
Which group has more students with test scores greater or equal to 80 marks?

Which group has more students with test scores less than 70 marks?

Which group has more students with test scores greater or equal to 70 marks andless than 90 marks?

256 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Diagram 4
iv.

Diagram 4 shows a histogram drawn for the test scores of the boys.
•

How many boys scored greater than or equal to 80 marks?

•

How many percent of boys scored greater than or equal to 60 marks but less than 70
marks?

Diagram 5v.

Draw a histogram for the girls’ test scores on diagram 5.
•

Compare the shapes of the two histograms in Diagrams 4 and 5.
Discuss the distribution of the boys’ and the girls’ test scores.

•

In which class interval, greater than or equal to and less than, do most students belong
to each group?

•

How many percent of students scored less than 60 marks in each group?

•

How many percent of students scored greater or equal to 75 marks in each group?

Chapter 5: Data Handling and Graphs | 257

Topic 3: Using Graphs in PPDAC Cycle
Standards:
Identifying appropriate graphs for problem solving in data handling using the PPDAC cyclei.
Analyse a problem situation and discuss the expected outcomes before collecting data toclarify the purpose of a survey
ii. Plan the survey for the intended purposeiii. Collect the data based on the purpose of the survey
iv. Use appropriate graphical representation which is most suitable for the purpose
v. Appreciate the use of graphs for making the conclusion
Sample Tasks for Understanding the Standards
Task 1: Choosing the Appropriate Graph to Handle Data
Diagram 1 shows an ice cream vending
machine. The Entrepreneur Club of a school
intends to set up such vending machines togenerate income for the club. A committee
was formed to discuss and make decisions
for setting up the machine. A decision madeby the committee was although there are 12
flavours of ice cream, 6 flavours only will bechosen to fill a machine. As such, a problem
faced by the committee is:“What 6 flavours are to be chosen?”i.

Diagram 1

The following questions concerning eachflavour of ice cream were raised during
the committee meeting.
① What is the unit price of each flavour?
② What is the number of students who
like each flavour?
③ What is the colour of each flavour?
•

Which question is directly relevant to
the problem?

•

What data will help the club choose
the 6 flavours?

•

How can the data be collected?

258 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelii.

Diagram 2 shows the five phases of the Problem Plan Data Analysis Conclusion (PPDAC)
cycle for data handling and a brief description of the five phases.

Source:1. MathsNZ Students https://students.mathsnz.com/9.10/pdfs/1.pdf
2. Data Education in Schools https://dataschools.education/

Diagram 2

iii.

•

Match each description with its corresponding phase.

•

Explain briefly, what the Entrepreneur Club committee needs to do in each of thephases.

The 12 ice cream flavours available are Vanilla, Mint, Chocolate, Orange, Mango, Coffee,
Mocha, Raspberry, Neapolitan, Coconut, Pecan, and Strawberry. The committee collected
the data on favourite flavours from 80 students and 20 teachers as shown in Diagram 3.
Vanilla

Mocha

Mocha

Raspberry

Chocolate

Vanilla

VanillaNeapolitan

Mango

Mint

VanillaRaspberry

Chocolate

VanillaRaspberry

Vanilla

Mango

Strawberry Pecan

Strawberry

Chocolate

Vanilla

Chocolate

Mango

Raspberry

Chocolate

Chocolate

Mocha

VanillaNeapolitan

VanillaStrawberry Chocolate

Vanilla

VanillaMocha

Coconut

Mango

Mango

Coconut

Mint

CoffeeMango

Neapolitan Mint

Pecan

Vanilla

Mocha

Strawberry

VanillaStrawberry Vanilla

Mango

Mint

Chocolate

Chocolate

Mango

Strawberry

Strawberry Mocha

Mango

Chocolate

VanillaStrawberry Mocha

OrangeRaspberry

VanillaOrangeCoconut

VanillaMint

Coconut

Chocolate

Raspberry

Vanilla

Chocolate

Mocha

Mango

Strawberry

Chocolate

Mango

Mango

Vanilla

VanillaRaspberry

CoffeeOrangeCoconut

Chocolate

CoffeeCoffee

Chocolate

CoffeeStrawberry

Chocolate

Vanilla

Chocolate

Strawberry Mocha

Diagram 3•

Build a table to show the number of people who like each flavour.

•

What graphical representation will you use to analyse the data? Explain your choice.

•Analyse the data using your choice of graphical representation.
What 6 flavours will you choose on behalf of the committee? Explain your reasons.

Chapter 5: Data Handling and Graphs | 259

iv.

Data handling using the PPDAC cycle is often used in making decisions in problem
situations. As an example, Penang Island is a popular tourist destination in Malaysia.
Its attractions include hawker food, street art, heritage buildings, clan jetties, and sandbeaches. Diagram 4 shows some of these attractions.

Diagram 4
Mr Zamri, an owner of a tour agency in Penang Island, intends to plan tour packages which
include visits to popular tourist spots, meals, accommodation, and transportation.
•

List down some data-relevant questions that Mr Zamri should ask to help him plan thetour packages?

•

Choose one question from your list.

•Explain what data should Mr Zamri collect to help him answer the question.



Explain briefly, how the data can be collected.

Collect some data related to your chosen question from your class of students.

260 | Mathematics Challenges for Classroom Practices at the Upper Primary Level•

Organise and represent your collected data using appropriate tables and graphs.
•v.

Analyse the data.

Make some recommendations to Mr Zamri based on your analysis of data.

Identify a contextual problem situation in your country.
•

Plan a project that engages the PPDAC cycle to solve the problem.

•

Implement the cycle.

•

Present your findings.

vi. To what extent can this task nurture appreciation toward the usefulness of data handlingamong your students?

Chapter 5: Data Handling and Graphs | 261

Topic 4: Applying Data Handling for Sustainable Living
Standards:
Applying data handling for sustainable development and appreciating the power of data handlingfor predicting the future.
i.
Read data related to sustainable development on SDGs and adopting positive views for the
betterment of the society
ii. Understand the idea of probability as a ratio and percentage in reading the data for situations
related to sustainable developmentiii. Experience implementing a project of reasonable size in data handling to achieve sustainable
development and appreciate the power of data handling
Sample Tasks for Understanding the Standards
Task 1: Read Data Related to Sustainability
Climate Change

Data retrieved from Climate Change Knowledge Portal
https://climateknowledgeportal.worldbank.org/country/malaysia/climate-data-historical

Diagram 1i.

The graph in Diagram 1 shows the monthly average temperature of Malaysia over two
periods of time.
•

Discuss the effect of global warming on the climate of Malaysia over the years.

262 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelii.

Search for the climate-change data of your country from the Climate Change Knowledge
Portal or your country’s meteorological department.
•

Draw suitable graphs to represent the climate change data of your country.

•

Compare the global warming effects between your country and Malaysia.

Land Use Change
Table 2 shows some information on the forest land area (in square km) in Cambodia andVietnam 30 years apart.
Table 2
Forest Land Area in Square Kilometres by Years
Year

Cambodia(square km)

Vietnam(square km)

1990

110 048

93 7602020

80 684146 431

Data retrieved from The World Bank Data Bank portal
https://databank.worldbank.org/source/world-development-indicators

The band graphs in Diagrams 2 and 3 show the percentage of forest and non-forest land by
area in Cambodia and Vietnam respectively, for the years 1990 and 2020.

Diagram 2

Chapter 5: Data Handling and Graphs | 263

Diagram 3i.

Discuss the changes in forest land area in the two countries based on the band graphs.

ii.

Approximately, what is the percentage of decrease in forest land in Cambodia from theyear 1990 to 2020?

iii.

•

What may be the causes for the decrease in forest land?

•

How does the decrease in forest land affect the well-being of Cambodians?

Approximately, what is the percentage of increase in forest land in Vietnam from the year
1990 to 2020?
•iv.

How is it possible that the percentage of forest land increases after 30 years?

How will you use this task to raise awareness of the importance of sustainable development?

264 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

Carbon Dioxide Emission
Table 3Global Annual CO2 Emission from Burningof Fossil Fuels and Cement Production
Year

Annual CO2 Emission(Billion tonnes)

1900

1.95

1910

3.03

1920

3.51

1930

3.92

1940

4.85

1950

6.00

1960

9.39

1970

14.90

1980

19.49

1990

22.75

2000

25.23

2010

33.34

2020

34.81

Table 3 shows the data on global annual
carbon dioxide emissions from theburning of fossil fuels for energy andcement production.
i.

Draw a line graph to show the change
in carbon dioxide emissions from1900 to 2020.

ii.

Describe the trend in the annual
emission of CO2 in the world.

iii.

Predict what may happen to theemission in 2030.

iv.

How does carbon dioxide emission
affect the environment?

Data retrieved from Our World in Data website.
https://ourworldindata.org/co2-emissions

Data Handling and Sustainable Livingi.

In what ways is data handling important in promoting sustainable living?

ii.

How will you use the tasks in this topic to create awareness of the importance of sustainable
development among your students?

