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Cambridge Lower Secondary
Mathematics
TEACHER’S RESOURCE 9
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Lynn Byrd, Greg Byrd & Chris Pearce
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
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© Cambridge University Press 2021
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without the written permission of Cambridge University Press.
First published 2014
Second edition 2021
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of URLs for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate. Information regarding prices, travel timetables and other
factual information given in this work is correct at the time of first printing but
Cambridge University Press does not guarantee the accuracy of such information
thereafter.
NOTICE TO TEACHERS IN THE UK
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It is illegal to reproduce any part of this work in material form (including
photocopying and electronic storage) except under the following circumstances:
(i)
where you are abiding by a licence granted to your school or institution by the
Copyright Licensing Agency;
(ii) where no such licence exists, or where you wish to exceed the terms of a licence,
and you have gained the written permission of Cambridge University Press;
(iii) where you are allowed to reproduce without permission under the provisions
of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for
example, the reproduction of short passages within certain types of educational
anthology and reproduction for the purposes of setting examination questions.
Disclaimer
Cambridge International copyright material in this publication is reproduced under licence
and remains the intellectual property of Cambridge Assessment International Education.
Test-style questions, answers and mark schemes have been written by the authors. These
may not fully reflect the approach of Cambridge Assessment International Education.
Third-party websites, publications and resources referred to in this publication have not
been endorsed by Cambridge Assessment International Education.
Projects and their accompanying teacher guidance have been written by the NRICH Team.
NRICH is an innovative collaboration between the Faculties of Mathematics and Education
at the University of Cambridge, which focuses on problem solving and on creating opportunities
for students to learn mathematics through exploration and discussion https://nrich.maths.org.
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
CONTENTS
Contents
Introduction6
7
How to use this series
8
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About the authors
How to use this Teacher’s Resource
10
About the curriculum framework
15
About the assessment
15
Introduction to thinking and working mathematically
16
Approaches to learning and teaching
22
Setting up for success
24
Teaching notes
Number and calculation
2
Expressions and formulae
3
Decimals, percentages and rounding
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1
25
33
49
Project Guidance: Cutting tablecloths
Equations and inequalities
5
Angles72
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Project Guidance: Angle tangle
6
Statistical investigations 85
7
Shapes and measurements
90
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Fractions99
Project Guidance: Selling Apples
9
Sequences and functions
112
10 Graphs120
Project Guidance: Cinema membership
11 Ratio and proportion
131
12 Probability137
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
13 Position and transformation
147
Project Guidance: Triangle transformations
14 Volume, surface area and symmetry
160
15 Interpreting and discussing results
169
Project Guidance: Cycle training
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Digital resources
The following items are available on Cambridge GO. For more information on how
to access and use your digital resource, please see inside front cover.
Active learning
Assessment for Learning
Developing learner language skills
Differentiation
Improving learning through questioning
Language awareness
Metacognition
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Skills for life
Letter for parents – Introducing the Cambridge Primary and
Lower Secondary resources
Lesson plan template and examples of completed lesson plans
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Curriculum framework correlation
Scheme of work
Thinking and working mathematically questions
Diagnostic check and answers
Mid-point test and answers
End-of-year test and answers
Answers to Learner’s Book questions
Answers to Workbook questions
Glossary
4
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CONTENTS
You can download the following resources for each unit:
Additional teaching ideas
Language worksheets and answers
Resource sheets
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End-of-unit tests and answers
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Introduction
Welcome to the new edition of our very successful Cambridge Lower Secondary Mathematics series.
Since its launch, Cambridge Lower Secondary Mathematics has been used by teachers and children
in over 100 countries around the world for teaching the Cambridge Lower Secondary Mathematics
curriculum framework.
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This exciting new edition has been designed by talking to Lower Secondary Mathematics teachers all
over the world. We have worked hard to understand your needs and challenges, and then carefully
designed and tested the best ways of meeting them. As a result, we’ve made some important changes
to the series. This Teacher’s Resource has been carefully redesigned to make it easier for you to plan
and teach the course.
The series still has extensive digital and online support, which lets you share books with your class.
This Teacher’s Resource also offers additional materials available to download from Cambridge GO.
(For more information on how to access and use your digital resource, please see inside front cover.)
The series uses the most successful teaching approaches like active learning and metacognition and
this Teacher’s Resource gives you full guidance on how to integrate them into your classroom.
Formative assessment opportunities help you to get to know your learners better, with clear learning
intentions and success criteria as well as an array of assessment techniques, including advice on self
and peer assessment.
Clear, consistent differentiation ensures that all learners are able to progress in the course with tiered
activities, differentiated worksheets and advice about supporting learners’ different needs.
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All our resources are written for teachers and learners who use English as a second or additional
language. They help learners build core English skills with vocabulary and grammar support, as well
as additional language worksheets.
We hope you enjoy using this course.
Eddie Rippeth
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Head of Primary and Lower Secondary Publishing, Cambridge University Press
It takes a number of people to put together a new series of resources and their comments, support
and encouragement have been really important to us.
We would like to thank the following people: Anna Cox, Jan Curry and Joan Miller for their support
for the authors; Lynne McClure for her feedback and comments on early sections of the manuscript;
Ethel Chitagu, Caoimhe Ní Dhónaill, Emma McCrea and Don Young as part of the team at
Cambridge preparing the resources. We would also like to particularly thank all of the anonymous
reviewers for their time and comments on the manuscript and as part of the endorsement process.
Lynn Byrd, Greg Byrd and Chris Pearce
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ABOUT THE AUTHORS
About the authors
Lynn Byrd
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Lynn gained an honours degree in mathematics at Southampton
University in 1987 and then moved on to Swansea University to do her
teacher training in Maths and P.E. in 1988.
She taught mathematics for all ability levels in two secondary schools in
West Wales for 11 years, teaching across the range of age groups up to
GCSE and Further Mathematics A level. During this time, she began
work as an examiner. In 1999, she finished teaching and became a senior
examiner, and focused on examining work and writing. She has written or
co-authored a number of text books, homework books, work books and
teacher resources for secondary mathematics qualifications.
Greg Byrd
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After university and a year of travel and work, Greg started teaching
in Pembrokeshire, Wales, in 1988. Teaching mathematics to all levels of
ability, he was instrumental in helping his department to improve GCSE
results. His innovative approaches led him to become chairman of the
‘Pembrokeshire Project 2000’, an initiative to change the starting point
of every mathematics lesson for every pupil in the county. By this time he
had already started writing. To date he has authored or co-authored over
60 text books, having his books sold in schools and colleges worldwide.
Chris Pearce
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Chris has an MA from the University of Oxford where he read
mathematics.
He has taught mathematics for over 30 years in secondary schools to
students aged 11 to 18, and for the majority of that time he was head of
the mathematics department.
After teaching he spent six years as a mathematics advisor for a local
education authority working with schools to help them improve their
teaching. He has also worked with teachers in other countries, including
Qatar, China and Mongolia.
Chris is now a full-time writer of text books and teaching resources for
students of secondary age. He creates books and other materials aimed
at learners aged 11 to 18 for several publishers, including resources to
support Cambridge Checkpoint, GCSE, IGCSE and A level. Chris has
also been an examiner.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
How to use this series
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All of the components in the series are designed to work together.
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The Learner’s Book is designed for students to use in
class with guidance from the teacher. It contains fifteen
units which offer complete coverage of the curriculum
framework. A variety of investigations, activities,
questions and images motivate students and help them
to develop the necessary mathematical skills. Each
unit contains opportunities for formative assessment,
differentiation and reflection so you can support your
learners’ needs and help them progress.
