Mathematics Challenges
for Classroom Practices
at theLOWER SECONDARY LEVEL
Based on SEAMEO Basic Education Standards:
Common Core Regional Learning Standards in Mathematics

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Editors:
GAN Teck HockISODA Masami
TEH Kim HongWith Support of:

Maitree lnprasitha (KKUJSupattra Pativisan (IPSTJ
Wahyudi & Sumardyono (SEAQIMJA Collaboration Project:%tretk~University ofTsukuba

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

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

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

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

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

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

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

ISBN: 978-967-930-055-0

or

The Director
SEAMEO RECSAM
Jalan Sultan Azlan Shah, Gelugor
11700 Penang, MALAYSIA
Phone: +604-652 2727 Fax: +604-658 7558
Email: director@recsam.edu.my
URL: http://www.recsam.edu.myiv | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

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

Dr ETHEL AGNES PASCUA-VALENZUELA
Director, SEAMEO Secretariat, Bangkok, Thailand

Foreword |v

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

Dr SHAH JAHAN BIN ASSANARKUTTY
Centre Director, SEAMEO RECSAM

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

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 the 21st century skills and OECD competency (2005)
in education. For SEA-BES-M, I had collaborated with Pedro Montecillo Jr.,Kim Hong Teh, and the late Mohd Sazali bin Hj. Khalid of RECSAM, along 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 iscompartmentalised between every content description. Therefore, a major contribution of SEABES-M to the world, particularly the SEAMEO countries is its clear description of the meaning of
higher order thinking for the mathematics curriculum standards to develop mathematical thinking.
After the publication, we recognized that there are difficulties for readers to understand the intended
meaning of every standard. This is mainly due to the fact that readers will interpret what is readbased on their curriculum knowledge and experience teaching mathematics of their own countries.
As such, it is crucial to develop a book series to be used as references for interpreting SEA-BES-M.
This book series is prepared, particularly for teacher educators, textbook authors, and curriculum
developers as well as teachers to understand higher order thinking for developing mathematicalthinking in their classrooms. In this book series, the authors had collaborated with leading researchersand educators in major teacher education institutions, CRME-IRDTP at KKU (Thailand), IPST
(Thailand) and SEAMEO QITEP in Mathematics (Indonesia) with the support of the coordinators.
Furthermore, the authors had made some minor revisions to the SEA-BES-M to align with theneeds of the Era of the 4th Industrial Revolutions to develop stakeholders and users of Artificial
Intelligence and Big Data in business as well as establishing a successful life under the realityof humanity with technology. The minor revision was made based on the curriculum reform
recommendation (2020) by APEC InMside Project with the purpose of promoting mathematical
capitalism under mathematical-statistical-informational sciences on the demands of the Era. It
is hoped that this book series is used in teacher education to develop new curriculum content
knowledge for teaching in this Era. I would like to acknowledge SEAMEO secretariat and centres
for their collaborations, especially Shah Jahan Bin Assanarkutty, the Director of RECSAM, who
made possible this publication. Last but not least, I would like to convey my sincere appreciation
to all contributing writers stated in the contributor list and Gakko Tosho (the Japanese Textbook
Publisher) which provided us innovative ideas.

MASAMI ISODA
Director of CRICED, University of Tsukuba, Japan

Preface | vii

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

Numbers and Algebra
Relations and Functions
Space and Geometry
Statistics and Probability

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

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

viii | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

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

Editors
GAN Teck HockISODA Masami
TEH Kim Hong

Acknowledgement | ix

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

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

TABLE OF CONTENTS
Foreword

iv

Prefacevii

Acknowledgementix

Table of contentsxAuthors, Contributing Members and Reviewers

xi

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

1

Chapter 2: Numbers and Algebra• 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 the Numbers with Square Roots• Solving Quadratic Equations

3

Chapter 3: Relations and Functions•
Extending Proportion and Inverse Proportion to Functions with Variables•
Exploring Linear Function in Relation to Proportions•
Exploring Simple Quadratic Function•
Generalising Functions

43

Chapter 4: Space and Geometry•
Exploring Angles, Construction and Designs in Geometry•
Exploring Space with its Components•
Exploring Ways of Argument for Proving and Its Application in Geometry

71

Chapter 5: Statistics and Probability•
Exploring Distribution with the Understanding of Variability•
Exploring Probability with Law of Large Numbers and Sample Space•
Exploring Statistics with Sampling

115

Bibliography

149

Appendix A – Framework for CCRLS in Mathematics

151

Appendix B – Terminologies for Mathematical Thinking and Processes

157

Appendix C – Strand on Mathematical Processes-Humanity (Key Stage 3)

170

Appendix D – Learning Standards for Key Stage 1

173

Appendix E – Learning Standards for Key Stage 2

185

Authors and Contributing Members | xi

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

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

xii | Mathematics Challenges for Classroom Practices at the Lower Secondary 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
Mariam Binti Othman (SEAMEO RECSAM, Malaysia)
Murugan A/L Rajoo (SEAMEO RECSAM, Malaysia)
Warabhorn Preechaporn (SEAMEO RECSAM, Malaysia)
Wan Noor Adzmin Binti Mohd Sabri (SEAMEO RECSAM, Malaysia)Chapter 1: Guide to the SEA-BES CCRLS Framework in Mathematics |1

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

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

2| Mathematics Challenges for Classroom Practices at the Lower Secondary Levelbe found in Appendix B. The standards for the strand on Mathematical Processes-Humanity is
attached in Appendix C to provide readers with challenging activities to promote metacognitive
thinking at different level of arguments to make sense of mathematics. In order to understand thedevelopment and progression of learning from the primary level to the secondary level (Key Stage3), the learning standards of Key Stage 1 and Key Stage 2 can be referred in Appendix D andAppendix E, respectively.
The interconnection of the three components is shown in Figure 2. The ultimate aim of the CCRLS
framework is to develop mathematical values, attitudes and human characters which are theessence of a harmonious society. This component is closely related to the affective domain ofhuman character traits which correspond to soft skills that can be developed through appreciation.
In relation to this, acquisition of mathematics contents as hard skills and reflection on the thinking
processes are needed to inculcate the capability of appreciation. The reflection is necessary forlearners to recognise their cognitive skills derived through the contents. Even though contents
appeared to be learned independently through acquisition, the mathematical thinking and process,and the appreciation of mathematical values, attitudes and habits for human character is possible
to be developed through reflective experiences.

CO
N

TE

XT
S

TS

N
CO

X
TE

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

Chapter 2: Numbers and Algebra |3

CHAPTER TWO
Numbers and Algebra
Topic 1: Extending Numbers to Positive and Negative Numbers
Standard 1.1:
Extending numbers to positive and negative numbers and integrate four operations into addition
and multiplication
i.
ii.
iii.
iv.

Understand the necessity and significance of extending numbers to positive and negative
numbers in relation to directed numbers with quantity
Compare numbers which is greater or less than on the extended number line and use absolutevalue for distance from zero
Extend operations to positive and negative numbers and explain the reason
Get efficiency on calculation in relation to algebraic sum

Sample Tasks for Understanding the Standards
Task 1: Positive and Negative Numbers in the Real World
Altitude or Depth of a Location

Diagram 1
Altitude is the vertical distance above sea level and depth is the vertical distance below sea
level.
Diagram 1 shows the cross sectional view of Mount Kinabalu to its nearest city, Kota Kinabalu
which is next to the South China Sea.
i.

Use numbers to indicate the altitude or depth of the following locations:
•
•
•
•
•

ii.

The summit of Mount Kinabalu
The eagle
Kota Kinabalu City
The sea turtle
The seabed of South China Sea

How can numbers be used to differentiate locations above and below the sea level?

4| Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Elevator with Positive and Negative Floors
Diagram 2 shows a button panel of anelevator. In this elevator, positive numbers
are used to label floors on and above theground, whereas negative numbers are used
to label basement floors as shown below.

Diagram 2i.

Are the positive and negative numbers
used correctly in labelling the floors?
Why or why not?

ii.

What modifications to the button panel
would you suggest?

iii.

Explain the reasons for your suggestions.

Chapter 2: Numbers and Algebra |5

Task 2: Comparing Positive and Negative Numbersi.

Diagram 3 shows five number lines used torepresent the numbers +1, +2, +3, +4, +5, –1,–2, –3, –4 and –5.
•

Describe how positive and negative
numbers are represented in a number
line.

•

With the use of real-world examples,
explain why +5 > +3, but –5 < –3.

Diagram 3ii.

Diagram 4 shows a number line for zero and positive numbers whereas Diagram 5 shows
an extended number line to include the negative numbers.

Diagram 4

Diagram 5
As shown in Diagram 4, any number on the number line is always 1 less than the number
to its right.
•
•
•

iii.

Does the same rule still apply for the extended number line?
Based on this rule, what can you conclude on using a number line to compare positive
and negative numbers?
Delete the wrong word in the ().
 All positive numbers are (less/greater) than zero.
 All negative numbers are (less/greater) than zero.
 All positive numbers are (less/greater) than negative numbers.
 For any two positive numbers, the one with larger absolute value is (smaller/
larger).
 For any two negative numbers, the one with larger absolute value is (smaller/
larger).

Why is number line a good tool to represent positive and negative numbers?

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

Task 3: Addition and Subtraction of Positive and Negative Numbers Using Number Line
Addition Using Number Line
+3

-1 0

1

-3

+2

2

3

+2

4

5

(a) (+3) + (+2) =-5 -4 -3 -2 -1 0

2

3

4

5

(c) (+3) + (-2) =i.

Diagram 6 shows four additions
involving positive and negative
numbers performed using numberlines.
•
Draw an arrow on each of thenumber lines to show the answer
 for each of the additions.

ii.

A student claims that “adding a negative
number is the same as subtracting thepositive part of the number.”•
Justify that this claim is always
true.

-3

-2

-2
1

2

(b) (-3) + (+2) =

+3

-1 0

1

On a number line, adding a positive or
negative number can be performed bymoving the number of steps toward thepositive (right) or the negative (left) direction.

-5 -4 -3 -2 -1 0

12(d) (-3) + (- 2) =Diagram 6

Subtraction Using Number Line

Subtraction is the inverse operation of(a) (+3) - (+2) =(a) (+3) - (+2)=
addition.
Thus, a subtraction can be thought

Think asThink
anasexample, (+3) – (+2) =  can either beof as an addition with a missing addend. As
thought of as(+2) += (+3)+ (+2)(+2)+ ==(+3)
(+3)

+2
-1 0

1

+2 +2
2

3

4

5

-1 0
-1

+3

1 2 3

4

+ (+2) = (+3)(+2) +  = (+3) or  + (+2) = (+3).
i.

Diagram+2 7 shows two subtractions
involving positive and negative-1 0 1 2 3 4 5numbers
performed using numberlines. +3•
Draw an arrow on each of thenumber lines to show the answer
 for the two ways of thinking foreach of the subtractions.

ii.

Perform the following subtractions
using number lines:•(+3) – (–2)•(–3) – (–2)

iii.

Which of the two ways of thinking about
subtraction do you find it easier toperform subtraction involving positive
and negative numbers using number
line? Why?

5

+3
+3

(b) (-3) - (+2) =Think as(+2) += (-3)+ (+2) = (-3)+2
-5 -4 -3 -2 -1 0

1

+2
2

-5 -4 -3 -2 -1 0

-3

-3

Diagram 7

1

2

Chapter 2: Numbers and Algebra |7

Task 4: Addition and Subtraction of Positive and Negative Numbers Using Playing Cards
Representing Numbers with Playing Cards

Diagram 8

Diagram 8 shows the shapes of the foursuits in a set of playing cards.
Black shapes (Spades & Clubs) are used
to represent positive numbers, and red
shapes (Hearts & Diamonds) are used torepresent negative numbers.
Diagram 9 shows +8 and –8 represented by
8 of Spades and 8 of Hearts respectively.

Diagram 9

When two or more cards are used torepresent a number, equal numbers of blackand red shapes will cancel off and become
zero. Diagram 10 shows three examples ofnumbers represented by two cards.
i.

What numbers are represented by thefollowing sets of cards?
••

Diagram 10•

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

Addition Using Playing Cards

When using playing cards, addition
can be performed by “putting in” cards.
Diagram 11 shows three examplesof addition performed using playing
cards.
ii.

Use playing card to perform thefollowing calculations:•(+8) + (–5)•(–4) + (–2)•(–1) + (+4)

Diagram 11
Subtraction Using Playing Cards

Subtraction is the inverse operationof addition. So, it can be performedby “taking away” cards. Diagram 12shows three examples of subtraction
performed using playing cards.
i.

Diagram 12

Use playing card to perform thefollowing calculations:•(+7) – (+2)•(+3) – (–6)•(–3) – (–8)•(–5) – (+7)

Chapter 2: Numbers and Algebra |9

Task 5: Comparing Number-Line Method and Playing-Card Method
Number lines and playing cards are two methods of performing addition and subtraction. These
two methods have some distinct similarities and differences in terms of the mathematical ideas
involved.
i.

Compare and contrast these two methods.

ii.

Which of these two methods do you think will be easier for your students? Why?

iii.

Which of these two methods do you think can help your students learn other mathematicalideas such as vector in their future grades?

Task 6: Addition and Subtraction Rules Involving Negative Numbers(+5) + (+2)
↓ 1 less(+5) + (+1)
↓ 1 less(+5) + 0↓ 1 less(+5) + (-1)

= (+7)↓ 1 less= (+6)↓ 1 less= (+5)↓ 1 less= (+4)

Diagram 13
A student was asked to find the sum (+5) + (-1) and he gave the answer +4. When asked to
explain his method to get the answer, the student’s reasoning is shown in Diagram 13. He
further explained that “adding a negative number is the same as subtracting the corresponding
positive number.”i.

Is the student’s reasoning valid? Why or why not?

ii.

Using the example (+5) – (–1) = (+6), how would you use the same reasoning to convinceother students that “subtracting a negative number is the same as adding the corresponding
positive number”?

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

Task 7: Application of Negative Numbers in Real WorldIn an international athletics competition, depending on its direction of blow, a tailwind isconsidered either assisting or opposing the effort of the runners. According to the International
Association of Athletics Federation (IAAF) Competition Rules, a tailwind of more than +2 m/s
will make the results not considered as a record although the results are considered valid forthat competition.
Table 1
Current World Records for Man’s 100 m Sprint
Record Holder & Country

i.

Time

9.58 sTailwind+0.9 m/s

Venue

World Athletics
Championship, Berlin,
GermanyDate

16 Aug 2009

Table 1 shows the current world records for 100 m sprint for the man category.
•
•
•

ii.

Usain Bolt, Jamaica

Why was the tailwind of +0.9 m/s considered as an advantage to Usain Bolt?
Usain Bolt’s run was the result of his own running speed assisted by the tailwind.
Calculate the resultant speed of Usain Bolt’s run.
Assuming the tailwind was pushing Usain Bolt’s whole body forward at the speed of+0.9 m/s. Calculate Usain Bold’s own running speed without the assistance of the
tailwind.

The current woman record holder for South America Continent is Rosângela Santos ofBrazil. Her record is 10.91 s with a tailwind of – 0.2 m/s.
•••

Why was this tailwind of – 0.2 m/s considered as a disadvantage to Rosângela Santo?
Calculate the resultant speed of Rosângela Santo with the influence of the tailwind.
Calculate the actual speed of Rosângela Santo without the influence of the tailwind.

Chapter 2: Numbers and Algebra | 11

Task 8: Real-World Multiplication and Division Involving Positive and Negative Numbers

Diagram 14
Diagram 14 shows a boy and a girl walking along a straight track. The boy walks to the rightfrom one end of the track with a speed of 5 km/h, and the girl walks in the opposite direction
from another end with a speed of 4 km/h.
Both the boy and the girl cross at location A and continue their walking.
i.

Use multiplication sentences to represent the respective distance of the boy and the girlfrom location A at the following times:•

2 hours after they crossed.

•

2 hours before they crossed[Hint: Consider the direction of movement (right or left) and time (after or before).]Diagram 15
ii.

Diagram 15 shows two instances of the boy and girl during the walk. Use division sentences
to represent the respective time for the two instances.

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

Task 9: Multiplication and Division Rules Involving Positive and Negative Numbers
Justify by the Distributive Rule of Multiplication(-3) x 0 = 0 … any number multiplied by 0 is 0(-3) x [(+2) + (-2)] = 0 … since (+2) + (–2) = 0i.

Expand by Distributive Rule,

Diagram 16 shows two different
approaches used by two students
to justify that (–3) × (–2) = (+6).
•[(-3) x (+2)] + [(-3) x (-2)] = 0(-6) + [(-3) x (-2)] = 0 … since (-3) × (+2) = (-6)∴ [(-3) x (-2)] = (+6) … since (-6) + (+6) = 0••

Which of these approaches doyou think is easier to convince
your students? Why?
Justify that (+3) × (–2) = (–6).
Justify that (–8) ÷ (–2) = (+4).

Justify by Permanence of Form(-3) x (+2)
= (-6)↓ 1 less↓ +3(-3) x (+1)
= (-3)↓ 1 less↓ +3(-3) x 0= 0↓ 1 less↓ +3(-3) x (-1)
= (+3)↓ 1 less↓ +3(-3) x (-2)
= (+6)
Diagram 16ii.

•

24 ÷ 3 × (–2)
= 24 ÷ (–6)
= –4
24 ÷ 3 × (–2)

= 24 × 1– × (–2)
31

= –(24 × – × 2)
3
= –16

Diagram 17

Two
students
calculated
24 ÷ 3 × (–2) in two different ways
and got different answers as shownin Diagram 17.

•
•

iii.

Which student is correct?
Explain your reasons.
What was the mistake made by
the student who got the wrong
answer?
How would you help yourstudents
to
avoid
such
mistake?

Calculate••••(–12) ÷ (–2) × 5
4 × (–5) ÷ 12(–36) ÷ (–8) ÷ (–3)
29– ÷ (– – ) ÷ 5
35

Chapter 2: Numbers and Algebra | 13

Task 10: Calculation Involving Algebraic Sumi.

A student claims that subtraction
involving positive and negative
numbers is easier to calculate if thesubtraction is change to addition.
Diagram 18 shows the explanation
given by the student based on two
examples.
•Is the student’s claim valid?
Why or why not?

ii.

Change the following into additiononly mathematics expressions andthen find the sum for each expression.
•(+8) – (+2)•(+6) – (–4)•(–9) + (–3) – (–5)•(–5) – (–2) – (–8)•(+2) + (–0.6) – (+1.8)(+2) – (+5) = (+2) + (–5)… because subtracting a positive
number is the same as adding thecorresponding negative number.
(+2) + (–5) is the algebraic sum of
two terms, the positive term is (+2)
and the negative term is (–5).
So, the sum of (+2) and (–5) is (–3).

(+2) – (–5) = (+2) + (+5)… because subtracting a negative
number is the same as adding thecorresponding positive number.

iii.

So the sum of (+2) and (+5) is (+7).
Diagram 18

iv.

Find the value of (a + b + c) for each of
the following sets of a, b and c.
•
a = 5, b = –8, c = –3•
a = –6.5, b = 2, c = 4
11•
a = –3 – , b = –4, c = – –22Another student claims that thecommutative property, a + b = b + a,
which only hold true for addition of
any real numbers a, b and c can also
be used to do calculation involving
subtraction. She gives the following
example to support her claim.
3–4+6=3+6–4

Then, she uses playing card to explain
her example as shown in Diagram 19.
•
Is the second student’s claim
valid? Why or why not?
v.

Diagram 19

Calculate•
12 – 18 +14•–12 + 3 – 4 + 6•
2 – 5 +7 – 1•–2.3 – 5.7•
1 – 1.8 – 0.4
1 2
1•– – + – – (– – )3

3

6

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

Topic 2: Utilising Letters for Algebraic Expressions and Equations
Standard 2.1:
Extending the utilisation of letters for general representation of situations and find ways to simplify
algebraic expressionsi.
Appreciate the utilisation of letter for general representation of situations to see the expression
as process and value
ii. Find ways to simplify expressions using distributive law and figural explanations, establish thecalculation with like and unlike letteriii. Acquire fluency of simplifying expression and appreciate it for representing the pattern ofsituation
Sample Tasks for Understanding the Standards
Task 1: Representing Number Patterns by Algebraic Expressions
Truss-Bridge Problem

Diagram 1i.

i.

Diagram 1 shows a truss bridge made up of many triangles.
15 steel beams are needed to build the following bridgewith 7 triangles.

•

How many steel beams are needed to build a truss
bridge with 11 triangles?

There are n number of triangles in the following truss bridge.

n∆(Note: n is an odd number.)•ii.

Write an algebraic expression in term of n to represent the number of steel beams
needed to build the bridge.

Diagrams 2, 3 and 4 show the reasoning of three students in solving the Truss-Bridge
Problem.
Student A’s Reasoning

Diagram 2

Chapter 2: Numbers and Algebra | 15

Student B’s Reasoning

Diagram 3
Student C’s Reasoning

Diagram 4
The three students viewed n to represent different things in their reasoning and ended up
in getting three different expressions in term of n.
Student A: n x 3 – (n – 1)
Student B: (n x 2) + 1
Student C: n + (n + 1)
•
•
•
iii.

What was n representing in Student A’s reasoning?
What was n representing in Student B’s reasoning?
What was n representing in Student C’s reasoning?

If the students have learned the simplification of algebraic expressions, they will seethat these three different expressions will be simplified to be a same expression, that is
2n + 1. However, if the students have not learned how to simplify algebraic expressions, itwill be difficult for them to view the three expressions as representing a same solution to
the Truss-Bridge problem.
•

Without simplifying the expressions, how would you help your students see that thethree different expressions are representing a same solution?

16 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 2: Simplifying Algebraic Expressions
i.

a kg

a kg

Diagram 5 shows 3 chairs each weighing
a kg.
•

a kg

•

Diagram 5ii.

Diagram 6 shows a floor mat which is 3 m
wide and a m long.
•
•
•

iii.
Diagram 6

a

7

3

7a

•

7+3

Diagram 7

iv.

3

6x +1
6x +1
=
= 3x +1
2
21Show how the distributive property
of multiplication can be used to verify
that 7a + 3a = 10a.
Show how the distributive property
of multiplication can be used to verify
that 7a – 3a = 4a.

Diagram 8 shows a student’s work whensimplifying an algebraic expression.
•

What is the student’s misconception?

•

Simplify 6x + 1 by using the following
2
methods.
 Draw a diagram.
 Use the distributive property ofmultiplication.

Diagram 8v.

What does 3a mean with respect tothe floor mat?
Write a multiplication sentence
involving 3a.
How does this multiplication sentence
relate to the addition sentence in (i)?

A student is having difficulty to understandthat 7a + 3a = 10a. Diagram 7 shows adrawing used by a teacher to help thestudent overcome the difficulty.
•

3aWhat does 3a mean with respect tothe chairs?
Write an addition sentence involving
3a.

A student claims that 3a + 2b = 5ab.
••What is the student’s misconception?
How would you help the student tocorrect the misconception?

Chapter 2: Numbers and Algebra | 17

Standard 2.2:
Thinking about set of numbers in algebraic expression with letters as variables and represent themwith equality and inequalityi.
Recognise numbers as positive and negative numbers, and explain integers as a part ofnumbersii. Represent a set of numbers using variables with equality and inequalityiii. Translate given sets of numbers on the number lines using interval and inequality notations
iv. Appreciate redefining of even and odd numbers using letters to represent different sets ofvariables
Sample Tasks for Understanding the Standards
Task 1: Sets of Numbers in Algebraic Expressions-4 -3 -2 -1 0 1

2

3 4

Diagram 1
The dark line segment in Diagram 1 represents the value of x for –2 ≤ x ≤ 3.
Given that x is an integer.
i.

Why is the diagram not correctly representing the values of x?

ii.

Make correction to the diagram.

iii.

What if x is a real number and –2 ≤ x < 3?

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

Task 2: Redefining Even and Odd Numbers
A and B are the sets of positive odd and evennumbers respectively.
••

A = {1, 3, 5, 7, 9, 11, 13, …}
B = {2, 4, 6, 8, 10, 12, 14, …}

Diagram 2 shows the first five odd and first
five even numbers represented by black dots.
1

3

5

7

9

i.
2

4

6

8

10

Diagram 2Given that n is an integer.
•
Write in terms of n, an algebraic
expression to represent
 Set A Set B•ii.

Prove that:•
The sum of two even numbers isalways even.
•
The sum of two odd numbers isalways even.

iii.

Diagram 3 shows a student’s confusion
when he read the statement about both
2n + 1 and 2n – 1 are odd numbers forany integer n.
•Using specific values of n, constructa table to verify the statement to thestudent.

iv.

In this case, the letter n functions
as a variable and the expression 2n
represents the condition of the set ofeven numbers, whereas the expression
2n + 1 and 2n – 1 represent the conditionof the set of odd numbers.
•
What is a variable?
•
Why is variable important in Algebra?

If n is an integer, then••2n is an even number.
both 2n + 1 and 2n – 1
are odd numbers.
Diagram 3

State clearly the possible values of nfor each algebraic expression in (i).

Chapter 2: Numbers and Algebra | 19

Standard 2.3 :Thinking about how to solve simple linear equationi.
Review the answers of equations from the set of numbers and thinking backward
ii. Know the properties of equations which keep the set of answers of equationiii. Appreciate the efficient use of the properties of equations to solve linear equation
iv. Use equations based on life situations to develop fluency, to solve equation, and interpret thesolution
Sample Tasks for Understanding the Standards
Task 1: Equality and Inequality
Diagram 1 shows two bags and a marbles. One
bag has x number of marbles in it and the otherbag has y number of marbles. All marbles are thesame.
Diagram 1

Diagram 2 shows three situations comparing thebags and marbles.
i.

Find the number of marbles in
•
•

ii.

Diagram 2

Bag x
Bag y

Describe your process of finding the solutionsfor (i).

20 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 2: Properties of Equalityi.

ii.

Solve the equations by inspection:•a + 2 = 6 Equation ①•
b – 2 = 6 Equation ②•
3x = 12 Equation ③y•=3
Equation ④
2

Diagram 3 shows the processes of solving the four equations in (i).
•Fill in the equation for each?

.

Diagram 3
iii.

The process of solving equations ① to ④ as illustrated in Diagram 3 reflect four properties
of equality.
•

State the four properties of equality.

Chapter 2: Numbers and Algebra | 21

Task 3: Applying Properties of Equation in Solving Equations

2x
+6=5
3
2x
3× +6=3×5
3

2x + 6 = 15

2x + 6 – 6 = 15 – 6
2x = 92x ÷ 2 = 9 ÷ 2
12

x = 4–

Diagram 4i.

What does it mean by solving an equation?

ii.

Diagram 4 shows the work of a student solving the equation

2x+ 6 = 5.
3•

What is the error made by the student?

•

How would you guide the student to realise that x = 4 – is a wrong solution?

•

How would you help the student to correct his error?

12

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

Topic 3: Algebraic Expressions, Monomials and Polynomials and Simultaneous Equations
Standard 3.1:
Thinking about the calculations of monomial and polynomials, simple casei.
Introduce terms, monomials and polynomials
ii. Introduce a number raised to the power of two as square, and a number raised to the power
third as cubeiii. Get fluency for the calculation of polynomials such as combining like terms and the use of thefour operations in simple cases
Sample Tasks for Understanding the Standards
Task 1: Monomials and Polynomial Expressions

Diagram 1

Diagram 1 shows a figure formed by a big semicircle, two smaller semicircles and a small circle.
Given that AOB is the diameter of the big semicircle and AO = OB = x cm.
i.

If the diameter of the small circle is 2y cm, find its area and circumference as monomial
expressions in terms of y.

ii.

Show that the outer perimeter of the shaded region is 2πx .

iii.
iv.

Find the area of the shaded region as a polynomial expression in terms of x and y.
If both x and y are doubled, what will happen to the outer perimeter and area of the shaded
region?

Chapter 2: Numbers and Algebra | 23

Task 2: Square and Cube
Diagram 2 shows a square metal sheet with side
length x units.
i.

Find the area of the metal sheet in term of x.

ii.

Calculate the area of the metal sheet for thefollowing values of x.
•
x = 10 cm•
x=½m•
x = 5.2 miii.

Identical metal sheets with area x2 square
units are stacked up to 10 cm, 20 cm andx cm high as shown in Diagram 3.

Diagram 2

10

•

x

20

•

Diagram 3

Find the volume of the three stacks ofmetal sheets in terms of x, respectively.
Calculate the volume of each stack ofmetal sheets for the same values of xin (ii).

Task 3: Calculation Involving PolynomialsStudent A’s Solution(4x + 1) – (x – 5)
= 4x + 1 – x – 5= 3x – 4
Student B’s Solution(4x + 1) – (x – 5)
= 4x + 1 – x + 5= 3x + 6= 9x
Diagram 4i.

Diagram 4 shows the solutions of two
students, A and B, when doing calculation
involving polynomials.
•

For each of the student work, show
where the mistake is and explain why.

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

Student A’s Method

=

=
=
=

x + 2yx–y–2
33(x + 2y)2(x – y)–6
63(x + 2y) – 2(x – y)6
3x + 6y – 2x – 2y6
x + 8y6Student B’s Method

=

=
=
=

x + 2y
x–y
–
2
3
1
1
(x + 2y) – (x – y)
2
3
1
1
1
x+y– x+ y
2
3
3
3
2
3
1
x– x+ y+ y
6
6
3
3
1
4
x+ x
6
3

i.

ii.

Diagram 5 shows two students’ methods to do
calculation involving polynomials.
•

Are the answers obtained using the two
methods the same or different? Explain
your reasons.

•

Which method do you think is easier foryour students? Explain your reasons.

Simplify.
•••–2(3x – 4) + 8x – 3
a + 5b3
4x – yx – 3y+32a–Diagram 5

Task 4: A Mathematical Trick
Diagram 6 shows the rule to play a trick using
the number cards 1 to 9.