Bibliography | 265

BIBLIOGRAPHY
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Ball, D. L., Thames, M. H. & Phelps, G. (2008). Content knowledge for teaching: What makes it special?
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Bassarear,T. & Moss, M. (2020). Mathematics for elementary school teachers (7th ed.). Cengage.
Bennett, A.B.Jr., Burton, L. J., & Ted Nelson, L. (2012). Mathematics for elementary teachers: A conceptual
approach. (9th ed.). McGraw Hill.
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teachers (11th ed.). Pearson.
Freudenthal, H. (1973). Mathematics as an educational task. Reidel Publishing Company.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel Publishing Company.
Freudenthal, H. (2010). Revisiting Mathematics Education [China Lectures]. Mathematics Education Library(9). Kluwer.
Isoda, M. & Olfos, R. (2020). Teaching Multiplication with Lesson Study: Japanese and Ibero-American
Theories for International Mathematics Education. Springer. DOI: 10.1007/978-3-030-28561-6
Isoda. M. & Murata, A. (Eds.). (2020), Study with your friends: Mathematics for elementary school(12 vols.). Gakko Tosho.
Isoda, M., Murata, A. & Yap, A. I. (Eds.). (2015). Study with your friends: Mathematics for Elementary School(12 vols). Gakko Tosho.
Isoda, M. & Murata, A. (Eds.). (2011). Study with your friends: Mathematics for Elementary School(12 vols). Gakko Tosho.
Isoda, M. (2013). Fraction for Teachers: Knowing What before Planning How to Teach. CRICED, University ofTsukuba.
Isoda, M. & Kagarigi, S. (2012). Mathematical thinking: How to develop it in the classroom. World Scientific.
DOI: 10.1142/8163
Isoda, M. (2018). Mathematisation: A theory for mathematics curriculum design. Proceedings of the International
Workshop on Mathematics Education for Non-Mathematics Students Developing Advanced Mathematical
Literacy 2018. (pp. 27-34).
https://www.researchgate.net/publication/347879094_Mathematization_A_Theory_for_Mathematics
Curriculum Design
Long, C. T., DeTemple, D. W. & Millman, R. S. (2015). Mathematical reasoning for elementary teachers(7th ed.). Pearson Education.
MacKay, R. J., & Oldford, R. W. (2000). Scientific Method, Statistical Method and the Speed of Light. Statistical
Science. 15(3), 254-278. https://doi.org/10.1214/ss/1009212817
Mason, J., Burton, L. & Stacey K. (1982). Thinking mathematically. Addison-Wesley.
Mangao, D. D., Ahmad, N. J. & Isoda, M. (Eds.). (2017). SEAMEO Basic Education Standards (SEA-BES):
Common Core Regional Learning Standards (CCRLS) in Mathematics and Science. SEAMEO RECSAM.
http://www.recsam.edu.my/sub_SEA-BES/images/docs/CCRLSReport.pdf

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Nolan, E. C., Dixon, J. K., Roy, G. J. & Andreasen, J. B. (2016). Making sense of mathematics for teaching.
Grades 6 – 8. Solution Tree Press.
Nolan, E. C., Dixon, J. K., Safi, F. & Haciomeroglu, E. S. (2016). Making sense of mathematics for teaching.
High school. Solution Tree Press.
O’Daffer, P., Charles, R., Cooney, T.,Dossey, J., & Schielack, J. (2005). Mathematics for elementary school
teachers. 3rd ed. Pearson Education.
Polya, G. (1945). How to solve it; A new aspect of mathematical method. Princeton University Press.
Sobecki, D., Bluman, A. G., & Schirck-Matthews, A. (2011). MATH in our world (2nd ed.). McGraw Hill.
Templeton, G. (2015). What are prime numbers, and why are they so vital to modern life? https://www.
extremetech.com/extreme/219570-what-are-prime-numbers-and-why-are-they-so-vital-to-modern-life

Websites of National Curriculum in Some SEAMEO Countries
Cambodia Mathematics Curriculum: Grades 1-9
https://www.mathunion.org/fileadmin/ICMI/files/Other_activities/Database/Cambodia/Math-Curiculum_
Grades_1_-_9.pdf
Malaysian Primary Mathematics Curriculum:
http://bpk.moe.gov.my/index.php/terbitan-bpk/kurikulum-sekolah-rendah/category/2-dlp-rendah
Malaysian Secondary Mathematics Curriculum:
http://bpk.moe.gov.my/index.php/terbitan-bpk/kurikulum-sekolah-menengah/category/319-kurikulummenengah
Philippines Mathematics Curriculum: Grades K to 10
https://www.deped.gov.ph/k-to-12/about/k-to-12-basic-education-curriculum/grade-1-to-10-subjects/
Thailand Mathematics Curriculum: Years 1 – 12https://www.futureschool.com/thailand-curriculum/#552f669b3e568
Singapore Mathematics Curriculum: Primary 1 – 6
https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/
mathematics_syllabus_primary_1_to_6.pdf
Singapore Mathematics Curriculum: Secondary One to Four
https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/2020-ntmaths_syllabus.pdf
Vietnamese Mathematics Curriculum: Years 1 to 12
https://www.futureschool.com/vietnam-curriculum/

Online Reference - Open Access
CRICED Online Content
https://www.criced.tsukuba.ac.jp/en/online-contents.html
Teaching Multiplication with Lesson StudyJapanese and Ibero-American Theories for International Mathematics Education
https://link.springer.com/book/10.1007/978-3-030-28561-6

Appendix A – Framework for CCRLS in Mathematics | 267

APPENDIX A
Framework for CCRLS in Mathematics
Nature of Mathematics
Mathematics has been recognised as a necessary literacy for citizenship and not only for 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 in establishing 21st-century skills through reviewing mathematics
as the science of patterns for future prediction and designing with big data which produces innovation not only
for technology advancement but also for business models.
Mathematics is an essential subject to establish common reasoning for the sustainable development ofsociety through viable arguments in understanding each other and developing critical reasoning as the habit ofmind. Mathematics should be learned as the basis for all subjects. For clarifying the framework in CCRLS onmathematics and by knowing the role of mathematics education, the humanistic and philosophical natures ofmathematics 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 toward Fermat’s intelligence and seek a way to be understood theexcellence of his finding on Pascal’s Triangles. If we read Pascal’s Pensées, we recognise how Pascal denied
Descartes’s geometry using algebra from the aspect of ancient Greece geometry. On the other hand, Descartes
tried to overcome the difficulties of ancient geometry through 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 the Continent. In that context, Johann
Bernoulli, a continental mathematician, posed a question in the journal about the Brachistochrone problem,the locus of the point on the circumference of the circle when it rotates on the line. No one replied andBernoulli extended the deadline for 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 understanding 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 and acommon scientific language. Plato and Aristotle are usually compared on this perspective. Plato believes
that the existence of the world of “idea” and mathematics existed in the world of “idea” in 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, the reasonable, harmony and beautifulness of amathematical system is usually felt. Aristotle tried to explain about reaching an idea from the “material” to the“form”. This explains that abstract mathematics can be understood with concrete materials using terms suchas “modelling”, “instruments”, “aembodiment”, “metaphor” and “change representation”. From both ontologicalperspectives, mathematics can be understood and acquired by anyone and if acquired, it serves as a common
scientific language which is used to express any subject. Once representing the ideas using the sharedcommon language, the world can possibly be perceived in the same view autonomously.
From the epistemological perspective, mathematics can be developed through processes which 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-examples. In another word, beyond the contradictionis the nature of the mathematical activity and it provides the opportunity to think mathematically for overcoming.
In this context, mathematics is not fixed but an expandable system that can be restructured through a processof dialectic in constructing viable arguments. Plato also used dialectic for reaching ideas with examples of268 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelmathematics. 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 reorganise mathematics with 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 the mathematical development of operations by the term reflective
abstraction. Reflection is also a necessary activity for mathematisation by Freudenthal. On materialism,
under Vygotskyian perspective, intermediate tools such as language become the basis for reasoning in themind. Under his theory, the high-quality mathematical thinking can be developed depending on high-quality
communication in mathematics classrooms. Dialectical-critical discussion should be enhanced in mathematics
class. From both the epistemological perspectives, mathematics can be developed through the processesof communication, problem-solving and mathematisation which include the reorganisation of mathematics.
Those processes are necessary to acquire mathematical values and ways of thinking through reflection.
Aims of Mathematics in CCRLS
The aims of mathematics in CCRLS for developing basic human characters, creative human capital, and wellqualified citizens in ASEAN for a harmonious society are as follows:•
Develop mathematical values, attitudes and habits of mind for human character,•
Develop mathematical thinking and be able to engage in appropriate processes,•
Acquire proficiency in mathematics contents and apply mathematics in appropriate situations.
Framework for CCRLS in Mathematics as shown in Figure 3 is developed based on the three components
with discussions of the humanistic and philosophical nature of mathematics. This framework also depicts theconcrete ideas of mathematics learning of the above aims.