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The Teacher’s Resource is the
foundation of this series and you’ll
find everything you need to deliver the
course in here, including suggestions for
differentiation, formative assessment
and language support, teaching ideas,
answers, unit and progress tests and
extra worksheets. Each Teacher’s
Resource includes:
•
A print book with detailed teaching
notes for each topic
•
Digital Access with all the material
from the book in digital form plus
editable planning documents, extra
guidance, worksheets and more.
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HOW TO USE THIS SERIES
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The skills-focused Workbook provides further practice
for all the topics in the Learner’s Book and is ideal for
use in class or as homework. A three-tier, scaffolded
approach to skills development promotes visible
progress and enables independent learning, ensuring
that every learner is supported.
Access to Cambridge Online Mathematics is provided with the
Learner’s Book. A Teacher account can be set up for you to create
online classes. The platform enables you to set activities, tasks and
quizzes for individuals or an entire class with the ability to compile
reports on learners progress and performance. Learners will see a
digital edition of their Learner’s Book with additional walkthroughs,
automarked practice questions, quickfire quizzes and more.
A letter to parents, explaining the course, is available to download
from Cambridge GO (as part of this Teacher’s Resource).
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
How to use this
Teacher’s Resource
Teaching notes
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This Teacher’s Resource contains both general guidance and teaching notes that help you to deliver
the content in our Cambridge Lower Secondary Mathematics resources. Some of the material is
provided as downloadable files, available on Cambridge GO. (For more information about how to
access and use your digital resource, please see inside front cover.) See the Contents page for details
of all the material available to you, both in this book and through Cambridge GO.
This book provides teaching notes for each unit of the Learner’s Book and Workbook. Each set of
teaching notes contains the following features to help you deliver the unit.
The Unit plan summarises the topics covered in the unit, including the number of learning hours
recommended for the topic, an outline of the learning content and the Cambridge resources that can
be used to deliver the topic.
Approximate
Outline of learning content
number of
learning hours
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Topic
1.1 Irrational 2
numbers
Understand that some numbers
cannot be written as fractions.
These numbers are called irrational
numbers. Square roots of 2 or 10
are examples.
Resources
Learner’s Book Section 1.1
Workbook Section 1.1
Additional teaching ideas Section 1.1
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Cross-unit resources
Language worksheet: 1.1–1.3
Diagnostic check
End of unit 1 test
The Background knowledge feature explains prior
knowledge required to access the unit and gives
suggestions for addressing any gaps in your learners’
prior knowledge.
Learners’ prior knowledge can be informally
assessed through the Getting started feature in the
Learner’s Book.
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Understand the hierarchy of natural numbers,
integers and rational numbers (Stage 8).
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HOW TO USE THIS TEACHER’S RESOURCE
The Teaching skills focus feature covers a teaching skill
and suggests how to implement it in the unit.
TEACHING SKILLS FOCUS
Active learning
There are some mathematical ideas that are not
explained in the introductory material.
Reflecting the Learner’s Book, each unit consists of multiple sections. A section covers a
learning topic.
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At the start of each section, the Learning plan table includes the learning objectives, learning
intentions and success criteria that are covered in the section.
It can be helpful to share learning intentions and success criteria with your learners at the start of a
lesson so that they can begin to take responsibility for their own learning.
LEARNING PLAN
Framework codes
9Ni.01
Learning objectives
Success criteria
• Understand the difference
between rational and
irrational numbers.
• Learners can explain the
difference between rational
and irrational numbers
written in decimal form.
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There are often common misconceptions associated with particular learning topics. These are listed,
along with suggestions for identifying evidence of the misconceptions in your class and suggestions
for how to overcome them.
How to identify
How to overcome
Ask learners to find the square root
of 2 using a calculator. Ask them if
this is exact.
Emphasise the fact that the
square roots of positive integers
that are not square numbers will
be irrational.
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Misconception
Thinking that calculators give exact
values for square roots or cube
roots.
For each topic, there is a selection of starter ideas, main teaching ideas and plenary ideas. You
can pick out individual ideas and mix and match them depending on the needs of your class. The
activities include suggestions for how they can be differentiated or used for assessment. Homework
ideas are also provided.
Starter idea
Main teaching idea
Getting started (10 minutes)
Irrational numbers (10 minutes)
Resources: Getting started exercise at the start of
Unit 1 in the Learner’s Book
Learning intention: To understand that there are
numbers on the number line that are not rational
numbers.
Description: Ask the learners to do the questions.
After a few minutes check the answers. Do this by
asking a learner to give the answer.
Resources: Calculators
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
The Language support feature contains suggestions for
how to support learners with English as an additional
language. The vocabulary terms and definitions from
the Learner’s Book are also collected here.
The Cross-curricular links feature provides suggestions
for linking to other subject areas.
LANGUAGE SUPPORT
Irrational number: a number on the number line
that is not a rational number
Rational number: any number that can be written as
a fraction
CROSS-CURRICULAR LINKS
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Many of the key words in this unit and in the
Learner’s Book will be used in different types of
businesses, in economics, engineering and science.
Guidance on selected Thinking and
working mathematically questions
Characterising and generalising
Learner’s Book Exercise 1.2, Question 14
Learners can see that multiplying by 10 is
characterised by increasing the index by 1. This works
for both positive and negative indices. A further
generalisation is that dividing by 10 decreases the
index by 1.
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Thinking and working mathematically skills are woven
throughout the questions in the Learner’s Book and
,
Workbook. These questions, indicated by
incorporate specific characteristics that encourage
mathematical thinking. The teaching notes for each unit
identify all of these questions and their characteristics.
The Guidance on selected Thinking and working
mathematically section then looks at one of the
questions in detail and provides more guidance about
developing the skill that it supports.
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Additional teaching notes are provided for the six
NRICH projects in the Learner’s Book, to help you make
the most of them.
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HOW TO USE THIS TEACHER’S RESOURCE
Digital resources to download
This Teacher’s Resource includes a range of digital materials that you can download from
Cambridge GO. (For more information about how to access and use your digital resource, please see
inside front cover.) This icon
indicates material that is available from Cambridge GO.
Helpful documents for planning include:
Letter for parents – Introducing the Cambridge Primary and Lower Secondary resources: a
template letter for parents, introducing the Cambridge Lower Secondary Mathematics resources.
• Lesson plan template: a Word document that you can use for planning your lessons. Examples of
completed lesson plans are also provided.
• Curriculum framework correlation: a table showing how the Cambridge Lower Secondary
Mathematics resources map to the Cambridge Lower Secondary Mathematics curriculum
framework.
• Scheme of work: a suggested scheme of work that you can use to plan teaching throughout
the year.
Each unit includes:
•
•
•
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•
Language worksheets: these worksheets provide language support and can be particularly
helpful for learners with English as an additional language. Answer sheets are provided.
Resource sheets: these include templates and any other materials that support activities described
in the teaching notes.
End-of-unit tests: these provide quick checks of the learner’s understanding of the concepts
covered in the unit. Answers are provided. Advice on using these tests formatively is given in the
Assessment for Learning section of this Teacher’s Resource.
Additionally, the Teacher’s Resource includes:
•
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•
Diagnostic check and answers: a test to use at the beginning of the year to discover the level that
learners are working at. The results of this test can inform your planning.