1

6

2

7

3

8

4

9

5

•

Pick any two cards.
Example: 3 and 5 .

•Form two 2-digit numbers using
the digits.
Example: 35 and 53.

•

Find the difference between thetwo numbers.
Example: 53 – 35 = 18
Diagram 6

On his first trial, a student picked 4 and 5 andobtained the following result:
54 – 45 = 9
On his second trial, the students picked 4 and
7 and obtained the following result:
74 – 47 = 27
Based on these results, the student concluded
that the difference between the two numbers isalways a multiple of 9.
i.

Prove that the student’s conclusion isalways true.
[Hint: Any 2-digit number ab can be
expressed as a polynomial 10a + b, wherea and b are any digit from 1 to 9.]

Chapter 2: Numbers and Algebra | 25

Standard 3.2:
Thinking about how to solve simultaneous equation in the case of linear equationsi.
Understand the meaning of solution of linear equations and simultaneous linear equations asa pair of numbers
ii. Know the substitution and elimination methods of solving simultaneous linear equations
iii. Get fluency for selecting the methods from the form of the simultaneous linear equations
iv. Appreciate simultaneous linear equation in the situations
Sample Tasks for Understanding the Standards
Task 1: Solving Simultaneous Linear Equations with Two Variables
Chicken-and-Sheep Problem
Table 1
Solving Chicken-and-Sheep Problem
x

y

Total
AnimalHeads

Old MacDonald had a chicken-and-sheep
farm. There are x number of chickens andy number of sheep in the farm.
i.

Total
AnimalFeet

One day, he counted all the animals
in his farm and announced to his wife
that:
“There are altogether 18 animalheads in our farm.”ii.

•

Write an equation in termsof x and y to represent therelationship.

•

Fill in Table 1 to show all thepossible values of x, y and total
number of animal heads.

Later, his wife counted all the animals
and announced to him that:“There are altogether 60 animal feetin our farm.”•

Write another equation in termsof x and y to represent thesecond relationship.

•Complete Table 1 to show all thepossible numbers of animal feet.

iii.

How many chicken and how many
sheep are there in Old McDonald’s
farm?

iv.

What does it mean by solving two
equations simultaneously?

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

Assume there are 9 chickens.
There will be 9 sheep.

18 heads and 54 feet
Not enough feet. Assume there are 5 chickens.
There will be 13 sheep.

18 heads and 62 feet
Too many feet. Assume there are 7 chickens.
There will be 11 sheep.

18 heads and 58 feet
Not enough feet. Assume there are 6 chicken.
There will be 12 sheep.

18 heads and 60 heads
Diagram 1v.

A student solved the Chicken-andSheep problem by first setting atentative number of chickens, thenproceeded to find the solution bya repeated process as shown in
Diagram 1.
•

Explain the student’s reasoningin solving the problem.

•

Using this student’s method,
solve
x+y=9
2x + y = 16

Chapter 2: Numbers and Algebra | 27

Apple-and-Orange Problem
In a supermarket in Bangkok, a packet of 2 apples and 1 orange is sold at 31 Bhat,
but a packet of 1 apple and 2 oranges is sold at 26 Baht.
What is the price of an apple and an orange respectively?
vi.

Solve the Apple-and-Orange problem by your own reasoning.

vii. Three students solved the Apple-and-Orange problem.
•
Diagrams 2 shows the reasoning of the first student in solving the problem.

take away

add intake away

add inDiagram 2
•Explain the first student’s reasoning.
Does your own reasoning used in solving the problem match the students’reasoning? If yes, explain how the reasoning matched? If no, represent yourreasoning with a diagram.

The second student represented his reasoning while solving the problem with adiagram as shown in Diagram 3 and explained it using a series of equations.

2x + y = 31+) x + 2y = 26
3x + 3y = 57

x + (x + y) = 31
x = 12x + y = 19
(x + y) + y = 26
y=7

Diagram 3

Explain how the equations are connected to the student’s reasoning.

28 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level•

The third student represented his reasoning while solving the problem with a diagram
as shown in Diagram 4.

Diagram 4
Explain the third student’s reasoning using a series of equations.

Guava-and-Mango Problemi.

Guavas and mangoes arepacked into seperate boxes.
Each box has a same numberof guavas or mangoes.

Diagram 5 shows a problem involving
the number of fruits in a box.
•

Somchai bought 3 boxes ofguavas and 5 boxes of mangoes.
He counted the fruits and foundthat there are altogether 86 fruits.
Then, he gave away 3 boxes ofmangoes to his parents and hehad 56 fruits left.
How many fruits in each box of guavas andmangoes, respectively?
Diagram 5

Set A
x + y = 12
		
2x + 3y = 28		ii.

Diagram 6 shows two sets ofequations.
•

When these two sets of equations
are solved simultaneously, what
can you say about the numberof solutions for each set of
equations?

•

What is the relationship between
the number of variables andthe number of equations whensolving a set of equations
simultaneously?

Set B
a + b + c = 15
a + 2b + 3c = 24

Diagram 6

Represent your reasoning insolving the Guava-and-Mango
problem with a diagram. Explain
your diagram.

Chapter 2: Numbers and Algebra | 29

Topic 4: Expansion and Factorisation of Polynomials
Standard 4.1:
Acquisition to see the polynomials in second degree with expansion and factorisation and use iti.
Use the distributive law to explain the formulae for expansion and explain them on diagrams
ii. Acquire proficiency for selecting and using the appropriate formulaeiii. Use the expansion formulae to factorise the second degree expression and recognize bothformulae with inverse operation
iv. Solve simple second degree equation using the factorisation and apply in life situations
Sample Tasks for Understanding the Standards
Task 1: The Distributive Property of Multiplicationi.

Diagram 1 shows the distributive property
involving the multiplication of any threenumbers a, b and c.
•

Verify that a × (b + c) = ab + ac holds
true for both positive and negative
numbers.

•

Draw another diagram to verify that(b + c) × a = ab + ac = a × (b + c).

𝑎  ×  (𝑏  + 𝑐)  =  𝑎𝑏 + 𝑎𝑐
Diagram 1

a × (b – c)
= a × [b + (–c)]= ab + (–ac)= ab – acii.

A student claimed that “since subtracting
a number is the same as adding thecorresponding negative number, thereforea × (b – c) = ab – ac.” The student’s
reasoning is shown in Diagram 2.
•

Diagram 2

Draw a diagram to support thestudent’s claim.

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

Task 2: Multiplication and Division of Algebraic Expressions
Multiplication and Divisionof Polynomials by Monomialsi.

(2x) × (3x + 5y) = 6x2 + 5y
Diagram 3

ii.
2a m

(10a2 + 6a) m2
Diagram 4

Multiplication of Polynomials
by Polynomialsi.

(a + b)(c + d) = ac + bd

When asked to multiply the monomial 2x
with the polynomial 3x + 5y, a student gave awrong answer as shown in Diagram 3.
•

What is the possible cause for the
student’s mistake?

•

Draw a diagram to help the student
correct the mistake.

Diagram 4 shows a plot of rectangular
land
with
length (2a) m andarea(10a2 + 6a) m2•

Write an algebraic expression relating thewidth of the land to its length and area.

•

Find the width of the land in terms of a.

When asked to multiply the polynomials(a + b) with (c + d), a student gave a wrong
answer as shown in Diagram 5.
•

Diagram 5

Let (c + d) = M.
Then, (a + b)M = aM + bM.
… then … err … ???

ii.

Another student tried to explain to the firststudent by drawing a diagram as shownin Diagram 6. However, this student got
stuck halfway and could not complete his
explanation.
•

Diagram 6

What is the possible cause for the
student’s mistake?

Complete the explanation using thediagram drawn by the second student.

Chapter 2: Numbers and Algebra | 31

Task 3: Expansion of Algebraic Expressions
Expansion Formulae

Diagram7
shows
theexpansionof(x + a) (x + b) = x2 + (a + b)x + ab diagrammatically.
i.

Explain the process of expansion using the
distributive property of multiplication.

ii.

Explain how the following expansion
formulae could be derive from the expansionof (x + a)(x + b).
•(x + a)2•(x – a)2•(x + a)(x – a)iii.

Diagram 7

iv.

Use the expansion formulae to expand•(2x + 3)(2x + 5)•
(3b + 2)2•
(2x – 3y)2•
(3c – 5)(3c + 5)•
(a – 3b)(–2a + 4b)
Expand•4(a + 2b – 3c)•(a + b)(x – y + 3)

Task 4: Factorisation of Algebraic Expressions

Diagram 8 shows how the polynomial
x2 + 3x + 2 is factorised to be (x + 1)(x + 2).

x2 + 3x + 2

x2

x

x

1

factorising
expandingDiagram 8i.

What does Diagram 8 tell you about
factorisation of polynomial expressions?

ii.

Use the cut-out pieces (x2-piece, x-piece,
1-piece) in Material Sheet 1 at the end of this
topic to factorise:• x2 + 5x + 6• 2x2 + 7x + 3

x1(x + 1)(x + 2)

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

Task 5: Factorise by Common Factor
i.
B

•

C
ii.

A

x2

x2

x

x

x

Factorising 2x2 + 3x
Diagram 9

Diagram 9 shows how the polynomial
2x2 + 3x is factorise using the cut-out pieces.

iii.

The factorisation of 2x2 + 3x involves acommon factor.
•

What is the common factor?

•

Use the cut-out pieces to factorise
2x2 + 6x.

There are two different ways to arrange thecut-out pieces in the factorisation of 2x2 + 6x.
•

iv.
v.

What is the value of A, B and C
respectively?

What is the same and what isdifferent about the results of these two
arrangements?

How would you use the cut-out pieces tofactorise 2x2 – 6x ?

Factorise•2ax – 6ay•
3x2 + 6xy•
x2y + xy2•
a2 – 4ab + 3a

Chapter 2: Numbers and Algebra | 33

Task 6: Factorise by Formulae(x + a) (x + b) = x2 + (a + b)x + ab
x2 +

i.

– 6

5x

–1 x 6 = –6
1 x (–6) = –6
–2 x 3 = –6
2 x (–3) = –6

Thestudent
found
four
possiblecombinations of numbers a and b to get
the product –6. He then decided that(–1) and 6 was the correct combinationand complete the factorisation as
x2 + 5x – 6 = (x – 1) (x + 6).

Diagram 10•ii.

(ax + b) (cx + d)
= acx2 + (ad + bc)x + bd
ax

b

cx

d
(ad + bc)x

Diagram 11

When using the expansion formula for(x + a) (x + b) to factorise the polynomial
expression x2 + 5x – 6, a student madea comparison between the formula andthe expression as shown in Diagram 10.

iii.

Did the student make the correct
decision? Explain your reasons.

Factorise•(x + a) (x + b)•
(x + a)2•(x – a)2•(x + a) (x – a)•
Use any of these expansion formulae
to factorise the following.
 4x2 – 9
 x2 – 6x + 9
 a2 + 10a + 25
 y2 – y – 12

From the expansion of (ax + b) (cx + d),
a student made a note on how to factorisesome second degree polynomials asshown in Diagram 11.
••

Explain the student’s method.
Factorise 2x2 + 5x + 3
 9x2 – 12x + 4

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

Material Sheet 1: Cut-Out Pieces ( x2-piece, x-piece, 1-piece )

2

x

x

x

x

x

x

x

x

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Chapter 2: Numbers and Algebra | 35

Topic 5: Extending Numbers with Square Roots
Standard 5.1 :Extending numbers with square roots and calculate the square roots algebraicallyi.
Define square root and discuss ways to estimate the nearest value of a square root by Sandwich
Theoremii. Understand that some square roots cannot be represented as fractions
iii. Compare square roots using number line and understand that the order does not change but
the differences between two consecutive square roots varied
iv. Think about multiplication and addition of square root and understand the algebraic way of
calculation which is similar to polynomial
v. Appreciate square roots in applying to situations in life
Sample Tasks for Understanding the Standards
Task 1: Square RootMeaning of Square

x cm

a cm

2

Diagram 1 shows a square with side lengtha cm and area x cm2.
i.
ii.

Diagram 1

How would you explain that x = a ?

We know that 4 = 2. Why is –2 also a squareroot of 4?

Plotting Square Roots on a Double
Number Line
Table 1
Estimated Values of

i.

x

x

x

0

0.0

1

1.0

2

1.4

3
4

2.0

5
6
7
8

2.8

9

3.0

10

3.2

Table 1 shows some positive estimatedvalues ofx to one decimal place for0 ≤ x ≤ 10 where x is an integer.
•

Complete the table.
(Use an electronic calculator to help you.)36 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Diagram 2
ii.

Diagram 2 show a double number line for 0 ≤ x ≤ 10. The arrows show the mapping of x
onto x for 0, 1, 2 and 4.
•
Complete the mapping of x onto x on the double number line for the other values of
x in Table 1.
•
What do you observe about the lengths between two consecutive dots on the x number line?
 the x number line?
 What could you conclude about the order and values of x ?

iii.

Between 8, and 9, which number is closer todecision.

Estimate Square Roots Using
Drawing

Diagram 3 shows a square with one of its diagonals.
The side length of the square is 1 m.
i.

1m

72.3 ? Explain how do you make the

ii.

iii.
Diagram 3

Show that the length of the diagonal is 2 m.
Prove that

Using a scale of 10 cm to represent 1 m, draw
the square on a graph paper.
•
•

iv.

2 cannot be express in the forma–b , for a, b are integers and b ≠ 0.

How would you use your drawing toestimate the value of 2 ?
Show that 1.4 < 2 < 1.5 on your drawing.

Draw another diagram to estimate the value of
5 to one decimal place. Check your estimate
with a calculator.

Chapter 2: Numbers and Algebra | 37

Estimate Square Roots using
Electronic Spread Sheet
Table 2Estimated Value of 2b2b2 = b1.00000

1.00000

2.25000

1.50000

1.96000

1.40000

2.102501.45000

If a, b and c are positive numbers and a < b < c, thena < b < c . This relationship can be used toestimate the values of square roots. For example,
since we know that 1 = 1 and 4 = 2, then
1 < 2 < 2.
i.

Table 2 shows some values of b and b2 for
1 < b < 2 and 1 < b2 < 4. As shown in the table,
1.40 < 2 < 1.45 So, the estimated value of 2
to one decimal place has to be 1.4.
•ii.

4.00000

Use an electronic spread sheet toconstruct Table 2. Continue to input thevalues of b to estimate the value of 2 tofour decimal places. Check your estimate
with an electronic calculator.

Construct another table to estimate the value of
3 to three decimal places. Check your estimate
with an electronic calculator.

2.00000

Task 2: Calculation Involving Square Rootsi.

A square root, x, is simplified if x cannot be divided by a perfect square other than 1. For
examples, 2, 3 and 5 are simplified, but 8 and 18 are not simplified.
•
Simplify each expression.

8

18

75 – 4 3•
What property of square roots is used to simplify the expressions?

ii.

Rationalise the denominator of each expression.
•
•

iii.

10
5
3
5+1

What properties of square roots are used to rationalise the denominator of each expressionin (ii)?

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

Task 3: Applying Square Roots to Situations in Life

Diagram 4 shows how the A series ofinternational paper sizes A1 to A8 arederived from paper size A0. Each size ishalf the paper size before it. For examples,
A1 is half the size of A0, A2 is half the sizeof A1, and so on.
Table 3 shows the measurements of the
A0 to A5 sizes given as (width x length) inmillimitres.
i.

Calculate the ratio of length : width foreach size.

ii.

Show that the ratios for the various
paper size are all approximately equal
to 2.
•

Diagram 4
Table 3
International A Series Paper Size
Sizewidth (mm) x length (mm)A0841 x 1188

A1

594 x 841

A2

420 x 594A3

297 x 420A4

210 x 297A5

148 x 210

Why is it so?

Chapter 2: Numbers and Algebra | 39

Topic 6: Solving Quadratic Equations
Standard 6.1 :Solving simple second degree equation using the factorisation and apply on the situationi.
Find the answers of simple second degree quadratic equations by substitution and explore bycompleting the square, quadratic formula and factorisation
ii. Get fluency to select the appropriate ways for solving quadratic equations
iii. Apply quadratic equations in life situations
Sample Tasks for Understanding the Standards
Task 1: Forming Quadratic Equations
Uncle Ben has a rectangular plot of land as
shown in Diagram 1. Given that the length ofthe land is x m and its perimeter is 48 m.

xm

Diagram 1

i.

What is the width of the plot of land in terms
of x?

ii.

If the area of the plot of land is 140 m2, write
an equation in the form of ax2 + bx + c = 0.

iii.

What do we need to do with the equation
if we want to find the length and width of
Uncle Ben’s land?

Task 2: Solving Quadratic Equations of Various Forms
Zero-Factor Propertyi.

Generally, a quadratic equation is in theform ax2 + bx + c = 0, where a, b, c are real
numbers and a ≠ 0 .
•Why is it that a cannot be zero?
•If b = 0, then c cannot be more thanzero. Explain with examples.
•
If c = 0, then the quadratic equation canbe solved by factorising its quadratic
expression. Explain with examples.

ii.

The zero-factor property, as stated in
Diagram 2, is an important idea in solving
quadratic equation by factorising its
quadratic expression.
•
Solve the quadratic equations
 5x2 – 10 = 0
 3x2 + 15x = 0
 2m2 – 3m – 5= 0

Zero-Factor Property

If a and b are real
numbers, with ab = 0,then a = 0 or b= 0 or
both.

Diagram 2

40 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Solving Quadratic Equations of theform (x + q)2 = r

2x

4x2

2x

i.

Diagram 3 shows a square with length 2xunits and area 4x2 square units.
•ii.

Any quadratic equation in the form
(x + q)2 = r may be solved using the idea ofsquare roots.
•

Area = (2x) x (2x) = (2x)2 = 4x2
Diagram 3x2 + 6x – 3 = 0 → x2 + 6x = 3

x2



iii.

Diagram 4

Solve the quadratic equations
 (a – 3)2 = 25
 (2x + 3)2 = 49
 x2 + 6x + 9 = 36

Any quadratic equation in the generalform x2 + bx + c = 0 may be solved if
it could be transformed into the form
(x + q)2 = r. Diagram 4 shows part of theprocess to transform x2 + 6x – 3 = 0 into the form
(x + q)2 = r .
•What is the number in  ?

•Explain how the quadratic equation
x2 + 6x – 3 = 0 is transformed into(x + 3)2 = 12.

•
x2 + 6x +  = (x + 3)2

What is the value of x, if the area of thesquare is 20 square unit?

•

In general, what number needs to be
added to the expression (x2 + bx) inorder to change it to be (x + q)2 ?
Transform each quadratic equation
into the form (x + q)2 = r
 x2 + 8x – 7 = 0
 x2 – 5x + 2 = 0
 3x2 + 6x – 9 = 0

Chapter 2: Numbers and Algebra | 41

Quadratic Formulax2 + bx + c = 0x2 + bx =bx2 + bx + ( 2– )2 =b
2(x + – )2 ==±b
2x =–– ±x=–cb2 – 4c4

b2 – 4c2–b ± b2 – 4c2

Diagram 5i.

ii.

Diagram 5 shows how a quadratic formula can be derived to solve any quadratic equationin the general form x2 + bx + c = 0.
•

Fill in the missing parts.

•

What can we say about the solutions of a quadratic equation x2 + bx + c = 0 if
 b2 – 4c > 0
 b2 – 4c = 0
 b2 – 4c < 0

Derive a general formula for solving ax2 + bx + c = 0

42 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 3: Real-World Problem Solving with Quadratic Equations

Uncle Manat has a piece of rectangular land. He wants
to create paths with equal width as shown in Diagram 6.
The rest of the land will be made into gardens.
Let x be the width of the paths.
The total area of the gardens is to be 11 781 m2.

Diagram 6

Diagram 7i.

Write a quadratic equation in terms of x to show the
relationship.

ii.

Solve the equation to find the width of the paths.

iii.

There are two solutions to the quadratic equation in(ii), but only one is the width of the path. Why is theother solution not the width of the path?

iv.

What if Uncle Manat wants to add another path ofequal width as shown in Diagram 7, but still keeps
the same total area for the gardens?

Chapter 3: Relations and Functions | 43

CHAPTER THREE
Relations and Functions
Topic 1: Extending Proportion and Inverse Proportion to Functions with Variables
Standard 1.1:
Extending proportion and inverse proportion to functions with variables on positive and negative
numbers
Extend proportions to positive and negative numbers, using tables and equations on situations
Plot set of points as graph for proportions defined in ordered pairs (x, y) in the coordinate plane
using appropriate scales preciselyiii. Introduce inverse proportion using tables, equations and graphs
iv. Introduce function as correspondences of two variables in situations
v. Explore the property of proportional function with comparison of inverse proportional function
vi. Appreciate proportion and inverse proportion functions in lifei.
ii.

Sample Tasks for Understanding the Standards
Task 1: Function
3x cm

Diagram 1 shows a rectangle with a width which isthrice its length, x.
x cm

i.

Given that y is the perimeter of the rectangle.
•

Express y in terms of x using an equation.
 What is the relationship between x andy?

•Complete the following table for the valuesof x and y.

Diagram 1x (cm)

Diagram 2y (cm)

5

10

15

20

25

30

40

•

Plot the graph of y against x on Diagram 2.

•

Explain why y is a function of x.

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

Diagram 3ii.

Diagram 3 shows the graph of p2 + x2 = 9.
•

Is p a function of x? Explain your reasons.

Chapter 3: Relations and Functions | 45

Task 2: Direct Proportions

Diagram 4
Diagram 4 shows a water tank with a height of 300 cm and its orthogonal-projection drawing.
The tank is being filled with water and the water level increases 2 cm per minute.
i.

Let the half-tank water level be the reference point and y cm as the water level after x
minutes.
•
What is the water level 15 minutes later?
•
What is the water level 10 minutes before?
•
Complete the following table.
x (min)ii.

y (cm)

–15

–10

–5

–10

0
0

5

10

10

•
•
•
•

y
When x ≠ 0, find the value of x for each order pairs (x, y).
Express y in terms of x as an equation.
What does it mean by y is directly proportional to x?
What is the constant of proportion for this case?

•

Plot the graph of y against x.

15

20

After the tank is fully filled, water is being removed where the water level decreases 3 cm
per minute. Again, let the half-tank level be the reference point.
•
Complete the following table for the removal of water.
x (min)•••y (cm)–30–20

Explain the meaning of y = 30 cm when x = –10 min
 y = –60 cm when x = 20 min

–10
30

0
0

10

20

–60

30

Plot the graph of y against x for the removal of water.
In this case, is y still directly proportional to x? Explain your reasons.

40

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

Task 3: Inverse Proportion
Diagram 5 shows a rectangle with width x cm andlength y cm.

x cm

i.
y cm

Diagram 5Given that the area of the reactangle is 18 cm2.
•

Express y in terms of x using an equation.

•

Explain why x ≠ 0 and y ≠ 0.

•ii.

Find the values of y for the values of x as
follows:
 x=¼
 x=½
 x=¾
 x=5

Complete the following table for the values of x
and y.
x (cm)iii.

iv.

y (cm)

18

2
9

3
6

4

5½

6

Study the order pairs (1, 18) and (2, 9). As thevalue of x increases 2 times, what can you say
about the value of y ?
•What about (1, 18) and (3, 6)?

•

Can you say the same for all other order
pairs?

What does it mean by y is inversely proportional
to x?
•

v.

1

What is the constant of proportion for thiscase?

Explain what will happen to the rectangle and thevalues of y when x < 0.
•••Extend the table to include the values of–6 ≤ x < 0.
Plot the graph of y against x for the domain–6 ≤ x ≤ 6.
Why is the point when x = 0 cannot beplotted on the graph?

Chapter 3: Relations and Functions | 47

Task 4: Application of Proportion
Proportional Relationship of a Circle
Diagram 6 shows three circles with diameters d1,
d2 and d3.
i.

From practical measurements, we know that

C1
CC
= d2 = d3 = k, where C1, C2 and C3 ared1
2
3

circumferences of the three corresponding
circles, and k is a constant.
•ii.
Diagram 6

What is the proportional relationshipbetween C and d ?

We also know that the formula to calculate thearea of a circle is A = πr2, where A is the area
and r is the radius of the circle.
•

What is the proportional relationshipbetween A and r?

Problem Solving with Weighing
Pak Agus wanted to use 300 pieces of iron nails but
he found it very troublesome to count the nails. So,
he decided to weigh the nails instead. He counted
10 identical nails of the same mass and weighed
the iron nails as shown in Diagram 7.
He then added more iron nails until he got a lumpof the iron nails.

25 gDiagram 7i.

The lump of iron nails was 800 g. Did Pak
Agus get enough pieces of iron nails? Justify
your answer.

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

Burning Insence StickDiagram 8 shows an insence stick burning at a
rate of –1.2 cm/min. After burning for few minutes,
the length of the insence stick was found to be
24 cm.
i.

What was the length of the insence stick
2 minutes before this?

ii.

What will the length of the insence stick be
5 minutes after this?

iii.

How many more minutes will it take theinsence stick to burn completely?

Diagram 8Interlocking Gears
Diagram 9 shows two gears, A and B that are
rotating while interlocking with each other. Gear
A has 20 teeth and rotates 8 times per second.
Gear A is fixed, but Gear B can be fitted withdifferent gears.

Diagram 9i.

If Gear B has 40 teeth, how many times will
it rotate in one second?

ii.

If we want Gear B to rotate 1 time in one
second, how many teeth must it have?

Chapter 3: Relations and Functions | 49

Walking Speed
Chamdren

Ali

Diagram 10 shows a moving walkway which
is 60 m long and moves at a speed of 1 m persecond.
As Ali was getting onto the moving walkway, he
saw his classmate, Chamdren, started walking
next to the moving walkway from the opposite
direction at a speed of 1.2 m per second.

Ali

Diagram 10

Diagram 11
Diagram 11 shows the graph that represent Ali’s movement.
i.

Plot another graph on Diagram 11 to represent Chamdren’s movement.

Use the graphs in Diagram 11 to answer questions (ii) to (iv).
ii.
iii.
iv.

20 s after they both started walking, how far apart were Ali and Chamdren?
After how many seconds of walking would Chamdren walk pass Ali?
When Chamdren arrived at the end of the moving walkway, how many m was Ali from theother end of the walkway?

50 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Topic 2: Exploring Linear Function in Relation to Proportions
Standard 2.1:
Exploring linear function in relation to proportion and inverse proportionsi.
Identify linear functions based on situations represented by tables and compare it withproportional functions
ii. Explore properties of linear function represented by tables, equations and graphs and compare
it with direct and inverse proportional functions
iii. Acquire fluency to translate the rate of change of a linear function represented in table, as
coefficient in an expression and gradient in a graphiv. Acquire fluency to translate y values of x = 0 in a table, constant in an expression, and yintercept in graphv. Apply the graphs of linear functions to solve simultaneous equations
vi. Apply the linear function for data representation on situations to determine best fit line
Sample Tasks for Understanding the Standards
Task 1: Linear Functioni.

Diagram 1
ii.

Given that y is the perimeter of the shape and z is its area.
•
Complete the following table for the domain 0 ≤ x ≤ 10.
x (cm)y (cm)

iii.

iv.

v.
vi.

A square with side x cm is joined to another square withside 1 cm to form a new shape as shown in Diagram 1.
•
What will happen to the shape if x = 1?
•
What if 0 < x < 1?
•
What if x = 0?

z (cm2)

0

¾

1

2

6

7½

9

10

Given that y is a function of x.
•
Express y as a function of x using an equation for the domains (a) 1 ≤ x ≤ 10 and(b) 0 ≤ x ≤ 1.
•
What do you notice about function y for the two domains?

Given that z is also a function of x.
•
Express z as a function of x using an equation for the domains (a) 1 ≤ x ≤ 10 and(b) 0 ≤ x ≤ 1.
•
What do you notice about function z for the two domains?
Sketch the graphs of y against x and z against x respectively.
•
What is the same and what is different about the two graphs?

What will happen to functions y and z if the values of x are extended to the domain x < 0?
Illustrate your answer with a sketch of the new shape.

vii. Given that y is a linear function but z is not.
•What is a linear function?

Chapter 3: Relations and Functions | 51

Task 2: Linear Functions and Proportional Functions
Water Temperature

Room temperature= 22 oC
Rate of heating= 12 oC per minute

Diagram 2 shows a kettle of water being heated up atroom temperature.
i.

Explain what does it mean by the increase in watertemperature is proportional to the time of heating.

ii.

Assuming y is the increment of water temperature ino
C and x is the time of heating in minute.
•
•
•

iii.

Diagram 2

Running Speed

Diagram 3

Express y as a function of x.
What is the constant of proportion?
What is the increment of water temperature
after 5 minutes of heating?

Given that the water temperature is po C afterx minutes of heating.
•••Express p as a function of x.
Why is function y different from function p?
Is p a linear function? Explain your reasons.

Diagram 3 shows a runner reaching the finishing line ina marathon run.
i.

A runner takes 6 hours to finish the run, what is his
average speed of running in km per hour?

ii.

A runner is running at an average speed of 8 km perhour. How many hours will he take to finish the run?

iii.

Explain what does it mean by the average speed isinversely proportional to the time of running.

iv.

Assuming x is the time in hours and s is the average
speed in km per hour for a runner to run the 42.2 km
marathon race.
•••

Express s as a function of x.
What is the constant of proportion?
Is s a linear function? Explain your reasons.

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

Task 3: Rate of Change of a Function
Linear Function
Table 1
Values of x and y of a Linear Function
x-increment

x
y

–2
–4

+1

–1

?

–1

+1 +½

+1 +1½

2

6½

0

?

½

3½

1½

?
?
y-increment

3?

Table 1 shows the values of x, y and the respective
x-increment of a linear function y for the domain–2 ≤ x ≤ 3 .
i.

ii.