Figure 3. CCRLS Framework for Mathematics and Aims of Mathematics Learning: Old edition

Appendix A – Framework for CCRLS in Mathematics | 269

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 to live
harmoniously in society. Mathematical values, mathematical attitudes and habits of mind are simultaneously
developed and inculcated through learning the content knowledge.
Essential examples of 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 proving isnecessary for 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 systems of mathematics.
Usefulness and simplicity are used in the 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 objects. Posing questions and providing explanations such as the “why”
and the “when” are the ordinary sequence for thinking mathematically. Changing representation to other ways
such as modelling can overcome running out of ideas in problem-solving. The mindset for trying to understandothers is the basis to explain one’s own ideas that are understandable by the rest with appreciation. Producing
a concept with a 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
is developed independently for those who appreciate life creatively, innovatively and harmoniously. Seeking
an easier and more effective manner of selecting appropriate tools is necessary. Mathematics is a subject
to challenge and experience competitiveness, appreciation with others, and 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 a mathematical way of reasoning in general which does not depend on specific concepts.
Mathematical activities are various types of activities such as problem-solving, exploration and inquiry.
Mathematical processes which include these components are necessary skills to use mathematics in ourlife, such as innovation in this society (e.g., Internet of Things (IoT)). In the context of education, competency
referring to mathematical processes is the basis for STEM and STEAM education as well as a basis for social
science and economic education.
Mathematical ideas serve as the basis of content knowledge related to promoting and developing mathematicalthinking. Some key ideas of mathematics are used as a special process. The fundamental ideas of set andunit lead to a more hierarchical and simple structural relationship. The ability to compare, operate, and perform
algorithms of related functions enables efficient ways of learning mathematics and solving problems in learners’
life with mathematics.
In the case of a set, it is a mathematical idea related to conditions and elements. It is related to activity ingrouping and distinguishing with other groups by conditions. For 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 categorising, we use intervals such as x > 0, x <0, and x = 0. This situation can be seen in the hyperbolic graph where y=1/x.
In the case of a unit, it is a mathematical idea that is related to a process to produce and apply the unit with
operations. In some cases, trying to find the common denominator is the way to find the unit of two given
quantities. A tentative unit such as arbitrary units can be set and applied locally whereas standard units areused globally. The combination of different quantities produces new measurement quantities such as distancewith respect to time produced speed. A square unit such as square centimetres is a unit of area.
STEM refers to Science, Technology, Engineering and Mathematics. STEAM refers to Science, Technology, Engineering, Arts
and Mathematics or Applied Mathematics.
1

270 | Mathematics Challenges for Classroom Practices at the Upper 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 recognised. 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 backwards in relation to ancient Greek term analysis.
Mathematical activities are ways to represent a mathematical process. The problem-solving process wasanalysed by Polya. He influenced problem-solving with various strategies. Technology enhances the activities
of conjecturing and visualising inquiries. Conceptualisation is done based on procedures such as theprocedure 3+3+3+3=12, which becomes the basis for 4x3. The proceduralisation of multiplication is done
through developing the multiplication table, the idea of distribution and memorising.

Content
For cultivating well-qualified citizens, content knowledge of mathematics is essential. The content ofmathematics is usually divided by the set of mathematics. However, developing human characters and creative
human capital should be developed through mathematical processes. Values, attitudes and habits of mind are
driving forces 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 related. 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 different definitions. 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 numbers, they are conceptually different. The definition in
Key Stage 1 is based on counting, Key Stage 2 is based on division and Key Stage 3 is based on algebraic
notation. Expressing such theoretical differences requires names of strands for content to be distinguished. In
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 | 271

the case of measurement, there is no strand name of measurement in Key Stage 3. Key Stage 1 relates to thequantity and setting of the units. In Key Stage 2, it extends to non-additive quantity beyond dimension. In Key
Stage 3, the idea of unit and measurement is embedded in every strand. For example, the square root in the
Numbers and Algebra strand is an irrational number which means unmeasurable, the Pythagorean Theorem inSpace and Geometry strand is used for measuring, proportional function in the Relations and Functions strand
is used for counting the number of nails by weight, and in Statistics and Probability strand, new measurement
units are expressed such as 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 of teaching.
The “Mathematical values, attitudes, habits for human character” component and the “Mathematical thinking
and processes” component cannot exist without the “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.

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:•
Explore a problem with curiosity in a situation and attempt to formulate mathematical problems•
Apply the mathematics learned, listen to other’s ideas and appreciate the usefulness, power andbeauty of mathematics•
Enjoy classroom communications on mathematical ideas in solving problems with patience anddeveloping perseverance•
Feel the excitement of “Eureka” with enthusiasm for the solutions and explanation of unknown
problems•
Think about ways of explanation using understandable representations such as language, symbols,
diagrams and notation of mathematics•
Discuss the differences in seeing situations before and after learning mathematics•
Explain, understand others and conclude mathematical ideas•
Explore ideas through inductive and deductive reasoning when solving problems to foster
mathematical curiosity•
Explore ideas with examples and counter-examples•
Feel confident in using mathematics to analyse and solve contextual problems both in school andin real-life situations•
Promote knowledge, skills and attitudes necessary to pursue further learning in mathematics•
Enhance communication skills with the language of mathematics•
Promote abstract, logical, critical and metacognitive thinking to assess one’s own and other’s work•
Foster critical reasoning for appreciating other’s perspectives•
Promote critical appreciation of the use of information and communication technology in mathematics•
Appreciate the universality of mathematics and its multicultural and historical perspectives

272 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

Those contexts are chosen for illustrating the interwoven links of the two components with contents. It looks
like methods of teaching, however, all three components are the subjects of teaching in 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 | 273

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274 | Mathematics Challenges for Classroom Practices at the Upper 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 | 275

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.

276 | Mathematics Challenges for Classroom Practices at the Upper 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 | 277

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.

Pattern,
Recursion andInvariantMathematics is the science of patterns, in other words, the science of finding
invariants. A pattern means the invariant, something with no change in repetition.
In the case of numbers, it is usually related to natural number sequences andsomething constant. For example, sequence patterns such as the arithmetic
sequence have the same difference and the geometric sequence has the sameratio. In a table, if the first difference is constant it is an arithmetic sequence, but if
the second sequence is constant it is a quadratic sequence. The table also showssome functional relationships between x and y. The pattern is usually found on
the table when given pairs of numbers are lined in an appropriate order. Lining
ordered pairs under natural numbers appropriately is necessary to find the pattern.
Recursion on natural number sequence is usually represented by the recurrence
formula. In mathematics, it is used for complete induction. In the programming,recursion means just repetition in a limited set on the natural number for therecurrence formula and it focuses on the part of complete induction if p(k) is true
then p(k+1) is true.
In the case of a figure, a pattern is usually found on the tessellation of the congruence
triangle. It is an appropriate repletion of the same triangle. If we tessellate it, we canfind the invariant properties of angles in relation to parallel lines. On the tessellated
design as a whole, we can also find translation, rotation and symmetry. We can alsofind the enlarged figure which shows the similarity of the figures. Mandelbrot set on
infinite geometry is the recursion of figures with the natural number sequence. On
the programming, we just realize finite cases on the screen.

278 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

An invariant is a stronger word in mathematics if we compared it with the usage
pattern just meant repetition in informatics such as programming because it isnecessary to be proved on the mathematics system.
Dynamic Geometry Software (DGS) provides ways to find something invariant
which should be proved. It is normal, nothing strange, that two lines meet at one
point if it is not parallel. However, it is very special, and strange, if three lines meetat one point because three lines usually produce a triangle. Thus, the constructionsof circumcenter, incenter and centroid on any triangles are very much amazing andshould be proven.
Graphing Tools (GT) for functions are the tools to represent invariant properties
in Algebra and Analysis. On f(x)=ax2+bx+c, if it is drawn by GT with parametersa, b, c, and fix b and c as any constant, and changes a real number, the graphof f(x) produces the family of functions on the screen. Any graph of the f(x) familyintersects with the y-axis on (0, c) and y=bx+c is the tangent line for any f(x).
To say these findings as invariant in mathematics, we should prove it on the chosenmathematical system. Various proofs are possible if we change the theory inmathematics.

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 generalcase to establish a stronger theory. Thus, so many cases, task sequence of every
unit in textbooks is progressed from special cases to general for generating simple
procedures, exceptionally the tasks for exercise and training the procedures whichusually 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 ‘for example’by and for themselves is a minimum requirement for teaching mathematics.

Appendix B – Terminologies for Mathematical Thinking and Processes | 279

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
to the whole set or situations. Three considerable cases or more will be theminimum to be considered inductive reasoning. To promote inductive reasoning,teachers usually provide the table for finding the patterns, however it is not theway 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 well ordering of the cause and get the effect, follow byrecording the data in a table. To promote inductive reasoning, teachers usuallyprovide the table first for finding the patterns, however it is not the way to develop
the inductive reasoning. It is the concern of students because to develop inductive
reasoning, students have to learn the ordering of natural numbers is vital because
it is the cause of effect and beautifulness of patters that is originated from theorder of natural number. Table is a tool for finding pattern but students cannot
produce their inductive reasoning by 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 of experiment in Science and Engineering.

280 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

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
for 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 thegiven context in process to establish mathematical representations in general.

Appendix B – Terminologies for Mathematical Thinking and Processes | 281

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.
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 from282 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

students to consider and fix the sets, orders, variables, correspondences and so onby 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
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.

Appendix B – Terminologies for Mathematical Thinking and Processes | 283

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 class cannot
be distinguished with the open-ended approach. The open approach is not necessary
to prepare the task sequence because it is characterised by an independent openended task. If teachers set the open-ended task independently, it is the first approach.
If teachers set the task sequence of open ended tasks for learning 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 leveland since 1943 for secondary level which was done by Prof. Shimada Shigeru andothers. Prof. Nobuhiko Nohoda was known as he theorized the open approachwith open-ended task and task sequence. Currently, Japanese textbooks until the
middle school level equipped the task sequence for problem 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 used284 | Mathematics Challenges for Classroom Practices at the Upper Primary Levelin 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.

Appendix B – Terminologies for Mathematical Thinking and Processes | 285

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

286 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

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.