Mid-point test and answers: a test to use after learners have studied half the units in the
Learner’s Book. You can use this test to check whether there are areas that you need to go
over again.
End-of-year test and answers: a test to use after learners have studied all units in the Learner’s
Book. You can use this test to check whether there are areas that you need to go over again, and
to help inform your planning for the next year.
Additional teaching ideas
Answers to Learner’s Book questions
Answers to Workbook questions
Glossary
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•
•
•
•
•
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
14
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ABOUT THE CURRICULUM FRAMEWORK
About the curriculum
framework
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The information in this section is based on the Cambridge Lower Secondary Mathematics (0862)
curriculum framework from 2020. You should always refer to the appropriate curriculum framework
document for the year of your learners’ examination to confirm the details and for more information.
Visit www.cambridgeinternational.org/lowersecondary to find out more.
The Cambridge Lower Secondary Mathematics (0862) curriculum framework from 2020 has been
designed to encourage the development of mathematical fluency and ensure a deep understanding
of key mathematical concepts. There is an emphasis on key skills and strategies for solving
mathematical problems and encouraging the communication of mathematical knowledge in written
form and through discussion.
At the Primary level, the framework is divided into three major strands:
•
Number
•
Geometry and Measure
•
Statistics and Probability
Algebra is introduced as a further strand in the Lower Secondary framework.
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Underpinning all of these strands is a set of Thinking and working mathematically characteristics
that will encourage learners to interact with concepts and questions. These characteristics are
present in questions, activities and projects in this series. For more information, see the Thinking
and working mathematically section in this resource, or find further information on the Cambridge
Assessment International Education website.
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A curriculum framework correlation document (mapping the Cambridge Lower Secondary
Mathematics resources to the learning objectives) and scheme of work are available to download
from Cambridge GO (as part of this Teacher’s Resource).
About the assessment
Information concerning the assessment of the Lower Secondary Mathematics (0862) curriculum
framework is available on the Cambridge Assessment International Education website: https://www.
cambridgeinternational.org/lowersecondary
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
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Introduction to
Thinking and working
mathematically
Thinking and working mathematically is an important part
of the Cambridge Lower Secondary Mathematics (0862)
course. The curriculum framework identifies four pairs
of linked characteristics: Specialising and Generalising,
Conjecturing and Convincing, Characterising and
Classifying, and Critiquing and Improving. There are many
opportunities for learners to develop these skills throughout
Stage 9. This section provides examples of questions that
require learners to demonstrate the characteristics, along with
sentence starters to help learners formulate their thoughts.
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Test an idea
ica
Use an example
Characterising
and
Classifying
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Specialising and generalising
Critiquing
and
Improving
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You can download a list of the Thinking and working
mathematically questions set in this stage and their
respective characteristics on Cambridge GO.
Conjecturing
and
Convincing
Specialising
and
Generalising
Specialising and
generalising
Say what would happen
to a number if …
Give an example
Specialising
Specialising involves choosing and testing an example to see if it satisfies or does not satisfy specific maths
criteria. Learners look at particular examples and check to see if they do or do not satisfy specific criteria.
The Thinking and Working Mathematically star © Cambridge International, 2018
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INTRODUCTION TO THINKING AND WORKING MATHEMATICALLY
Example:
a
Use a calculator to find
i ( 2 + 1) × ( 2 − 1)
ii ( 3 + 1) × ( 3 − 1)
iii ( 4 + 1) × ( 4 − 1)
b Continue the pattern of the multiplications in part a.
Learners are specialising when they use the given examples to identify a pattern in the answers.
SENTENCE STARTERS
• … is the only one that …
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• I could try …
• … is the only one that does not …
Generalising
Generalising involves recognising a wider pattern by identifying many examples that satisfy the same
maths criteria. Learners make connections between numbers, shapes and so on and use these to
form rules or patterns.
Example:
c Generalise the results to find ( N + 1) × ( N − 1) where N is a number.
d Check your generalisation with further examples.
Learners are generalising when they express the pattern in an algebraic form and then choose further
examples to check its accuracy.
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SENTENCE STARTERS
• I found the pattern … so …
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Conjecturing and convincing
Talk maths
Make a statement
Conjecturing and
convincing
Persuade someone
Share an idea
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Conjecturing
Conjecturing involves forming questions or ideas about mathematical patterns. Learners say what
they notice or why something happens or what they think about something.
Example:
The table shows the maximum daytime temperature in a town over a period of 14 days.
Maximum daytime
temperature (°C)
28
26
30
Number of cold
drinks sold
25
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It also shows the number of cold drinks sold at a store each day over the same 14-day period.
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31
28
34
29
32
27
27
24
25
23
26
24
28
27
29
26
30
29
33
31
27
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Without looking at the values in the table, do you think there will be positive, negative, or no
correlation between the maximum daytime temperature and the number of cold drinks sold?
Explain your answer.
Learners are conjecturing when they read the description of the context of the question and use this to
make a prediction about what type of correlation will be shown without looking at the actual data.
SENTENCE STARTERS
• I think that …
• I wonder if …
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Convincing
Convincing involves presenting evidence to justify or challenge mathematical ideas or solutions.
Learners persuade people (a partner, group, class or an adult) that a conjecture is true.
Example:
Is it possible to estimate the number of cold drinks sold at the store when the temperature is 44 °C?
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Explain your answer.
SENTENCE STARTERS
• This is because …
• You can see that …
• I agree with … because …
• I disagree with … because …
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INTRODUCTION TO THINKING AND WORKING MATHEMATICALLY
Characterising and classifying
Spot a pattern
Organise into group
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Characterising and
classifying
Say what is the same and
what is different
Characterising
Characterising involves identifying and describing the properties of mathematical objects. Learners
identify and describe the mathematical properties of a number or object.
Example:
Copy and complete this table.
Triangle
Square
Pentagon
Hexagon
Octagon
Number of lines of
symmetry
3D prism
Number of planes of
symmetry
Triangular
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2D regular polygon
Square
Pentagonal
Hexagonal
Octagonal
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Learners are characterising when they identify the number of lines of symmetry of each 2D regular
polygon and the number of planes of symmetry of each 3D prism.
SENTENCE STARTERS
• This is similar to … so …
• The properties of … include …
Classifying
Classifying involves organising mathematical objects into groups according to their properties.
Learners organise objects or numbers into groups according their mathematical properties. They
may use Venn and Carroll diagrams.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Example:
Sort these cards into groups that have the same answer.
A 3 × 0.09
B 30 × 0.05
E 500 × 0.03
F 5 × 0.03
J 0.005 × 30
D 0.005 × 3
G 0.3 × 5
K 0.03 × 0.5
H 0.3 × 0.5
L 0.5 × 3
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I 0.003 × 5
C 0.3 × 0.07
Learners are classifying when they sort the cards into groups that have the same answer.
SENTENCE STARTERS
• … go together because …
• I can organise the … into groups according to …
Critiquing and improving
Consider the advantages
and disadvantages
and correct if required
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Evaluate the
method used
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Critiquing and
improving
Critiquing
Critiquing involves comparing and evaluating mathematical ideas for solutions to identify
advantages and disadvantages. Learners compare methods and ideas by identifying their advantages
and disadvantages.
Example:
Arun and Zara simplify the expression 6 x5 ÷ 3x 2 like this.
Arun’s method.
6x5 ÷ 3x2
6 ÷ 3 = 2 and x5 ÷ x2 = x5 – 2 = x3
So, answer is 2x3.
Zara’s method.