11

When x increases from –2 to –1, the x-incrementis +1. What is the corresponding y-increment?
y-increment

Let the ratio x-increment = k.
•Find the value of k for x = –2 to x = –1.

•

Verify that the value of k is a constant.

•

The constant k is known as the rate ofchange of y with respect to x. Explain themeaning of rate of change.

•

iii.

The 6 ordered pairs of (x, y) from Table 1 areplotted on a graph as shown in Diagram 4.
These 6 points are then joined to form a straightline as shown in Diagram 5.
•

The 6 points plotted are finite, but a straightline is formed by infinite points. Although
the 6 points look like forming a straight
line, can we say that the graph of linear
function y plotted against x is a straightline? Explain your reasons.

•Justify that the graph is a straight line byusing similarity of triangles.
[Hint: Take a common point such as A on
the line. Then, consider corresponding
angles of various similar triangles going
through point A.]

Diagram 4•
iv.
Diagram 5

Find all other corresponding values ofy-increment and k for the x-increment
shown in Table 1.

Find the gradient of the graph of linear
function y.

Express y in terms of x using an equation.

Chapter 3: Relations and Functions | 53

Nonlinear Function
Table 2
12
Inverse Proportion y = x

x

y

…
…

-3

-2

-1

0
–

1

2

3

…
…

This task is to investigate the rate of change of an inverse proportion function.
i.

•What is the constant of proportion?

•

Complete Table 2.

•

Find the rate of change of y with respect to x for the following intervals of x:
 x = –3 to x = –2
 x = –2 to x = –1
 x = 1 to x = 2
 x = 2 to x = 3

•
•
ii.

12

Given an inverse proportion y = x .

12

Verify that the rate of change of y = x is not a constant.
12

Why is y = x considered a nonlinear function? Explain your reasons.

Is each of the following statements true or false? Explain your reasons.
•All linear functions are direct proportions.

•

All nonlinear functions are not direct proportions.

•

A linear functions is not an inverse proportion.

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

Task 4: Graphs of Linear Functions
Table 3
Values of x and y for Three Linear Functionsxy1 = 2xy2 = 2x + 6
y3 = 2x – 4

…

…
…

–4

…

–3

–2

–1

0

1

–6

–4

–2

–2
4

0
6

2

8

2

3

4

5

…
…

…
…

Any linear function can be represented in the form y = ax + c. Table 3 shows some values of x
and y for y1 = 2x, y2 = 2x + 6 and y3 = 2x – 4.
i.

Complete the table.

ii.

Plot the graphs of y1, y2 and y3 on Diagram 5.
•
What does the value of a and c tell us respectively?
•
What will happen to the graph when a = 0?

iii.
iv.

A student claims that “when a increases positively, the graph y = ax +4 will rotate counterclockwise.” What does the student’s claim mean?
Plot the graphs of y4 = –x +4, y5 = –2x +4, and y6 = –3x +4 on Diagram 6.
•
What will happen to the graph when a decreases negatively?

Diagram 5

Diagram 6

Chapter 3: Relations and Functions | 55

Task 5: Using Graphs of Linear Functions to Solve Simultaneous Equationsi.

Given a system of linear equations
x + 2y = 20
3x – y = 4•

Solve the simultaneous equations.

•

Diagram 7 shows the graphs of the two
equations. A and C are two points on thelines 3x – y = 4 and x + 2y = 20 respectively.
If A and C move in the directions as shownby the arrows, the two points will meet at
B, which is the intersection point of the twolines. The x-coordinates of A and C are aand c, respectively.
 What is the y-coordinate of A in termsof a?
 Find the y-coordinate of A as the valueof a changes as follows:a = 2⅓ → a = 3⅓ → a = 4⅓
 What is the y-coordinate of C in termsof c?
 Find the y-coordinate of C as the valueof c changes as follow:
c=6→c=5→c=3
 What will happen to the values of a andc when A and C meet at B? What about
the y-coordinates of A and C?

Diagram 7

•
•

ii.

Diagram 8

Find the coordinates of B.

Compare the coordinates of B with thesolution of simultaneous equation in (i).
What could you conclude?

Diagram 8 shows the graphs of x = 3 andy = 3. Both x = 3 and y = 3 are straight lines on
the graph but not both are linear functions.
•

Why is y = 3 a linear function but x = 3 isnot?

56 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 6: Application of Linear Functions
Scientific Investigation

Diagram 9 shows a beaker of water at room
temperature is being heated. Table 4 shows thewater temperature, y ºC, after it was heated forx minutes.
i.

Based on Table 4, plot all the points (x, y)
on a piece of graph paper. Use a scale of
2 cm = 10 °C for y-axis and 2 cm = 1 min for
x-axis.

ii.

Assuming y is a linear function of x, draw thestraight line of best fit for the points. Use thegraph to answer the following questions.
•
How many minutes after heating will thewater temperature reach 80 ºC?.
•
Find the rate of increase in watertemperature with respect to time.
•
Find the equation of the line of best fit.

i.

Diagram 10 shows the relationship ofdistance (y metres) and time (x seconds) for
a 100-metre race between two friends, Anh
Dung and Bao.
•
Explain what happen at the start?
•
What is the running speed (in m/s) of
Anh Dung and Bao respectively?
•
At what time and distance from the start
did Bao overtake Anh Dung?
•
At what time did Bao complete the race?
•
How many metres away from theending point was Anh Dung when Bao
completed the race?

ii.

5 seconds after the race started, Chinh ridea bicycle to chase after the two friends along
the same track at a speed of 10 m/s. Draw
the graph that represent Chinh’s movement
on Diagram 10.
•
Did Chinh manage to overtake (a) Anh
Dung, and (b) Bao respectively? If yes,at what distance from the ending point?

Diagram 9
Table 4
Water Temperature After x Minutes of
Heating
x
(min)

0

1

2

3

4

y
(ºC)

25.2

35.3

45.0

57.1

68.2

Study of Movement

Bao
Anh Dung

Diagram 10

Chapter 3: Relations and Functions | 57

Determining the Best Deal
Printing Company

A

Printing Cost
0

10

RM

RM100 for printing each book.

+

60

RM

B
There is an initial payment of RM100 and then RM60 for
printing each book.

0

40

RM

C

If the number of books is not morethan 60, the cost is RM400 no matter
how many books are printed.

Diagram 11
RECSAM wants to print a guide book for Mathematics teachers. Diagram 11 shows thequotations from three printing companies.
Let the cost of printing x copies of books be y ringgit (RM).
i.

Express y in terms of x for each of the quotations.

Plot the graphs of y against x for the three quotations, then use the graphs to answer thefollowing questions.
ii.

If RECSAM wants to print 25 copies of the book, at what cost will each company offer?

iii.

Which company offers the cheapest cost for printing•
15 copies of the books?
•
35 copies of the books?
•
55 copies of the books?

58 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Topic 3: Exploring Simple Quadratic Function
Standard 3.1:
Exploring quadratic function y = ax2 in relation to linear functioni.
Identify the quadratic function on situations using tables and compare it with linear function
ii. Explore properties of quadratic function using tables, equations as well as graphs andcomparing it with linear functioniii. Apply the quadratic function on situations in daily life and appreciate it
Sample Tasks for Understanding the Standards
Task 1: Simple Quadratic Function, y = ax2

Diagram 1i.

A tennis ball was rolled on a flat runway. The stroboscopic photograph in Diagram 1 shows
the position of the tennis ball every second after it was rolled. Let y be the distance in metre
travelled by the tennis ball after x seconds.
Table 1Distance Travelled for Flat Runway
x (s)

0

1

y (m)

0

0.3

2

3

4

5

6

•

Complete Table 1 for the flat runway.

•

Plot the graph of y against x.
 Is y proportional to x? Explain your reasons.

•Write an equation to represent the relationship of y as a function of x for the flat runway.

•

Where will the position of the tennis ball be after it rolled for 7 s?

Chapter 3: Relations and Functions | 59

Diagram 2ii.

Later, the runway was tilted and the tennis ball was let go to roll down the slope. The
stroboscopic photograph in Diagram 2 shows the position of the tennis ball every second
after it was let go.
Table 2
Distance Travelled for Tilted Runway
x (s)

0

1

y (m)

0

0.1

2

3

4

•

Complete Table 2 for the tilted runway.

•

Plot the graph of y against x.
 Is y proportional to x? Explain your reason.

•

Plot the graph of y against x2.
 Is y proportional to x2? Explain your reason.

•

Write an equation to represent the relationship of y as a function of x for the tilted
runway.

•Where would the position of the tennis ball be after it rolled for 5 s?

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

Task 2: Properties of Quadratic Function
Table 3
Quadratic Function y = x2
x

x2
A
B

0

1

2

–

–

2

0
–

1
1

4
3

3
9
5

4

16
7

5

25

6

36

7

49

8

3x2
A
B

0

1

–

–

0
–

3
3

2

12
9
6

3

27
15

4

48
21

5

75

6

7

i.

The numbers in row A are derived
from the numbers in row x2. Study
the numbers 1, 3, 5, 7 in row A
carefully.
•
How are the numbers 1, 3, 5,
7 derived from the numbers inrow x2?
•Complete row A.

ii.

The numbers in row B are derived
from the numbers in row A by asimilar way.
•
How is the number 2 in row B
derived from the numbers in
row A?
•
Complete row B.
•
What do you notice about the
numbers in row B?

64

Table 4
Quadratic Functionx

Table 3 shows the values of x2, A and B
for the interval 0 ≤ x ≤ 8.

8

Table 4 shows the values of 3x2, A and B
for the interval 0 ≤ x ≤ 8.
iii.

The numbers in rows A and B are
derived the same way as in Table 3.
•
Complete Table 4•
What do you notice about the
numbers in row B?

iv.

Construct a similar table for thequadratic function y = 5x2.
•
What will the number in row B
be?
•Explain your reasons.

v.

What will the number in row B be forthe following quadratic functions?
•
•
•

vi.

1
2

y = – x2 ?
y = –2x2

y = 2x2 + 3

Why does the number in row B
becomes a constant for a particularquadratic function?

Chapter 3: Relations and Functions | 61

Task 3: The Graph of Function
The graph of quadratic function y = ax2 isalways in the shape of a parabola irrespectiveof the values of a.
i.

Diagram 3 shows two graphs of y = ax2
for a > 0.
•
Match each of the graphs with thecorresponding function. Justify yourdecision.


Diagram 3••

y = 3x2

Diagram 4

1

y = 2– x2
y = 2x2

What will happen to the graph whena < 0?
What will happen to the function andits graph when a = 0?

ii.

The parabolas shown in Diagram 3 may
look different. However, the two parabolas
are basically a same shape because allparabolas are similar shapes.
•
Prove that the two parabolas are
similar.
[Hint: Consider enlargement of theparabolas with origin as the centre ofenlargement.]iii.

Diagram 4 shows the graphs of y = 3x2
passing through the origin. A, B and C are
three points on the graph.
•
What will happen to the values of x
and y as point A moves in the direction
indicated by the arrows toward point
C passing through point B and theorigin?
•At what value of x is y minimum?
•
What is the minimum value of y?
•
What is the rate of change of y withrespect to x between points A and B?
•
What will happen to this rate ofchange as point A moves towards C?

62 | Mathematics Challenges for Classroom Practices at the Lower Secondary Leveliv.

y = 3x2

Diagram 5 shows the graph of y = 3x2
again but this time a slanting line is drawn
to intersect with the graph at points P and
Q. Another line parallel to the x-axis isdrawn to intersect with the graph at points
R and S. The x-coordinates of point P, Q, R,
S are p, q, r, s respectively.

•

What is the numerical relationships
between the absolute values |p| and|q|?

•

Explain your reasons.

•

Diagram 5v.

Diagram 6 shows the graph of y = –3x2
passing through the origin. A and B are
two points on the graph.

•

What will happen to the values of x
and y as point A moves in the direction
indicated by the arrows toward point
B passing through the origin?

•

What is the maximum value of y?

•

What will happen to the rate of changeof y with respect to x as point A moves
toward point B?

•

y = –3x2

Diagram 6v.

What is the numerical relationshipbetween the absolute value |r| and|s|?

At what value of x is y maximum?

Compare the graphs of quadratic function
y = ax2 with linear function y = ax. Describe
the differences between the two graphsfrom the following aspects:•••

Shape of the graph, when (a) a > 0,(b) a < 0.
Change in the value of y as thevalue of x changes, when (a) a > 0,(b) a < 0.
Rate of change of y with respect to x
when (a) a > 0, (b) a < 0.

Chapter 3: Relations and Functions | 63

Task 4: Applications of y = ax2 in Real World

Diagram 7
When you follow behind a car during a driving journey, it is important to keep a safe stopping
distance. Stoping distance is the distance it takes you to bring your car to a complete stop in an
emergency. As show in Diagram 7,
Stopping
distance=

Reaction
distance+

Braking
distance

Reaction distance is the distance your car travels between something happening on the roadup ahead and you reacting to it. This distance is directly proportional to the speed of your car, x.

where a is your reaction time.

Reaction distance = ax

Braking distance is the distance your car travels after you hit the brakes before it comes to acomplete stop. This distance is directly proportional to the square of the speed of your car, x2.
Braking distance = kx2,

where k is a constant of proportion depending on the breaking system of your car.
i.

Given that your reaction time is 0.6 s. When your car is running at a speed of 40 km perhour, its’ breaking distance is 10 m.
••

If your car is travelling at 90 km per hour, calculate the safe stopping distance.
What if your car is travelling at 110 km per hour?

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

Topic 4: Generalising Functions
Standard 4.1:
Generalising functions with various representations on situationsi.
Distinguish domain, range and intervals and is appropriately use for explaining function
ii. Use various situation for generalizing ideas of functions such as moving point A and moving
point B with timeiii. Compose a graph as a function of two or more graphs with different domains in a situation
iv. Introduce situations of step-functions with graph for generalization the idea of function whichcannot be represented by equation
Sample Tasks for Understanding the Standards
Task 1: Domain, Range and Interval of a Quadratic Function

Diagram 1 shows the graph of y = x2 – 4x + 3 for–2 ≤ x ≤ 4.
i.

ii.

iii.

Diagram 1

What is the interval of x and range of y foreach of the following domains?
•–2 ≤ x ≤ 4•–2 ≤ x ≤ 1•
1≤x≤4
What is the minimum value of y for each of
the following domains?
•–2 ≤ x ≤ 4•–2 ≤ x ≤ 1•
1≤x≤4

What is the maximum value of y for each of
the following domains?
•••–2 ≤ x ≤ 4–2 ≤ x ≤ 11≤x≤4

Chapter 3: Relations and Functions | 65

Task 2: Domain, Range and Interval of a Direct Proportion Function

Diagram 2 shows a square ABCD with side
length 8 cm. A point P is moving from A to
D through B and C along sides AB, BC andthen CD.
Given y is the area of triangle APD.
i.

Diagram 2

ii.

Express y as a function for the domain
•
0<x≤8
•
8 ≤ x ≤ 16•
16 < x ≤ 24

Sketch the graph of function y for thedomain 0 < x ≤ 16?

Task 3: Domain and Range of an Inverse Proportion Function

Given an inverse proportion function y = –2– .
x

i.

ii.

Diagram 3

Sketch the graph of function y on
Diagram 3 for the domain 1 ≤ x ≤ 10.
Consider the domain 0 < x ≤ 1.
•
As x is approaching 0, what will
happen to the value of y?

y is approaching 0, what
happen to the value of x?

iii.

As
will

iv.

What is the range of y for the domain
0 < x ≤ 10?

v.
vi.

Sketch the graph of function y on
Diagram 3 for the domain –10 ≤ x ≤ –1.
Consider the domain –1 ≤ x ≤ 0.
•
As x is approaching 0, what will
happen to the value of y?
•
As y is approaching 0, what will
happen to the value of x?

vii. What is the range of y for the domain–10 ≤ x < 0?

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

Task 4: Step Functions

Consider the domain 0 ≤ x ≤ 5 where x isa real number and y is the value when x is
rounded to the nearest whole number.
Diagram 5 shows a double number
line mapping some values of x to thecorresponding values of y. Examples
shown areDiagram 50→0i.

0.5 → 1
1.3 → 1
1.5 → 2
2.7 → 3Map each of the following values of x
to its correcponding value of y.
2,
0.2,
3.2,

3,
0.8,
3.8,

4,
1.9,
4.4,

5,
2.3,
4.6.

ii.

Draw the graph of y against x for thedomain on Diagram 6.

iii.

This type of function y is also known
as a step function. Explain why y is afunction of x.

iv.

Another way to represent a step
function is by listing out the valuesof y for different intervals of x. For
example, for the domain 0 ≤ x < 2.5 , ycan be represented as follow:

Diagram 6•

Complete this way of representing
function y for the domain0≤x≤5.

Chapter 3: Relations and Functions | 67

Task 5: Functions in Real WorldCar Parking Fee
40 sen for every½ hour or part thereof

Diagram 7 shows the rate of parking fee charged
by Penang Island City Council.

Diagram 7i.

A student think that since it charges 40 sen,which is RM0.40 for every ½ hour, so theparking fee is directly proportional to parking
time. He then plotted the graph as shownin Diagram 8 to represent the relationshipbetween parking fee, y and time x.
•

What is wrong with the graph? Why is
it wrong?

Diagram 8ii.

Diagram 9 shows the graph of parking fee,y in RM against parking time, x in hourplotted for the interval 0 < x ≤ 1.
••

Diagram 9Complete the graph for the interval
1 < x ≤ 5.

Is y a function of x? Explain yourreasons.

68 | Mathematics Challenges for Classroom Practices at the Lower Secondary Leveliii.

Diagram 10

Diagram 10 shows the graph of parking
fee, y against parking time, x of a private
car park in Penang Island for the interval
0 < x ≤ 7.
•

Based on the graph, explain the rateof parking charges at this private car
park.

•

Uncle Ben parked his car at this private
car park for 4 hours 25 minutes. How
much is the parking charge?

•

Ms Aishah has RM20 only with her.
How long can she park her car at thisprivate car park?

Rotating Wheels
E'

valve
↓

F'

C'

G'

B'
H'

Diagram 11

D'

Diagram 12

A'

Diagram 13

Diagram 11 shows a bicycle wheel with its valve, A, at the ground level. The wheel is rotating
counter-clockwise to move to the left as shown by the arrows. Given that the radius of the rimof the wheel is 50 cm and the valve rotates at a constant speed of 1 revolution in every 4 s.
i.

What will happen to the location of the valve as the wheel move?

ii.

Diagram 12 shows seven other positions of the valve, B, C, D, E, F, G, and H, as the wheel
moves.
•
What is the time taken for the valve to move from
 A to C?
 A to E?
 A to G?
 A to B?
 A to H?
•
What is the vertical distance of the valve from the ground level at position C?
 position E?
 position G?

iii.

Diagram 13 shows another eight positions of the valve, A’, B’, C’, D’, E’, F’, G’ and H’.
•What is the time taken for the valve to move from
 A to A’?
 A to F’?

Chapter 3: Relations and Functions | 69

iv.

Let the vertical height of the valve from the ground level be y cm after it starts to rotate
x seconds from A. Diagram 14 shows part of the graph of y against x for the domain
0 ≤ x ≤ 4.

Diagram 14
In this graph, the coordinates (x, y) when the valve is at positions A, A’, B, B’ and C are
already plotted. At A and C , y = 0 cm and y = 50 cm respectively. However, with the helpof Diagram 14, the coordinates at A’, B, and B’ are plotted without the need to know thevalues of y.
•
•
•
v.

Explain how this can be done.
Plot the coordinates for all other positions of the valve.
Complete the graph by joining all the coordinates plotted with a smooth curve.

Explain why y is a function of x for the domain 0 ≤ x ≤ 4.
••

If we continue drawing the graph for the domain 4 < x ≤ 8, how would the graph looklike?
Is y still a function of x for the domain 0 ≤ x ≤ 8? Explain your reasons.

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

Chapter 4: Space and Geometry | 71

CHAPTER FOUR
Space and Geometry
Topic 1: Exploring Angles, Construction and Designs in GeometryStandard 1.1:
Exploring angles to explain simple properties on the plane geometry and do the simple geometrical
Construction
i.
ii.
iii.

Explain how to determine the value of angles using the geometrical properties of parallel lines,
intersecting lines, and properties of figures
Use ruler and compass to construct a simple figure such as perpendicular lines and bisectors
Appreciate the process of reasoning that utilizes the properties of angles and their congruency
in simple geometrical constructions

Sample Tasks for Understanding the Standards
Task 1: Lines and Angles
Lines, Rays and Line Segments
Diagram 1 shows three basic figures that passes
through points A and B, a line, a ray and a line
segment.
i.

Explain the similarities and differences
between a line, a ray and a line segment.

Diagram 1
Vertical Angles

Diagram 2 shows two intersecting lines, l and m.
i.

∠a and ∠b are a pair of vertical angles.
••
Diagram 2

ii.

What is the relationship between ∠a and ∠c ∠b and ∠c
Explain why ∠a =∠b?

∠c and ∠d are another pair of vertical angles.
•
Prove that ∠c =∠d .

72 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Corresponding Angles and Alternate
Interior Angles

Diagram 3 shows two parallel lines, l1, l2 and anintersecting line, m.
i.

∠a and ∠p are a pair of corresponding angles.
••

Diagram 3

ii.

•

of



Name the three pairs of corresponding
angles.

State the relationships among thesepairs of corresponding angles.

Explain why ∠d = ∠r.

There is another pair of alternate interior
angles in Diagram 3. Name the pair ofalternate interior angles and state theirrelationship.

In Diagram 4, line l is parallel to line m.
••

Diagram 4

There are three other pairs
corresponding angles in Diagram 3.

∠d and ∠r are a pair of alternate interior angles.
•iii.

What is the relationship between ∠a and∠p ?

Find ∠a, ∠b and ∠c.

Explain your reasoning in finding ∠a, ∠b
and ∠c by using the properties of angle
such as parallel lines and intersection.

Chapter 4: Space and Geometry | 73

Task 2: Sum of Interior Angles

i.

Diagram 5 shows a triangle with interiors
angles ∠a, ∠b and ∠c.

•

Prove that ∠a + ∠b + ∠c = 180°

[Hint: Use the properties ofintersecting with parallel lines.]lines

Diagram 5ii.

Diagram 6 shows a pentagon with interior
angles ∠a, ∠b, ∠c, ∠d and ∠e.
•Find the sum ∠a + ∠b + ∠c + ∠d +∠e.

Diagram 6iii.

Diagram 7 shows a concave quadrilateralwith interior angles ∠a, ∠b ∠c and ∠d.
•Prove that ∠a +∠b +∠c +∠d = 360°.

[Hint: Use sum of interior angles of atriangle.]
Diagram 7

iv.

Diagram 8 shows a concave pentagon withinterior angles ∠a, ∠b, ∠c, ∠d, and ∠e.
•

Diagram 8

Find the sum ∠a + ∠b + ∠c + ∠d +∠e.

74 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 3: Determining a Location

Diagram 9
Diagram 9 shows a system to determine location in a plane. P, the centre of the concentric
circles is taken as the reference point. Two basic measurements used to determine a location
are the distance from P and the angle from ray PQ.

ii.

Find the animals located at•
60 m from P•
45 m from P•
240° from ray PQ•
150° from ray PQ

iii.

State the location of the cow, the deer and the panda respectively.

i.

What animal is located at 45 m from P and 150° from ray PQ?

v.

If there is an animal located at 180° from ray PQ, what can you say about the distance of
the animal from P?

vi.

In this system, why are both distance and angle important in determining a location?

iv.

In a treasure hunt game, what would you do for each of the following instructions?
•Stand at the tree. One treasure is 10 m from the tree.
•
Stand at the tree and look at the mountain top. Another treasure is 45° from thisdirection.

Chapter 4: Space and Geometry | 75

Task 4: Geometrical Constructions

CP

CQ

i.

Diagram 10 shows a line segment PQ. Twocircles, CP and CQ are constructed with P and
Q as the centre respectively. Both circles
have equal radii, which is the length of PQ.
R is an intersection point of CP and CQ.
•••

Diagram 10

CB

CA

ii.

Diagram 11

CN

iii.

What is the value of ∠PQR ?

Why is line segment PQ perpendicularto AB?

Diagram 12 shows a line segment AB
and a point P on the segment. A circle,
CP is constructed with P as its centre. CPintersects with AB at points M and N. Then,
two other circles with equal radii, CM andCN with centre at M and N respectively are
constructed. R and S are two intersection
points of CM and CN.
•

Diagram 12

Why is ∠PQR = ∠PRQ = ∠QPR ?

Diagram 11 shows a line segment AB and apoint P not on the segment. Two circles CA
and CB with centre A and B respectively are
constructed. The radius of CA is AP and theradius of CB is BP. Q is an intersection pointof CA and CB.
•CM

Why is PQ = PR = QR?

Why is line segment RS perpendicularto AB?

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

CA

CB

iv.

Diagram 13 shows a line segment AB.
Two circles with equal radii, CA and CB
are constructed with the centre A and B
respectively. R and S are two intersection
points of CA and CB. M is an intersection
point of AB and RS.
•••

Diagram 13v.

•
Diagram 14

vi.

CQ

•
Diagram 15

What does line RS do to line segment
AB?

What relationship exists between line
segments
 AP and AQ?
 BP and BQ?
 CP and CQ?
 DP and DQ?

What can you conclude about the
distance from any point on the bisector
to P and Q respectively?

Diagram 15 shows an angle ∠AOB. A circle,
CO, is first constructed with O as its centre.
CO intersects with line segments OA and OB
at points P and Q respectively. Then, twoother circles with equal radii, CP and CQ are
constructed with P and Q as the centres. OR
is a ray joining O to the intersection point of
CP and CQ.
•COWhy is ∠AMR = 90° ?

Diagram 14 shows a line segment PQ and
its perpendicular bisector l. A, B, C and D
are four points on l.
•CP

Why is AM = MB ?

Why is ∠AOR =∠BOR?

What does ray OR do to AOB?

Chapter 4: Space and Geometry | 77vii. Diagram 16 shows an angle ∠POQ and its
angle bisector l. A, B and C are three points
on l.
•What relationships exist between the
distances from point A, B, C to thetwo sides of the angle, OP and OQ
respectively?

Diagram 16viii. Diagram 17 shows a line segment AB.

A

B

•

Given C is a point such that
∠CAB = 60° and ∠CBA = 30°. Usecompass and ruler only to construct
ABC.

•

Given P is a point such that ∠PAB = 45°
and ∠PBA = 90°. Use compass andruler only to construct ABP.

Diagram 17ix.

Diagram 18 shows a circle with its centre
unmarked. T is a point on the circle.
•

What properties of a circle can be used
to locate the centre of the circle?

•

Use compass and ruler only to locate
the centre of the circle on Diagram 18.
Mark and label the centre as O.

•

Diagram 18

The tangent of a circle is perpendicular
to its radius that passes through thepoint of tangency. Use compass andruler only to construct the tangent of
the circle at point T on Diagram 18.

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

Task 5: Application of Geometrical Construction in Real World

Diagram 19
Diagram 19 shows a groundwater well pump and the locations of three villagers’ houses in arural village in Cambodia.
i.

Mr Ratha and Mr Chantha want to construct a groundwater well together for their families
use. They want it to be built at a location which will be the same distance from the twohouses.
•Help the two villagers to identify the possible locations to construct the well by doingsome geometrical constructions on Diagram 19.
•
Which location will be the nearest to their houses?

ii.

When Mr Vutha heard about his two neighbours’ plan, he asks to join in the constructionof the well. Now, the three of them are having a little problem in identifying the location for
the well that will be the same distance from the three houses.
•
Help the three villagers to identify the best location for the well by doing somegeometrical constructions on Diagram 19.

iii.

If a fourth villager wants to join in the construction of the well, there will be a condition thatneed to be fulfilled in order to find a location that is the same distance from four houses.
•
What is the condition?
•
What if there are five houses or more?

Chapter 4: Space and Geometry | 79

Standard 1.2:
Exploring the relationship of figures using congruency and enlargement for designsi.
Explore the congruence of figures through reflection, rotation and translation and explain the
congruency using line of symmetry, point of symmetry and parallel lines
ii. Explore similarity of figures with enlargement using points, ratio, and correspondencesiii. Enjoy using transformations in creating designs
Sample Tasks for Understanding the Standards
Task 1: Transforming Geometric Figures
i.

①

②

③

Diagram 1

Diagram 1 shows a geometric figure and its
three images. Each image is the result of only
one transformation.
•
Explain why enlargement cannot be
involved in creating any of the three
images.
•Which image is the result of a translation?
Justify your decision.
•
Which two images are the result of areflection? Justify your decision. Then,
describe each reflection clearly.
•
All three images are the results of arotation. Describe each rotation.
 A student claims that every image
has more than one angle of rotation.
How is this possible?
•
•

ii.

Diagram 2

Each of the geometric figures has twolines of symmetry. Identify the lines ofsymmetry for each figure.
Is there any figure that has a point ofsymmetry? If any, where is the point?

Three black pentagons and three whitepentagons are tiled as shown in Diagram 2.
•
What is the minimum number oftransformation to move pentagon ① to fit
onto ②, ⑤, and ⑥, respectively?
 Describe how it can be done.
•

What is the minimum number oftransformation to move pentagon ④ to fit
onto ⑥?
 Describe how it can be done.

•

What is the minimum number oftransformation to move pentagon ③ to fit
onto ⑥?
 Describe how it can be done.

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

Task 2: Isometric Transformations

Reflection, translation and rotation are isometrictransformations.
i.

In Diagram 3, teddy bear ① has undergone
translation, reflection and rotation respectively.
Study teddy bear ① and its three images.
••

Diagram 3

ii.

Identify the transformation that has
moved teddy bear ① onto teddy bears②, ③ and ④, respectively.
What has changed and what remains
unchanged between teddy bear ① andeach of its images?