Appendix C – Mathematical Values | 287

APPENDIX C
Mathematical Values

Values indicate the direction that we seek. Thus, they set the direction of thinking. On values, in mathematics,
generalisable and expandable ideas are usually recognised as strong ideas. Proving is usually necessary inmathematics for seeking reasonableness. Harmony and beautifulness are described not only in relation tomathematical arts but also in the science of patterns and the basic structure of the system of mathematics.
Usefulness and simplicity are used in the selection of mathematical ideas and procedures. Here, each term
does not mean an independent category but is related to a feeling such as ‘Simple is beautiful in Mathematics.’
Mathematical Values seeking for:
Generality and Expandability
School mathematics seeks to establish general theory as well as University mathematics. For example, in thecase of multiplication, the following learning process is going through seeking general operation with extension.
At the initiation stage of multiplication, if students acquired the multiplication table from Row 1 to Row 9, they
are released from accumulation (repeated addition) and began to learn multiplication as the binary operation.
However, without extension of numbers for multiplication such as Row 10, Row 11, and so on, they still haveto do repeated addition. Shall they have to memorise all of them? It is impossible! If they acquire column
multiplication, they can multiply any digits with their acquired knowledge of the multiplication table. The column
multiplication can extend to decimals if they are able to manage the position of the decimal point on the baseten place value notation. On these extensions and generalizations of multiplication, the accumulation form
functioned to learn multiplication as repeated addition at the initiation, however, in the next stage, the column
form under the place value system functioned to alternate the accumulation to the column multiplication asa binary operation. These are the process to seek generality and expandability in the case of multiplication.
For learning the value to seek generality and expandability, teachers are necessary to begin from the ungeneral and before-expansion cases, even though teachers knew the general and the final extended forms.
As long as teachers try to teach their general knowledge, students will never have the opportunity to learn
these values. A good teacher can design the process of generalisation and expansion and seeks to teach thevalues through reflection and appreciation. If students appreciate, they can learn the values and seek them byand for themselves.

Reasonableness and Harmony
Mathematics is reasonable. Good teachers usually ask why for developing students who will explain the reason
by and for themselves by using what students already learned. Here, a logical-reasonable explanation for using
learned knowledge is not always deductive, but more analogical and inductive is related to numbers at the
primary level. Significance and objective are also considered as the reason why. For teaching reasonableness,
students may have to feel something strange or good first and explain it by using the learned knowledge orsomething sharable. An unreasonable situation is necessary for creating the opportunity to learn the reason
itself. This meant good teachers know what is unreasonable for students with a comparison of what students
already learned and plan the process to recognise the unreasonable and to reach appropriate reasons by using
what already learned. In the problem-solving approach, the unknown problem itself is set by such a teacher
to provide students with the feeling of unknown or unreasonableness. However, on the given appropriate task
sequence designed by good teachers, students are able to use what they already learned for the unknown
problem. Thus, on the well-designed task sequence, teachers are possible to ask students why.
Mathematics is a harmonious/harmonised subject: how do you explain it? In Ancient Greece, harmony is aname of the subject of mathematics represented by the Pythagorean music scale developed by the ratio, 1 to2, 2 to 3 and 3 to 4. Buildings were also constructed by the special ratio in Ancient Greece. Some ratios wereused to represent beautifulness (Eros) in the Era.

288 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

In the case of music, we have to consider the overtone for the development of scale and code. The height ofscale (sound) is developed by a kind of multiple however dividing real strings (code) itself for the developmentof the musical instruments themselves are division as the multiple of reciprocal. It was the origin of music asa subject of education in Europe. In current mathematics, harmony has very much limited use such as the
harmonic series as for the extension of overtone. On another story, the current music scale itself is defined as
equal temperaments which are the products of multiple to produce a harmonised code in orchestration and it
is not the ratios on the Pythagorean scale itself because it produces special growl in a special case.
Here, as for mathematics education, harmony as a seeking value is metaphorically used such as the harmony
produced by the current music orchestration even if it is not the usage of current mathematics. Firstly, at
the moment, mathematical notions are proved on a system, it is recognised that already existed from the
beginning in a system: It is called pre-supposed/established harmony likely the providence such as the god
already planned before. Up to the final moment to be proved in a system, the notions do not have a clear
position, however at the proved moment, it is a part of the system based on mathematical structure. Even
though they are not proved in one system (theory), they can be used hypothetically as true. Even, each ofthem is proved on the different systems (theories), not one system, and used at the same time as long as it
works. However, it looks something strange until re-setting the harmonised position in a system like the historyof negative and imaginary numbers.
In music, different scale/temperament system produces different sounds however the music looks the same.
For example, Current Buch players use the instruments under equal temperaments even though they useddifferent cords. We are listening to and playing different Bach music however we still recognise it as Bach.
Metaphorically, it implies that we recognise mathematical notions on the different systems and use themharmoniously through proven under different systems/theories, even though they should be proved in asystem, later. Indeed, the school mathematics curriculum is the possible sequence of local theories which
includes several contradictions among them, however, we are possible to accept them harmoniously. In thismetaphorical usage, a teacher needs to prepare the class based on various mathematical systems/theories
which students already learned.
Secondly, with more metaphorical usage in a mathematics classroom, each student looks at a player to produce
mathematical notions harmoniously with collaboration. Current orchestration is possible through each player
and conductor’s contributions and collaborations. Each player is a necessary component to produce a kindof resonance of code. Metaphorically, it is a kind of verbal and non-verbal communication in the mathematics
classroom and each student’s idea has the role to produce harmonised reasoning as a whole and to beappreciated the likely resonance of code. Even though there is a student who does not talk verbally, he or she
is contributing by non-verbal manners such as attitude and eye line and so on. A well-harmonised lesson canbe seen through such attitudes and so on.
Usefulness and Efficiency
Mathematics usually alternates the meanings which are necessary for reasoning to the procedure which
automates and compresses the reasoning. If this alternation is recognised and done by students, they areable to learn the efficiency of the procedure. If teachers only teach the final procedure from the beginningwithout any meaning, students lost the opportunity to learn efficiently. Even though teachers take over theopportunity from students, teachers are able to teach the usefulness of the application problem after they teachthe procedure. In the previous example of multiplication, accumulation is necessary at the beginning but if wememorise the multiplication table at once and used it for column multiplication it is unnecessary to consider theaccumulation. What good things you learned about column multiplication can be explained if they learned theaccumulation before column multiplication. What are the good things you learned about positive and negative
numbers? These questions are firstly answered by usefulness and efficiency even though there are othervalues. For answering these questions, students might have the opportunity to compare before and after.

Appendix C – Mathematical Values | 289

Simpler and Easier
These values are similar to the value of usefulness and efficiency, however, simpler is usually used for theselection of procedure and setting the mathematical form. In the development of the procedure, we chose
simpler forms and so on. For example, numbers are read from the largest place value. Thus, children usually
try to add from the largest place at the beginning. However, on the column addition, if it has the carrying, they
have to rewrite the number several times in the process. If they decide to add from the lowest place, it becomes
simple. For explaining the procedure, we use a diagram such as a proportional number line for making easierto produce the expression and interpret the meaning. If students can draw a diagram for an explanation of themeaning, it must be easier to understand.

Beautifulness
In mathematics teaching, beautifulness uses several occasions with various values such as reasoning is simple
and thus beautiful. If we find structure/pattern/invariant, it is beautiful. If we make a line with equal signs in theoperation of the equation easier to see, then it is beautiful. For students to feel the beautifulness, teachers
need to set students to feel the ugly/not beautiful situation and change it to feel beautiful. For example, it is justa heap of coins before the arrangement of the same coins. However, if the coins are arranged in vertical bars
as in ordering natural numbers, we can alternate the numbers to the height. and if the coins are regrouped by
the height of 10 coins, we can easily count by 10. Comparisons before and after are necessary to discuss how
beautiful is the order.
Beautifulness is also discussed in relation to specific mathematical ideas. In number and operation, the answerof operation should be a number, thenthan

, thusis a number even if it is

In algebra, the general form of the quadratic function isof the quadratic function islooks more beautiful
and the standard form

. If we do not know how these two forms are beautiful, we cannot

operate a quadratic function. In a figure, symmetry is usually found in the tessellation of a figure. Symmetryis the word to represent the beautifulness of geometry and algebra. On the sequence, the recursion on thenumbered sequence is beautiful because it is a representation of invariant pattern.

290 | Mathematics Challenges for Classroom Practices at the Upper Primary Level

APPENDIX D
Strand: Mathematical Process – Humanity for Key Stage 2
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 explanations for mathematical problems
and modelling. The ability to connect and reason mathematical ideas would trigger 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 part ofthe enjoyable activities. Through the consistent use of representations such as diagrams, the 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 the consistency of representations
Enjoying tiling with various figures and blocks
Producing valuable explanations 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

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 volume1i.
ii.

Pose questions to develop a 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 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 volume2
Give the formula for the area and volume for counting units
Use measurement for communication in daily life

1
This is connected to the three strands, Extension of Numbers and Operations, Measurement and Relations, and Plane Figures
and Space Figures.
2
Euclidean algorithm is a method of finding the largest common divisor of two numbers.

Appendix D – Strand on Mathematical Processes-Humanity (Key Stage 2) | 291

Using ratio and rate in situations3
i.
ii.
iii.

Understand division as partitive (between different quantities) and quotative (between the 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 situations4i.
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 the consistency of representations5
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 areas 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 blocks6
i.
ii.
iii.
iv.