6x5
5–2
3
1
3x2 = 2x = 2x
So, answer is 2x3.
2
20
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INTRODUCTION TO THINKING AND WORKING MATHEMATICALLY
a Critique Arun’s and Zara’s methods? Whose method do you prefer? Why?
Learners are critiquing when they are shown two different ways to answer a question and they are
asked to decide which method they prefer and to explain why. They need to be able to follow the
working shown, and choose the method that they think is the best.
SENTENCE STARTERS
Improving
PL
E
• the advantages of … are … and the disadvantages are …
Improving involves refining mathematical ideas to develop a more effective approach or solution.
Learners find a better solution.
Example:
This is how Sofia and Marcus work out 2.6 ÷ 10−2.
Sofia
Marcus
1
1
10– 2 = 10 2 = 100
1
100
2.6 ÷ 100 = 2.6 × 1
M
= 2.6 × 100
= 260
6 26 26
2.6 = 2 10 = 10 = 10 1 = 26 × 10 – 1
10 –1
–1
–2
26 × 10 ÷ 10 = 26 × 10 –2
= 26 × 10 – 1– – 2
= 26 × 10 1 = 260
Can you think of a better method to use to divide a decimal by 10 to a negative power?
Discuss your answers with other learners in your class.
SA
Learners are improving when they are shown two different methods for working out a division and are then asked to
think of a better method. They can then discuss their methods with other learners to find the best method.
SENTENCE STARTERS
• It would be easier to …
• … would be clearer and easier to follow …
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Approaches to
learning and teaching
Active learning
PL
E
The following are the teaching approaches underpinning our course content and how we understand
and define them.
Active learning is a teaching approach that places student learning at its centre. It focuses on how
students learn, not just on what they learn. We as teachers need to encourage learners to ‘think
hard’, rather than passively receive information. Active learning encourages learners to take
responsibility for their learning and supports them in becoming independent and confident learners
in school and beyond.
Assessment for Learning
M
Assessment for Learning (AfL) is a teaching approach that generates feedback which can be used to
improve learners’ performance. Learners become more involved in the learning process and, from
this, gain confidence in what they are expected to learn and to what standard. We as teachers gain
insights into a learners’s level of understanding of a particular concept or topic, which helps to
inform how we support their progression.
Differentiation
SA
Differentiation is usually presented as a teaching approach where teachers think of learners as
individuals and learning as a personalised process. Whilst precise definitions can vary, typically the
core aim of differentiation is viewed as ensuring that all learners, no matter their ability, interest or
context, make progress towards their learning intentions. It is about using different approaches and
appreciating the differences in learners to help them make progress. Teachers therefore need to be
responsive, and willing and able to adapt their teaching to meet the needs of their learners.
Language awareness
For many learners, English is an additional language. It might be their second or perhaps their third
language. Depending on the school context, students might be learning all or just some of their
subjects in English.
For all learners, regardless of whether they are learning through their first language or an additional
language, language is a vehicle for learning. It is through language that learners access the learning
intentions of the lesson and communicate their ideas. It is our responsibility as teachers to ensure
that language doesn’t present a barrier to learning.
22
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APPROACHES TO LEARNING AND TEACHING
Metacognition
Metacognition describes the processes involved when students plan, monitor, evaluate and make
changes to their own learning behaviours. These processes help learners to think about their
own learning more explicitly and ensure that they are able to meet a learning goal that they have
identified themselves or that we, as teachers, have set.
Skills for Life
These six key areas are:
Creativity – finding new ways of doing things, and solutions to problems
Collaboration – the ability to work well with others
Communication – speaking and presenting confidently and participating effectively in meetings
Critical thinking – evaluating what is heard or read, and linking ideas constructively
Learning to learn – developing the skills to learn more effectively
Social responsibilities – contributing to social groups, and being able to talk to and work with
people from other cultures.
M
•
•
•
•
•
•
PL
E
How do we prepare learners to succeed in a fast-changing world? To collaborate with people from
around the globe? To create innovation as technology increasingly takes over routine work? To
use advanced thinking skills in the face of more complex challenges? To show resilience in the
face of constant change? At Cambridge we are responding to educators who have asked for a way
to understand how all these different approaches to life skills and competencies relate to their
teaching. We have grouped these skills into six main Areas of Competency that can be incorporated
into teaching, and have examined the different stages of the learning journey, and how these
competencies vary across each stage.
Cambridge learner and teacher attributes
This course helps develop the following Cambridge learner and teacher attributes.
Cambridge teachers
Confident in working with information and
ideas – their own and those of others.
Confident in teaching their subject and
engaging each student in learning.
Responsible for themselves, responsive to
and respectful of others.
Responsible for themselves, responsive to
and respectful of others.
Reflective as learners, developing their
ability to learn.
Reflective as learners themselves,
developing their practice.
Innovative and equipped for new and future
challenges.
Innovative and equipped for new and future
challenges.
Engaged intellectually and socially, ready to
make a difference.
Engaged intellectually, professionally and
socially, ready to make a difference.
SA
Cambridge learners
Reproduced from Developing the Cambridge learner attributes with permission from Cambridge
Assessment International Education.
More information about these approaches to learning and teaching is available to download from
Cambridge GO (as part of this Teacher’s Resource).
23to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Setting up for success
Our aim is to support better learning in the classroom with resources that allow for increased learner
autonomy, whilst supporting teachers to facilitate learner learning.
Through an active learning approach of enquiry-led tasks, open ended questions and opportunities
to externalise thinking in a variety of ways, learners will develop analysis, evaluation and problem
solving skills.
•
•
•
•
•
PL
E
Some ideas to consider to encourage an active learning environment:
Set up seating to make group work easy.
Create classroom routines to help learners to transition between different types of activity
efficiently, e.g. move from pair-work to listening to the teacher to independent work.
Source mini white boards, which allow you to get feedback from all learners rapidly.
Start a portfolio for each learner, keeping key pieces of work to show progress at
parent-teacher days. This could be used to record discussions with learners or for your
learners to select pieces of work on which they want to reflect.
Have a display area with learner work and vocabulary flashcards.
Planning for active learning
We recommend the following approach to planning. A blank Lesson Plan Template is available to
download to help with this approach.
Planning learning intentions and success criteria: these are the most important feature of the lesson.
Teachers and learners need to know where they are going in order to plan a route to get there.
2
Plan language support: think about strategies to help learners overcome the language demands
of the lesson so that language doesn’t present a barrier to learning.
3
Plan starter activities: include a ‘hook’ or starter to engage learners using engaging and
imaginative strategies. This should be an activity where all learners are active from the start of
the lesson.
SA
M
1
4
Plan main activities: during the lesson, try to: give clear instructions, with modelling and written
support; co-ordinate logical and orderly transitions between activities; make sure that learning is
active and all learners are engaged; create opportunities for discussion around key concepts.
5
Plan assessment for Learning and differentiation: use a wide range of Assessment for Learning
techniques and adapt activities to a wide range of abilities. Address misconceptions at
appropriate points and give meaningful oral and written feedback which learners can act on.
6
Plan reflection and plenary: at the end of each activity, and at the end of each lesson, try to: ask
learners to reflect on what they have learnt compared to the beginning of the lesson; extend
learning; build on and extend this learning.
7
Plan homework: if setting homework, it can be used to consolidate learning from the previous
lesson or to prepare for the next lesson.
To help planning using this approach, a blank Lesson plan template is available to download from
Cambridge GO (as part of this Teacher’s Resource). There are also examples of completed lesson plans.