•

What does it mean by all the teddy bears
in Diagram 3 are congruent figures?

•

What is an isometric transformation?

Diagram 4 shows a bird and its image after
being rotated.
•

Use compass and ruler to locate andmark the centre of rotation.

•

Measure and state the angle of rotation.

Diagram 4iii.

Diagram 5 shows a bird and its image after
being reflected.
•

Diagram 5

Use compass and ruler to draw and markthe axis of reflection.

Chapter 4: Space and Geometry | 81

Task 3: Non-Isometric Transformation

Diagram 6
The three elephants in Diagram 6 are similar figures.
i.

Explain the meaning of similar figures.

ii.

What transformation is involved here?

iii.

What is point C? Why is it important?

iv.

Given that the size of elephant ③ is 2 times the size of elephant ②, and the size ofelephant ① is ½ the size of elephant ②.
•••

If the tail of elephant ① is 15 cm long, how long is the tail of elephant ③?

What is the ratio between the size of elephant ③ and the size of elephant ①?

If the ear of elephant ② has an area of 2500 cm2, what is the area of the ear ofelephant ③?

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

Diagram 7v.

Diagram 7 shows three Japanese torii gates created by enlarging the smallest gate.
•

Locate and mark the centre of enlargement on Diagram 7.

Diagram 8
vi.

Diagram 8 shows another three torri gates as a results of enlarging the smallest gate.
•

Locate and mark the centre of enlargement on Diagram 8.

Chapter 4: Space and Geometry | 83

Task 4: Creative Designs Using Transformations

Diagram 9
Transformation is often found in the creation of creative visual art.
i.

The tessellation of the white and black birds in Diagram 9 is a creative design produced
using transformation. As we can see, both the black and white birds are congruent figures.
•

What transformation takes bird ① onto bird ② ?

Diagram 10
ii.

Diagram 10 shows another tessellation of white and black birds.
••

What one transformation takes bird ① onto bird ②?
What two transformations take bird ③ onto bird ④?

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

①

②

③

④

⑤

⑥

Diagram 11iii.

Symmetry is commonly seen in traditional native designs found in different cultural items
such as textiles and sculptures. Diagram 11 shows six native designs from the ASEAN
countries.
•

Draw all the lines of symmetry for each of the designs.

•

Identify all the rotational symmetry for each of the designs, if any.

•

Identify the point of symmetry for each of the designs, if any.

Chapter 4: Space and Geometry | 85

Topic 2: Exploring the Space With Its Components
Standard 2.1:
Exploring space by using the properties of planes, lines and their combinations to form solidsi.
Explore the properties produced by planes, lines and their combinations, such as parallel lines
produced by intersection of parallel planes with another plane
ii. Produce solids by combining planes such as nets and motion such as rotation, reflection andtranslationiii. Recognise the space of an object based on its properties and projection in life
Sample Tasks for Understanding the Standards
Task 1: Lines and Planes in Spacei.

Diagram 1

ii.

Diagram 2

Diagram 1 shows three points A, B, C andthree lines passing through point A on a sameplane.
•How many other lines passing throughpoint A are there altogether?

•

Of all the lines passing through point A,
how many line will also pass through pointB?

•

Is there a line that passes through all
three points A, B and C?
 Why or why not?

•How many minimum points are required
to determine a line?

Diagram 2 shows three points P, Q and R inspace. Two planes passing through both
points P and Q are shown.
•

How many planes passing through both
points P and Q are there altogether?

•

Of all the planes that passing throughpoints P and Q, how many plane will alsopass through point R?

•

A minimum of three points in space are
required to determine a plane. What isthe condition of the three points?

86 | Mathematics Challenges for Classroom Practices at the Lower Secondary Leveliii.

Diagram 3 shows a camera tripod.
•

Diagram 3

iv.

Diagram 4 shows a cuboid and four lines, l1, l2,
m1 and m2 that pass through four of its edges.
•

Why are l1 and l2 called parallel lines?

•

Lines l1 and m1 intersect with each otherat a point because they are not parallellines. However, lines l2 and m2 are notparallel lines but they will not intersectwith each other. Explain why l2 and m2 will
not intersect with each other.
[Note: l2 and m2 are known as a pair ofskew lines.]•

What is the condition for two non-parallellines in space to intersect with each other?

Diagram 4v.

Diagram 5

Why does a camera tripod have 3 legs?

Diagram 5 shows three planes, P, Q and R.
Planes P and Q are parallel, whereas plane R
intersects with planes P and Q.
•

Why are planes P and Q called parallel
planes?

•When plane R intersects with planes P
and Q, two intersecting lines are formed.
 What is the positional relationshipbetween the two intersecting lines?
 What if planes P and Q are notparallel?

•

How can we define the distance between
two parallel planes such as planes P andQ?

•

How can we define the angle between two
intersecting planes such as planes P andR?

Chapter 4: Space and Geometry | 87

Diagram 6vi. Diagram 6 shows a cuboid ABCDEFGH placed on plane APQD. M is a point on edge CD.
Given that line FB is perpendicular to plane ABCD and ∠FBC = 90°.
•••

What is ∠FBP, ∠FBA, ∠FBD and ∠FBM, respectively?

What is the positional relationship between lines GC, HD and EA with line FB?
What is the positional relationships between lines GC, HD and EA with plane APQD,
respectively?

(a)

(b)
Diagram 7vii. In Diagram 7(a), a rectangular card ABCD is folded along EF such that EF ∥ AB ∥ CD.
Then, part of the card, EDCF is turned to form two planes intersecting at EF as shown in
Diagram 7(b). P, Q and R are three points on lines AB, EF and DC, respectively such that
PQ ∥ AE ∥ BF and QR ∥ ED ∥ CF.
•••

What is the measure of ∠DEF?

What is the positional relationships between ∠PQR, ∠BFC and ∠AED?
What is the measures of ∠PQE and ∠RQF respectively?

88 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Diagram 8viii. Diagram 8 shows a solid made up of a cuboid ABCDEFGH as the base with a rectangularpyramid EPRHT on top of the cuboid. TS is the perpendicular line from the apex of the
pyramid to its base. Line TQ is perpendicular to line PR. M is a point on edge FG such that
QM ∥ PF ∥ RG and N is a point on edge BC such that NM ⊥ FG.
•
•
•
•
•
•

Which edges are parallel lines with PR?

Which edges are intersecting lines with PR?
Which edges are skew lines with PR?

State the distance from point E to (a) point A, (b) line FB, and (c) face BCGF, respectively.

State the distance from point T to (a) point P, (b) line PR, and (c) base EPRH of the pyramid,
respectively.
State the angle between base EPRH of the pyramid and (a) line TE, (b) line TQ, (c) face PRT,
respectively.

Chapter 4: Space and Geometry | 89

Task 2: Solids of One-Directional Movement

Diagram 9i.

Diagram 9 shows a cuboid generated from square ABCD and a cylinder generated fromcircle P.
•

Describe how square ABCD could be moved to generate the cuboid.

•

Describe how circle P could be moved to generate the cylinder.

•

What transformation is involved in generating the cuboid and the cylinder?

ii.

Diagram 10 shows a triangle ABC on a plane
P and line l is perpendicular to plane P.
•

If ABC moves vertically up along line

l from A to D, what solid will be createdby the movement of ABC?

•

Diagram 10

Sketch the solid.

90 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 3: Solids of Revolutioni.

Diagram 11 shows a cone generated from
ABC and a cylinder generated fromrectangle ABCD.
•

Describe how ABC could be moved
to generate the cone.

•

Describe how rectangle ABCD could
be moved to generate the cylinder.

Diagram 11ii.

Diagram 12 shows a semicircle and a line l.
•

Diagram 12iii.

Diagram 13 shows a trapezium, a square
and a circle with a line l.
•

Diagram 13

What solid will be generated if thesemicircle is revolved once about axisl?

Sketch the solid that will be created by
the following revolutions.
 Revolving trapezium Ⓐ once
about axis l.
 Revolving square Ⓑ once aboutaxis l.
 Revolving circle Ⓒ once aboutaxis l.

Chapter 4: Space and Geometry | 91

Task 4: Projections of Points, Line Segments and Faces

Diagram 14 shows a triangular pyramid withbase ABC and apex P. A'B'C' is a cross sectionof the pyramid. The projections of AB and A'B'
intersect at point C''. Likewise, the projectionsof AC and A'C' intersect at point B''. This makes
B''C'' the line of intersection of plane AC''B'' andplane A' C''B''.
i.

Given the projections of BC and B'C
intersect at A''.
••

Diagram 14

Where will A'' be?
Verify your answer on Diagram 14.

ii.

How can we define the angle between thetwo intersecting planes, plane AC''B'' andplane A' C''B''?

iii.

What will points A, B' and C become,respectively?
•
•
•

iv.

What will line segments AB, BB', A'C' andB'C' become, respectively?
•
•
•

v.

Taking the view from the perspectiveof point P.
Taking the view from the perspectiveof point C''.
Taking the view from the perspectiveof point B''.

Taking the view from the perspectiveof point P.
Taking the view from the perspectiveof point C''.
Taking the view from the perspectiveof point B''.

What will faces ABC, A'B'C', ABB'A', and
BCC'B' become, respectively?
•••

Taking the view from the perspectiveof point P.
Taking the view from the perspectiveof point C''.
Taking the view from the perspectiveof point B''.

92 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 5: Projection of Solids

Top view

Side view

Top view

Solid ①

Solid ②
Front view

Front view
Top view

Side view

Top view

Solid ③

Side view

Solid ④

Front view
Top view

Side view

Front view
Side view

Top view

Solid ⑤

Side view

Solid ⑥

Front view

Front view

Diagram 15
Diagram 15 shows the projections of 6 solids.
i.

What solid does each projection represent?

ii.

Sketch the solid for each projection.

Chapter 4: Space and Geometry | 93

Task 6: Nets of Solids

Diagram 16i.

Diagram 16 shows the nets of 4 solids. Identify and sketch each solid.
ii.

Top view

Diagram 17 shows a net folds up to
be a solid tetrahedron.
•Side view

Net of tetrahedron

Match the top, front and sideviews with the following drawings.

Front view

Diagram 17iii.

Diagram 18 shows a net folds up to
be a solid octahedron.
•

Draw the top, front and side viewsof the octahedron.

Diagram 18
iv.

Diagrams 19 and 20 show the nets of two regular polyhedrons.
•

Make a copy of the nets.

•

Cut out the nets and then build the regular polyhedrons.

94 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Diagram 19

Chapter 4: Space and Geometry | 95

Diagram 20

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

Task 7: Surface Area and Volume of Solidsi.

Diagram 21 shows a candy
box in the shape of a cube
and its net. Each edge ofthe square base is 8 cm.
•

Calculate the surface
area and volume of the
candy box.

Diagram 21

ii.

Diagram 22 shows anothercandy box in the shape of aprism and its net. The baseof the prism is a regularhexagon. Each edge of thehexagon is 4 cm and theheight of the prism is 20 cm.
•

Calculate the surface
area and volume of the
candy box.

Diagram 22iii.

Diagram 23 shows a gift box
in the shape of a pyramid
and its net. The base of the
pyramid is a square with 12
cm edge. The distance fromits apex to the edge of thesquare base is 10 cm.
•

Diagram 23

Calculate the surface
area and volume of thegift box.

Chapter 4: Space and Geometry | 97

Diagram 24
iv.

Diagram 24 shows a tin of canned food and its nets. The circular base of the tin has adiameter of 10 cm and its height is 5 cm.
Calculate the surface area and volume of the tin.

12

cm

•

Diagram 25v.

Diagram 25 shows the net of a gift box.
•

Sketch the shape of the gift box.

•

Find r, the radius of the circular base.

•Calculate the surface area and volume of the gift box.

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

Diagram 26vi.

A gift box has the shape shown in Diagram 26. The height of the gift box is 10 cm, thelonger edges are 20 cm long, and the short edges of the square corner cut-outs are each
5 cm long. Part of the net of the gift box is also shown in Diagram 26.
•

Complete the net of the gift box.

•

Calculate the surface area and volume of the gift box.

Chapter 4: Space and Geometry | 99

Topic 3: Exploring the Ways of Argument for Proving and Its Application in GeometryStandard 3.1:
Exploring properties of congruency and similarity on plane geometryi.
Explore ways of arguments using the congruence of two triangles and appreciate the logic ofargument in simple proving
ii. Explore ways of arguments using the similarity of two triangles based on ratio and angles andappreciate the logic of arguments in simple provingiii. Explore the proof of the properties of circles such as inscribed angles, intercepted arcs
iv. Appreciate proving through making the order of proven propositions to find new propositions
Sample Tasks for understanding the standards
Task 1: Congruent and Similar Figures

Diagram 1
Diagram 1 shows a quadrilateral ABCD being rotated to become quadrilateral EFGH andenlarged to become quadrilateral PQRS.
ABCD is said to be congruent to EFGH, but similar to PQRS.
i.

Explain the similarities and differences between congruent figures and similar figures.

100 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 2: Proof Using Congruence Conditions for Triangles
Diagram 2 shows the three conditions forcongruence of triangles:① All three pairs of corresponding sides
are equal (Side-Side-Side).
② Two pairs of corresponding sides andthe angle between them are equal(Side-Angle-Side).
③ Two pairs of corresponding angles
and the side between them are equal(Angle-Side-Angle).
Triangles ABC and PQR are congruent
if any of the conditions is fulfilled.
i.

Diagram 2

In the figure shown in Diagram 3,
AB ∥ CD and AM = DM. Line segments
AD and BC intersects at point M. Two
students, Antonio and Buwan, tried towrite the proof for CM = MB using a
congruence condition for triangles.
Antonio’s Written Proof
Since ∠BAM = ∠CDM ,∠AMB = ∠CMD, and AM = DM,therefore, AMB CMD
So, CM = MB.
Buwan’s Written Proof
Given that AB ∥ CD and alternate
interior angles formed by parallel lines
are equal, so∠BAM = ∠CDM … Supposition ①
Since vertical angles are equal, so∠AMB = ∠CMD … Supposition ②

Also given AM = DM … Supposition ③
From suppositions ①, ②, and ③,
because of the Angle-Side-Angle
congruence condition,
AMB CMD.
Diagram 3

Since the corresponding sides
in congruent triangles are equal,
therefore CM = MB.
••

Which proof is easier tounderstand? Why?
Prove that AC ∥ BD.
(State all the suppositions clearlyin your proof.)

Chapter 4: Space and Geometry | 101

ii.

In the figure shown in Diagram 4,
AD = BC and AC = BD.
••

Prove that ∠DAC = ∠DBC.

(State all the suppositions clearlyin your proof.)

Diagram 4iii.

Diagram 5 shows a quadrilateral ABCD.
Given that AB ∥ DC and AD ∥ BC. The
diagonals AC and BD intersect at point
M.
•
•
•

Diagram 5

Using any of the congruence
conditions for triangles, prove that
∠ADB = ∠ACB.

Prove that ∠ABC = ∠ADC.

Prove that AB = DC and AD = BC.
Prove that M is the mid-point ofboth diagonals.
(State all the suppositions clearlyin your proof.)102 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 3: Proof Using Conditions for Similarity of Triangles

Diagram 6 shows the three conditions for
similarity of triangles:

Condition ① : AB:PQ = AC:PR= BC:QR① The ratios of all three pairs ofcorresponding sides are equal.
② The ratios of two pairs of corresponding
sides and the angle between them areequal.
③ Any two pairs of corresponding angles
are equal.
Triangles ABC and PQR are similar if
any of the conditions is fulfilled.

Condition ② : AB:PQ = AC:PR and ∠a =∠p

Condition ③ : ∠a =∠p and ∠b =∠q
Diagram 6i.

Diagram 7

Diagram 7 shows a parallelogram
ABCD. R is a point on AB and S is theintersection point of DR and AC. Provethat ARS is similar to CDS

Chapter 4: Space and Geometry | 103

i.

cm
10
cm

cm

m
6c

12

20

In the figure shown in Diagram 8, C is
the intersection point of line segments
AD and BE. Given BC = 6 cm,
AC = 12 cm, CE = 10 cm and CD = 20 cm.
•

Prove that ABC is similar to DEC.

•

Prove that line segment AB is parallel toline segment ED.

Diagram 8ii.

Diagram 9 shows a right-angled triangle
ABC.
Given that P and Q are the
mid-points of AC and BC, respectively.
Youarerequired
to
prove
that
(a) AB = 2PQ, (b) ABC is similar to PQC, and(c) area of PQC = ¼ × area of ABC.
•

Which of the three proofs should you
work out first? Explain your reasons.

•Work out the three proofs.

Diagram 9i.

Diagram 10

Diagram 10 shows a tree and a man
standing under the sun at a certain time of aday. The shadow of the tree was 16.8 m andthe shadow of the man was 1.2 m.
•

Explain how and why the idea of similar
triangles can be used to find the heightof the tree.

•

If the height of the man is 1.7 m, what is
the height of the tree?

104 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 4: Properties of a Circle

Diagram 11i.

Diagram 11 shows three cases of the positional relationships between the inscribed angle∠AMB subtended from minor arc AB to point M and the corresponding central angle ∠AOB
of a circle with centre O. In all three cases we can prove that ∠AMB = ½ ∠AOB.
•••

Which case will you prove first? Explain your reasons.
State clearly your order of proving the three cases.
Work out the three proofs according to your sequence.

ii.

Diagram 12 shows a circle with centre O. ∠p,∠q and ∠r are three inscribed angles subtended
from minor arc AB to points P, Q and R,
respectively.
•

Using the results from the proofs in (i),prove that ∠p =∠q = ∠r.

Diagram 12iii.

Diagram 13

Diagram 13 shows a circle with centre O anddiameter AB. Using the results from the proofsin (i), prove that ∠APB = 90°.

Chapter 4: Space and Geometry | 105

iv.

Diagram 14 shows a circle with centre O and A,
B, C, D are four points on the circle.
•

Prove that ∠ADC = 180° – ∠ABC.

Diagram 14v.

Diagram 15 shows a circle with centre O. AB
and CD are two minor arcs with equal lengths.
•

Prove that ∠APB = ∠CQD.

Diagram 15

vi.

Diagram 16 shows a circle with centre O. AB and
CD are two minor arcs where the relationship of
the arc lengths is AB = 2CD.
•••

Diagram 16

Prove that ∠AMB = 2∠CND.
If AB = 3CD, make a conjecture about therelationship between ∠AMB and ∠CND.
Prove your conjecture.

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

Standard 3.2:
Exploring Pythagorean theorem in solving problems in plane geometry and spacesi.
Explore the proving of Pythagorean theorem using diagram and use it in solving problems
involving plane figures
ii. Apply Pythagorean theorem on prism by viewing the figures through facesiii. Explore the situations for simple trigonometry using special angles in relation to the Pythagorean
theoremiv. Appreciate the use of Pythagorean theorem in life
Sample Tasks for Understanding the Standards
Task 1: Proving Pythagorean Theorem
Prove by Tiling Pattern
Diagram 1 shows a tiling pattern with threedarken squares.
i.

Explain how this tiling pattern could be
used to prove the Pythagorean Theorem.

Diagram 1

Algebraic Proof
Diagram 2 shows a right-angle triangle ABC
with sides a, b and c. Three other right-angle
triangles congruent to ABC are added asshown to form a bigger square CHFD.
i.

Use the figure in Diagram 2 to prove thata2 + b2 = c2.

(Hint: Consider the relationship between
the area of ABGE and the area of CDFH.)

Diagram 2

Chapter 4: Space and Geometry | 107

Geometrical Proof by Euclid
Diagram 3 shows a figure used by ancientmathematician Euclid to prove the Pythagorean
Theorem. In the diagram, CVW is a straight line
parallel to AG.
i.

Valid arguments involving a2 can be built
up based on the figure. Euclid argued thatthe area of BEC = BFV based on theanswers to the following questions:•

Why is ACD a straight line?

•

Why is the area of BEC = BEA?

•••

Diagram 3ii.

•

Diagram 4

Why is the area of BFC = BFV?
Why is the
BFWV = a2?

area

of

rectangle

In a similar manner, Diagram 4 can be used
to build up the arguments involving b2.
•iii.

Why is the area of BEA = BFC?

Explain why ACH = AGV.

Explain why the area of rectangle
AGWV = b2.

Explain why c2 = a2 + b2.

108 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 2: Application of Pythagorean Theorem

Diagram 5i.

A teacher was given the task to constructa football field as shown in Diagram 5.
In order to ensure that all corners of the
field are 90°, he used the 3-4-5 method
to construct the corners. He first made a12-m loop with three knots A, B and C tomake intervals of 3 m, 4m and 5 m on theloop. He then stretched the loop to forma triangle with sides 3 m, 4 m and 5 m asshown in Diagram 6.

•

Prove that ∠BAC = 90°.

ii.

A timber log with a diameter of 40 cm is
to be cut into a timber block with a squarecross section as shown in Diagram 7.

•

What is the maximum side length of thesquare in order to cut out the thickest
timber block?

Diagram 6

40cm
Diagram 7

Chapter 4: Space and Geometry | 109iii.

Diagram 8 shows an equilateral triangle
ABC with each side 8 cm.

•

Find the area of ABC.

iv.

Diagram 9 shows a circle centre O withradius 5 cm. AB is a chord 2 cm from thecentre of the circle.

•

Find the length of chord AB.

v.

Diagram 10 shows the locations of point
A(–1, 3) and point B(2, –2)•

Find the distance between A and B.

Diagram 8

Diagram 9

Diagram 10

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

13
cm

vi.

Diagram 11 shows a circular cone with abase radius of 5 cm and a generatrix oflength 13 cm.
•

Find the height and the volume of the
cone.

5 cm
Diagram 11vii. Diagram 12 shows a rectangular prism witha dimension of 8 cm × 15 cm × 17 cm asshown. AG is a diagonal of the prism.
•

Find the length of AG.

Diagram 12viii. Diagram 13 shows a cube with side length
a cm. DF is a diagonal of the cube.
•

Prove that the ratio of its diagonal to its
DF
side,
= 3.
AB

Diagram 13

Chapter 4: Space and Geometry | 111

ix.

Diagram 14 shows a rectangular prism box
with a beetle at vertex D. The beetle starts
to crawl in a straight line towards edge BC
and it then turns to crawl in a straight line
until it reaches vertex F as shown.
A student argues that if the beetle crawls
towards vertex B in a straight and thencrawls along edge BF as shown in Diagram
15, the distance travelled will be theshortest.

Diagram 14•

Prove that the student’s argument iswrong.
[Hint: Open up the box at face BCGF
so that ADGF becomes a rectangle.]•

Find the shortest distance travelled by
the beetle from D to F.

Diagram 15

n
bo

rib

x.

Diagram 16 shows a cylinder with a baseradius of 5 cm, height 12 cm and generatrix
AB.
A ribbon ties around the cylinder goesaround once from point A to point B as
shown in the diagram.
•

Diagram 16

Find the shortest length of ribbon,
rounding your answer to the nearest
tenth.
(Let π = 3.14)

112 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 3: Simple Trigonometry with Special Angles

i.

Diagram 17 shows a right-angle triangle ABC with an angle
θ. Given that 0°<θ<90°. the length of sides BC, AC and AB
are a, b and c, respectively.
•

Why are sinθ, cosθ and tanθ called the trigonometricalratios with respect to ABC?

θ
Diagram 17

Diagram 18ii.

Diagram 18 shows three equilateral triangles, ABC, PQR and XYZ with sides 2 units,
4 units, 6 units, respectively. Given that BJ, QK and YL are perpendicular bisectors to AC,
PR and XZ respectively.
•

Find the length of sides AJ, PK and XL, respectively.

•

Find the length of sides BJ, QK and YL, respectively.

•

Find each of the following trigonometrical ratios using the measures in ABC, PQR
and  XYZ respectively.
 sin60°
 cos60°
 tan60°••Show that the each of the trigonometrical ratios sin60°, cos60°, tan60° and found fromthe three right-angle triangles with different sizes is a constant.
Calculate values of sin60°, cos60°, and tan60°, rounded to four decimal places.

Chapter 4: Space and Geometry | 113

iii.

Diagram 19 shows the situation when the three rightangle triangles are overlapped.
•

Based on this diagram, explain why each of thetrigonometrical ratios sin60°, cos60°, and tan60° is
a constant ratio?

•

Will each of the trigonometrical ratios for angles
other than 60° also a constant? Explain yourreasoning.

Diagram 19
iv.

Diagram 20 shows the same three triangles in Diagram 18.

Diagram 20
•
•

•
v.

Explain why ∠ABJ = ∠PQK = ∠XYL = 30°.

Find each of the following trigonometrical ratios using the measurements in ABC,
PQR and XYZ respectively.
 sin60°
 cos60°
 tan60°

Explain why each of the trigonometrical ratios sin30°, cos30° and tan30° is also a
constant.

What conclusion can you make about the ratios sinθ, cosθ and tanθ for any angle θ, where0 < θ < 90°? What if 0 = θ? What if 0 = 90°?

114 | Mathematics Challenges for Classroom Practices at the Lower Secondary Levelvi.

What relationship exists between the following trigonometrical ratios?
•••

sin30° and cos60°
sin60° and cos30°
tan30° and tan60°

Diagram 21vii. Diagram 21 shows a square ABCD with sides 1 unit and AC is its diagonal.
••••

Explain why ∠BAC = ∠BCA = 45°.
Find the length of side AC.

Show that sin45° = cos45° =2.
2

Find tan45°.

viii. From your answers in (ii), (iv) and (vii), verify each of the following trigonometric identities.
•
•
•
ix.

sin230° + cos230° = 1
sin245° + cos245° = 1
sin260° + cos260° = 1

Using the Pythagorean Theorem, prove that for the case of a right-angled triangle with anacute angle θ, sin2θ + cos2θ = 1.

Chapter 5: Statistics and Probability | 115

CHAPTER FIVE
Statistics and Probability
Topic 1: Exploring Distribution with the Understanding of Variability
Standard 1.1:
Exploring distribution with histograms, central tendency and variability
i.
ii.
iii.
iv.
v.

Use histogram with different class intervals to show different distribution of the same set ofdata.
Identify alternative ways to show a distribution such as dot plot, box plot and frequency polygon
Investigate central tendencies such as mean, median, mode 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

Sample Tasks for Understanding the Standards
Task 1: Describing a Set of Data Using Central Tendency and Dispersion
Representative Value of a Set of Data
Two classes of students, Class A and Class B, took a Mathematics test.
i.

Table 1 shows the test scores of 23 students from Class A.
•

Draw a dot plot to represent the test scores of this class of students on Diagram 1.

•

At a glance, what important information about the distribution of the test scores could
you get from the dot plot?

Diagram 1

116 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Table 1
Test Scores of Class A•

Student No.

Score

A1

69

A2

98

A3

64

A4

78

A5

98

A6

73

A7

62

A8

69

A9

98

A10

68

A11

68

A12

74

A13

72

A14

65

A15

70

A16

69

A17

72

A18

65

A19

74

A20

70

A21

98

A22

66

A23

72

•

Find the following measures of central tendency of
the test scores.


Mean



Mode



MedianFind the following measures of dispersion for thetest scores.


Range



Inter-quartile range•

In reporting the performance of Class A, theteacher decided to pick 98 to be the representativescore of the class since it is the highest score.
Is the score 98 appropriate to be chosen as therepresentative score of the class? Explain yourreasons.

•

In your opinion, what value of score is mostappropriate to represent the scores of this class ofstudents? Justify your choice.

Chapter 5: Statistics and Probability | 117

Table 2
Test Scores of Class B

ii.

Student No.

Score

B1

60

B2

61

B3

61

B4

64

B5

64

B6

65

B7

65

B8

66

B9

67

B10

67

B11

68

B12

68

B13

68

B14

68

B15

68

B16

69

B17

69

B18

69

B19

70

B20

70

B21

71

B22

71

B23

72

B24

73

B25

73

B26

74

B27

76

B28

78

Table 2 shows the test scores of another 28
students from Class B in ascending order.
•iii.

Find the following measures of central
tendency and dispersion for the test scores.


Mean, mode and median

Range and inter-quartile rangeIn your opinion, which of these measuresof central tendency is most appropriate
to represent the scores of Class B?
Justify your choice.

What do these measures of dispersion
tell you about the distribution of Class
B’s test scores?

Compare the mean scores of Class A and Class
B.
•

Based on this comparison, which classperformed better in the test? Justify yourdecision.

•

How confident are you with the decision?
Explain your reasons.

•

What can you conclude on the use ofmeasures of central tendency and dispersion
to represent a set of data? Explain briefly.

118 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Comparing Two Sets of Data Using Box Plot

Diagram 2i.

Diagram 2 shows the box plot for the test scores of Class A from Table 1.
•

Draw a box plot for the test scores of Class B from Table 2 on Diagram 2.

•Compare the two box plots.

•What can you say about the two classes’ test scores from the aspect of central
tendency and dispersion?



Based on this comparison, which class performed better in the test? Justify yourdecision.

Is this decision same as the decision based on comparison of mean scores in theprevious sub-task? Explain briefly.

Which set of data has a greater variability? How does the variability affect the choiceof representative value for the set of data?