Appreciate producing parallel lines with a 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
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 a discussion
Assessing the appropriateness of explanations using representations such as generality, simplicity
and clarity
Use other’s ideas to produce a better understanding
Use inductive reasoning for extending formulae

Performing activities of grouping and enjoying representing with Venn diagrami.
ii.

Use the idea of the Venn diagram for social study
Understand classifications based on characteristics and represent them by using Venn diagrams

Ratio and proportion bridge multiplication and division in situation of two quantities with reference to Extension of Numbers
and Operations and Measurement and Relations.
4This is a bridge to the Extension of Numbers and Operations and Measurement and Relations.
5
Bridge to the three strands, Measurement and Relations, Plane Figures and Space Figures and Data Handling and Graphs.
6
Connected to the two strands, Measurement and Relations, and Plane Figures and Space Figures.
3292 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

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 a conclusion with justification based on data analysis

Preparing sustainable life with number sense and mathematical representations7v.
vi.

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 developmentvii. Estimate the efficient use of resources in situations
viii. Maximise the use of resources through an appropriate arrangement in a space such as a room
ix. 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 notebooks to exchange learning experiences with each other such as in mathematics
journal writingvii. Use protractors, triangular compasses, straight edges, and clinometers for drawing and measuringviii. Use the idea of proportionality to use mechanisms such as rotating once and moving 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 sharethem with 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 life7
It is related to Numbers and Operations, Pattern and Data Representations and Quantity and Measurement all under Key
Stage 1.

Appendix E – Learning Standards for Key Stage 1 | 293

APPENDIX E
The four strands on content learning standards and the strand on Mathematical Process-Humanity of Key
Stage one
Key Stage 1
Key Stage 1 (KS1) serves as the foundation of knowledge covering the basic facts and skills developed
through simple hands-on activities, manipulation of concrete objects, pictorial and symbolic representations.
This stage focuses on arousing interest, enjoyment and curiosity in the subject through exploration of patterns,
characterisation, identification and describing shapes, performing the four fundamental operations, identifying
its algorithm, and understanding basic mathematical concepts and skills experienced in daily life. Calculationsof quantities will also be established to carefully and wilfully understand the attribution of objects used to makedirect and indirect comparisons.

Strand: Numbers and Operations
The number is introduced with situations, concrete objects, pictorial, symbolic representations and extended
based on knowledge and skills learned. Ways of counting and distributions are extended to addition,
subtraction, multiplication and division. The base ten number system is the key to extending the numbers andoperations for standard algorithms in vertical form. Also, various procedures of calculations and algorithms arefocused on. Models and diagrams are used for extensions instead of concrete materials themselves. Number
sense will be developed through the establishment of fluency in calculations with connection to situations andmodels. Fractions and decimals are introduced with manipulatives.
Topics: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 Forms for Addition and Subtraction and Acquiring Fluency of Standard Algorithms
Introducing Multiplication and Produce 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

Introducing Numbers up to 120
Enjoying counting orally and manipulatively with number names, without symbolic numeralsi.
Develop fluency in the order of number names and use them based on situations1
ii. Set the initial object for counting, the 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
Understanding and using the cardinality and ordinality of numbers with objects and numerals2 through activities
of grouping, corresponding and ordering, and developing number sense3 and appreciating its beautifulnessi.

Group objects for counting with conditions such as cups, flowers, and rabbits in situations andintroduce numerals

The denomination such as 3 cups, 2 cups, and one cup is described at strand on the Quantity and Measurement
Inclusive in reading and writing of numerals
3
Units for counting that describe the Quantity and Measurement
12294 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

ii.
iii.
iv.
v.
vi.

Obtain fluency in counting concrete objects and understand counting on, and recognise the necessityof zeroCompare different sets by one-to-one correspondence and recognize larger, smaller or equal withappreciation in drawing paths between objects.
Compose and decompose numbers for strengthening the number sense4 and cardinality
Understand the difference between ordinal and cardinal numbers and use them appropriately insituations and challenge the mixed sequence.
Acquire number sense5 and appreciate the beautifulness of ordering numerals with and without
concrete objects

Introducing the base-ten system with groupings of 10 and extending numbers up to 1206.
i.
ii.
iii.
iv.
v.

Extend numbers to more than 10 with base-ten manipulative representing numbers in ones andtens, and appreciate the base-ten numeration system.
Extend the order sequence of numbers to more than 10 in relation to the size of ones and tens andcompare numbers using numerals in every place value.
Introduce number lines to represent the order of numbers and for comparison starting from zero andcounting by ones, twos, fives, and tens.
Enjoy various ways of the distribution of objects with counting such as playing cards, and explain it
and enhance the number sense
Draw a diagram for representing the size of the number with base-ten blocks such as a flat (square)for a unit of hundred and a rectangular bar for a unit of ten

Introducing Addition and Subtraction
Understanding situations for addition up to 10 and obtaining fluency of using addition in situations
i.
ii.
iii.

Introduce situations (together, combine, and increase) for addition and explain it orally withmanipulative to define addition for operation
Develop fluency in addition expressions using a composition of numbers for easier calculation withnumber sense for the composition of numbers
Apply addition with fluency in learners’ life

Extending addition to more than 10 and obtaining fluency of using addition in situations
i.
ii.
iii.
iv.

Extend addition situations and think about how to answer using the idea of making 10 withdecomposition and composition of numbers
Explain the idea of addition with place value using base-ten blocks
Develop fluency in addition expressions to more than 10 for easier calculation
Apply addition fluency in learners’ life.

Understanding situations for subtraction up to 10 and obtaining fluency of using subtraction in situations
i.
ii.
iii.

Introduce subtraction situations (remaining, complement, and difference) and explain orally withmanipulative to define subtraction for operation7
Develop fluency in subtraction expressions using the decomposition of numbers for easier calculation
Apply subtraction fluency in learners’ life

Extending subtraction to more than 10 and obtaining fluency in` using subtraction in situationsi.
ii.

Extend subtraction with situations and think about how to answer using the idea of 10 with additionand subtraction of numbers (composition and decomposition of numbers)
Explain the idea of subtraction in place value using base ten blocks

Relationship of composing and decomposing numbers become the preparation for addition and subtraction for inverse operation
Number pattern is discussed under Pattern and Data Representations
6
For discussing the difference of hundred twenty is not twelve ten in English
7
Distinguish minuend and subtrahend
45Appendix E – Learning Standards for Key Stage 1 | 295

iii.
iv.

Develop fluency in subtraction expressions to more than 10 for easier calculation
Apply subtraction fluency in learners’ daily life

Utilising Addition and Subtraction
Utilising addition and subtraction in various situations and understanding their relationships
i.
ii.
iii.
iv.

Understand the difference between addition and subtraction situations with tape diagrams
Explain subtraction as an inverse of addition situations with tape diagrams
Understand addition with three numbers, subtraction with three numbers and combination of additionand subtraction situations
Apply addition and subtraction in various situations such as in ordering numbers

Extending Numbers with Base-Ten System Up to 1 000 000 Gradually
Extending numbers using base-ten system up to 1 000
i.
ii.
iii.
iv.
v.

Experience counting of 1 000 by using various units and appreciate the necessity of the base tensystemExtend the order of numbers to more than 1 000 in relation to the size of ones, tens and hundreds
Use a partial number line to compare the size of numbers through a translation of the size of every
digit with the appropriate scale
Represent appropriate diagram to show the size of numbers without counting such as three ofhundreds mean 30 of tens and visualise the relative size of numbers
Represent larger or smaller numbers by the symbol of inequality

Extending numbers using a base-ten system up to 10 000
i.
ii.
iii.

Visualise the 10 000 by using thousand, hundred, ten and one as units
Extend the order sequence of numbers to more than 10 000 in relation to the size of ones, tens,
hundreds and thousands
Use a number line with an appropriate scale to the show size of numbers and the relative size of
numbers while focusing on the scale

Extending numbers using base-ten system up to 1 000 0008
i.
ii.
iii.

Extend numbers up to1 000 000 and learn the representation of the place value for grouping every
3-digit numeral system up to a million
Write large numbers using grouping of a 3-digit numeral system9 such as thousand as a unit andcompare numbers in relation to it
Develop number sense such as larger and smaller based on comparison of place values throughvisualisation of the relative size of numbers

Producing Vertical Form Addition and Subtraction10 and Acquiring Fluency in Standard Algorithms
Thinking about the easier ways for addition and subtraction and producing vertical form algorithms
i.

Think about easier ways of addition or subtraction situations and use models with base-ten blocks
meaningfully for representing the base-ten system

One million is too big for counting and is introduced only for learning the three-digit system
3-digit numeral system such as 123 times thousand equals the same way of reading plus thousand. In the case of Chinese, thefour-digit numeral system is used.
10
Understanding the relationship between addition and subtraction is discussed under Pattern and Data Representations
89

296 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

ii.
iii.
iv.

Produce and elaborate efficient ways and identify the standard algorithms11 in relation to the baseten system with appreciation
Explain the algorithms of borrowing and carrying with regrouping of base-ten models
Acquire fluency in addition and subtraction algorithms

Acquiring fluency in standard algorithms for addition and subtraction and extending up to 4-digit numbersi.
ii.

Extend the vertical form addition and subtraction through the extension of numbers and appreciate
the explanation using the base-ten block model
Develop fluency in every extension up to 3-digit numbers and simple case for 4-digit numbers

Developing number sense12 for estimation13 and using a calculator judiciously for addition and subtraction
i.
ii.
iii.