For more guidance on setting up for success and planning, please explore the Professional Development
pages of our website www.cambridge.org/education/PD
24
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1 NUMBER AND CALCULATION
Unit plan
Topic
PL
E
1 Number and
calculation
Approximate
Outline of learning content
number of
learning hours
Resources
1.1 Irrational 2
numbers
Understand that some numbers
Learner’s Book Section 1.1
cannot be written as fractions.
Workbook Section 1.1
These numbers are called irrational
Additional teaching ideas Section 1.1
numbers. Square roots of 2 or 10
are examples.
1.2 Standard 2
form
Write large and small numbers in
standard form using positive and
negative powers of 10.
Learner’s Book Section 1.2
Workbook Section 1.2
Additional teaching ideas Section 1.2
1.3 Indices
Work with positive, negative
and zero powers of any positive
integer. Use index laws for
multiplication and division.
Learner’s Book Section 1.3
Workbook Section 1.3
Additional teaching ideas Section 1.3
M
2
SA
Cross-unit resources
Language worksheet: 1.1–1.3
Diagnostic check
End of unit 1 test
BACKGROUND KNOWLEDGE
For this unit, learners will need this
background knowledge:
• Understand the hierarchy of natural numbers,
integers and rational numbers (Stage 8).
• Use positive, negative and zero indices, and
the index laws for multiplication and division
(Stage 8).
• Understand the relationship between squares
and corresponding square roots, and cubes and
corresponding cube roots (Stage 7).
In this unit, learners will learn how to recognise
rational and irrational numbers. They will extend
their knowledge of numbers to using and
understanding numbers written in standard form.
They will also extend their understanding of indices
to include using the index laws for multiplication
and division with negative indices.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
TEACHING SKILLS FOCUS
responsibility for their own learning and to be less
dependent on the teacher.
Active learning reflection
At the end of the unit think about how you
responded to questions from learners. Did you tell
them the answer? Or did you ask learners questions
that would help them to think through the problem
themselves and find the solution?
PL
E
Active learning
There are some mathematical ideas that are not
explained in the introductory material. Instead
they are developed through questions in the
exercises. This gives learners the opportunity to
be more active in their learning and to think things
out for themselves. This is an important way to
help learners to be more confident, to take more
1.1 Irrational numbers
LEARNING PLAN
Framework codes
9Ni.01
Success criteria
• Understand the difference
between rational and
irrational numbers.
• Learners can explain the
difference between rational
and irrational numbers
written in decimal form.
• Use knowledge of square and
cube roots to estimate surds.
• Learners can use known
square numbers to estimate
the square root of 150.
M
9Ni.04
Learning objectives
LANGUAGE SUPPORT
SA
Irrational number: a number on the number line
that is not a rational number
Rational number: any number that can be written
as a fraction
Surd: an irrational square root or cube root
The examples of irrational numbers will be the
square roots and cube roots of natural numbers.
The word surd is used to indicate the square root
or cube root of a number. Encourage learners to
use the word surd in discussions when appropriate.
Common misconceptions
Misconception
How to identify
How to overcome
Thinking that calculators give exact
values for square roots or cube
roots.
Ask learners to find the square root
of 2 using a calculator. Ask them if
this is exact.
Emphasise the fact that the square
roots of positive integers that
are not square numbers will be
irrational.
26
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1 NUMBER AND CALCULATION
Starter idea
They will see 2.645 751 311 or similar. Note that the
number of decimal places can vary with different
calculators.
Getting started (10 minutes)
Resources: Getting started exercise at the start of Unit 1
in the Learner’s Book
Main teaching idea
Irrational numbers (10 minutes)
Learning intention: To understand that there
are numbers on the number line that are not
rational numbers.
Resources: Calculators
Description: Ask ‘What does rational number mean?’
Agree on two points:
5
16
7
15
1
7
15
17
decimal form for 12 , 18 , 3 and 6 .
5
16
Other examples of irrational numbers are the cube roots
of any number that is not a cube number. Ask learners
to decide whether the following six numbers are rational
or irrational.
25; 250; 3 343 ; 3 81; 62.5 ; 6.25
Learners should work in pairs. Check the answers after
a minute or two. After this activity, learners can start
Exercise 1.1.
Answers: 25 = 5 rational; 250 = 15.811… irrational;
3
343 = 7 rational; 3 81 = 4.326… irrational;
62.5 = 7.905… irrational; 6.25 = 2.5 rational
M
• You can write a rational number as a fraction.
• The decimal expression will either terminate or have
a repeated sequence of one or more digits.
Ask learners to use a calculator if necessary to find the
Answers:
Ask ‘Is this a rational number?’, ‘Does the decimal
number eventually terminate?’, ‘Is there a repeating
sequence of digits?’ The answer to each question is no.
The proof of this is too advanced for learners at this
stage, but you can explain to them that the square root
of any positive integer that is not a square number
(1, 4, 9, 16, …) will be similar to this. It has a decimal
expansion that does not terminate and does not have a
repeating pattern. Since it is not rational it is called an
irrational number.
PL
E
Description: Ask the learners to do the questions. After
a few minutes check the answers. Do this by asking a
learner to give the answer. Then ask them to explain
why. Use this to check that learners are familiar with
the prior knowledge required for this unit. This includes
the concept of a rational number, square roots and cube
roots, positive integer indices and the index rules for
multiplication and division (positive indices only).
Since 22 = 4 and 32 = 9 then, as you can see, 2 < 7 < 3.
12 = 12.3125 (this decimal terminates).
SA
7
18 = 18.466 666 6… (here the digit 6 repeats).
15
1
3 = 3.142 857 14… (here there is a sequence of
7
6 repeating digits 142 857)
15
17
6 = 6.882 352 941… (a calculator does not show
enough digits to see the repeating pattern. Explain that
there is in fact a pattern of 16 repeating digits and
15
·
·
6 = 6.882 352 941 176 470 5 where the sequence from
17
8 to 5 is repeated).
Now ask learners to use a calculator to find 7.
Differentiation ideas: For more confident learners,
ask them to find the squares of successive decimal
approximations to 7 = 2.645 751 3…
They will find:
2.62 = 6.76; 2.652 = 7.0225; 2.6462 = 7.001 316;
2.64582 = 7.000 257 64
Beyond this the answers will be rounded because of the
limit of the calculator display.
Ask ‘What do you notice?’ They should see that the
answers get closer to 7, but the number of decimal digits
increases by two each time. This makes it likely that the
decimal value of the square root will not terminate.
Plenary idea
Summary (5 minutes)
Resources: None
Description: Ask learners to draw a diagram to show
the relationship between integers, rational numbers and
irrational numbers. Can they do it?
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Give them a couple of minutes to discuss this in pairs.
Then ask for suggestions.
One way is to draw a Venn diagram like this:
Guidance on selected Thinking and
working mathematically questions
Specialising and generalising
Learner’s Book Exercise 1.1, Question 10
R
Learners are given several examples and asked to
identify a pattern in the answers. They need to express
this in an algebraic form and then choose further
examples to check its accuracy. An algebraic proof is
beyond the ability of learners at this stage, but they can
use examples to justify their generalisation.
PL
E
I
Homework ideas
The rectangle represents all the numbers on the number
line. The set I is the integers. The set R is the rational
numbers. The irrational numbers are outside R.
Other diagrams could be possible as long as they show
the real numbers divided into two with the integers as
a subset of the rational numbers. Every number on the
number line must be rational or irrational.