Chapter 5: Statistics and Probability | 119

Task 2: Using Histogram to Investigate Trends in Data

Table 3
Hand Circumference of Lower Secondary
Male (M) and Female (F) Students
Student Hand (cm) / Student Hand (cm) /No.
GenderNo.
Gender1
17.2 / F39
21.0 / M2
17.5 / F40
21.1 / M3
18.1 / F
41
21.1 / M4
18.2 / F
42
21.3 / F5
18.3 / F
43
21.3 / M6
18.5 / F44
21.4 / F7
18.5 / F45
21.4 / M819.0 / F46
21.7 / F
919.1 / F47
21.8 / F
10
19.2 / F48
22.0 / M
11
19.2 / M49
22.3 / F
12
19.3 / F50
22.3 / M
13
19.3 / F
51
22.3 / M
14
19.3 / F
52
22.3 / M
15
19.4 / F
53
22.4 / M
16
19.4 / F
54
22.5 / M
17
19.5 / F
55
22.5 / M
18
19.5 / F56
22.6 / F
19
19.5 / M57
22.6 / F20
19.6 / F58
22.7 / M
21
19.6 / F59
22.7 / M
22
19.6 / F60
22.8 / M
23
19.7 / F
61
22.8 / M24
19.8 / F62
22.9 / M25
19.8 / M
63
22.9 / M
26
20.0 / F
64
23.1 / M
27
20.0 / F65
23.1 / M28
20.2 / M66
23.5 / M29
20.3 / F67
23.6 / F30
20.3 / F68
23.6 / M69
20.3 / M
23.6 / M
31
70
20.5 / F23.7 / M
32
71
20.5 / M
23.8 / F
33
34
20.5 / M72
23.8 / M35
20.6 / M73
23.8 / M74
20.8 / F24.5 / M3675
20.9 / F24.9 / M
37
21.0
/
M
38
-

Diagram 3
The circumference of a hand as
shown in Diagram 3 is one of the basic
measurements to determine glove size. A
manufacturer investigated some data on
hand circumference in order to produce
hand gloves with appropriate sizes forschool students.
i.

Table 3 shows the raw data of handcircumference measured in cm, in
ascending order, obtained from 75
lower secondary male (M) and female(F) students.
•

Find the following measures ofcentral tendency and dispersionof hand circumference for thisgroup of students.


Mean, median and mode

Range and inter-quartilerange

120 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Table 4Frequency Distribution of Hand Circumferenceof Lower Secondary Students
Hand
Circumference,x cm
17 ≤ x < 18
18 ≤ x < 1919 ≤ x < 20

Frequency
1 cm
interval2
518

20 ≤ x < 2112

23 ≤ x < 2410

21 ≤ x < 2210

24 ≤ x < 252

22 ≤ x < 23

16

Diagram 4

2 cm
intervalii.

The raw data from Table 3 are grouped
into classes with 1 cm and 2 cm ofinterval respectively. The frequency
distributions for the students’ handcircumference with the two different
class intervals are shown in Table 4.
•

Which class will 20.6 cm belong
to?

•Which class will 22.0 cm belong
to?

•

Based on Table 4, find the(a) mean, (b) median, (c) mode,(d) range and (d) inter-quartile
range of the students’ handcircumference based on the data
for the two different class intervals,
respectively.

•Comparethese
valuesofmeasures of central tendency anddispersion from those obtained
from Table 3.
 What do you observe about
each of the values? Explain
briefly.
 If the values are different,
which values do you think
are more appropriate torepresent the set of data?

7
30
26
12

Diagram 5

Chapter 5: Statistics and Probability | 121iii.

The two frequency distributions of students’ hand circumference are displayed using
histogram with class interval 2 cm and 1 cm, respectively as shown in Diagram 4 andDiagram 5.
•

Compare the two histograms.


What is the same and what is different about the information on the students’ handcircumference that you can read from the two histograms?

•

What characteristic of the data shown in Diagram 5 indicates that the glove sizes of themale and female students may need to be different in term of measurement?

•

What can you say about the effect of changing the size of class interval on thecharacteristics of the data shown by the histograms?

Table 5Frequency Distribution of Hand Circumference
for Male and Female Students
Hand Circumference,x cm
17 ≤ x < 18
18 ≤ x < 19

FrequencyMale-

19 ≤ x < 203

22 ≤ x < 2313

20 ≤ x < 2121 ≤ x < 22
23 ≤ x < 24
24 ≤ x < 25

5Femaleiv.

Based on the information from
Diagram5,
themanufacturer
decides to have two different setsof measurement for the male andfemale students glove sizes. Hence,the data for the male and female
students were separated as shown in
Table 5.
•Completethefrequency
distribution for the female
students in Table 5.

•

Based on Table 5, find the(a) mean, (b) median, (c) mode,(d) range and (e) inter-quartile
range of the male and female
students’ hand circumference,
respectively.

6
82

122 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Diagram 6v.

Diagram 7

Diagram 6 shows the histogram of the male students’ hand circumference.
•

Draw the histogram of the females’ hand circumference on Diagram 7.

•

Compare the two histograms.

•

Based on the comparison, what can you say about the distributions of the male andfemale students’ hand circumference?

Diagram 8

Chapter 5: Statistics and Probability | 123

vi.

Diagram 8 shows the frequency polygon of the male students’ hand circumference.
•

Draw the frequency polygon for the girls’ hand circumference on Diagram 8.

•

Compare the two frequency polygons. What can you read from them?

•

What is the advantage of frequency polygon in analysing trend as compare tohistogram?

vii. The manufacturer decided to make three different sizes, Small, Medium, and Large, for thehand gloves. However, two different sets of measurement for each size will be determined
for the male and female students.
•

What measurement of each size would you recommend for the male and female
students? Justify your decision.

•

What proportions of the different sizes would you recommend the manufacturer toproduce? Justify your decision.
viii. Diagram 9 shows the measurement of footlength used to determine shoe size. In aproject to determine appropriate shoe sizes
for school students, the foot lengths of 170
male and female students were measured.
Table 6 shows the frequency distribution of
the foot lengths.

Diagram 9
Table 6Frequency Distribution of Foot Length of
School Students
Foot Length, x cm

Frequency

20 ≤ x < 215

21 ≤ x < 2217

24 ≤ x < 2514

27 ≤ x < 2818

30 ≤ x < 315

22 ≤ x < 2325

25 ≤ x < 2622

28 ≤ x < 2914

31 ≤ x < 322

23 ≤ x < 2415

26 ≤ x < 2729

29 ≤ x < 30

4•Draw the histogram for the foot lengths
on Diagram 10.

•

Analyse the histogram.
 What characteristic of the data
indicates that there may be aneed to analyse further on thefrequency distribution?

•

What do you think is the appropriate
next step of analysing the data?
Explain your reasons.

124 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Frequency

Grade 9 Students' Foot Length, in cm

Diagram 10

Task 3: Relationships Among Mean, Median and Mode

Diagram 11i.

Diagram 11 shows three histograms used to represent one symmetric distribution and two
skewed distributions of three sets of data.
••

Match the histogram with the following relationships.
 Mode < Median < Mean Mode = Median = Mean Mean < Median < Mode
Justify your choice for each relationship.

Chapter 5: Statistics and Probability | 125

Diagram 12ii.

Diagram 12 shows the frequency polygons of the distributions of three set of data.
•
Match the frequency polygons with the following distributions.
 Symmetric Right skewed
 Left skewed•

In what ways will skewed distribution affect the measures of central tendency anddispersion for a set of data?

•

What will you suggest to reduce the effects?

Task 4: Variability Arises From Data Distribution and Informal Statistical Inferencesi.

Variability is an essential element of statistical thinking due to the fact that it arises
everywhere in statistics. It represents the conception of variation, which may not be
defined mathematically at a certain stage of learning, before it is defined formally as a
specific measurement. For this standard, it is considered as an observable characteristicof an entity which describes how much variation is present in data and how spread out thedata are. It is disccussed at every stage of learning statistics, depending on what is alreadylearnt and what is not known yet by the learners at each stage. In addition, through theuse of informal statistical inference, which is trying to make inference without using formal
knowledge and procedures of statistics, any related discussion becomes open ended
until variation is defined as a formal concept mathematically at a later stage. As such,informal statistical inference plays an important role in school curriculum toward developing
statistical reasoning among students. Furthermore, the informal nature of discussion canalso bring fun and interest to the learning of statistics in school curriculum. Thus, in this eraof artificial intellegence and data science, discussion on variability and informal statisticalinference is inevitable in school statistical education curriculum.
As such, one implicit aim of this topic is to help students develop understanding of variability
arises from variation in data distribution through the formation of informal statisticalinferences.
•

In your opinion, to what extend are the tasks in this topic able to promote a better
understanding of variability arises from variation in data distribution throughinformal statistical inferences among lower secondary school students.

126 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Topic 2: Exploring Probability With Law of Large Numbers and Sample SpaceStandard 2.1:
Exploring probability with descriptive statistics, law of large numbers and sample spacei.
Experiment with tossing coins and dice to explore the distribution of the relative frequency andunderstand the law of large numbers
ii. Use the idea of equally likely outcomes to infer the value of a probability
iii. Analyse sample space of situations represented by a table to determine the probability and use
it for predicting occurrence.
iv. Use various representations such as table, tree diagram, histogram and frequency polygon forfinding probability
v. Analyse data related to issues on sustainable development and use probability to infer andpredict future events
Sample Tasks for Understanding the Standards
Task 1: Probability From Empirical Trials
Empirical Probability

Heads

Tails
Diagram 1

Diagram 1 shows the two sides of a new 10-piso coin of the Philippines.
i.

In an experiment of tossing the coin, the frequencies of getting heads for different numberof total trials are recorded as shown in Table 1. The difference in frequencies between two
consecutive total number of trials, fn – fn–1 , and its ratio to the difference in total number oftrials, Nn – Nn–1, are calculated.
Table 1
Ratio of Difference in Frequency to Difference in Total TossesnTotal Number ofTosses, N

Frequency ofTossing Heads, f

1

50

23

2

100

45

3

200

4

fn – fn–1

fn – fn–1

Nn – Nn–1

104

59

100

0.590

400

201

97

200

0.485

5

600

307

106

200

0.530

6

800

412

105

200

0.525

7

1000

522

110

200

0.550

8

1200

628

106

200

0.530

9

1400

718

10

1600

810

11

1800

896

12

2000

1002

-

22

-

50

Nn – Nn–1-

0.440

Chapter 5: Statistics and Probability | 127••i.

Complete the table.
Do you notice any convergence in the value of the ratio , as the total trials increases?

The relative frequencies of tossing heads for different number of total trials are also
calculated as shown in Table 2.
Table 2
Relative Frequency of Tossing Heads•
ii.

Total Number of
Trials, N

Frequency ofTossing Heads, f

Relative Frequency ofTossing Heads,

50

23

0.460

100

45

0.450

200

104

0.520

400

201

0.503

600

307

0.512

800

412

0.515

1000

522

0.522

1200

628

0.523

1400

718

1600

810

1800

896

2000

1002

Complete the table.

Diagram 2 shows a part of the line graph plotted for the results in Table 2.

Diagram 2

128 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level•

Complete the line graph.

•

What happen to the relative frequency as the number of trials increases?

•

Compare what happen to the relative frequency and the ratio Nn – Nn–1 , as the total
trials increases.

•iv.

fn – fn–1

Explain what happen to the relative frequency as the number of trials increases with
the Law of Large Numbers.

Table 3 shows the results of three students each tossed a fair coin 100 times.
Table 3Results of Tossing a Fair Coin 100 Times•Student

Heads

Tails

A

61

39

B

42

58

C

50

50

When asked to comment on these results, Chee Seng says:“Student C gets the most accurate result because in 100 trials, we expect to get 50 heads and50 tails. Student A’s result is very not accurate and since he gets much more heads than tails, if
he continue to toss the coin for another 10 times, he should get more tails than heads in orderto balance out his result. Similarly, Students B should also get more heads in his next 10 trials.
What is wrong with Chee Seng’s comment? Why is it wrong?
v.

Diagram 3 shows a die and its six faces.
•

Carry out an experiment of throwing a die.

•Record your result of gettingin Table 4.

Table 4
Result of Throwing a Die
Total Numberof Trials

Frequency ofThrowing

Relative Frequencyof Throwing

50
100
200
300
400

Diagram 3

500•

What is the probability of throwing?

Chapter 5: Statistics and Probability | 129

Empirical Probability in Real Worldi.

ii.

Clinical trials have found a drug to have a success rate of 92% in curing patients infectedwith COVID-19. In the coming year, 14 290 COVID-19 patients will be treated with the
drug.
•

How many patients would be expected to be cured?

•

How certain are you with the expected number of cured patients? Explain briefly.

•

Based on your answer, make a more reasonable estimate of the number of cured
patients. Explain your reasoning.

Table 5 shows the number of live births in Malaysia from years 2013 to 2018.
Table 5Birth Statistics of Malaysia, 2013 to 2018
Year

Total

Male

Female2018

501 945259 582

242 3632017

508 685

262 575

246 1102016

508 203262 755

245 4482015

521 136

269 255

251 8812014

511 865

264 396247 4692013

503 914260 725

243 189

Source: Department of Statistics Malaysia Official Portal.
Available online at: https://www.dosm.gov.my/•

What is the relative frequency of giving birth to (a) a baby boy, and (b) a baby girl in
Malaysia from years 2013 to 2018?

•

In this period of time, if a mother in Malaysia gives birth to 2 babies, what is theprobability that the babies will be a boy and a girl?

•If another mother gives birth to 3 babies, what is the probability that the babies will be
all girls?

130 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Diagram 4iii.

The histogram in Diagram 4 shows the distribution of body mass of 400 babies at birth.
•

Find the probability that a baby is born with a body mass of less than 3.0 kg.

•

Find the probability that a baby is born with a body mass of 3.0 kg or more, but lessthan 4.0 kg.

Diagram 5
iv.

The frequency polygon in Diagram 5 shows the distribution of relative frequencies of the
400 newly born babies’ body mass from the histogram in Diagram 4.
•

Find the probability that a baby born with a body mass of less than 3.0 kg.

•

Find the probability that a baby is born with a body mass of 4.0 kg or more, but lessthan 4.4 kg.

Chapter 5: Statistics and Probability | 131

Task 2: Probability From Theoretical Perspective
Tossing Two Fair Coins
Table 6
Sample Space of Tossing Two Coin, A and B

Table 6 shows the sample space of tossing two coins, A and B. Assuming each side of the coin
is equally likely to occur.
i.

Explain the meaning of sample space.

ii.

What does it mean by “equally likely to occur”?

iii.

What is the probability of each of the following events?
•
Two heads•
One heads and one tails, in any order•Not two tails

iv.

The two coins are tossed 1000 times.
•
How many times do you expect to get two heads?
•
How certain are you with your answer? Explain briefly.

v.

A fair coins is tossed 4 times and the sequences of the outcomes are recorded. Compare
the probability of obtaining the following sequences: HHHH and HTTH.
•

In a discussion about the question, a student explain that:“The probability of getting HTTH is higher than the probability of getting HHHH because inreal life, it is very difficult for the same thing to happen 4 times continuously.”What is wrong with the student’s reasoning? Explain briefly.

132 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Tossing Two Fair Dice

Diagram 6
Diagram 6 shows a fair die and its six faces.
Table 7
Sample Space of Tossing Two Dice#2
#1
(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 6)(3, 1)(3, 2)(3, 3)(3, 4)(3, 5)(3, 6)(2, 1)(4, 1)(5, 1)(6, 1)(2, 2)(4, 2)(5, 2)(6, 2)(2, 3)(4, 3)(5, 3)(6, 3)(2, 4)(4, 4)(5, 4)(6, 4)(2, 5)(4, 5)(5, 5)(6, 5)(2, 6)(4, 6)(5, 6)(6, 6)

Table 7 shows the sample space of tossing two fair dice together.
i.

What does it mean by the dice are “fair”?

ii.

Find the probability of the following.
•
Both the tossed numbers are the same•
The sum of the tossed numbers is 10•
The sum of the tossed numbers is less than 6iii.

The sum of the two tossed numbers could be 2 to 12, inclusively.
•
Which sum has the greatest probability?

iii.

Three fair dice are rolled. What is the probability of each of the following events?
•
Three event numbers•
Two even and one odd numbers•
No even number at allChapter 5: Statistics and Probability | 133

Tossing a Fair Die and a Fair Coin

Diagram 7
Diagram 7 shows a fair die and a fair coin. The die and coin are tossed together.
i.

Show the sample space using•
a table•a tree diagramii.

Find the probability of getting•
number ‘6’ and ‘heads’•
an even number and ‘tails’

Picking Marbles Randomly

Diagram 8i.

Diagram 8 shows a bag with 10 black marbles. You pick one marble at random from thebag.
•

What does it mean by picking “at random”?

•

What is the probability of you getting a black marble?

•

The marble picked is then put back to the bag and you repeat picking a marble following
the same procedure for a total of 10 times.
 How many black marbles do you expect to get?
 How certain are you with your answer? Explain your reasons.

134 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Diagram 9
ii.

iii.

Diagram 9 shows a bag with 7 black marbles and 3 white marbles. You pick one marble at
random from the bag.
•

What is the probability of you getting a black marble?

•

The marble picked is then put back to the bag and you repeat picking a marble following
the same procedure for a total of 10 times.
 How many black marbles do you expect to get?
 How certain are you that you will get exactly your expected number of black
marbles? Explain your reasons.

You pick two marbles together from the bag in Diagram 8. This situation is the same as
picking one marble first and then pick a second marble without putting the first marble
back to the box.
•

If the first marble picked is white, what is the probability of getting a black marble inthe second pick?

•If the first marble picked is black, what is the probability of getting a white marble inthe second pick?

•

What is the probability that you will pick one white and one black marbles in any order?

•

What is the probability that both marbles will be white?

•

What is the probability that both marbles will be black?

Diagram 10
iv.

Diagram 10 shows two boxes, A and B, with black and white balls. Agus is to pick a ballat random from one of the boxes. If he pick a black ball, he will win a prize. So, he has todecide which box to pick the ball from. He thinks to himself:•“There are more black balls in box A. So, I will be more likely to pick a black ball from A thanfrom B.”
What is wrong with Agus’s reasoning?

Chapter 5: Statistics and Probability | 135

Spinning Two Spinners

Diagram 11
Diagram 11 shows two spinners. Each spinner board is divided into three regions, A, B and C.
Assuming each spinner moves freely when it is spun.
i.

Spinner #1 is spun and stopped.
•ii.

What is the probability that it will fall in region A, B and C, respectively?

Spinner #2 is spun and stopped.
•

A student claims: “Since there are 3 possible outcomes, so the probability for the spinner to1fall in region A is 3 .”
iii.

What is wrong with the student’s reasoning? Explain briefly.

Both spinners are spun one after another.
•

Draw a tree diagram to show the sample space.

•

What is the probability of each of the following events?
One spinner falls in region A and one spinner falls in region C.

Both spinners fall in region B.

Both spinners fall in region C.

None of the spinners fall in region A.

136 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 3: Not Equally Likely Outcomes
Tossing a Biased Coin

Heads

Tails
Diagram 12

Diagram 12 shows an ancient coin used in India around the end of the 5th century. The coinis biased and each side of the coin is not equally likely to occur when it is tossed. Given the3
probability of tossing heads is .
5i.

The coin is tossed once.
•
What is the probability of tossing tails?

Tossing acoin
2 times

Diagram 13ii.

The tree diagram in Diagram 13 shows the sample space of tossing the coin twice insuccession. What is the probability of tossing each of the following events?
•
Two heads•
One heads and one tails, in any order•Not two tailsiv.

The coin is tossed three times in succession. What is the probability of tossing each of thefollowing events?
•
Three tails•
One head and two tails in any orderChapter 5: Statistics and Probability | 137

Task 4: Misconception About Probabilityi.

On the eve of a test, Alejandro tells his parent that he has a 50% chance of passing thetest. He goes on to explain that:“There are only two possible outcomes. Either I pass the test, or I fail. So, the probability of me
passing the test is 1 , which is 50%!”
2

•
ii.

What is wrong with Alejandro’s reasoning?

Ahmed is playing a board game. He needs to toss a die and get “6” to start moving his
counter. However, after 8 times of tossing, he is still unable to get a “6” to start the game.
Based on this experience, Ahmed concludes that:“It is very hard to get “6”. I think the probability of getting a “6” is lower than other numbers.”•iii.

What is wrong with Ahmed’s reasoning?

Aroon tosses a fair coin 8 times and gets 8 heads in a row. Before the next toss, he thinks
to himself:“Since I have got 8 heads in a row, I will have a better chance to get a tails than a heads this time.”•

iv.

What is wrong with Aroon’s reasoning?

Analu takes a quiz with 10 ‘true or false’ questions. After the quiz, he tells his friends:
1“I just guess the answers for all the questions. Since the probability that my guess is right is for2each question, I am certain that I will get 5 right answers.”•v.

What is wrong with Analu’s reasoning?

Liverpool FC is to play against Manchester United in a charity football match. Adriel says
to his friends:“Liverpool FC can either win, lose or draw, so the probability that Liverpool will win is 1 .”3•

What is wrong with Adriel’s reasoning?

138 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 4: Application of Probability in Real World
Draw Lots
Three brothers want to go for a movie but they only have enough money to buy one ticket. After
discussion, they agree to draw lots in order to decide who will be the lucky person to see themovie. So, they prepare three pieces of paper as shown in Diagram 14.

O

X

X

Diagram 14
The papers are folded and put into a box. The three brothers then take turn to draw a pieceof paper from the box without looking into the box. The person who draw the paper with the O
mark will win the ticket and get to see the movie.
i.

The three brothers think to themselves.
First Brother:“The first person to pick will have two pieces of paper with the “O” and the chances topick “O” is higher. So, I better let my two brothers pick first.”
Second Brother:“The first person to pick will have the chance to pick “O” first. So, I better pick first before
the “O” is picked by my two brothers.”
Third Brother:“Whether I pick first or last, my chance of winning the ticket will still be the same. So,
there is no need to rush. I better let my two brothers to draw the lots first.”•

Which brother reason correctly? Support your answer with some calculation ofprobability.

•

What if they have enough money to buy two tickets?

Chapter 5: Statistics and Probability | 139

A Game of Choices

Diagram 15
You are playing a game with your friend to win a prize. There are three boxes in front of you
as shown in Diagram 15. Your friend will put the prize into one of the boxes without you seeing
it. Then, you will make an initial choice by choosing one box that you think has the prize in it,
without opening it. At that point, your friend will open a box with no prize from among the twoboxes that you did not choose. You will then have to decide whether to stick with your original
choice, or switch your choice to the remaining unopened box.

Box 1
Diagram 16
The tree diagram as shown in Diagram 16 illustrates the case of you choosing Box 2 initially.
i.

ii.

Let’s say you choose Box 2 initially.
•

If the prize is in Box 1, then your friend is left with only one choice. Explain why.

•

If the prize is in Box 2, then your friend is left with two choices. Explain why.

•

If the prize is in Box 3, then your friend is also left with only one choice. Explain why.

Should you switch to change your choice? Explain your reasons.

140 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

False Positives and False Negatives in Medical Test
Statistical data shows that 2% of senior citizen from a region are infected with COVID-19.
i.

ii.

A particular COVId-19 test gives a correct positive result with a probability of 0.95 when thevirus is present, but gives an incorrect positive result with a probability of 0.15 when thevirus is not present (false positive).
•

What is the probability of the test to give an incorrect negative result when the virus ispresent (false negative)?

•

What is the probability of the test to give a correct negative result when the virus is
not present?

A senior citizen from that region, Uncle Sam is tested positive by the test.
•iii.

Another senior citizen from the same region, Auntie Pamela is tested negative to the test.
•iv.

What is the probability that Uncle Sam is really infected by the virus?

What is the probability that Auntie Pamela is really not infected by the virus?

If 1000 senior citizen from the region are tested positive by the test, how many of them doyou expect to really have not infected by the virus?

Note. You may refer to readings on Bayes’ Theorem in order to solve this problem. Although
this theorem may not be found in your country lower secondary school mathematics curriculum,
knowledge on it is a worthwhile enhancement for teachers in order to understand the decision
making process in artificial intelligence and data science.

Chapter 5: Statistics and Probability | 141

Predicting the Weather
You plan to go outing with your family members in one non-rainy day next week. The TV station
has just broadcast the weather forecast for next week as shown in Table 6. In addition, the odds
in favour of raining on Monday is also shown.
Table 6
Whether Forecast for Next Week
Day

Probability of Raining

Odds in favour of Raining

Monday30%

3:7

Tuesday60%

Wednesday

75%

Thursday60%

Friday85%

Saturday90%

Sunday50%i.

Find the odds in favour of raining on each of the remaining day for next week and complete
Table 6.

ii.

Based on the data in Table 6, at a glance, should you cancel your plan? Justify yourdecision.

iii.

Find the odds in favour of raining for 7 successive days in next week.
•
Should you revise your decision in (ii)? Explain your reasons.

Task 4: Variability and Uncertainty in Samplingi.

For this standard, variability is considered from the perspectives of uncertainty due
to sampling in determining measures of likelihood. A good understanding of how thisuncertainty could affect the measures of likelihood is crucial for a better understanding ofprobability.
As such, one implicit aim of this topic is to help students develop understanding of variability
arises from uncertainty in sampling through the formation of informal statistical inferences.
•

In your opinion, to what extend are the tasks in this topic able to promote a better
understanding of variability arises from uncertainty in sampling through informal
statistical inferences among lower secondary school students.

142 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Topic 3: Exploring Statistics with SamplingStandard 3.1:
Exploring sampling with the understanding of randomnessi.
Discuss the hidden hypothesis behind sample and population
ii. Use randomness to explain samplingiii. Analyse the data exploratory such as dividing the original into two for knowing better data
representations and discuss appropriateness such as regrouping
iv. Appreciate data sampling in a situation with sustainable development
Sample Tasks for Understanding the Standards
Task 1: Population and Sampling Variability
Census Survey and Sample Survey
Diagram 1 shows the meaning of census survey
and sample survey.
i.

Compare the strengths and weaknesses ofcensus survey and sample survey.

ii.

Would a census or sample survey bemore appropriate for each of the following
investigations? Explain your reasons.

iii.

Diagram 1•

The length of time an electric light bulb
produced by a factory will last•

The causes of car accidents in a district•

Public opinions on a newly installedfacility in a townshipThe purpose of a sample is to provide anestimate of a particular characteristic of thetotal population.
•

What is a good sample?

•

How to choose a good sample?

Chapter 5: Statistics and Probability | 143

Estimating Population Characteristic From Sample Characteristic
A school intends to organise a mathematics seminar for all its 200 lower secondary students.
The school conducted a census survey where all students from the population were asked
whether or not they are in favour of the seminar and their replies were shown in Diagram 2,
where Y indicates a “yes’ and N indicates a “no”.
1st student → Y Y Y N Y N Y N N N
N Y N Y Y Y N Y Y Y
41st student → N N Y N N N N Y Y N
N N N Y N Y Y Y N Y

81 student → N Y Y Y Y Y Y Y Y YstN Y Y Y Y N Y N Y Y

121st student → N Y N Y Y Y N Y Y YY N N Y N Y N Y Y Y

161st student → N N N Y Y Y Y Y Y YN Y Y Y Y N N Y Y N

Y N N Y Y N Y Y Y N
Y N Y N Y Y Y Y Y Y ← 40th student
Y N Y N Y N N N Y N
Y Y Y Y Y N Y Y Y Y ← 80th student
N N N Y N Y Y N Y Y

Y N N N Y N Y Y Y Y ← 120th student
Y Y Y Y N Y Y N Y Y
N Y Y Y Y Y Y Y Y Y ← 160th student
N Y N Y N N Y Y N N
Y N Y Y N N N N Y Y ← 200th student

Diagram 2i.

Find the proportion of students from the population who are in favour of the seminar.

ii.

Find the proportion of the (a) first 10, and (b) last 10 students who are in favour of theseminar.
•
Are these two good samples? Explain your reasons.

iii.

Generate a set of 20 random numbers between 1 and 200 inclusively. Use this set of
random numbers to select a random sample of 20 students from the population.
•
Find the proportion of students of this sample who are in favour of the seminar.
•
Generate another two sets of 20 random numbers to select two other random samplesof 20 students.
•
Find the proportion of students who are in favour of the seminar in the two cases
above.
•
Compare the results of the three samples. Comment briefly on the results of yourcomparison.
[Note. 1. Random numbers can be generated easily by the lucky-draw method. Write thenumbers 1 – 200 separately on 200 pieces of cards. Place all the cards face
down, mix them well, and then draw 20 cards to generate a set of 20 random
numbers between 1 – 200 inclusively.
2. Random numbers can also be generated from a random number table, acalculator or an electronic spreadsheet.

144 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Task 2: The Effect of Sample Size
Diagram 3 shows the heights, in cm, of a population of 120 lower secondary students from aschool.

1st student → 152

146

162

185

169

158

149

183

161

173

173

180

184

167

166

183

181

156

151

180 ← 20th student

21st student → 170

176

175

184

173

167

177

156

162

167

178

175

164

182

147

161

177

147

152

154 ← 40th student

41st student → 180

153

161

152

166

152

180

183

168

184

166

165

158

161

156

176

179

179

159

170 ← 60th student

61st student → 183

180

166

172

181

169

175

158

183

155

157

173

184

174

181

157

163

164

163

157 ← 80th student

81st student → 158

148

155

145

178

161

158

168

164

156

166

185

156

157

160

171

175

167

160

170 ← 100th student

101st student → 180

175

155

153

147

174

173

156

168

184

147

181

179

183

154

160

165

175

167

170 ← 120th student

Diagram 3i.

Find the mean height of the population.

ii.

Randomly select a sample of 10 students.
•
Find the mean height of the sample.
•
How close is the sample mean to the population mean? Explain your answer.

Chapter 5: Statistics and Probability | 145

iii.