Develop number sense for mental arithmetic with estimation for addition or subtraction of numbers
Identify necessary situations to use calculators judiciously in real life.
Appreciate the use of calculators in the case of large numbers for finding the total and the difference

Introducing Multiplication and Produce Multiplication Algorithm
Introducing multiplication and mastering multiplication table
i.
ii.
iii.
iv.

Understand the meaning of multiplication14 situations with models using the idea of addition anddistinguish from the common addition to find the total number
Produce a multiplication table in the case of counting by 2 and 5 with array diagrams, pictures orblock models and extend it until 9 and 1 with an appreciation of patterns15
Develop a sense for multiplication through mental calculation with fluency
Use multiplication in daily life, differentiating multiplication in various situations with the understandingthat any number can be a unit for counting in multiplication

Producing multiplication in vertical form and obtaining fluency
i.
ii.
iii.
iv.
v.
vi.
vii.

Think about easier ways of multiplication in the case of numbers greater than 10 using array diagramsand block models
Develop multiplication in vertical form using multiplication table, array, model, and base-ten systemwith appreciation
Extend the multiplication algorithm to 3-digit times 2-digit numbers
Obtain fluency in the standard algorithm for multiplication
Use estimation with the multiplication of tens or hundreds in life
Compare the multiplication expressions which is larger, smaller or equivalent
Appreciate the use of calculators sensibly in life in the case of large numbers

Introducing Division and Extending It to Remainder
Introducing division with two different situations and finding the answers by multiplication
i.
ii.
iii.
iv.

Understand division with quotative and partitive division for distribution situations
Think about how to find the answer to division situations by distribution using diagrams, repeatedsubtractions and multiplication
Obtain fluency to identify answers of division through the inverse operation of multiplication
Appreciate the use of multiplication table for acquiring mental division

Various algorithms are possible and there is no one specific form because depending on the country, the vertical form itself is
not the same. Here, the standard algorithm means the selected appropriate form.
12
Money system is discussed under Measurement and Relations
13
Rounding numbers are treated in key stage 2 under Measurement and Relations
14
Meaning of area is described in Measurement and Relations
15
Multiplication row of 1 is not a repeated addition
11

Appendix E – Learning Standards for Key Stage 1 | 297

Extending division into the case of remainders and using division for distribution in daily situations
i.
ii.
iii.

Extend division situations with remainders and understand division as a repeated subtraction withremainders.
Obtain fluency in the division and apply it in daily situations
Understand simple cases of the division algorithm

Introducing Fractions and Extending to Addition and Subtraction of Similar Fractions
Introducing simple fractions such as halves, quarters and so on using paper folding and drawing diagrams
i.
ii.
iii.
iv.

Introduce simple fractions using paper folding and drawing diagrams in the context of part-whole
relationship
Use “a half of” and “a quarter of” in a daily context such as half a slice of breadCount a quarter for representing one quarter, two quarters, three quarters, and four quarters
Compare and explain simple fractions in the case where the whole is the same

Extending fractions using tape diagram and number line to one, and think about how to add or subtract similar
fractions for producing a simple algorithm
i.

ii.
iii.
iv.

Extend fractions to more than one unit quantity for representing the remaining parts (unit fraction)such as measuring the length of tape, recognising the remaining parts as a unit measure of length,and understanding proper and improper fractions
Appreciate fractions with quantities in two ways; firstly, the whole is a unit of quantity and secondly,
based on the number of unit fraction
Compare fractions in the case where the whole is the same and explain it with a tape diagram or
number line, and develop fraction number sense such as with quantities and so on
Think about how to add or subtract similar fractions with a tape diagram or number line and produce
a simple algorithm with fluency

Introducing Decimals and Extending to Addition and Subtraction
Introducing decimals to tenths, and extending addition and subtraction into decimals
i.
ii.
iii.
iv.

Introduce simple decimals to tenths by remaining part such as using a tape diagram with appreciation
Compare the size of decimal numbers on a number line with the idea of place value
Extend addition and subtraction of decimals utilising the place value system in the vertical form up
to tenths
Think about appropriate place value for applying addition and subtraction in life

Strand: Quantity and Measurement
Attributes of objects are used to make direct and in-direct comparisons., The non-standard and standard units
are also used for comparison. Counting activities denominate units of quantities such as cups for volumes, armlength and hand-spans for length. Standard units such as metre, centimetre, kilogram and litre are introduced.
Time and durations which are not the base ten system are introduced. Money is not a complete model fora base-ten system. The concept of conservation of quantities will be established through the calculation ofquantities. The sense of quantity is developed through the appropriate selection of measurement tools.
Topics:Comparing Size, Directly and Indirectly, Using Appropriate Attributes and Non-Standard Units
Introducing Quantity of Length and Expand It to DistanceIntroducing 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298 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

Comparing Size, Directly and Indirectly, Using Appropriate Attributes and Non-Standard Units
Comparing and describing quantity using appropriate expression
i.
ii.
iii.

Compare two objects directly by attributes instead of stating in length and amount of water such as
longer or shorter and less or more
Compare two objects indirectly using non-standard units to appreciate the unification of units
Use appropriate denomination16 of quantity (such as number of cups) for counting and appreciate
the usage of units for quantity in a suitable context

Introducing Quantity of Length and Expanding to DistanceIntroducing centimetre for length and extending to millimetres and metre
Compare the length of different objects and introduce centimetre with a calibrated tape17 of one
centimetre
ii. Demonstrate equivalent length with addition and subtraction such as part-part wholeiii. Extend centimetre to millimetre to represent remaining parts with ideas of equally dividing and theidea of making tens
iv. Extend centimetre to metre to measure using a metre stickv.
Estimate the length of objects and select appropriate tools or measuring units for measurement withfluencyvi. Convert mixed and common units of length for comparison18
vii. Convert mixed and common units of length when adding or subtracting in acquiring the sense forquantity		Introducing distance for the extension of length
i.

i.
ii.
iii.

Introduce kilometres to measure distance travelled using various tools and appreciate the experiencesof measuring skills
Distinguish the distance travelled and the distance between two places on the map
Compare mixed units of length with an appropriate scale on a number line

Introducing Quantity of Mass and Its Measurement and Operation
Introducing gram for mass and extending to kilogram and tons
Compare the mass of different objects directly using balance and introduce gram
Demonstrate equivalent mass with addition and subtraction such as part-part whole
Extend gram to kilogram, measure with a weighing scale
Extend kilogram to metric ton through the relative measure (such as 25 children, each weighing 40
kilograms)v.
Estimate the mass of objects and select appropriate tools or measuring units for measurement withfluencyvi. Convert mixed and common units of mass for comparisonvii. Convert mixed and common units of mass for addition and subtraction in acquiring the sense ofquantity
i.
ii.
iii.
iv.

Denomination is necessary for learning the group of counting. It also describes pattern and data representations and number
and operation both of Key Stage 1
17
The plane tape can be used for direct comparison and indirect comparison by marking. If the tape is scaled by a non-standard
unit, we can use it for measurement. If the tape is scaled by one centimetre we can define the length of the centimetre
18
Which one is longer, 2 m 3 cm or 203 mm?
16

Appendix E – Learning Standards for Key Stage 1 | 299

Introducing Quantity of Liquid19 Capacity and Its Measurement and Operation
Introducing litre for capacity of liquid and extending to millilitre
i.
ii.
iii.
iv.
v.
vi.

Compare the amount of water in different containers and introduce litre with measuring cups of 1 litre
Demonstrate equivalent capacity with addition and subtraction such as part-part whole
Extend litre by decilitre/100-millilitre cup for representing remaining parts with ideas of equallydividing and making 10, and extend until millilitre
Estimate the capacity of containers and select the appropriate measuring unit
Convert mixed and common units of capacity for comparison
Convert mixed and common units of capacity for addition and subtraction in acquiring the sense ofquantity

Introducing Time and Duration, and Its Operation
Introducing analogue time and extending to duration
i.
ii.
iii.

Tell and write analogue time of the day corresponding with different activities in daily life such as
morning, noon, afternoon, day and night.
Show time by using a clock face with an hour hand and a minute hand
Understand the relative movement of clock hands

Extending clock time to a duration of one day20i.
ii.

Introduce duration in hours and minutes based on the beginning time and end time of activities
Express time and duration on a timeline, and understand duration as the difference between two
distinguished timesiii. Addition and subtraction of duration and time
iv. Extend time and duration to secondsv.
Convert mixed and common units of duration for comparison
vi. Estimate the duration of time and select an appropriate measuring unit for measurement with fluency
and appreciate the significance of time and duration in life
vii. Appreciate the difference in time depending on the area (time zone) and the seasons
Introducing Money as Quantity
Introducing money as quantity and use as the model of the base-ten system21
i.
ii.
iii.
iv.