Set suitable parts of Workbook Section 1.1 as
homework. Marking should be done by learners at the
start of the next lesson. Any help or discussions with
any problems should take place immediately.
Assessment ideas
There are a number of opportunities in this section
where learners are working in pairs. Working in pairs
encourages learners to discuss what they are doing.
This then leads to clarification of ideas and selfassessment. Learners often find it easier to say they do
not understand to another learner than to a teacher
in a more public forum. Do not forget to use the idea
of traffic lights or thumb up, thumb down if you
want to get a quick assessment of understanding of a
particular concept.
M
Assessment ideas: Ask the learners to check each
other’s diagrams and assess whether they are correct.
Workbook, Section 1.1.
SA
1.2 Standard form
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ni.03
• Understand the standard
form for representing large
and small numbers.
• Learners can convert
between different notations
for large and small numbers
from a scientific context.
28
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1 NUMBER AND CALCULATION
LANGUAGE SUPPORT
Scientific notation: the same as standard form
Standard form: a way of writing very large or very
small numbers in the form a × 10n, where 1 ⩽ a < 10
and n is an integer
Mathematicians use the term standard form.
Scientists use the term scientific notation.
The meaning is identical.
Common misconceptions
Not understanding that for a
number written in standard form,
the value of a in a × 10n is always 1
or more but less than 10.
Starter idea
Powers of 10 (5 minutes)
Resources: None
How to identify
How to overcome
Check that answers in Exercise 1.2
are written correctly.
Always emphasise this point in
discussion, giving examples of
correct and incorrect forms.
PL
E
Misconception
Description: On the board, write ‘22’. Ask learners for
the value. Repeat with 23, 24, etc. Ask ‘How do you work
out each successive number?’. Learners should see you
multiply the previous number by 2.
Now, on the board, write ‘3.8 × 105’. Point out that this
is written in standard form. Ask learners to write this
number out in full. It is 380 000.
Repeat with an arbitrary choice of other numbers
written in standard form, asking learners to write down
the answer and then checking they are correct. Learners
could do this in pairs first, using peer assessment.
A common error is to assume that the index is the
number of 0s to be written on the end. Look for this
error and make sure it is corrected.
M
Repeat with powers of 10. In this case multiplying
by 10 is easy. This is often described as ‘add a 0’, but
emphasise what you are actually doing is multiplying
by 10. This is not an addition!
Explain that numbers written in this way are said to be
in standard form.
Finally, say that standard form is sometimes called
scientific notation when it is used in a scientific context.
Standard form and scientific notation are identical.
Main teaching idea
Differentiation ideas: For learners that are struggling
with this concept, use examples of a similar type.
For example, you might use 6.2 × 106 and then 6.28 × 106
and then 6.289 × 106 to clarify how many 0s are needed.
SA
Ask some reverse questions such as ‘What power of 10
is 100 000?’
Standard form (10 minutes)
Learning intention: To learn how to write large numbers
in standard form.
Resources: None
Description: On the board, write ‘6.38 × 10’. Ask for the
answer. It is 63.8 of course.
Next, on the board, write ‘6.38 × 102’. Ask for the
answer. Make sure that learners realise that this could
also be written as ‘6.38 × 100’ or ‘6.38 × 10 × 10’, three
different ways of writing the same thing.
Continue in the same way with ‘6.38 × 103’ and then
‘6.38 × 104’.
Point out that in each case you started with a number
with a single digit in front of the decimal point and that
digit was not zero.
For learners who have understood this concept, you
could give a number between 0 and 100 times a power
of 10 and ask for it to be written in standard form.
For example, 45 × 103 = 4.5 × 104.
Plenary idea
Recap (5 minutes)
Resources: None
Description: Ask learners to give examples of large
numbers where it is useful to write the numbers in
standard form. There were examples in Exercise 1.2.
29to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Learners might also think of examples in other
subject areas such as science (astronomical masses and
distances) or geography (populations).
CROSS-CURRICULAR LINKS
Standard form is used to write large and small
numbers in science. In that context it is called
scientific notation. Learners might already have
seen examples. Science teachers could give
suggestions of examples that you could include
in lessons.
Ask a similar question about small numbers. A science
teacher might be able to suggest examples here with
which the learners will be familiar.
Homework ideas
PL
E
Assessment ideas: You could use this as an opportunity
for learners to write particular real examples in standard
form and extended form to check that they can do
this accurately.
Guidance on selected Thinking and
working mathematically questions
Characterising and generalising
Learner’s Book Exercise 1.2, Question 14
Learners can see that multiplying by 10 is characterised
by increasing the index by 1. This works for both
positive and negative indices. A further generalisation is
that dividing by 10 decreases the index by 1. Continuing
this theme, multiplying or dividing by 1000 increases or
decreases the index by 3. A similar result holds for other
powers of 10.
Workbook, Section 1.2.
Set suitable parts of Workbook Section 1.2 as
homework. Marking should be done by learners at the
start of the next lesson. Any help or discussions with
any problems should take place immediately.
If you have discussed 106 = 1 million and 109 = 1 trillion
you could ask learners to research the names of larger
powers of 10.
Learners can complete the poster started in the plenary
activity in the Additional teaching ideas.
Assessment ideas
M
There are opportunities for peer assessment in the
activities and in some of the questions in Exercise 1.2.
1.3 Indices
SA
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ni.02
• Use positive, negative and
zero indices, and the index
laws for multiplication and
division.
• Learners know that 50 = 1
1
2
and that 2−3 = 3 and that
82 ÷ 85 = 8−3.
LANGUAGE SUPPORT
There is no new vocabulary in this section.
Learners should be familiar with the word ‘index’ in
this mathematical context. They should also know
that the plural of index is indices.
The word index has other more generic meanings
in English. Emphasise the specific meaning here.
30
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1 NUMBER AND CALCULATION
Common misconceptions
How to identify
How to overcome
Look at answers to questions
in Exercise 1.3 and ask specific
questions in class discussion.
Emphasise how unexpected this result
is when you first introduce it, but point
out how it follows the pattern.
Thinking that a negative power
will give a negative answer.
Look at answers to questions in
Exercise 1.3.
Point out that any (integer) power of a
positive integer is a positive number.
There will not be any negative
answers.
Starter idea
PL
E
Misconception
Thinking that n0 = 0 when n is a
positive integer.
A reminder about powers (5 minutes)
Resources: Calculator (optional)
Description: On the board, write ‘24’ and ‘42’. Ask
learners to find the value of each.
They should see that both are equal to 16. Check that
learners remember that 24 means 2 × 2 × 2 × 2. Reinforce
this with other examples if necessary.
As you move to the right, the index on the top row
increases by 1 and the number on the bottom row is
multiplied by 3.
Ask ‘How does the pattern work when you move from
right to left?’. In this case the index on the top row
decreases by 1 and the number on the bottom row is
divided by 3.
Now put two more columns on the left.
This is an example where ab = ba. In this case a = 2
and b = 4.
Main teaching idea
Negative powers (20 minutes)
SA
Learning intention: To extend the range of definitions of
powers to include zero and negative integers.
Resources: None
Description: Show this table. This table is also in the
Learner’s Book. Leave space to extend the table to the left.
32
33
34
9
27
81
Ask learners ‘What numbers go in the two empty
columns? Explain why.’
They should get this:
33
34
35
36
9
27
81
243
729
Ask ‘What numbers go in the two empty columns to
continue the pattern?’.