Table 1 shows the data obtained from 16 samples each with 10 students’ heights randomly
selected from the population. The mean heights for sample #1 to sample #10 have been
calculated.
Table 1
Sample Mean of Students’ Height in cm With Sample Size n = 10
SampleStudent Height

Sample Mean

#1

148, 161, 173, 152, 174, 175, 155, 170, 166, 175

164.9

#2

173, 180, 153, 168, 170, 163, 168, 184, 183, 170

171.2

#3

158, 177, 161, 152, 161, 179, 180, 153, 175, 156

165.2

#4

184, 165, 154, 179, 160, 178, 158, 157, 147, 147

162.9

#5

173, 157, 184, 181, 158, 183, 185, 175, 157, 156

170.9

#6

168, 151, 156, 173, 178, 157, 180, 154, 158, 152

162.7

#7

183, 180, 173, 157, 175, 178, 158, 146, 183, 161

169.4

#8

173, 175, 154, 184, 160, 169, 164, 167, 180, 155

168.1

#9

156, 177, 157, 179, 146, 180, 157, 184, 183, 151

165.0

#10

170, 166, 174, 152, 163, 161, 155, 179, 162, 147

162.9

#11

164, 156, 165, 164, 160, 158, 174, 167, 171, 146#12

181, 182, 145, 185, 160, 184, 185, 152, 147, 166#13155, 149, 176, 183, 168, 158, 147, 164, 175, 152#14158, 180, 152, 148, 158, 160, 174, 155, 173, 167#15

175, 166, 156, 156, 157, 166, 161, 163, 180, 171#16

175, 147, 173, 166, 183, 173, 147, 181, 156, 164

#17
#18
•

Calculate the mean heights for sample #11 to sample #16 and fill in the appropriate
columns in Table 1.

•

Select two more random samples (#17 and #18) each with 10 students and fill in thedata in Table 1.

•

Calculate the mean heights for these two random samples and complete Table 1.

146 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

iv.

Table 2 shows the mean heights obtained from 16 samples each with 20 students’ heights
randomly selected from the population.
Table 2
Sample Mean of Students’ Height in cm With Sample Size n = 20••v.

SampleMean Height(cm)

SampleMean Height(cm)

SampleMean Height
(cm)

#1

163.7

#7

165.2

#13

168.9

#2

165.8

#8

168.4

#14

168.6

#3

167.7

#9

163.9

#15

167.2

#4

163.7

#10

164.7

#16

169.2

#5

170.4

#11

167.8

#17

#6

162.6

#12

163.8

#18

Select two more random samples (#17 and #18) each with 20 students’ heights fromthe population.
Find the sample means for samples #17 and #18 to complete Table 2.

Complete the frequency distribution table for the sample means with sample size n = 10
from Table 1 and sample size n = 20 from Table 2.
Table 3Frequency Distribution Table for Sample Means With Sample Sizes n = 10 and n = 20
Sample Means, x cm

Frequency
Sample Size n = 10

Sample Size n = 20

145 ≤ x < 150
150 ≤ x < 155
155 ≤ x < 160
160 ≤ x < 165
165 ≤ x < 170
170 ≤ x < 175
175 ≤ x < 180
•
•
•
vi.

180 ≤ x < 185

Draw the histograms of sample size n = 10 and n = 20 respectively,Compare the two histograms. What do you observe about the distribution of sample
means for the two different sample sizes?
What effect does the sample size have on estimating the mean of the population?

Generally, how can we increase the reliability of an estimate of a population characteristic
based on sampling survey?

Chapter 5: Statistics and Probability | 147

vii. Diagram 4 shows the eggs produced bya farm in one day. Ten random samples
each with 20 eggs were selected andthe mean mass, in g, for each sample ofeggs was calculated. Table 4 shows the
sample means arranged from smallest tolargest.
•Diagram 4

Based on the data, state whether
each of the following statements istrue or false.


Table 4
Mean Mass in g of 10 Samples of Eggs With
Sample Size n = 20
58.6 g60.9 g61.3 g62.8 g63.2 g63.3 g63.4 g63.5 g64.0 g64.2 gThe mean for the population isat least 58.6 g.
The mean for the population is
not more than 64.2 g.
Among the sample means,
there is a value which is exactly
the same as the mean of the
population.
The mean of the population isclose to 63 g.
If we increase the sample size
to 40, the sample mean would
be more reliable as an estimate
for the population mean.

Task 3: Application of Sample Survey
Capture-Recapture Method

Diagram 5i.

Diagram 5 shows a fish pond belongs
to a farmer Ko Aung Myint. He wants toestimate the number of fish in his pond.
At 10 different locations of the pond, hecaught 60 fish and each fish was “tagged”
and then released back to the pond. A
week later, 80 fish was caught randomly
from the pond and it was found that 12 ofthem had tags.
•

What is the population?

•

What is the sample?

•

Estimate how many fish there are in
the pond.

148 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Quality Control of Factory Products

Diagram 6 shows your favorite dark chocolatebar. The weight on the wrapping indicates
100 g. To avoid customer complaints andlawsuits, the manufacturer has to make surethat 98% of all chocolate bars weigh 100 g or
more.
i.

Diagram 6

As a quality control check , 84 chocolatebars were randomly selected from a batchof chocolate bars produced by a factoryand 2 of them were found to weigh lessthan 100 g.
•

Estimate how many percent of the
chocolate bars are up to the standard
specified on the wrapping?

Task 4: Variability and Random Samplingi.

For this standard, variability is considered from the perspective of random nature of sample
survey. As such, one implicit aim of this topic is to help students develop understanding ofvariability arising from randomness of samples through the formation of informal statisticalinferences.
•

In your opinion, to what extend are the tasks in this topic able to promote a better
understanding of variability arising from randomness of sample through informal
statistical inferences among lower secondary school students?

Bibliography | 149

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Billstein, R., Libeskind, S., & Lott, J. (2013). A problem solving approach to mathematics for elementary school
teachers (11th ed.). Boston, MA: Pearson.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: D. Reidel.
Freudenthal, H. (2010). Revisiting Mathematics Education [China Lectures]. Mathematics Education Library(9). Dordrecht: Kluwer.
Fujii, T. & Matano, H. (Eds.). (2012a). Mathematics international grade 7. Tokyo, Japan: Tokyo Shoseki.
Fujii, T. & Matano, H. (Eds.). (2012b). Mathematics international grade 8. Tokyo, Japan: Tokyo Shoseki.
Fujii, T. & Matano, H. (Eds.). (2012c). Mathematics international grade 9. Tokyo, Japan: Tokyo Shoseki.
Isoda, M. & Olfos, R. (2020). Teaching Multiplication with Lesson Study: Japanese and IberoAmerican Theories for International Mathematics Education. Cham, Switzerland: Springer.
DOI: 10.1007/978-3-030-28561-6
Isoda. M. & Murata, A. (2020), Study with your friends: mathematics for elementary school (12 vols.). Tokyo,
Japan: Gakko Tosho.
Isoda, M. & Tall, D. (Eds.). (2019a). Junior high school mathematics: 1. Tokyo, Japan: Gakko Tosho.
Isoda, M. & Tall, D. (Eds.). (2019b). Junior high school mathematics: 2. Tokyo, Japan: Gakko Tosho.
Isoda, M. & Tall, D. (Eds.). (2019c). Junior high school mathematics: 3. Tokyo, Japan: Gakko Tosho.
Isoda, M. & Kagarigi, S. (2012). Mathematical thinking: How to develop it in the classroom. Singapore: 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 (7thed.). Boston, MA: Pearson Education.
Mason, J., Burton, L. & Stacey K. (1985). Thinking mathematically. Workingham, England: 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
Nolan, E. C., Dixon, J. K., Roy, G. J. & Andreasen, J. B. (2016). Making sense of mathematics for teaching.
Grades 6 – 8. Bloomington, IN: Solution Tree Press.
Nolan, E. C., Dixon, J. K., Safi, F. & Haciomeroglu, E. S. (2016). Making sense of mathematics for teaching.
High school. Bloomington, IN: Solution Tree Press.

150 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Polya, G. (1945). How to solve it; a new aspect of mathematical method. Princeton, NJ: Princeton University
Press.
Shield, M., Broderick, S. & Adamson, S. (2001). Mathematics for Queensland. Year 11A. Victoria, Australia:
Oxford University Press.
Shield, M. & Adamson, S. (2002). Mathematics for Queensland. Year 12A. Victoria, Australia: Oxford University
Press.
Websites of National Curriculum in Some SEAMEO Countries
Cambodia Mathematics Curriculum: Grade 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
https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/2020express_na-maths_syllabuses.pdf
Vietnamese Mathematics Curriculum: Years 1 to 12
https://www.futureschool.com/vietnam-curriculum/Appendix A – Framework for CCRLS in Mathematics | 151

APPENDIX A
Framework for CCRLS in Mathematics
Nature of Mathematics
Mathematics has been recognised as a necessary literacy for citizenship and not only living economically but
also to establish a society with fruitful arguments and creations for better living. It has been taught as a basic
language for all academic subjects using visual and logical-symbolic representations. In this informationsociety, mathematics has increased its role to establish 21st century skills through reviewing mathematics as
the science of patterns for future prediction and designing with big data which produces innovation not only
for technology advancement but also for business model. Mathematics is an essential subject to establish
common reasoning for sustainable development of society through viable argument in understanding eachother and develop critical reasoning as the habits of mind. Mathematics should be learned as basis for all
subjects. For clarifying the framework in CCRLS on mathematics and by knowing the role of mathematics
education, the humanistic and philosophical natures of mathematics are confirmed as follows:
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 recognize competitive
attitude of Blaise Pascal to Fermat’s intelligence and seeking the way to be understood on his excellence of his
finding on Pascal’s Triangles. If we read the Pascal’s Pensées, you recognise how Pascal denied Descartes
geometry using algebra from the aspect of ancient Greece geometry. On the other hand, Descartes tried toovercome the difficulties of ancient geometry by algebra. If you read the letter from Descartes to Elisabeth, yourecognise how Descartes appreciated and felt happy the Royal Highness Elisabeth used his ideas of algebrain geometry. Despite being a princess, Elisabeth had been continuously learning mathematics in her life.
There were discussions on who developed calculus between Britain and Continent. On that context, Johann
Bernoulli, a continental mathematician, posed a question on the journal about the Brachistochrone problem,
locus of the point on circumference of the circle when it rotates on the line. No one replied and Bernoulli
extended the deadline of the answer and asked Newton to reply. Newton answered it within a day. Finally, six
contributions of the appropriate answer including Newton and other Continental mathematicians were accepted.
All those stories show that mathematics embraces the humanistic nature of proficiency for competitivenessand understand others for sharing ideas.
Philosophical nature of mathematics can be explained on ontological and epistemological perspectives. On
the ontological perspective, mathematics can be seen as a subject for universal understanding and common
scientific language. Plato and Aristotle are usually compared on this perspective. Plato believes that theexistence of the world of “idea” and mathematics existed in the world of “idea” on Platonism. On this context,
mathematical creation is usually explained by the word “discover” which means taking out the cover from which
it has already existed. At the moment of discovery, reasonable, harmony and beautifulness of mathematicalsystem 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 such as “modelling”,“instruments”, “embodiment”, “metaphor” and “change representation”. From both ontological perspectives,
mathematics can be understood and acquired by anyone and if acquired, it serves as a common scientificlanguage which is used to express in any subjects. Once representing the ideas using the shared common
language, the world can possibly be perceived in the same view autonomously.
On 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. On the context of Hegel, a member of German idealism, Imre Lakatos explained the developmentof mathematics through proof and refutation. In his context, mathematics is not fixed but an expandable systemthat can be restructured through a process of dialectic in constructing viable arguments. Plato also used dialectic
for reaching ideas with examples of mathematics. The origin of dialectic is known as the origin of indirect proof.
In education today, dialectic is a part of critical thinking for creation. Parallel perspectives for mathematicaldevelopments are given by George Polya and Hans Freudenthal. For the discovery of mathematics, Polyaexplained mathematical problem solving processes with mathematical ideas and mathematical ways of thinking
in general. Freudenthal enhanced the activity to reorganise mathematics by the term mathematisation.

152 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Genetic epistemologist Jean Piaget established his theory for operations based on various theories, including
the discussion of Freudenthal and explained mathematical development of operations by the term reflective
abstraction. Reflection is also a necessary activity for mathematisation by Freudenthal. On materialism,
under 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 the highquality communication in mathematics classrooms. Dialectical-critical discussion should be enhanced in themathematics class. From both the epistemological perspectives, mathematics can be developed throughthe processes of communication, problem solving and mathematisation which include reorganization ofmathematics. Those processes are necessary to acquire mathematical values and ways of thinking throughreflection.
Aims of Mathematics in CCRLS
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 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.

Appendix A – Framework for CCRLS in Mathematics | 153

Mathematical Values, Attitude and Habits for Human Character
For cultivating basic human characters, values, attitudes and habits of mind are essentials to be developed
through mathematics. Values are basis for setting objectives and making decisions for future directions.
Attitudes are mindsets for attempting to pursue undertakings. Habits of mind are necessary for soft skills tolive harmoniously in the society. Mathematical values, mathematical attitudes and mathematical habits of mind
are simultaneously developed and inculcated through the learning of the content knowledge.
Essential examples on values, attitudes and habits of mind are given in Figure 3. On mathematical values,
generalisable and expandable ideas are usually recognised as strong ideas. Explaining why a proving isnecessary in mathematics is a way of seeking reasonableness. Harmony and beautifulness are described
not only in relation to mathematical arts, but also in the science of patterns and system of mathematics.
Usefulness and simplicity are used in selection of mathematical ideas and procedures.
On mathematical attitudes, “seeing and thinking mathematically” means attempting to use the mathematics
learned for seeing and thinking about the objects. Posing questions and providing explanation such as the“why” and the “when” are ordinary sequence for thinking mathematically. Changing representation to otherways such as modelling can overcome the running out of ideas in problem solving. The mindset for trying tounderstand others is the basis to explain one’s own ideas that is understandable by the rest with appreciation.
Producing a concept with definition operationally is a manner of mathematics.
On mathematical habits of mind for citizens to live, mathematical attitudes and values are necessary forreasoning critically and reasonably. Appreciating and respecting other ideas is also necessary. Mathematics isdeveloped independently for those who appreciate life creatively, innovatively and harmoniously. Seeking theeasier and effective manner of selecting appropriate tools is necessary. Mathematics is a subject to challenge
and experience competitiveness, appreciation with others, develop the mindset for lifelong learning, personal
development and social mobility.
Mathematical Thinking and Processes
For developing creative human capital, mathematical ideas, mathematical ways of thinking and mathematicalactivities are essential. Mathematical ideas are process skills involving mathematical concepts. Mathematicalthinking is mathematical way of reasoning in general which does not depend on specific concepts. Mathematicalactivities are various types of activities such as problem solving, exploration and inquiry. Mathematicalprocesses which include these components are necessary skills to use mathematics in our life, such as
innovation in this society (e.g. Internet of Things (IOT)). In the context of education, competency referring tomathematical processes is the basis for STEM and STEAM1 education as well as basis for social science andeconomy education.
Mathematical ideas serve as the basis of content knowledge related to promoting and developing mathematicalthinking. Some key ideas of mathematics are used as special process. The fundamental ideas of set and unit
lead to a more hierarchical and simple structural relationship. The ability to compare, operate, and perform
algorithm of related functions enables efficient ways of learning mathematics and solving problem in learners’
life with mathematics.
In the case of set, set is a mathematical ideas is related with conditions and elements. It is related with activity
in grouping and distinguishing with other groups by conditions. Example, 3 red flowers and 4 white flowers
become 7 flowers, if we change the condition of the set by not considering the colours. “A” and non A” is a
simple manner to distinguish sets with logical reasoning. For categorizing, we use intervals such as x > 0, x <0, x = 0. This situation can be seen in the hyperbolic graph where y=1/x.
In the case of unit, it is a mathematical idea that is related with the process to produce and apply the unit with
operations. On some cases, trying to find the common denominator is the way to find the unit of two given
quantity. Tentative unit such as arbitrary units can be set and applied locally whereas standard units are used
globally. In the combination of different quantities it produces new measurement quantity such as distance withrespect to time produce speed. Square unit such as square centimetres is a unit for area.
STEM refers to Science, Technology, Engineering and Mathematics. STEAM refers to Science, Technology, Engineering, Arts
and Mathematics or Applied Mathematics.
1

154 | Mathematics Challenges for Classroom Practices at the Lower Secondary 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 backward in relation to ancient Greek term analysis.
Mathematical activities are ways to represent mathematical process. Problem solving process was analysedby Polya. He influenced problem solving with various strategies. Technology enhances the activities
of conjecturing and visualizing for inquiries. Conceptualization is done based on procedures such as theprocedure 3+3+3+3=12, become the basis for 4x3. The proceduralisation of multiplication is done throughdeveloping the multiplication table, idea of distribution and memorizing.
Content
For cultivating well qualified citizens, content knowledge of mathematics is essential. Content of mathematics
is usually divided by the set of mathematics. However, for developing human characters and creative humancapitals, it should be developed through the mathematical processes. Values, attitudes and habits of mind are
driving force for engagement in mathematical processes. Thus, without involving human character formations
with mathematical process skills, content knowledge of mathematics cannot be realised. The content is divided
into three stages in CCRLS and every stage has four strands. Between the stages, the names of the strands
are directly connected and those on the standard level are well connected too. The names of strands for every
key stage are as follows:
Key Stages

		Strands

Key Stage 1 		Numbers and Operations
			Quantity and Measurement
			Shapes, Figures and Solids
			
Pattern and Data Representations
Key Stage 2 		Extension of Numbers and Operations
			Measurement and Relations
			Plane Figures & Space Figures
			
Data Handling and Graphs
Key Stage 3 		Numbers and Algebra			
Relations and Functions
			Space and Geometry
			Statistics and Probability
In every stage, four content strands are mutually related2. Between the key stages, all strands in different
key stages are mutually related. The same content strand names are not used to indicate development andreorganisation beyond each stage. For example, “Numbers and Operations” in Key stage 1, “Extension of
Numbers and Operation” in Key stage 2, and “Numbers and Algebra” in Key stage 3 are well connected.
These names of the content strands show the extension and integration of contents. For example, even andodd numbers can be taught at any stage with the different definition. At Key stage 1 even numbers can be
introduced as “counting by two” which does not include zero. In Key stage 2, it can be re-defined by a number
divisible by two. Finally, in Key stage 3 it can be re-defined as a multiple of two in integers which includes zero.
Although we use the same name as even number, they are conceptually different. The definition in Key stage
1 is based on counting, Key stage 2 is based on division while Key stage 3 is based on algebraic notation3.
Expressing such theoretical differences requires name of strands for content be distinguished. In the case ofStrands 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 compartmentalisation through categorisation ofcontent.
2

Appendix A – Framework for CCRLS in Mathematics | 155

measurement, there is no strand name of measurement in Key stage 3. Key stage 1 relates with quantity andsetting the units. In Key stage 2, it extends to non-additive quantity beyond dimension. In Key stage 3 the ideaof unit and measurement is embedded in every strand. For example, square root in Numbers and Algebra
strand is an irrational number which means unmeasurable, Pythagorean Theorem in Space and Geometry
strand is used for measuring, proportional function in Relations and Functions strand is used for counting thenumber of nails by weight, and in Statistics and Probability strand, new measurement units are expressed suchas quartile for boxplot.
Context to Link the Three Components
Three components in Figure 3 should be embedded in every key stage as standards for the content ofteaching. “Mathematical values, attitudes, habits for human character” component and “Mathematical thinking
and processes” component cannot exist without “Content” component. The first two components can betaught through teaching with the content. For teaching those three components at the same time, context is
introduced as shown in Figure 4.

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 attempting 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 anddevelop 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

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.
3

156 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

•
•
•
•
•
•
•

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 on the use of information and communication technology in mathematics
Appreciate the universality of mathematics and its multicultural and historical perspectives

Those contexts are chosen for illustrating the interwoven links of the two components with contents. It looks
like methods of teaching, however all the three components are the subjects of teaching on the contexts.
Extracted and edited from:
SEAMEO Basic Education Standards (SEA-BES): Common Core Regional Learning Standards (CCRLS) in
Mathematics and Science. SEAMEO RECSAM. (Mangao, Ahmad, & Isoda, 2017, p. 2-11)

Appendix B – Terminologies for Mathematical Thinking and Processes | 157

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 isreflected through activities that require us to think about membership (elements
and conditions) 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 thathas the structures of equality, order (greater, less than), and operations, which
is developed and extended throughout the curriculum from natural number tocomplex number: the set of complex numbers does not have the structure oforder.

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 gramfor situations involving measurement. Mathematically, unit is used to indicate anumber by mapping it with the quantity in a situation.
In a situation, it can be fixed based on the context of comparison, which caneither be direct or indirect comparison. In this context, a remainder or a difference
from a comparison can be used for fixing a new arbitrary unit for measurement
which is a fraction of the original unit. This process of determining a new unit is
the application of Euclidean algorithm for finding the greatest common divisor.
For the base-10 place value number system, every column is defined by theunits 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158 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

such as ones, 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 “per centos” 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

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.
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 topromote logic for transitivity which includes syllogism.
Comparison of multiple denominated numbers with different unit quantities on
the same 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, suchas 520 cm and 512 cm.  Furthermore, comparison of expressions which has thesame answers on the same operation such as 2 + 4, 3 + 3 and 4 + 2 can be used
to find rules and patterns.  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 comparisonof fractions such ad 1/2 and 1/3, we have to find the unit fraction 1/6 which canmeasure 1/2 and 1/3: 1/6 is the common denominator for 1/2 and 1//3. The
algorithm to find the unit fraction as the common denominator is called ‘reductionof fraction’.
On numbers, the relationship of two numbers can be equal, greater or less. The
numbers set up to real numbers is a total/linear order set, thus two numbers canbe compared on real number set. However, complex number as an extension ofreal number cannot be compared directly because it has two dimension.
On the number line as real number, the size of number (distance) is defined by
the difference. 1 = 2 – 1 = 3 – 2 = 4 – 3 =...: Here the difference is the value ofAppendix B – Terminologies for Mathematical Thinking and Processes | 159

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 theplane such as complex plane, even though the number is not simply ordered,
the size of number (distance) is defined by Pythagorean theorem. By using thisdefinition, I1I = I(√2/2)(1 + i)I = IiI = I(√2/2)(1 – 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 theunit of 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 the linearity. In Physics, ‘db’ is the size of volume which is produced by thecommon logarithm of sound pressure. ‘db’ is fitting well for human impression of
the size of sounds on its linearity. It is known as Weber-Fechner’s law that humansenses are proportional to the logarithm of stimulus. On science, logarithmic
scale is used for Semi-log graph and Log-log graph for demonstrating thelinearity even it is an exponential phenomenon. Logarithm produce the scale toillustrate multiplicative phenomena as an additive phenomena.
Operations

Addition, Subtraction, multiplication and division are four basic arithmetic
operations. These are binary operations involving any two numbers with symbols
of operations, +, –, × and ÷. Polynomial expressions such as 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, anoperation can be represented by the manipulation of concrete objects as well
as expressions. However, the manipulating process is not the same as theoperation 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.

Algorithm

Algorithm is a set of sequential activity at the special situation to produce
a particular solution 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 place value. 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 on theirculture, however it can be created 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.

160 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Fundamental
Principles

Fundamental principles are the rules which is related with mathematicalstructures and 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 are both positive and negative numbers, then the four expressions can beseen as the same.
There are also other fundamental principles for arithmetic operations at theelementary level such as the followings:
1 + 9 = 7↓+1 ↓–1
2 + 8 = 7

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
and other representations. On the extension of numbers and operations,principles are used for the discussion of the permanence of form (see thepermanence of form).
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 partof a shape, 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 possibilityof escribed circles. In addition, parallel lines are necessary to derive the area
formula for 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 16 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 any algebraic symbolic form.
In Education, the form is not a limited expression but includes the patterns andthe permanence of form can be used in various occasions. Especially, it is used
for the 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 | 161

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 are used as hypothesis and it makes possible to apply
it to the 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 necessityof zero (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 natureof representation which can be produced by specified symbols and operations.
Different representations have the different nature, use different symbols andoperations. Every representation has the limitation of interpretation on its nature,
thus, thinking by using only one specified representation provides the limitation ofreasoning and understanding. If one type of representation is translated to anothertype of representation, then the representation can be interpreted in other ways.
If the idea of specified representation with certain embedded nature is translated
into various representations, a rich and comprehensive meaning and use will
be produced. However, for making the translation meaningful, it is necessary toknow 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
in other representations and translate, at first, as well as between expressions
and situations.

Mathematical Ways of Thinking
Mathematical ways of thinking support the student’s thinking process by and for themselves.
Generalisation andSpecialisation

Generalisation is to consider the general under the given conditions of situations.
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 are usually unclear in textbooks, however
if students are asked to consider other cases such as by saying ‘for example’,the conditions become clear. In the upper grades, variable, domain, range,
parameter, discrete, continuous, zero, finite and infinite become the terms of set
to consider the conditions for general.
Specialisation is to consider the thinkable example on situations and necessary
to find the hidden conditions. Considering general with thinkable example iscalled generalisable-special example. In mathematics, general theory is usuallystronger than local theory. It is one of the objectives for generalisation andspecialisation.
In mathematical inquiry, the process is usually on going from special cases toa general case to establish a stronger theory. Thus, task sequence of every
unit in textbooks is usually progressed from special cases to general for
generating simple procedures, exceptionally the tasks for exercise and training

162 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

the procedures which usually progress from general to special because students
already learned the general.
For students engaging in generalisation and specialisation by and for themselves,
students have to produce examples. Thus, developing students who say‘for example’ by and for themselves is a minimum requirement for teachingmathematics.
Extension andIntegration

Extension means extend the structure beyond the known set.
The product of multiplication is usually increased if both the factors are naturel
numbers. However if we extend it into fractions and decimals, there are cases
where the products decrease.
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 aninevitable nature of mathematics curriculum and its learning. Even though it isa source of difficulty for students to learn mathematics, it is the most necessary
opportunity to think mathematically and to justify the permanent ideas to beextended. 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 isa kind of nets which connect local theory of mathematics as knots even though
their strings/paths include various inconsistencies with contradictions. It is a longterm principle for task sequence to establish relearning on the spiral curriculum.
In order to develop the way of thinking for extension and integration, teachers
need efforts to clarify the repetition of both, the same patterns for extensions
and the reflections after every extensions in classroom. For example, there
are repetitions for the extensions of numbers from key stage 1 to 3. On every
extension of numbers, there are discussions of the existence of numbers withquantity, the comparison with equality, greater and lesser, and constructing
operations of numbers with the permanence of form. Through the reflection of
every repetition, students are able to learn what should be done on the extensionof numbers.

Inductive,
Analogicaland Deductive
reasoning

These three reasoning are general ways of reasoning as for the components
of logical 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 ofcases to the whole set or situations. Three considerable cases or more willbe the minimum to be considered inductive reasoning. To promote inductive
reasoning, teachers usually provide the table for finding the patterns, however
it is not the way to develop the inductive reasoning. To promote inductive
reasoning, students need to consider various possible parameters in asituation and choose two or further parameters by fixing other parameters.
Then, consider the relationship in the situation for knowing the cause andeffect, and subsequently, set the ways to check the cases by well orderingof the cause and get the effect, follow by recording the data in a table. It isthe concern of students because to develop inductive reasoning, students

Appendix B – Terminologies for Mathematical Thinking and Processes | 163

have to learn the ordering of natural numbers is vital because it is the cause ofeffect and beautifulness of patters that is originated from the order of natural
number. Table is a tool for finding pattern but students cannot produce theirinductive 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.
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 case it is called abduction. Most of the reasoning to findways of solution for unknown-problem solving by using what we already knew
is analogical 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 well established, analogical reasoning may functions as a metaphor forunderstanding. Many teachers explains operations by using diagrams. It appears
meaningful to provide the hint for solving but most students cannot use the hintby and for themselves because students do not recognise the similarity as intheir 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
for proving 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 students do it just by recognising
the structure of expression intuitively without explaining why. For clarifying thereason, teachers are necessary to ask why?
For developing the three reasoning, knowing the objectives of reasoning arenecessary. Inductive reasoning is applied to find general hypothesises. Analogicalreasoning is applied to challenge unknown problem solving. Deductive reasoning
is applied to explain or proof in general on local system.

164 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Abstracting,concretising andembodiment

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 objects. 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 metaphors of abstract ideas.
Embodiment functions in both abstracting and concretising. When abstract ideas
can be concretised, it implicates that 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 limited only
for corresponding contents.

Objectifying
operations forsymbolising andestablishing new
operations onmathematisation

A mathematical representation can be characterised by its symbols andoperations with specified purpose and context. In the process of mathematisation,
lower level operational matters are usually objectified for new symbolising andits operations.
Until key stage 2, numbers do not mean the positive and the negative numbers.
The number in red on financial matter is large if the number is ‘large’. The numberin black on financial matter is large if the number is ‘large’. Here, the meaningof ‘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 haveto alternate new symbols and operations. We represent the red number by thenegative symbol ‘–’ and the black number by the positive symbol ‘+’ and integrate
the one direction for comparison 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 the ordered pairs
between two sets. Mapping is a relationship and ordering is also a relationship.
Equivalence class is a relationship in which axiom is defined by reflective,
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. In education, the second set isusually numbers. Relational thinking is the thinking which can be represented byvarious representations such as graphs. It is the activity for students who do notknow such representations, and teachers have to teach such representations on
the necessity of students to engage in their activity: It is a welcome for students

Appendix B – Terminologies for Mathematical Thinking and Processes | 165

to create their necessary informal representation such as graph and diagram bythemselves.
Historically, characteristics of functional thinking was described by Hamley(1934). He defined functional thinking by four elements: Class, Order, Variable
and Correspondence, Class is a set which has possibility to include equivalencein operations such as 5 can be seen as the value of 1+4, 2+3, 3+2, 4+1. Orderis discussed in a set and between sets. Variables are domain and range which
are not the same as the origin set and the destination set. Correspondence isdiscussed between domain and range.
Functional thinking is useful to predict and control a situation. In elementarylevel, proportionality and operations are used for functional thinking on thisobjective. For knowing the changing properties of each function, tables andgraphs are useful. Teachers usually provide the tables and the graph papers
from the beginning of experiment for finding patterns and properties of function.
However, this is not the way to develop functional thinking because teachers
take over the opportunities from students to consider and fix the sets, orders,variables, correspondences and so on by themselves.
Rate of change is used to check if a function has linearity. In the case if it isconstant, it is a liner function. If not, otherwise. The limit of the rate of change forfinding the tangent is the definition of differentiation.
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 xof the function for calculus. These are hypothetic-heuristic reasoning beginningfrom the 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 isallowed in a written form. On this reason, the unknown x is able to be written in
Algebra and the limit of x is able to be written in Calculus. However, until such
reformations of mathematics, analysis, a method of heuristics, do not allow to
be written in a part of theory. It is a reason why some mathematics textbooks
looks very difficult to understand because they have to be written in the form ofdeductive 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 processwith various solutions as well as misconception by using what already learned,

166 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

include the heuristics such as thinking backward and so on. In the standardsof key stage 1 and 2, addition and subtraction are inverse operation, andmultiplication and division are inverse operation as for verification of answers on
operations. Such ideas 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 wasformalised as the proof by contradiction in mathematics. It is also the way ofanalysis for thinking backward. 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 counter example is the component for the proof bycontradiction. Though the communication of both objective and way of reasoning
such as thinking backwards by using what students already learned, we are able
to develop students’ mathematical thinking.