Introduce units of money using notes and coins and determine the correct amount of money
Use counting by fives and so on for the base-10 system
Appreciate the fluency in the calculation of money with all the four operations
Appreciate number sense for the conversion and transaction of money in daily life

Strand: Shapes, Figures and Solids
Basic skills of exploring, identifying, characterising and describing shapes, figures and solids are learned
based on their features. Activities such as paper folding enable the exploration of various features of shapes.
Identification of similarities and differences in shapes and solids enables classification to be done for defining
figures. Using appropriate materials and tools, relationships in drawing, building and comparing the 2D shapes
and 3D objects are considered. Through these activities, the skills for using the knowledge of figures and solids
will be developed. The compass is introduced to draw circles and mark scales of the same length.
The density which explains the relationship between mass and liquid capacity is usually learned in science at a later stage. In
the case of CCRLS Science, it starts in Key Stage 2 such as 1 cubic centimetre of water is equivalent to 1 gram.
20
Calendar is possible in the keys stage 1 under Pattern and Data Representation
21
Coins and notes are dependent on the country. Some countries use currency units of twenty and twenty-five which are in coins
or notes. These forms are not appropriate for the model of the base-10 system
19

300 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

Topics:Exploring shapes of objects
Characterising shapes for figures and solids
Explaining positions and directions

Exploring Shapes of Objects
Exploring shapes of objects for finding their attributes
i.
ii.
iii.
iv.
v.

Roll, fold, stack, arrange, trace, cut, draw, and trace objects (blocks such as boxes, cans and so on)
for knowing their attributes
Use attributes of blocks for drawing the picture by tracing shapes on the paper and explain how todraw it with the shapes
Create patterns of shapes (trees, rockets and so on) by using the attributes and recognise thecharacteristics of shapes22
Appreciate functions of shapes of objects in learners’ life
Appreciate the names of shapes in daily life by using one’s mother tongue

Characterising the Shapes for Figures and Solids
Describing figures with characters of shapes
i.
ii.
iii.
iv.

Use characteristics of shapes for understanding figures (quadrilateral, square, rectangle and triangle,
right angle, same length)
Introduce line and right angle with relation to activities such as paper folding and use it for describing
figures with simple properties, such as a triangle has 3 lines
Classify triangles by specific components, such as side, vertex and angle (right-angled triangle,
equilateral, isosceles) and then know the properties of each classification
Reorganising rectangular shapes and squared shapes as figures by using the right angle and lengthof sidesDescribing solids with characteristics of shapes
i.
ii.
iii.

Use the characteristics of shapes to understand solids such as boxes can be developed by six
rectangular parts with simple properties
Develop boxes with the properties
Appreciate solids around daily life by considering the functions of the solids

Drawing a circle and recognising the sphere based on the circle
i.
ii.
iii.
iv.
v.

22

Think about how to draw a circle and find the centre and radius
Draw a circle with an instrument such as a compass
Enjoy drawing pictures using the function of circles such as SpirographFind the largest circle of the sphere with a diameter and identify the sphere by its centre and radius
Appreciate circles and spheres in daily life such as manholes, and the difference between a soccerball and a rugby ballPattern of shapes is discussed in Key stage 1 under Pattern and Data Representations

Appendix E – Learning Standards for Key Stage 1 | 301

Explaining Positions and Directions
Exploring how to explain a position and direction
i.

ii.
iii.

Identify simple positions and directions of an object accurately using various ways such as in myperspective, in your perspective in the classroom, and the left, right, front, back, west, east, north,south and with measurement
Draw the map around the classroom with consideration of locations
Design a game to appreciate the changing of positions and directions in a classroom

Strand: Pattern and Data Representations
Various types of patterns such as the number sequence and repetition of shapes are considered. The sizeof pictures can be represented by the number sequence. Tessellation of shapes and paper folding can be
represented by the repetition of shapes. Exploration of patterns and features is also considered to represent
the data structure in our life using pictographs and bar graphs. Patterns and features produce the meaningof data and represent mathematical information. Patterns are represented by diagrams and mathematicalsentences which are also used for communication in identifying and classifying situations to produce meaningfulinterpretations.
Topics:Using Patterns under the Number Sequence
Producing Harmony of Shapes using Patterns
Collecting Data and Represent the Structure

Using Patterns under the Number Sequence23
Arranging objects for beautiful patterns under the number sequence
i.
ii.
iii.
iv.
v.
vi.

Know the beautifulness of patterns in cases of arranging objects based on number sequence
Arrange objects according to number sequence to find simple patterns
Arrange expressions such as addition and subtraction to find simple patterns
Express the representation of patterns using placeholders (empty box)
Enjoy the arrangement of objects based on number sequence in daily life
Find patterns on number tables such as in calendars24

Producing harmony of shapes using patterns25
Arranging tiles of different or similar shapes in creating harmony
i.
ii.
iii.
iv.
v.

Know the beautifulness of patterns in cases of arranging the objects based on shapes, colours andsizes
Arrange objects according to shapes, colours and sizes to show patterns
Arrange boxes according to shapes, colours and sizes to create a structure
Arrange circles and spheres for designing
Enjoy the creation based on different shapes, colours and sizes in daily life

Number sequence will be discussed in Key Stage 1 under Numbers and Operations
Time and duration are discussed in key stage 1 under Quantity and Measurement
25
Harmony of shapes will be discussed in Key Stage 1 under Shapes, Figures and Solids
2324302 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

Collecting data and representing the structure
Collecting data through categorisation for getting information
i.
ii.
iii.

Explore the purpose of why data is being collectedGrouped data by creating similar attributes on the denomination26 of categories and counting them(check mark and count)
Think about what information is obtained from the tables with categories and how to use it

Organising the data collected and representation using pictograms for easy visualisation
i.
ii.
iii.

Produce the table and pictograms from collected data under each category
Interpretation of tables and pictograms as a simple conclusion about the data being presented.
Appreciate pictograms through collecting data and adding data in daily activities in learners’ life

Representing a data structure by using a bar graph to predict the future of communities
i.
ii.
iii.

Understand how to draw bar graphs from a table using data categories and sort the graph forshowing its structure
Appreciate ways of presenting data such as using tables, pictograms and bar graphs with sorting forpredicting their future communities
Appreciate the use of data for making a decision

Strand: Mathematical Process - Humanity
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 a number
sense that is essential to support explanations of contextual scenarios and mathematical ideas. Mathematicalways of posing questions in daily life are also necessary to learn at this stage. The ability to select simple,
general and reasonable ideas enables effective future learning. The application of number sense provides afacility for preparing sustainable life. The use of ICT tools and other technological tools provides conveniencein daily life. At the initial stage, concrete model manipulation is enjoyable, however, drawing a diagram is mostnecessary for explaining complicated situations by using simple representation.
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 and appropriate representations
Selecting simple, general and reasonable ideas which can apply to future learning
Preparing sustainable life with a number sense
Utilizing 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 situations27
i.
ii.
iii.
iv.
v.
vi.
vii.

26
27

In addition, pose questions for altogether and increase situations
In subtraction, pose questions for remaining and differences in situations
In multiplication, pose questions for the number of groups in situations
In division, pose questions for partition and quotation in situations
Enjoy questioning by using a combination of operations in various situations
In operations, pose questions to find easier ways of calculation
Use posing questions for four operations on measurements in daily life

Denomination will be learned in Key stage 1 under Quantity and Measurement
It is related to Numbers and Operations and Quantity and Measurement both in Key Stage 1.

Appendix E – Learning Standards for Key Stage 1 | 303

Enjoying measuring through setting and using the units in various situations28
i.
ii.
iii.
iv.
v.

Compare directly and indirectly
Set tentative units from difference 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 ten29
i.
ii.
iii.
iv.
v.
vi.

Show increasing and decreasing patterns using blocks
Show based ten system using blocks, the unit cube is 1, the bar stick is 10 and the flat block
represents 100
Explain the addition and subtraction algorithm in vertical form using a base-ten block model
Explain multiplication table with grouped blocks
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 colours30
i.
ii.
iii.

Appreciate producing beautiful designs with various shapes and finding the pattern to explain it
Reflect, rotate and translate to produce patterns
Cut and paste various shapes and colours to form the box and ball such as developing the globe
from a map

Explaining ideas using various and appropriate representations31
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, containers and paper folding forsharing ideas
Make a decision on how to explain the figures and the solids by using manipulative objects ordiagrams or only verbal explanation
Explain patterns using diagrams, numbers, tables and expressions with a blank box
Ask questions using terms such as why, how, what, if and if not, and reply using examples and ‘forexample’ in discussions
Change the representation and translate it appropriately into daily life

Selecting simple, general and reasonable ideas which can apply to future learning32
i
ii
iii
iv

Discuss the argument for the easier ways for addition and subtraction algorithms 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 to division

Applying number sense33 acquired in Key Stage 1 for preparing sustainable life34
iii

Use mathematics for the minimum and sequential use of resources in situations
Estimate for efficient use of resources in situations

It is related to Quantity and Measurement and Pattern and Data Representations both in Key Stage 1.
It is related to Pattern and Data Representations and Numbers and Operations both in Key Stage 1.
30
It is related to Shapes, Figures and Solids and Pattern and Data Representations both in Key Stage 1.
31
It is related to all strands in Key Stage 1.
32
It is related to Numbers and Operations and Pattern and Data Representations both in Key Stage 1.
33
It is related to Numbers and Operations, Pattern and Data Representations and Quantity andMeasurement all in included in Key Stage 1.
34
Sustainable development goals were crafted at the 70th Session of the United Nations General Assembly and indicated themas universal value in education.
28
29

304 | Mathematics Challenges for Classroom Practices at the Upper Secondary Leveliii
iv

Maximize the use of resources through an 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 such as
clocks35
i
ii
iii
iv
v
vi
vii
viii

Use calculators for poly addition in situations
Use mental calculations for estimations
Use a balance scale to produce equality and inequality
Use cups, tapes, stopwatch, and weighing scales for measuring distances and weights
Use calculators to explain the process of calculation by solving backwards and understand therelationship of addition and subtraction, and multiplication and division.
Enjoy using notebooks to exchange learning with each other such as mathematics journal writing
Enjoy presentations with board writing
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 them with others
Prepare and present ideas using posters to promote good practices in the neighbourhood
Listen to other’s ideas and ask questions for better creation
Utilise information, properties and models as a basis for reasoning
Utilise practical arts and outdoor studies to investigate local issues for improving the welfare of life

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.
35

Appendix F – Learning Standards for Key Stage 3 | 305

APPENDIX F
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 focuses 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 byexpressions 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 equivalents and solving equations with fluency. Substitution, addition and subtraction ofequations 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 integrating 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 are greater or less than on the extended number line and use absolutevalue for the distance from zero
Extend operations to positive and negative numbers and explain the reason
Get efficiency on calculation in relation to the algebraic sum

Utilising Letters for Algebraic Expressions and Equations
Extending the utilisation of letters for a general representation of situations and finding ways to simplify
algebraic expressionsi.