Learners should see that this is the pattern.
M
Ask learners to work in pairs to try to find another pair
of integers for which this is the case. This will check that
they are confident about calculating powers. In fact,
there are no others. Learners might suspect this quite
quickly. Proving it is difficult. This is a reminder to the
learners that not every problem has a solution.
32
30
31
32
33
34
35
36
1
3
9
27
81
243
729
31 = 3 will seem sensible but learners are usually
surprised by 30 = 1. They will say things such as ‘How
can multiplying no 3s make 1?’ Emphasise the fact that
you are choosing the value on the basis of the pattern.
Now you add more empty columns.
30
31
32
33 34
35
36
1
3
9
27
243
729
81
Ask ‘What now?’. The indices continue to decrease by
1 to give −1, −2, etc. Dividing by 3 on the bottom gives
1
3
1
3
1
9
1
9
1 ÷ 3 = and ÷ 3 = and ÷ 3 =
1
.
27
Leave the answers as fractions. Do not write them as
decimals. This is the table now.
32
33
34
35
36
9
27
81
243
729
31to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
3−3
3−2
3−1
30
31
32
33
34
35
1
27
1
9
1
3
1
3
9
27
81
243 729
1
9
1
3
1
3
1
3
36
The table shows that 3−1 = = 1 and 3−2 = = 2 and
3−3 =
1
1
= .
27 33
A negative power of 3 is the reciprocal of the
corresponding positive power.
1
5
So 5−2 = 2 =
1
1
1
and 2−5 = 5 = and 80 = 1.
25
2
32
Ask learners to give some other examples similar to this.
Write the examples on the board.
Differentiation ideas: If learners find this difficult, draw
up a similar table with powers of 4. Put in a few positive
powers to start. Then work through to 0 and negative
powers. Emphasise that the pattern is the same.
Plenary idea
Conjecturing
Learner’s Book Exercise 1.3, Question 17
This question is different from those that learners have
done previously. They need to develop strategies for
this kind of task. Encourage learners to discuss their
strategy, using suitable vocabulary when making their
conjecture. In part a they might recognise that 8 is a
power of 2. This will give them a way into the question.
Part b follows on from part a. In part c they should
recognise 27 as a power of 3, and so on.
PL
E
Explain that this is a general result that holds for any
positive number, not just 3.
Guidance on selected Thinking and
working mathematically questions
Check your progress (10 minutes)
Resources: ‘Check your progress’ exercise at the end
of Unit 1
Workbook, Section 1.3.
Set suitable parts of Workbook Section 1.3 as
homework. Marking should be done by learners at the
start of the next lesson. Any help or discussions with
any problems should take place immediately.
Assessment ideas
The exercise includes questions that will help learners to
understand how the rules they know for multiplication
and division of positive integer powers can be extended
to include negative powers. Encourage learners to make
their own assessments of how well they have understood
this. They can use their answers to the questions to
do this.
M
Description: Give the learners about 10 minutes to
answer the questions in the ‘Check your progress’
exercise in the Learner’s Book. Then go through the
questions, taking answers from learners and asking them
to explain their reasoning where appropriate.
Homework ideas
SA
Assessment ideas: Learners can check their answers
with a partner and assess accuracy. Use a quick trafficlight self-assessment (green = confident, yellow = a
few uncertainties, red = little understanding) to see if
learners are ready to move on.
32
Original material
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We are working with Cambridge Assessment International Education towards endorsement of this title.
2 EXPRESSIONS AND FORMULAE
Unit plan
Topic
PL
E
2 Expressions and
formulae
Approximate
number of
learning hours
2.1 Substituting 1–1.5
into expressions
2.2 Constructing 1–1.5
expressions
1–1.5
Resources
Use order of operations
with algebraic terms and
expressions, including integer
powers.
Learner’s Book Section 2.1
Workbook Section 2.1
Resource sheet 2.1
Additional teaching ideas Section 2.1
Represent situations either
Learner’s Book Section 2.2
in words or as an algebraic
Workbook Section 2.2
expression, and move between
Resource sheet 2.2
the two.
Additional teaching ideas Section 2.2
Understand how to manipulate
algebraic expressions when
applying the laws of indices.
Learner’s Book Section 2.3
Workbook Section 2.3
Resource sheet 2.3
Additional teaching ideas Section 2.3
Understand how to manipulate
algebraic expressions when
expanding the product of two
algebraic expressions.
Learner’s Book Section 2.4
Workbook Section 2.4
Resource sheet 2.4
Additional teaching ideas Section 2.4
M
2.3 Expressions
and indices
Outline of learning content
1–2
SA
2.4 Expanding
the product
of two linear
expressions
2.5 Simplifying
algebraic
fractions
1–1.5
2.6 Deriving and 1–1.5
using formulae
Understand how to manipulate
algebraic expressions when
simplifying algebraic fractions.
Learner’s Book Section 2.5
Workbook Section 2.5
Resource sheet 2.5A
Resource sheet 2.5B
Additional teaching ideas Section 2.5
Represent situations either
in words or as a formula, and
manipulate to change the
subject of a formula.
Learner’s Book Section 2.6
Workbook Section 2.6
Resource sheet 2.6
Additional teaching ideas Section 2.6
Cross-unit resources
Language worksheet: 2.1–2.6
End of unit 2 test
33to publication.
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
BACKGROUND KNOWLEDGE
• Understand that a situation can be represented
either in words or as a formula and be
manipulated using inverse operations to change
the subject of a formula. (Stage 8).
The focus of this unit is to extend learners’
knowledge and understanding of the algebra
skills learned in Stage 8, especially manipulating
algebraic expressions and the understanding of the
order of operations.
PL
E
For this unit, learners will need this background
knowledge:
• Understand that the laws of arithmetic and
order of operations (four operations, squares
and cubes) apply to algebraic terms and
expressions (Stage 8).
• Understand how to manipulate algebraic
expressions by applying the distributive law with
a single term (Stage 8) and factorising (Stage 8).
• Understand that a situation with linear integer or
fractional coefficients can be represented either
in words or as an algebraic expression (Stage 8).
TEACHING SKILLS FOCUS
Reflection
At the end of Unit 2, ask yourself:
• Did learners have useful discussions that solved
issues one of them was having?
• Did a variety of learners do the explaining –
or did you rely on just one or two learners?
• Did the learners that helped other learners
understand the work better themselves because
of the help they gave?
• Did learners that received help from others
benefit from it or did they then need help/advice
from you?
• Are all learners that require help getting it?
SA
M
Active learning
Throughout the six sections of Unit 2, if learners
do not understand or they continue to get the
same type of question incorrect, ask another
learner to explain/help. It is important that, initially,
you also listen to the explanation/help given by
another learner to check it is of good quality. You
might feel, however, that your learners have had
a lot of practice at this skill already and that these
discussions can happen without you being present.
Remind learners that the key to being successful in
this type of learning is that there is no judgement.
The learner asking for help and the learner giving
help are both learning and improving.
2.1 Substituting into expressions
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ae.01
• Understand that the laws
of arithmetic and order of
operations apply to algebraic
terms and expressions
(four operations and
integer powers).
• Learners can use the correct
order of operations in
algebraic expressions.
34
Original material
© Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
2 EXPRESSIONS AND FORMULAE
LANGUAGE SUPPORT
Counter-example: an example that shows a
statement is not true
Encourage learners to explain to you why they
set out their working in the way that they did.