Mathematical Activities
Mathematical Activities usually explain the teaching and learning process but also embedded the mathematicalideas and thinking. The style of teaching approaches can enhance mathematical activities but do not necessarily
make them clear.
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 usually pose 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 focuses on heuristics: It
is usually observed unexpectedly and to plan it inevitably is not easy. On thisdifficulty, the problem solving tasks are usually provided in two types. The first
type focuses on mathematical modelling from the real world. The second typefocuses 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, thetasks given by teachers are planned for students where they are able to learn thecontent, mathematical ideas and ways of thinking. For teachers, solving the tasks
itself is not the objective of their classes but students reveal the objectives ofteaching by teachers through recognising problematic as an unknown and finding
the way of solutions. For students to be able to learn by and for themselves, it isnecessary to plan the class with the preparation of future learning as well as byusing learned knowledge. Textbooks such as the Japanese textbooks equippedtask sequence for this purpose. In such textbooks, heuristics is not an accidental
matter but a purposeful matter because every task in the textbook will be solved
by utilising the already learned representation and so on. It is called a ‘guided

Appendix B – Terminologies for Mathematical Thinking and Processes | 167

discovery’ under ZPD because it expects well-learned students on the learning
trajectory and never expects the genius students on producing unknown ideas.
For observers, the way of teaching by the problem solving approach in a classcannot be distinguished with the open-ended approach. The open approach is
not necessary to prepare the task sequence because it is characterised by anindependent open-ended task. If teachers set the open-ended task independently,
it is the first approach. If teachers set the task sequence of open ended tasks
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 level and since 1943 for secondary level. Currently,
Japanese textbooks until the middle school level equipped the task sequence forproblem solving approach.
Even though problem solving in education resemble activities of authentic
mathematicians, it is actually not the same because problems of mathematicians
are usually unsolvable beyond decades, while tasks in classroom can be
explained by teachers who posed the problems. When teachers refer problem
solving in education, it includes various objectives such as developing
mathematical ideas, thinking, values and attitudes. These terminologies are usedin education to develop students who learn mathematics by and for themselves
while mathematicians only use some of them. On teacher education, if teachers
only learned the content of mathematics, they may lack the opportunity to learn
the necessary terminology. If teachers do not know it, the higher order thinking
becomes a black box which cannot be explained.
Exploration andInquiry

Exploration has been enhanced in finding hypothesis by using technology suchas Dynamic Geometry Software and Graphing Software. These software provide
the environment for students to explore easily. Exploration of environment
produces hypothesis.
Inquiry which includes exploration orients to the justification and proving throughreflections.
Both explorations and inquiry enhance questioning by students. Thus, theprocess and finding will be depending on students’ questioning sequence, notlikely 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
the world. Mathematical model is hypothetically set by using mathematics torepresent the situation of the problem. Mathematical answers based on themodel 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168 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

means hypothetical approach for the real world by using universal mathematicallanguage.
Mathematisation has two usages: The first usage is for science and engineeringin 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 ingeneral. The second usage is used in education as the principle of mathematics
curriculum sequence which enhances reorganisation of mathematicalexperiences. It includes the process of extension and integration based on theprior learned knowledge.
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
system but also the various ways of explanations. Counter example is a way of
refutation. Counter example is meaningful because it sets off the reasoning fromwhich if the conclusion is true.

Conceptualisation
andProceduralisation

Conceptual knowledge is the knowledge to explain the meaning and used forthe 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 formulate the new procedural knowledge
for convenience and the exercises produce proficiency of procedure which iscalled proceduralisation.
In mathematics, the proposition format ‘if…, then…’ is the basic format torepresent knowledge, however in school mathematics, it is impossible to makeclear the proposition from the beginning because ‘if’ part can be clarified later.
For example, ‘number’ changes its meaning several times in school curriculum.
In multiplication, products become large if it is […] number. Until numbers areextended to decimal, […] part cannot be learned. On this problematic, it isnormal that students meet the difficulty in their learning and challenge to produce
appropriate knowledge if they do have a chance to over-generalise knowledge
for knowing […] part.
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.

Appendix B – Terminologies for Mathematical Thinking and Processes | 169

Representations
and Sharing

A mathematical representation consists of symbols, operations, and objective(context). Solving equations in algebra can be seen as specific ordered elementsof every equation which has the same answer. Order of equations shows thecontext that it is the process of solving. Operation of equations between an
equation to another equation can be explained by the property of equality. Eachequation is a symbolic 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
to encourage for their explanation by themselves but every explanation is
independent until knowing the hidden ideas as in the comparison. Other
students cannot re-present it until the ways of drawing (operation) is shared.
Thus, comparison of various representations is a part of the process to recognise
ideas for producing symbol and operations, and translations as in mathematicalrepresentations.
In mathematics, different representations use different symbols and operations.
If there is a rule for correspondence between symbols and operations on different
representations, it can be translated and produce rich meanings. It is the way to
produce a mathematics system.
In mathematics, representation system should be defined universally, whichis the product of convention by mathematicians. For teachers, it looks far for
students’ activity in classroom however it is the opportunity for students toreinvent the representation and its system through considering the why andhow. For example, to produce meter as the measurement quantity includes
such activities: There are historical episodes on why and how ‘m’ was definedby Condorcet and others in the Middle of the French Revolution. On the
Area of Engineering, Informatics and Science, applied mathematicians usually
try to produce new measurements based on their necessity of research
to conceptualise the idea mathematically and operationally: Setting themeasurable quantity is a part of mathematical modelling for real-world
problem solving because if it is measurable we can apply known mathematics.

170 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

APPENDIX C
Strand: Mathematical Processes – Humanity for Key Stage 3
Critical argument in mathematics is enhanced through communication with others beyond Key Stage 2. The
proposed challenging activities will promote metacognitive thinking at different level of arguments to makesense of mathematics. Translating real life activities into mathematical models and solving problems using
appropriate strategies are emphasised in functional situations. The process of doing mathematical activities
involves patience that develops perseverance in learners and takes responsibility of one’s own learning. At thisstage, the habitual practice of self-learning will eventually develop confidence, thus, opportunity for challenges
to extend mathematics and the ability to plan sequence of future learning are also enhanced.
Standards
Enjoying problem solving through various questioning for extension of operations into algebra, space andgeometry, 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 probability1i.

Pose questions to extend numbers and operations into positive and negative numbers, algebraic
operations, and further extension into polynomial operations, and numbers with square roots
ii. Pose questions to solve linear equations, simultaneous equations and simple quadratic equations
iii. Pose assumptions in geometry as objects of argument and proof
iv. Pose questions to transform three dimensional objects into two dimensional shapes and vice versav.
Pose questions in relations and functions for knowing properties of different types of function
vi. Pose questions to exploratory of data handling for knowing structure of distribution
vii. Pose questions that apply PPDAC in relation to statistical problem solving
viii. Pose questions in relation to “equally likely” events
ix. Pose assumptions to discuss hypothesis based on sample and population
x. Pose conjecturing such as if x increases and y decreases then it is inverse proportion
Enjoying measuring space using calculations with various formulae2
i.
ii.
iii.
iv.
v.
vi.

1
2

Extend the number line to positive and negative numbers and compare the size of numbers with theidea of absolute value
Derive the square root using unit squared paper through the idea of area
Explain the expansion of polynomials using area diagrams
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

Connected to the three strands, namely Numbers and Algebra, Relations and Functions and Space and Geometry.
Connected to the three strands, namely Numbers and Algebra, Relations and Functions and Space and Geometry.

Appendix C – Strand on Mathematical Processes-Humanity (Key Stage 3) | 171

Producing proof in geometry and algebra
i.
ii.
iii.
iv.

Have an assumption through exploration and produce propositions
Justify the proposition using examples and counter examples for understanding
Rewrite propositions from sentences to mathematical expressions by using letters and diagrams
Search the ways of proving by thinking backward from the conclusion and thinking forward from thegivenv.
Show the proof and critique for the shareable and reasonablenessvi. Deduce another propositions in the process of proving and after proving using what if and what if notvii. Adapt ways of proving to other similar propositions of proof
viii. Explain the written proof in geometry and algebra by the known
ix. Revise others’ explanation meaningfully
Utilising tables, graphs and expressions in situations3i.
ii.

Explore the properties of functions by using tables, graphs and expressions and establish the fluency
of connections among them for interpreting functions in context
Analyse distribution of raw data by using tables, graphs and expressions in daily life

Using diagrams for exploring possible and various cases logically4
i.
ii.
iii.
iv.

Use number line with inequality to identify range and set
Use circle to identify relationship between the circumference and the central angle (acute, obtuse
and right)
Use rectangle and rotating a point on the side rectangle to draw the graph of the area
Use tree diagram for thinking about all possible cases sequentially

Exploring graph of functions by rotation, by symmetry and by translation of proportional function5
i.
ii.
iii.
iv.

Use the slope of a graph for the proportional function to rotate the graph or to determine the point ofintersection
Explore to know the nature of two simultaneous equations by using translation
Use the y-axis, x-axis and y = x as the line of symmetry to explore proportional function
Explain the graph of linear function by translation of proportional function.

Understanding ways for extension of numbers6
i.
ii.
iii.

Extend the numbers based on the necessity of solving equations such as x + 5 = 3 and x2 = 2, andshow examples for demonstrating the existence such as on the number lines, and understand it as
set
Compare the size of number and identify how to explain the order of numbers and its equivalence
Extend operations for keeping the form7 beyond the limitations of meaning8

Connected to the strand on Relations and Functions.
Connected to the two strands on Relations and Functions and Space and Geometry5
Connected to the two strands on Numbers and Algebra and Relations and Functions
6
Connected to the strand on Numbers and Algebra
7
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).
8For the extension of numbers to positive and negative numbers, beyond the limitations of meaning such as subtract a smaller
number from a larger number. For the extension of numbers to irrational number, beyond the limitation of meaning such as
rational number is a quotient (value of division).
3
4172 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Designing models for sustainability using mathematics9
i.
ii.
iii.
iv.

Discuss and utilise probabilities in life such as weather forecasting for planning
Design cost saving lifestyle models through comparison of data such as cost of electricity, water
consumption, and survey
Plan emergency evacuation such as heavy raining and landslide where the calculations on theamount of water in barrel per minute exceeds the maximum standards
Forecast the future with mathematics

Utilising ICT tools as well as other technological tools
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.

Use dynamic geometry software for assumption, specialisation and generalisation
Use graphing tool for comparison of the graph and knowing properties of function
Use data to analyse statistics with software
Use internet data for the discussion of sustainable development
Use calculators for operations at necessary context
Use projector for sharing ideas such as project survey, reporting and presentation
Use the idea of function to control mechanism
Use ICT tools for conjecturing and justifying to produce the object of proving.

Promoting creative and global citizenship for sustainable development of society using mathematics
i.
ii.
iii.
iv.
v.
vi.

9

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

Appendix D – Learning Standards for Key Stage 1 | 173

APPENDIX D
The four strands of content learning standards and the strand on Mathematical Process-Humanity of Key
Stage oneKEY 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, identify its
algorithm, and understanding basic mathematical concepts and skills experienced in daily life. Calculation ofquantities will also be established to carefully and wilfully understand the attribution of objects that are used tomake direct and indirect comparison.
Strand: Numbers and Operations
Number is introduced with situations, concrete objects, pictorial, symbolic representations and extended basedon knowledge and skills learned. Ways of counting and distributions are extended to addition, subtraction,
multiplication and division. Base ten number system is the key for extending the numbers and operations
for standard algorithms in vertical form. Also, various procedures of calculations and algorithms are focused.
Models and diagrams are used for extension instead of concrete materials itself. Number sense will be
developed through the establishment of fluency of calculations with connection to situations and models.
Fractions and decimals are introduced with manipulative.
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 numerals;
i.		
ii.

Develop the fluency of the order of number names and use it based on situations1
Set initial object for counting, direction of counting and recognize the end object with one to one
correspondence

Understanding and using the cardinality and ordinality of numbers with objects and numerals2 through activities
of grouping, corresponding and ordering, and develop number sense3 and appreciate 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, one cup is describe at strand on the Quantity and Measurement
Inclusive in reading and writing of numerals
3
Unit for counting that describe the Quantity and Measurement
12174 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

ii.
iii.
iv.
v.
vi.

Obtain fluency of 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 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 in ordering numerals with and without
concrete objects

Introducing base ten system with groupings of 10 and extend numbers up to 1206
i.
ii.
iii.
iv.
v.

Extend numbers to more than 10 with base ten manipulative representing numbers in ones and tens,
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 numeral 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 number sense
Draw diagram for representing the size of number with base ten blocks

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 of addition expressions using composition of numbers for easier calculation withnumber sense for 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 of 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 of subtraction expressions using decomposition of numbers for easier calculation
Apply subtraction fluency in learners’ life

Extending subtraction to more than 10 and obtaining fluency of 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 D – Learning Standards for Key Stage 1 | 175

iii.
iv.

Develop fluency of 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 size of numbers through translation of the size of every digit
with 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 symbol of inequality

Extending numbers using 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 number line with appropriate scale to show size of numbers and relative size of numbers while
focusing on the scale

Extending numbers using based ten system up to 1 000 0008
i.
ii.
iii.

Extend numbers up to 1 000 000 and learn the representation of the place value for grouping every
3-digit numeral system up to million
Write large numbers using grouping of 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 relative size of numbers

Producing Vertical Form Addition and Subtraction10 and Acquiring Fluency of 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 base ten system

One million is too big for counting and is introduced only for learning of 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, fourdigit numeral system is used.
10
Understanding the relationship of addition and subtraction is discussed under Pattern and Data Representations
89

176 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

ii.
iii.
iv.

Produce and elaborate efficient ways and identify the standard algorithms11 in relation to base tensystem with appreciation
Explain the algorithms of borrowing and carrying with regrouping of base ten models
Acquire fluency in addition and subtraction algorithms

Acquiring fluency of standard algorithm for addition and subtraction and extend up to 4 digit numbersi.
ii.

Extend the vertical form addition and subtraction through the extension of numbers and appreciate
the explanation using base ten block model
Develop fluency of every extension up to 3-digit numbers and simple case for 4 digit numbers

Developing number sense12 for estimation13 and using 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.

Understand the meaning of multiplication14 as a model of repeated addition situations and distinguishfrom the common addition
Produce multiplication table in the case of counting by 2 and 5 with array diagrams, pictures or baseten block models and extend it until 9 and 1 with appreciation of patterns15
Develop number sense for multiplication through mental calculation with fluencyiv. Use multiplicationin daily life, differentiating multiplication in various situations with understanding that any number can
be a unit for counting in multiplication

Producing multiplication in vertical form and obtaining fluencyi.

Think about easier ways of multiplication in the case of numbers greater than 10 using array diagramsand base ten block models
ii. Develop multiplication in vertical form using multiplication table, array, model, and base ten systemwith appreciationiii. Extend multiplication algorithm to 3-digits times 2-digits numbers
iv. Obtain fluency of standard algorithm for multiplicationv.
Use estimation with multiplication of tens or hundreds in learners’ life
vi. Compare multiplication expressions which is larger, smaller or equivalentvii. Appreciate the use of calculator judiciously in life in the case of large numbers
Introducing Division and Extending It to Remainder
Introducing division with two different situations and find 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 of division situations by distributions using diagrams, repeatedsubtractions and multiplication
Obtain fluency to identify answers of division through 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, vertical form itself is not
the same. Here, standard algorithm means the selected appropriate form.
12
Money system is discussed under Measurement an 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 D – Learning Standards for Key Stage 1 | 177

Extending division into the case of remainders and use division for distribution in daily situations
i.
ii.
iii.

Extend division situations with remainders and understand division even it has the same as repeatedsubtraction with different remainders
Obtain fluency in division and apply it in daily situations
Understand simple cases of 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 manipulative such as paper folding and drawing diagrams in thecontext 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 so on
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 simple algorithm
i.
ii.
iii.
iv.

Extend fractions to more than one unit quantity for representing the remaining parts such asmeasuring the length of tape, recognising the remaining parts as a unit measure of length, andunderstand proper and improper fractions
Appreciate fractions with quantities in two ways; firstly 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 tape diagram or
number line, and develop fraction number sense such as 1/10 with quantities and so on
Think about how to add or subtract similar fractions with tape diagram or number line and produce
simple algorithm with fluency

Introducing Decimals and Extend to Addition and Subtraction
Introducing decimals to tenths, and extend addition and subtraction into decimals
i.
ii.
iii.
iv.

Introduce simple decimals to tenths by remaining part such as using tape diagram with appreciation
Compare 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 vertical form up totenthsThink about appropriate place value for applying addition and subtraction in life
Strand: Quantity and Measurement

Attributes of objects are used to make direct and indirect comparison, the non-standard and standard units arealso 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 for baseten system. The concept of conservation of quantities will be established through the calculation of quantities.
The sense for quantity is developed through appropriate selection of measurement tools.
Topics:Comparing Sizes Directly and Indirectly Using Appropriate Attributes and Non-Standard Units
Introducing Quantity of Length and Expand to DistanceIntroducing Quantity of Mass for its Measurement and Operation
Introducing Quantity of Liquid Capacity for its Measurement and Operation

178 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Introducing Time and Duration and its Operation
Introducing Money as a Quantity
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 using attributes instead of stating in length and amount of water suchas 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 suitable context

Introducing Quantity of Length and Extend to DistanceIntroducing centimetre for length and extend to millimetre and meter
i.
ii.
iii.

Compare length of different objects and introduce centimetre with calibrated tape17 of one centimetre
Demonstrate equivalent length with addition and subtraction such as part-part whole
Extend centimetre to millimetre to represent remaining parts with ideas of equally dividing and idea
of making tens
iv. Extend centimetre to metre to measure using meter stickv.
Estimate length of objects and select appropriate tools or measuring unit 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.
ii.
iii.

Introduce kilometre to measure distance travelled using various tools and appreciate the experiencesof measuring skills
Distinguish distance travelled and the distance of two places on the map
Compare mixed units of length with appropriate scale on a number line

Introducing Quantity of Mass and Its Measurement and Operation
Introducing gram for mass and extend to kilogram and metric ton
i.
ii.
iii.
iv.
v.

Compare 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 weighing scale
Extend kilogram to metric ton through relative measure (such as 25 children, each weigh 40 kilogram)
Estimate mass of objects and select appropriate tools or measuring unit 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 forquantity

Denomination is necessary for learning the group of counting. It’s also describe 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 non-standard unit
we can use it for measurement. If the tape is scaled by one centimeter we can defined the length of centimetre
18
Which one is longer, 2 m 3 cm or 203 mm?
16

Appendix D – Learning Standards for Key Stage 1 | 179

Introducing Quantity of Liquid19 Capacity and Its Measurement and Operation
Introducing litre for capacity of liquid and extend to millilitre
i.
ii.
iii.
iv.
v.
vi.

Compare 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 capacity of containers and select 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 forquantity

Introducing Time and Duration, and Its Operation
Introducing analogue time and extend 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 clock face with hour hand and minute hand
Understand the relative movement of clock hands

Extending clock time to 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 time line, and understand duration as the difference between twotimesiii. 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 duration of time and select appropriate measuring unit for measurement with fluency andappreciate 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 it as the model of base ten system21
i.
ii.
iii.
iv.

Introduce unit of money using notes and coins and determine the correct amount
Use counting by fives and so on for base ten system
Appreciate fluency in 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 exploration of various features of shapes.
Identification of similarities and differences in shapes and solids enable classification to be done for defining
figures. Using appropriate materials and tools, relationship in drawing, building and comparing the 2D shapes

The density which explains the relationship between mass and liquid capacity is usually learned in Science at later stage. In thecase of CCRLS Science it starts in Key Stage 2 such as 1 cubic centimeter 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 unit of twenty and twenty-five which are in coins
or notes. These forms are not appropriate for the model of base 10 system
19180 | Mathematics Challenges for Classroom Practices at the Lower Secondary Leveland 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 with the same length.
Topics:Exploring shapes of objects
Characterising the 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.

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 or by tracing the shape on the paper and explain
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

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 relations to activities such as paper folding and use it for describing
figures with simple properties such as triangle has 3 lines
Appreciate the names of figures in daily life by using mother-tongue such as “segitiga” and “segiempat” in Malay and “tatsulok” in Tagalog
Classify triangles by specific properties, such as the sides, angles and vertexes (right triangle,
equilateral, isosceles)

Describing solids with characteristics of shapes
i.
ii.
iii.

Use the characteristics of shapes, understanding solids such as boxes can be developed by six
rectangular parts with simple properties
Develop boxes with their properties
Appreciate functions of solids around daily life

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 compass
Enjoy drawing pictures using the function of circles such as SpirographFind the largest circle through cutting the sphere and recognise the sphere by the centre and radius
Appreciate functions of circles and spheres in daily life such as the difference between soccer balland rugby ballPattern of shapes is discussed in Key stage 1 under Pattern and Data Representations

Appendix D – Learning Standards for Key Stage 1 | 181

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 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. Size of pictures
can be represented by number sequence. Tessellation of shapes and paper folding can be represented by therepetition of shapes. Exploration of patterns and features are also considered to represent the data structure
in our life using pictographs and bar graphs. Patterns and features produce meaning of data and represent
mathematical information. Patterns are represented by diagrams and mathematical sentences which are also
used for communication in identifying and classifying situations to produce meaningful interpretations.
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, colour and sizes to create structure
Arrange circles and spheres for designing
Enjoy the creation based on different shapes, colour 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

182 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Collecting Data and Represent the Structure
Collecting data through categorisation for getting information
i.
ii.
iii.

Explore the purpose of why data is being collected.
Grouped data by creating similar attributes on the denomination26 of categories and count 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 represent using pictogram for easy visualisation
i.
ii.
iii.

Produce the table and pictograms from collected data under each categories
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 bar graph to predict the future of communities
i.
ii.
iii.

Understand how to draw bar graph from table using data categories and sort the graph for showing
its structure
Appreciate ways of presenting data such as using tables, pictograms and bar graphs with sorting forpredicting their future communities
Appreciate the using of data for making decision
Strand: Mathematical Process – Humanity

Enjoyable mathematical activities are designed to bridge the standards in different strands. Exploration ofvarious number sequence, skip counting, addition and subtraction operations help to develop number sensethat is essential to support explanation of contextual scenarios and mathematical ideas. Mathematical waysof posing questions in daily life are also necessary learned at this stage. Ability to select simple, general andreasonable ideas enables effective future learning. Application of number sense provides facility for preparing
sustainable life. The use of ICT tools and other technological tools provide convenience in daily life. At initialstage, concrete model manipulation is enjoyable, however drawing diagram is most necessary 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 for future learning
Preparing sustainable life with number sense
Utilising ICT tools such as calculators as well as other tools such as note book and other instrument such as
clocksPromoting 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 situations, pose questions for ‘altogether’ and ‘increase’In subtraction situations, pose questions for ‘remainder’ and ‘difference’In multiplication situations, pose questions for number of groups
In division situations, pose questions for partition and quotient
Enjoy questioning in various situations by using combination of operations
In operations, pose questions to find the easier ways of calculation
Use posing questions for four operations on measurements in daily life

Denomination will be learned in Key stage 1 under Quantity and Measurement
It is related with Numbers and Operations and Quantity and Measurement both in Key Stage 1.

Appendix D – Learning Standards for Key Stage 1 | 183

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, a unit cube is 1, a bar stick is 10 and a flat block represents
100
Explain addition and subtraction algorithm in vertical form using base ten block model
Explain multiplication table with number of grouped blocks
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 to produce beautiful designs with various shapes and find the pattern to explain it
Reflect, rotate and translate to produce patterns
Cut and paste various shapes and colours to form the box and ball such as develop the globe frommap

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, container and paper folding for sharing
ideas
Make decision on how to explain figures and solids by using manipulative objects or diagrams or
only verbal explanation
Explain patterns using diagrams, numbers, tables and expressions with blank box
Ask questions using terms such as why, how, what, if and if not, and reply using examples and ‘forexample’ in discussion
Change the representation and translate it appropriately in daily life

Selecting simple, general and reasonable ideas which can apply for future learning32
i
ii
iii
iv

Discuss the argument for easier ways for addition and subtraction algorithm in vertical form
Extend the algorithm to large numbers for convenience and fluency
Use the pattern of increase in multiplication table for convenience
Use multiplication tables for finding answers of 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 with Quantity and Measurement and Pattern and Data Representations both in Key Stage 1.
It is related with Pattern and Data Representations and Numbers and Operations both in Key Stage 1.
30
It is related with 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 and Measurement, all 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

184 | Mathematics Challenges for Classroom Practices at the Lower Secondary Leveliii
iv

Maximize the use of resources through appropriate arrangement in space
Understand equally likely of resources in situations

Utilising ICT tools such as calculators as well as other tools such as notebook and other instrument such as
clocks35
i
ii
iii
iv
v
vi
vii
viii

Use calculators for multi addition in situations
Use mental calculations for estimations
Use a balance scale to produce equality and inequality
Use cups, tapes, stop watches, and weighing scales for measuring distances and weights
Use calculators to explain the process of calculation by solving backward and understand therelationship of addition and subtraction, and multiplication and division.
Enjoy using notebooks to exchange learning from 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 the notebooks and journal books to record and find good ideas and share with others
Prepare and present ideas using posters to promote good practices in neighbourhood
Listen to other’s ideas and asking questions for better creation
Utilise information, properties and models as basis for reasoning
Utilise practical arts and outdoor studies to investigate local issues for improving 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 E – Learning Standards for Key Stage 2 | 185

APPENDIX E
The four strands of content learning standards and the strand on Mathematical Process-Humanity of Key
Stage Two
KEY STAGE 2
Key Stage 2 (KS2) can be learned based on the key stage 1. This stage provides the extension of numbers
and operations, measurement and relations, plane figures and space figures as well as data handling andgraphs. This stage enable the extension of the four operations to daily use of numbers such as decimal andfraction and allows the use of mathematical terminologies, performing investigations and establish the groundfor analysing, evaluating and creating learners’ life. Appreciating the beauty of the structure of mathematics will
enable them to enjoy and sustain their learning which provides basis for key stage 3.
Strand: Extension of Numbers and Operations
Numbers are extended to multi-digits, fractions and decimals. Multiplication and division algorithms arecompleted with fluency. Fraction becomes numbers through the redefinition as a quotient instead of part - whole
relationship. Multiplication and division of decimals and fractions are also explored to produce procedures forcalculation. Various representations are used to elaborate and produce meaning for the calculation. Number
sense such as approximating numbers, relative size of numbers and values are enhanced for practicalreasoning in the appropriate context of life.
Topics:Extending Numbers with Base Ten up to Billion and also to Thousandths with Three Digit Numeral System
Gradually
Making Decision of Operations on Situations with Several Steps and Integrate Them in One Expression andThink about the Order of Calculations and Produce the Rule (PEMDAS)
Producing the Standard Algorithm for Vertical Form Division with Whole Numbers
Extending the Vertical Form Addition and Subtraction with Decimals to Hundredths
Extending the Vertical Form Multiplication and Division with Decimals and Find the Appropriate Place Valuesuch as the Product, Quotient and Remainder
Using Multiples and Divisors for Convenience
Introducing Improper and Mixed Fractions and Extending to Addition and Subtraction of Fractions to Dissimilar
Fractions
Extending Fractions as Numbers and Integrate
Extending Multiplication and Division to Fractions
Extending Numbers with Base Ten Up to Billion and also to Thousandths with Three Digit Numeral
System Gradually
Extending numbers using base ten system up to billion1 with three digit numeral system2iii

Adopt the three digit numeral system, extend numbers up to billion with the idea of relative size of
numbers
Compare numbers such as larger, smaller with base3 ten system of place values through visualisationof relative size of numbers using cubes, plane(flats), bar(longs) and unit

Billion is too large for counting and it is introduced in the three digit system under relative size of number
In British system, it is referred as short scale
3
Metric system names of units is discussed under measurement and relations
1
2

186 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Extending decimal numbers to hundredths, and to thousandths4
i.
ii.
iii.

Use the idea of quantity and fractions, extend decimal numbers from tenths to hundredths
Compare decimal numbers such as larger, smaller with base ten system of place value
Adopt the ways of extension up to thousandths and so on, and compare the relative sizes

Making Decision of Operations on Situations with Several Steps and Integrate them in One Expression
and Think about the Order of Calculations and Produce the Rule (PEMDAS)
Finding the easier ways of calculations using the idea of various rules of calculations5 such as the associative,
commutative and distributive rule
i.
ii.
iii.
iv.