Appreciate the utilisation of letters for a general representation of situations to see the expression
as a process and value306 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

ii.
iii.

Find ways to simplify expressions using distributive law and figural explanations, establish thecalculation with like and unlike letters
Acquire fluency in simplifying an expression and appreciate it for representing the pattern of thesituation

Thinking about a set of numbers in algebraic expression with letters as variables and representing 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 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 to equations from the set of numbers and think backwards
Know the properties of equations which keep the set of answers of the equation
Appreciate the efficient use of properties of equations to solve linear equation
Use equations based on life situations to develop fluency, solve equations, and interpret the solution

Algebraic Expressions, Monomials and Polynomials and Simultaneous Equations
Thinking about the calculations of monomials and polynomials for simple case1
i.
ii.
iii.

Introduce terms, monomials and polynomials
Introduce a number raised to the power of two as a square, and a number raised to the third power
as a cube
Get fluency in 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 solutions of linear equations and simultaneous linear equations as a pairof numbers
Know the substitution and elimination methods of solving simultaneous linear equations
Get fluency in 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 in selecting and using the appropriate formulae
Use the expansion formulae to factorise the second-degree expression and recognise both formulaewith the inverse operation
Solve the simple second-degree equations using the factorisation and apply it in life situations

Extending Numbers with Square Roots
Extending numbers with square roots and calculating the square roots algebraicallyi.
Define square root and discuss ways to estimate the nearest value of a square root by the Sandwich
Theorem
1

Simple case may vary from and depending on the countries based on the mapping of curriculum.

Appendix F – Learning Standards for Key Stage 3 | 307

ii.
iii.
iv.
v.

Understand that some square roots cannot be represented as fractions
Compare square roots using a 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 roots 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 equations using factorisation and apply on the situation
i.
ii.
iii.

Find the answers to 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 a system of equations. Functional relations are treatedamongst situations, tables, and equations of function are introduced based on patterns and relations withalgebraic representation in Key Stage 2 and Key Stage 3. The solution of a simple equation is done byequivalence deduction based on algebra learnt earlier. Two variables in simultaneous equations as a simple
system of equations are solved by substitution and additive-subtractive methods. Three representations, table,
equation and graph, are used as methods to analyse the properties of every function. Proportion and inverse
proportions are redefined with those representations mentioned. The proportional function is extended to linefunctions. The comparisons of inverse proportion and line functions are made clear by the property of linearity
with a ‘constant ratio of change’. The concept of proportion is extended to the function of y = ax2. Ways oftranslations between table and equation, equation and graph, and 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 a set of points as a 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 a 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

308 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

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 them withproportional functions
Explore properties of the linear function represented by tables, equations and graphs and compare
it with direct and inverse proportional functions
Acquire fluency to translate the rate of change of a linear function represented in the table, as acoefficient in 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
in a graph
Apply the graphs of linear functions to solve simultaneous equations
Apply the linear function for data representation on situations to determine the best-fit line

Exploring Simple Quadratic Function
Exploring quadratic function y = ax2 in relation to a linear function
i.
ii.
iii.

Identify the quadratic function on situations using tables and comparing it with a linear function
Explore properties of a quadratic function using tables, equations as well as graphs and compare itwith a linear function
Apply the quadratic function to situations in daily life and appreciate it

Generalising Functions
Generalising functions with various representations6 of situations
i.
ii.
iii.
iv.

Distinguish domain, range and intervals and is appropriately used for explaining the function
Use various situations for generalising ideas of functions such as moving point A and moving point
B with a time
Compose a graph as a function of two or more graphs with different domains in a situation
Introduce situations of step-functions7 with a graph for generalisation of the idea of a function whichcannot be represented by an equation

Strand: Space Geometry
Space and Geometry provide ways of reasoning for exploring properties in geometry and produce ways ofargument to explain justifications of visual reasoning. The calculations of angles are not just simple calculation
but also ways of using geometric propositions to justify answers by explaining why it is correct based on basic
properties. By explaining the relationship of figures using transformation, the properties of congruency anddescribing similarity are identified and described. Finding the value of angles and building arguments forproving are means for developing the habit of reasoning in the properties of plane figures. The conditionsof congruence and similarities, properties of circles, are also used to explain and prove the appropriatenessof geometric conjectures in relation to triangles, quadrilaterals, and circles. Dynamic geometric software aswell as a simple compass and ruler are used for conjecturing. It shows general ideas from consistency invariations. The counterexample is also found as a 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Appendix F – Learning Standards for Key Stage 3 | 309

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 doing 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 a 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 a line of symmetry, point of symmetry and parallel lines
Explore the similarity of figures with enlargement using points, ratios, 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 the 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 of theargument 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 spacesi.

Explore the proving of the Pythagorean theorem using a diagram and use it in solving problems
involving plane figures

8
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.
Mean, median and quartile are fixed depending on the data. However, mode changes depending on the class.

310 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

ii.
iii.
iv.

Apply the Pythagorean theorem on the 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 the Pythagorean theorem in life10

Strand: Statistics and Probability
Data handling is 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 data using
Information Communication and Technology (ICT) is enhanced. The histogram shows different dispersion if
we change the class. Probability is introduced as a ratio with the law of large numbers. Sample space withthe assumption 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 can be seen relatively and produce frequency distribution polygon. The difference between thesample and the population is discussed. Boxplots with quartiles are an extension of the median and the range
is used for comparisons of distributions. Using skills of statistics and probability make problem solving insituations possible. Analysing and identifying the trends in situations for making decisions is necessary suchas issues for sustainable living.
Topics:Exploring Distribution with the Understanding of Variability
Exploring Probability with the 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 histograms with different class intervals to show a different distribution of the same set of data
Identify alternative ways to show distribution such as dot plots, box-plot and frequency polygons
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 the Law of Large Numbers and Sample Space
Exploring probability with descriptive statistics, the 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 the probability12
Analyse sample space of situations represented by a table to determine the probability and use it forpredicting the occurrenceUse various representations such as tables, tree diagrams, histograms and frequency polygons forfinding the 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Appendix F – Learning Standards for Key Stage 3 | 311

Exploring Statistics with Sampling
Exploring sampling with the understanding of randomness
i.
ii.
iii.
iv.

Discuss the hidden hypothesis behind the 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
The critical argument in mathematics is enhanced through communication with others beyond Key Stage 2.
These proposed challenging activities will promote metacognitive thinking at a different level of arguments tomake sense of mathematics. Translating real-life activities into mathematical models and solving problems
using appropriate strategies are emphasised in functional situations. The process of doing mathematicalactivities involves patience that develops perseverance in learners and takes responsibility for one’s own
learning. At this stage, the habitual practice of self-learning will eventually develop confidence, thus, theopportunity for challenges to extend mathematics and the ability to plan sequences of future learning is alsoenhanced.
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 the structure of distributions
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 the hypothesis based on the sample and populationx.
Pose conjecturing such as if x increases and y decreases then it is an inverse proportion
Enjoying measuring space using calculations with various formulae14
i.

13
14

Extend the number line to positive and negative numbers and compare the size of numbers with theidea of absolute value

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.

312 | Mathematics Challenges for Classroom Practices at the Upper Secondary Level

ii.
iii.
iv.
v.
vi.

Derive the square root using unit squared paper through the idea of the area
Explain the expansion of polynomials using area diagrams
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 to understanding
Rewrite propositions from sentences to mathematical expressions by using letters and diagrams
Search the ways of proving by thinking backwards from the conclusion and thinking forward fromthe givenv.
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’ explanations 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 the context
Analyse the 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 a circle to identify the relationship between the circumference and the central angle (acute,obtuse and right)
Use a rectangle and rotate 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 pointof intersection
Explore to know the nature of two simultaneous equations by using translation
Use the y-axis, x-axis and as the line of symmetry to explore the proportional function
Explain the graph of the linear function by translation of the 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 numbers and identify how to explain the order of numbers and their 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

Appendix F – Learning Standards for Key Stage 3 | 313

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 rain and landslide where the calculations on the amountof water in barrel per minute exceed 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 a graphing tool for comparison of the graph and knowing the properties of a function
Use data to analyse statistics with software
Use internet data for the discussion of sustainable development
Use calculators for operations in the necessary context
Use a projector for sharing ideas such as project surveys, reporting and presentation
Use the idea of function to control a mechanism
Use ICT tools for conjecturing and justifying to produce the object of proving.

Promoting creative and global citizenship for the 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.