Find the easier ways of addition and subtraction and use it, if necessary, such as answer is the sameif add same number to the subtrahend and minuend
Find the easier ways of multiplication and division and use them in convenient ways such as 10times of multiplicand produce the product 10 times
Use associative, commutative and distributive rules of addition and multiplication for easier ways of
calculation, however commutative property does not work in subtraction and division
Appreciate the use of simplifying rules of calculations

Thinking about the order of calculations in situations and produce rules and order of operations
i.
ii.
iii.

Integrate several steps of calculation into one mathematical sentence
Produce the rule of PEMDAS and apply it to the several steps situation
Think about the easier order of calculation and acquiring fluency of PEMDAS and rules withappreciation

Producing the Standard Algorithm Using Vertical Form Division with Whole Numbers
Knowing the properties of division and use it for easier way of calculationi.
ii.

Find the easier ways of division and use it, if necessary, such as answer is the same if multiplying
the same number to the dividend and divisor
For confirmation of answer of division, use the relationship among divisor, quotient and remainderand appreciate the relationshipKnowing the algorithm of division in vertical form and acquiring fluency
i.
ii.
iii.
iv.

Know the division algorithm with tentative quotient and confirm the algorithm by the relationshipamong divisor, quotient and remainderInterpret meaning of quotient and remainder in situations
Acquire fluency for division algorithm up to 3-digit whole number divided by 2-digit
Think about situations with or without remainder in relation to situations for quotative and partitive
division

Under the three digit system, if we teach until thousandths we can extend by three digit
In measurement and relations, Use of constant sum, difference, product and quotient are described.( e.g. 25 -21 = 4, 26 – 22
= 4, 27 – 23 = 4)
6
Discussion of decimals to hundredths is related to the use of money. It is a minimum requirement. If teaching to hundredths,
further extension of place value can be understood.
45

Appendix E – Learning Standards for Key Stage 2 | 187

Extending the Vertical Form Addition and Subtraction with Decimals to Hundredths
Extending the vertical form addition and subtraction in decimals to hundredthsi.
ii.

Extend the vertical form addition and subtraction to hundredths6 place and explain it with models
Appreciate the use of addition and subtraction of decimals in their life

Extending the Vertical Form Multiplication and Division with Decimals and Find the Appropriate Place
Value Such as Product, Quotient and Remainder
Extending the multiplication from the whole number to decimal numbers
i.
ii.
iii.
iv.

Extend the meaning of multiplication with the idea of measurement by the number of unit length
for multiplication of decimal numbers and use diagrams such as number lines to explain them withappreciation in situations
Extend the vertical forms multiplication of decimals up to 3 digits by 2 digits with consideration of thedecimal places step by stepObtain fluency using multiplication of decimals with sensible use of calculators in learners’ life
Develop number sense in multiplication of decimals7 such as comparing sizes of products before
multiplying

Extending the division from the whole number to decimal numbers
i.
ii.
iii.
iv.
v.
vi.

Understand how to represent division situations using diagrams such as number lines, and extend
the diagram of decimal numbers for explaining division by decimal numbers
Extend the division algorithm in vertical form of decimal numbers and interpret the meaning ofdecimal places of quotient and remainder with situations
Acquire fluency in division algorithm of decimals up to 3 digits by 2 digits with consideration ofdecimal places step by stepObtain fluency using division of decimals with sensible use of calculators in learners’ life
Develop number sense of division in decimals such as comparing sizes of quotient before multiplying
Distinguish the situations with decimal numbers of multiplication and division

Using Multiples and Divisors for Convenience
Using multiples and divisors for convenience with appreciation to enrich number sense
i.
ii.
iii.
iv.
v.
vi.

Understand set of numbers by using multiples and divisors
Find common multiples and appreciate the use in situations, and enrich number sense with figuralrepresentations such as arrangement of rectangles to produce a square
Find common divisor and appreciate its use in situations, and enrich number sense with figuralrepresentations such as dividing a rectangle into pieces of square
Understand numbers as composite of multiplication of numbers as factors8.
Appreciate ideas of prime, even and odd numbers in situations using multiples and divisors
Acquire the sense of numbers to see the multiples and divisors for convenience

Introducing Improper and Mixed Fractions and Extending to Addition and Subtraction of Fractions to
Dissimilar Fractions
Extending fractions to improper, mixed and equivalent fractions
i.
7
8
9

Extend fractions to improper and mixed fractions using number line of more than one by measuringwith unit fraction9

Applying the idea of multiplication into ratio, percent and proportion is discussed in Measurement and Relations.
This idea is related to the strand on Measurement and Relations in key stage 2 at the area of rectangle.
Extension of fraction more than one is done by using the fraction with quantity for situation such as 43 m

188 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

ii.
iii.

Find ways to determine equivalent fractions with number lines and with the idea of multiple ofnumerator and denominator
Compare fractions using number line and the idea of multiple

Extending addition and subtraction of similar fractions to improper and mixed fractions, and dissimilar fractions
i.
ii.
iii.

Extend addition and subtraction of similar fractions to proper and mixed fractions with explanations
using models and diagrams
Extend addition and subtraction into dissimilar fractions with explanations using diagrams andcommon divisors
Acquire fluency of addition and subtraction of fractions with appreciation of idea to produce the samedenominators

Extending Fractions as Numbers and Integrate10
Seeing fractions11 as decimals and seeing decimals as fractions
i.
ii.
iii.

See fractions as decimals using division and define quotient from the case of indivisible
See decimals as fractions such as hundredths is per hundred
Compare decimals and fractions and ordering them on a number line

Extending Multiplication and Division to Fractions
Extending multiplication to fraction
i.
ii.
iii.

Extend multiplication situations to fraction using diagrams such as number lines, and find the simple
algorithm for the multiplication of fractions step by step
Acquire fluency with multiplication of fractions
Develop number sense12 of multiplication of fractions such as comparing sizes of products before
multiplying

Extending division to fraction
i.
ii.
iii.

Extend the division to fraction with situations using diagrams such as number lines step by step
Acquire fluency with division of fractions
Develop number sense of division of fractions such as comparing sizes of quotients before dividing
Strand: Measurement and Relations

Additive quantity such as angles, areas and volume and relational quantities such as population density, andspeed are introduced. Additive quantity can be introduced by establishment of the standard unit which is thesame way as the Quantity and Measurement of Key Stage 1. Relations of quantities in situations are discussedwith patterns such as sum is constant, difference is constant, product is constant and quotient is constantusing tables and represented by mathematical sentences and letters. Proportion and ratio are introduced withrepresentations of diagrams, graphs and tables for multiplication, and connected with decimals and fractions.
Percent is introduced with diagrams in relation to ratio and proportion. Relational quantity is produced bydifferent quantities with understanding of ratio. Area of a circle is discussed through a proportional relationshipbetween radius and the circumference. Ideas of ratio and proportion are fluently applied for real world problem
solving.

10

Selecting the appropriate denomination of quantities and units for fraction in the context ( 23 m is two of 13 m and the whole is

1m, howeve 3 m is 3x( 1 )m; the structure is the same as tens is ten of units and discussed under Measurement and Relations
3

3

Fraction as ratio is introduced in Measurement and Relations
12
Applying the idea of the multiplication of fraction into ratio, proportion, percentage and base is discussed in Measurement
and Relations
11

Appendix E – Learning Standards for Key Stage 2 | 189

Topics:Introducing Angle and Measuring it
Exploring and Utilising Constant Relation
Extending Measurement of Area in Relation to Perimeter
Extending Measurement of Volume in Relation to Surface
Approximating with Quantities
Extending Proportional Reasoning to Ratio and Proportion
Producing New Quantities Per Unit
Investigating the Area of a Circle
Exchanging Local Currencies in ASEAN Community
Extending the Relation of Time and Use of Calendar
Converting Quantities in Various System of Units
Showing Relationship Using Venn Diagram
Introducing Angle and Measuring It13
Introducing angle by rotation, enabling measure and acquire fluency using the protractor
i.
ii.
iii.
iv.
v.

Compare extent of rotation and introduce degree as a unit for measuring angle
Recognise right angle is 90 degrees, and adjacent angle of two right angles is 180 degrees, and 4
right angles is 360 degrees
Acquire fluency in measuring angles using the protractor
Draw equivalent angles with addition and subtraction using multiples of 90 degrees
Appreciate measurement of angles in geometrical shapes and situations in life14

Exploring and Utilising Constant Relation
Exploring equal constant relation with utilisation of letters to represent placeholders15
i.
ii.
iii.
iv.

Explore two possible unknown numbers such that their sum (or difference/product/quotient) isconstant,16 for example □ + ∆ = 12 (□ and ∆ are placeholders).
Use letters instead of placeholder17 (empty box)to derive equivalent relation
Understand the laws for operations (e.g. associative, commutative and distributive etc.) to explain
the simpler way of calculation
Appreciate the use of diagrams such as number lines and area to represent relation when finding
solutions

Extending Measurement of Area in Relation to Perimeter
Introducing area and produce formula for the area of a rectangle
i.
ii.
iii.
iv.
v.

Compare extent of area and introduce its unit, and distinguish it from perimeter
Introduce one square centimetre as a unit for area and its operation using addition and subtraction
Investigate area of rectangles and squares and produce the formula of area18
Extend square centimetre to square metre and to square kilometre for measure of large areas
Convert units and use appropriate units of area with fluency

Right angle is learned at key stage 1 in Shapes, Figures and Solids for explaining the properties of figures
Conservation of angles will be re-learnt in triangle under key stage 2 Plane Figures and Space Figures
15
The idea for the use of Numbers and Operations (key stage 2) in finding the easier ways of calculations with the idea of rulesof calculations
16
Constants of multiplication and division, corresponds proportionality in multiplication table at key stage 1 under number andoperations. Constant of addition and subtraction are treated in key stage 1 under Pattern and Data Representation
17
Place holder is introduced in key stage 1
18
Multiplications were studied in Key stage 1 Numbers and Operations
13
14

190 | Mathematics Challenges for Classroom Practices at the Lower Secondary Levelvi. Draw the equivalent size of rectangular area based on a given area with the composite numbers19
vii. Appreciate the use of areas in learners’ daily life such as comparing of land sizes.
Extending area of a rectangle to other figures to derive formulae
i.
ii.
iii.
iv.

Explore and derived formula for the area of a parallelogram by changing its shape to rectangle
without changing its area
Explore and derived formula for the area of a triangle by bisecting a rectangle into two triangles
without changing its area
Appreciate the idea of changing or dividing shapes of rectangle, parallelogram, or/and triangle forderiving the area of other figures
Use formulae to calculate areas in daily life

Extending Measurement of Volume in Relation to Surface
Introducing volume from area and derive formula for cuboid
i.
ii.
iii.
iv.
v.
vi.

Compare the extent of volume and introduce its unit, and distinguish it from surface
Introduce one cubic centimetre as unit for volume and its addition and subtraction
Investigate volume of cuboid and cube and produce the formulas
Extend cubic centimetre to cubic metre to measure large volume
Convert units and use appropriate units of volume with fluency
Appreciate the use of volume in their life such as comparison of capacity of containers

Extending volume of cuboid to other solid figures to derive formula
i.
ii.
iii.

Extend the formula for the volume of a cuboid as base area x height for exploring solid figures suchas prism and cylinder
Extend the formula for the volume of a prism and a cylinder to explore and derive the volume
formula of pyramid and cone
Use the formulas to calculate volume in daily life

Approximating with Quantities
Approximating numbers with quantities depending on the necessity of contexts
i.
ii.
iii.

Understand the ways of rounding such as round off, round up and round down
Use rounding as approximation for making decision on the quantity with related context
Critique over approximation beyond the context with a sense of quantity such as based on relative
size of a unit

Extending Proportional Reasoning to Ratio20 and Proportion
Extending proportional reasoning to ratio and percenti.
ii.

Understand ratio as a relationship between two same quantities or between two different quantities(the later idea is rate)21
Express the value of ratio by quotient such as the rate of two different quantities22

The idea of composite numbers such as 2 times 10 equals 5 times 4 is related to factors in extending the numbers and operations
at the same key stage 2.
20Band graph and pie chart for representing ratio are discussed under key stage 2 under the strand Data Handling and Graphs
21
Ratio of different quantities is a rate. Ratio of the same quantities is a narrow meaning of ratio.
22
The value of fraction as a ratio is not necessary a part of a whole in situations. Fraction as a ratio is usually used in the context
of multiplication situation, where denominator is the base or a unit for comparison.
19

Appendix E – Learning Standards for Key Stage 2 | 191

iii.
iv.
v.

Understand percent as the value of ratio with the same quantities23
Understand proportion with ratio
Apply the rule of three24 in using ratio

Extending proportional reasoning to proportion
i.
ii.
iii.
iv.
v.

Extend proportional reasoning to multiplication tables as an equal ratio and understandproportions
Understand proportion using multiple and constant quotient, not changing the value of ratio25
Demonstrate simple inverse proportion by constant product26
Express proportion in mathematical sentence using letters and graph27
Use properties of proportionality to predict and explain phenomenon in daily life

Producing New Quantities by Per Unit
Producing new quantities by per unit with the idea of average such as population density and speed
i.
ii.
iii.
iv.
v.
vi.

Introduce average as a unit for distribution and comparison of different sets of values
Introduce population density with the idea of average and appreciate it for comparison
Introduce speed with the idea of average and appreciate it for comparison
Appreciate using diagrams such as number lines and tables to decide the operations situations ofmeasurement per unit quantity
Comparing on the context of different quantities with the idea of average as rate28
Apply the idea of measurement per unit quantity in different context29

Investigating the Area of Circle
Areas of a circle are discussed through the relationship between the radius and circumference.
i.
ii.
iii.
iv.

Investigate relationship between the diameter of a circle and its circumference using the idea of
proportion
Investigate area of a circle by transforming into a triangle or parallelogram and find the formula of thecircleEstimate the area of a inscribed and out scribed shapes based on known formula of areas30
Enjoy to estimate the area of irregular shapes with fluency in life

Percent is used in Data Handling and Graphs
Rule of three is the method on the table to find one unknown term from the three known terms using proportional reasoning
such as;
			a
c
			?
d
23
24

Enlargement is discussed in Key stage 2 under Plane Figures and Space Figures. The graph is treated at key stage 2 under the
strand Data Handling and Graphs
26
Proportion and Inverse proportion is necessary in Key stage 3 in science
27This will be discussed in detail in same key stage 2 under Data Handling and Graphs
28
On number and operations key stage 2, rate is the value of division as quotient
29
Using measurement per unit quantity with fluency to make logical judgment in daily life, refer to Key stage 2 Data Handlingand Graphs
30
Relationship on polygons and circles are discussed in Key stage 2 under Plane Figures and Space Figures25

192 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

Exchanging Local Currency with Currency in ASEAN Community
Exchanging local currency in ASEAN community with the idea of rate
i.
ii.
iii.

Extend the use of ratio for currency exchange (rate of exchange)
Apply the four operations for money in appropriate notation in life
Appreciate the value of money

Extending the Relation of Time and Use of Calendar in Life
Extending the relation of time and use of calendar in life
i.
ii.
iii.

Convert time in 12-hour system with abbreviation a.m. and p.m. to 24-hour system and vice versa
Investigate the numbers in calendar to relate days, weeks, months and year using the idea of number
patterns
Appreciate the significance of various calendars in life

Converting Quantities in Various System of Units
Converting measurement quantities based on international and non-international system with the idea of base
10
i.
ii.
iii.
iv.
v.

Convert measurement system of metre and kilogram with prefixes deci-, centi-, and milli-, and withdeca-, hecto-, and kiloConvert measurement system of litre with cubic centimetre
Convert measurement system of area using are (a) and hectare (ha) with square meter
Convert measurement of local quantities with standard quantities
Understand the unit system with power, such as metre, square metre and cubic metre

Showing Relationship Using Venn Diagram
Using Venn diagram to show relationships of numbers and figuresi.
ii.

Show relationship of squares, rectangles, rhombus, parallelogram, trapezoid and quadrilateral byusing a Venn diagram
Show relationship of numbers
Strand: Plane Figures and Space Figures

Through tessellation, figures can be extended through plane figures. Parallelogram and perpendicular lines aretools to explain properties of triangles and quadrilaterals as plane figures. They are also needed for identifying
and recognising symmetry and congruency. Plane figures are used to produce solids in space and vice versa.
Opening faces of solids would produce plane figures which are referred as nets. Activities related to building
solids from plane figures are emphasized and encouraged to facilitate finding the area of a circle throughnumerous sectors of the circle to construct a rectangle. Circles are used for explaining the nets of cylinders.
Topics:Exploring Figures with their Components in the Plane
Exploring Solids with their Components in Relation to the Plane
Exploring Figures with Congruence, Symmetry and Enlargement

Appendix E – Learning Standards for Key Stage 2 | 193

Exploring Figures with their Components in the Plane
Exploring figures with their components in the plane and use properties
i.
ii.
iii.
iv.

Examine parallel lines and perpendicular lines by drawing with instruments
Examine quadrilaterals using parallel and perpendicular lines, and identify parallelogram, rhombus,
trapezium through discussion
Find properties of figures through tessellations such as a triangle where the sum of the angles is 180
degree, a straight angle
Extend figures to polygons and expand it to circles with knowing and using its properties

Exploring Solids with their Components in Relation to the Plane
Exploring rectangular prisms and cubes with their components
i.
ii.
iii.
iv.

Identify relationship among faces, edges and vertices for drawing sketch
Explore nets of rectangular prism and find the corresponding position between components
Explore the perpendicularity and parallelism between faces of a rectangular prism
Explain positions in rectangular prisms with the idea of 3 dimensions

Extending rectangular prism to others solids such as prisms and cylinders
i.
ii.
iii.

Extend the number of relationships among faces, edges and vertices for drawing sketch
Explore nets of prisms and cylinders, and find the corresponding position between components
Distinguish prism and cylinder by the relationship of their faces

Exploring Figures with Congruence, Symmetry and Enlargement
Exploring the properties of congruence
i.
ii.
iii.

Explore properties of figures which overlap and identify conditions of congruency with corresponding
points and sides
Draw congruent figures using minimum conditions and confirm by measuring angles and sides
Appreciate the power of congruent figures by tessellation

Exploring the properties of symmetry
i.
ii.
iii.

Explore the properties of figures which reflect and identify conditions of symmetry with line and its
correspondence
Draw symmetrical figures using conditions in appropriate location
Appreciate the power of symmetry in designs

Exploring the properties of enlargement31
i.
ii.
iii.

Explore properties of figures in finding the centre of enlargement in simple case such as a rectangle
Draw enlargement of rectangle using ratio (multiplication of the value of ratio)32
Appreciate the power of enlargement in interpretation of map
Strand: Data Handling and Graphs

The process of simple data handling is introduced through data representation such as using table, bar graph,line graph, bar chart and pie chart. Graphs are utilised depending on the qualitative and quantitative data
used such as bar graph is for distinguishing and counting in every category. The discussion of producing the31
32

General case will be discussed in key stage 3, space and geometry
Ratio and rate is discussed in key stage 2 under measurement and relations

194 | Mathematics Challenges for Classroom Practices at the Lower Secondary Levelline graph includes taking data at specific intervals, suitable scale used and slope. Histogram is necessary
for interpreting the data representation of social study and science, and is also used as a special type of bar
graph. Average is introduced based on the idea of ratio for making the dispersion of bar chart even, and used
for summarising and comparing data on a table. Problem-Plan-Data-Analysis-Conclusion (PPDAC) cycles areexperienced through the process of data handling by using those data representation skills. Those skills arenecessary for learning of sustainable development.
TopicsArranging Tables for Data Representations
Drawing and Reading Graphs for Analysing Data
Using Graphs in PPDAC Cycle (Employing the Problem-Plan-Data-Analysis-Conclusion) appropriately
Applying Data Handling for Sustainable Development
Arranging Tables for Data Representations
i.
ii.
iii.

Explore how to collect multi category data based on a situation
Explore how to arrange and read multi category data on appropriate tables.
Appreciate the using of multi category tables in situations

Drawing and Reading Graphs for Analysing Data
Drawing and reading line graphs for knowing the visualised pattern as tendency
i.
ii.
iii.
iv.

Introduce line graphs based on appropriate situations such as rainfall, temperature and others
Distinguish line graph from bar graph for observation such as increase, decrease, and no-change
Introduce the graph of proportion using the idea of line graph and read the gradient by constantratio33
Appreciate the line graph in various situations

Drawing and reading band graph and pie chart for representing ratio in a whole.34
i.
ii.
iii.

Explore how to scale the bar or circle for representing ratio or percent
Use the band graph and pie chart for comparison of different groups
Appreciate the band graph and pie chart in a situation.

Reading histogram35 for analysing frequency distribution.
i.
ii.
iii.

Draw a simple histogram36 from frequency table on situations
Read various histograms for analysing data distribution
Use mean37 to compare different groups in the same situation with histograms

Using Graphs in PPDAC38 Cycle Appropriately
Identifying appropriate graphs for data handling in PPDAC cyclei.

Critique a situation and discuss the expectation before taking data for proper clarification of theobjective

Proportions are learnt in the key stage 2 under Measurement and Relations
Ratio is learnt under Key stage 2 Measurement and Relations
35
How to draw histogram is discussed in Key stage 3 under Statistics and Probability. Reading histograms is necessary in Social
studies and science36
Using ICT for drawing graph will be mentioned in mathematical activities.
37
Mean is introduced as average in key stage 2 under Measurement and Relations
38
PPDAC itself will be described in the mathematical activities later.
33
34

Appendix E – Learning Standards for Key Stage 2 | 195

ii.
iii.
iv.
v.

Plan the survey for the expectation
Take the data based on the objective of the situation
Use appropriate graphical representation which is most suitable for the objective
Appreciate the use of graphs before making a conclusion

Applying Data Handling for Sustainable Living
Applying data handling for sustainable development39 and appreciate the power of data handling for predicting
the future.
i.

ii.
iii.

Read data related to sustainable development such as emergency preparedness and resiliency,
food and energy security, world weather warming, inclusion and human connectivity in society, andlifelong learning in the changing society such as TVET (Technical and Vocational and Training) andadopting positive views for changing the society.
Understand the idea of probability as ratio and percentage in reading the data for situations related
to sustainable development such as weather report and risk analysis
Experience a project of reasonable size in data handling purposes for sustainable development andappreciate the power of data handling
Strand: Mathematical Processes – Humanity

As a follow up of Key Stage 1, activities are designed to enable an appreciation of knowledge and skills learned
and the ways of learning such as applying knowledge of number sense to solve daily problems. Mathematicalprocesses such as communication, reasoning are used to provide explanation for mathematical problems
and modelling. The ability to connect and reason mathematical ideas would trigger an excitement among
learners. Discussions of misconceptions are usually enjoyable and challenging. Mathematics learning usually
begins from situations at Key Stage 1. In Key Stage 2, the development of mathematics is possible throughthe discussions for the extension of the forms. Appreciation of ideas and representations learned become partof the enjoyable activities. Through the consistent use of representations such as diagrams, application oflearning becomes meaningful.
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 relations in situations
Establishing the idea of proportion to integrate various relations with consistency of representations
Enjoying tiling with various figures and blocks
Producing valuable explanation based on established knowledge, shareable representations and examples
Performing activities of grouping and enjoy representing with Venn diagram
Experiencing PPDAC (Problem-Plan-Data Analysis-Conclusion) cycle and modelling cycle in simple projects
in life
Preparing sustainable life with number sense and mathematical representations
Utilizing 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 volume40i.
ii.

Pose questions to develop division algorithm in vertical form using multiplication and subtraction
Pose questions to develop multiplication and division of decimal numbers using the idea of
proportionality with tables and number lines

This standard which is related to SDG as inter subject content between social studies and science
This is connected to the three strands, Extension of Numbers and Operations, Measurement and Relations, and Plane Figures
and Space Figures.
39
40

196 | Mathematics Challenges for Classroom Practices at the Lower Secondary Level

i.
ii.
iii.
iv.
v.
vi.

Pose questions to develop multiplication and division of fractions using the idea of proportionality
with tables, area diagrams and number lines
Pose questions to extend multiplication and division algorithm in vertical form to decimal numbers
and discuss about decimal points
Pose questions to use decimals and fractions in situations
Pose questions to use area and volume in life
Pose questions to use ratio and rate in life
Pose conjectures based on ideas learned such as when multiplying, the answer becomes larger

Enjoying measuring through settings and using the area and volume in situations
i.
ii.
iii.
iv.

Compare directly and indirectly areas and volumes
Set tentative units from difference for measuring area and volume41
Give the formula for the area and volume for counting units
Use measurement for communication in daily life

Using ratio and rate in situations42
i.
ii.
iii.

Understand division as partitive (between different quantities) and quotative (between same quantity)
in situations
Develop the idea of ratio and rate utilizing the idea of average and per unit with tables and number
lines
Communicate using the idea of population density and velocity in life

Using number lines, tables, and area diagrams for representing operations and relations in situations43
i.
ii.

Represent proportionality on number lines with the idea of multiplication tables
Use number lines, tables, and area diagrams for explaining operations and relations of proportionality
in situations

Establishing the idea of proportion to integrate various relations with consistency of representations44
i.
ii.
iii.
iv.
v.
vi.

Use the idea of proportion as the relation of various quantities in life
Identify through the idea of proportion using tables, letters, and graphs
Adopt the idea of proportion to angles, arcs and area of circles
Adopt the idea of proportion to area and volume
Adopt the idea of proportion to enlargement
Use ratio for data handling such as percent and understand the difficulties to extend it to proportion

Enjoying tiling with various figures and blocks45
i.
ii.
iii.
iv.

Appreciate to produce parallel lines with tessellation of figures
Explain the properties of figures in tessellations by reflections, rotations and translations
Develop nets from solids and explain the properties of solids by each of the component figures
Use the idea of tiling for calculating the area and volume

Producing valuable explanations based on established knowledge, shareable representations and examplesi.
ii.

Establish the habit of explanation by referring to prior learning and ask questions using terms suchas why, how, what, if and if not, and reply using examples and ‘for example’ in discussion
Assessing the appropriateness of explanations using representations such as generality, simplicity
and clarity

Euclidean algorithm is a method of finding the largest common divisor of two numbers.
Ratio and proportion bridge multiplication and division in situation of two quantities with reference to Extension of Numbers
and Operations and Measurement and Relations.
43
This is a bridge to the Extension of Numbers and Operations and Measurement and Relations.
44
Bridge to the three strands, Measurement and Relations, Plane Figures and Space Solids and Data Handling and Graphs.
45
Connected to the two strands, Measurement and Relations, and Plane Figures and Space Solids
4142Appendix E – Learning Standards for Key Stage 2 | 197

iii.
iv.

Use other’s ideas to produce better understanding
Use inductive reasoning for extending formulas

Performing activities of grouping and enjoy representing with Venn diagrami.
ii.

Use the idea of Venn diagram for social study
Understand classifications based on characteristics and represent by using Venn diagrams

Experiencing PPDAC (Problem-Plan-Data-Analysis-Conclusion) cycle and modelling cycle in simple projects
in life
i.
ii.
iii.
iv.

Understand the problem of context
Plan appropriate strategies to solve the problem
Gather data and analyse using appropriate methods and tools
Draw conclusion with justification based on data analysis

Preparing sustainable life with number sense and mathematical representations46
i.
ii.
iii.
iv.
v.

Use minimum and sequential use of resources in situations
Use data with number sense such as order of quantity and percentage for the discussion of mattersrelated to sustainable development
Estimate the efficient use of resources in situations
Maximize the use of resources through appropriate arrangement in a space such as a room
Understand “equally likely” of resources in situations

Utilising ICT tools as well as notebooks and other technological tools
i.
ii.
iii.
iv.
v.
vi.

Use internet data for the discussion of matters related to sustainable development
Distinguish appropriate or inappropriate qualitative and quantitative data for using ICT
Use calculators for organizing data such as averageUse calculators for operations in necessary context
Use projectors for sharing ideas as well as board writing
Enjoy using of notebooks to exchange learning experiences between each other such as inmathematics journal writingvii. Use protractors, triangular compasses, straight edges, clinometers for drawing and measuringviii. Use the idea of proportionality to use mechanism such as rotate once and move twice (wheels,
gears)
ix. Use various tools for conjecturing and justifying
Promoting creative and global citizenship for sustainable development of community using mathematics
i.
ii.
iii.
iv.
v.

Utilise notebooks, journal books and appropriate ICT tools to record and find good ideas and sharewith others
Prepare and present ideas using posters and projectors to promote good practices in the community
Listen to other’s ideas and ask questions for better designs
Utilize information, properties, models and visible representations as the basis for reasoning
Utilize practical arts, home economics and outdoor studies to investigate local issues for improving
welfare of life

It is related to Numbers and Operations, Pattern and Data Representations and Quantity and Measurement all under Key
Stage 1.
46

The Southeast Asian Ministers of Education Organisation
Regional Centre for Education in Science and Mathematics(SEAM EO RECSAM, Malaysia)1}trJ1l*~

University '!f'Tsukuba

University ofTsukuba, Japan

ISBN 978-967-930-055-0

I II IIIII

9 789679 300550

Mathematics Challenges
for Classroom Practices
at theLOWER SECONDARY LEVEL
Based on SEAMEO Basic Education Standards:
Common Core Regional Learning Standards in Mathematics
The Southeast Asian Ministers of Education Organisation
Regional Centre for Education in Science and Mathematics(SEAM EO RECSAM, Malaysia)1}trJ1l*~

I'
I
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University '!f'Tsukuba

I
I
I

University ofTsukuba, Japan

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Editors:
GAN Teck HockISODA Masami
TEH Kim HongWith Support of:
Maitree lnprasitha (KKUJSupattra Pativisan (IPSTJ
Wahyudi & Sumardyono (SEAQIMJA Collaboration Project:ISBN 978-967-930-055-0

I II IIIII

9 789679 300550%tretk~University ofTsukuba

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