Make sure they use language which shows they
understand how to use the order of operations.
Common misconceptions
How to identify
How to overcome
Not remembering the
order of operations.
Question 2.
The worked example and Question 1 should be
enough of a reminder, but discussion of errors in
Question 2 will reinforce the correct techniques.
Being confused by
questions that contain
several negative numbers.
Most of the questions in
Exercise 2.1.
Most of the questions test learners’ ability to
deal effectively with negative numbers. Holding
regular class discussions about difficulties gives
some learners time to explain to others how they
approach the various questions. This is an excellent
way to develop deeper knowledge and more
effective communication skills in learners.
PL
E
Misconception
Starter idea
Main teaching idea
Code words (5–10 minutes)
Resources: Note books, Learner’s Book Getting
started exercise
Learning intention: To repeatedly practise with a variety
of types of expressions to substitute into.
Description: After a few reminders, learners should
have little difficulty with the Getting started questions.
Before learners attempt the questions, discuss what
they remember about indices. On the board, write
‘102 × 105’. Ask for suggestions of how to answer [beware
the suggestion of multiplying the indices, leading to an
Resources: Resource sheet 2.1
SA
M
Getting started (3–5 minutes)
5
answer of 1010]. Repeat with ‘102 ’.
10
If some learners need prompting with Question 4, ask
what factorising means. If necessary, remind learners
that it is the highest common factor that is outside the
brackets.
Remind learners that this is not a test. It is designed to
help learners prepare for Unit 2. It is good practice to
allow learners to attempt the questions as individuals,
but to discuss answers/problems in pairs/small groups.
Description: Set this activity when learners are near
the end of, or once they have completed Exercise 2.1.
Learners can work individually or in small groups,
depending upon ability. This activity can be done in class
or set as a homework. Give each learner or group a copy
of Resource sheet 2.1.
Learners start by working out a numerical answer to an
expression from the ‘Code words’ table. Then they find
that number in the ‘Values of letters’ table. They write
down the corresponding letter above the expression
they started with. They continue until they have found a
letter for each expression. This reveals the message.
Answers: EXPRESS YOURSELF
Differentiation ideas: Ask less confident learners to make
sure they write down their working in their note books to
allow for checking with peers. If you think learners will
have difficulty with accuracy, let learners work in groups
in class, regularly stopping to compare answers.
35to publication.
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Plenary idea
No brackets (5 minutes)
Resources: Mini white boards
Description: On the board, copy/display these
six questions:
Work out the value of each expression when a = 2,
b = −3, c = 4 and d = −5.
b+d
2
2d − b
3
ad − 10
4
d 2 + ab
5
20 + b3
6
4b + 4
+c
2
Ask learners to show their substitutions and to work out
the answer for each.
Assessment ideas: Peer-mark. Ask learners to pay
attention to the initial substitution when checking their
partner’s work.
Answers:
1 −3 + −5 = −8
3 2 × −5 − 10 = −20
2 2 × −5 − −3 = −7
4 (−5)2 + 2 × − 3 = 19
5
6
20 + (−3)3 = −7
If a learner’s answer does not show the two sides are
equivalent it is not usually helpful to try to work out the
mistake made. Instead, they should start again and be
more careful.
Homework ideas
PL
E
1
learners to show working and not to rush. If learners
work in pairs, they can work out one line then compare,
before attempting the next line then compare again,
etc. Emphasise that too many workings is better than
too few!
4 × −3 + 4
+4=0
2
As Section 2.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Resource sheet 2.1 can be used as a homework at the
end of Exercise 2.1 or as revision for a class test. See the
main teaching idea in these notes.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
M
Guidance on selected Thinking and
working mathematically questions
Workbook, Section 2.1.
Specialising and convincing
Learner’s Book Exercise 2.1, Question 10
SA
Both sides of the equation are so large that it is sensible
to work out each side separately. Tell learners that
there are a lot of individual calculations to be made,
almost all involving negative numbers. If they make
one mistake, they will get the wrong answer. Encourage
Assessment ideas
Use Question 7 as a class ‘test’. If learners can answer
these questions, they obviously understand how to ‘use
the correct order of operations in algebraic expressions’.
Asking learners to answer Question 7 on a separate
piece of paper will show evidence of their success at this
learning objective.
2.2 Constructing expressions
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ae.03
• Understand that a situation can be
represented either in words or as
an algebraic expression, and move
between the two representations
(including squares, cubes and roots).
• Learners can move between
situations represented in
words or as an algebraic
expression.
36
Original material
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We are working with Cambridge Assessment International Education towards endorsement of this title.
2 EXPRESSIONS AND FORMULAE
LANGUAGE SUPPORT
In terms of: refers to the letters you use to
represent unknown numbers in an expression
For classes that have difficulty with language skills,
working through Question 1 carefully is essential.
It might be useful for learners to discuss and help
one another in pairs or small groups, ensuring full
understanding, before moving on to Question 2.
Common misconceptions
Incorrectly using the order of
operations after substitution.
Starter idea
How to identify
How to overcome
Questions 5 and 7.
Self- and peer-checking once a question has
been completed. Incorrect answers will often be
due to incorrect use of the order of operations.
PL
E
Misconception
Area and perimeter (2–5 minutes)
If required, repeat using algebra:
x
y
Perimeter = y + x + y + x = 2x + 2y
Area = x × y = xy
M
Resources: Mini white boards if you decide to ask
learners questions
Description: You might need to remind learners of the
difference between perimeter and area, although work
done in Section 2.2 should mean this is unnecessary.
A quick example without using algebra works well with
less confident learners.
On the board, draw/display this rectangle:
Discuss methods of working out the perimeter, e.g.
adding all lengths in turn as shown, or 4 × 5.
Discuss methods of working out the perimeter, e.g.
adding all lengths in turn as shown, or, 2 × x + 2 × y,
2 × (x + y).
3cm
5cm
SA
Perimeter = 5 + 3 + 5 + 3 = 16 cm
Area = 5 × 3 = 15 cm2
Discuss methods of working out the perimeter, e.g. adding
all lengths in turn as shown, or 2 × 3 + 2 × 5, 2 × (3 + 5).
If required, repeat with a square:
5 cm
5cm
Perimeter = 5 + 5 + 5 + 5 = 20 cm
Area = 5 × 5 = 25 cm2
This may be a convenient time to ask learners what they
can tell you about the inverse of 52 = 25, i.e. the square
root of 25 cm2 is 5 cm
And with a square:
x
x
Perimeter = x + x + x + x = 4x
Area = x × x = x2
Main teaching idea
Rods (10–15 minutes)
Learning intention: To understand that a situation can
be represented in words and as an algebraic expression,
and move between the two representations.
Resources: Note books, Learner’s Books
37to publication.
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Description: Set this activity any time after Question 6.
Learners can work individually or in small groups,
depending upon ability.
Ask learners to look again at the four coloured rods in
Question 6.
Assessment ideas: Depending upon your class, you
might want learners to look at one rectangle at a time
and to check answers as you work through, or you might
ask learners to answer all four questions, dealing with all
issues at the end.
Ask them to find different ways to make a total length
of 12x + 6. They might include the two ways used in part
a ii of Question 6.
Guidance on selected Thinking and
working mathematically questions
Learners should show that they have a correct
combination of rods by setting out their working as in
the box shown alongside Question 6.
Conjecturing and convincing
PL
E
Learner’s Book Exercise 2.2, Question 9
Answers: 6 green, 4 green + 1 red + 1 yellow,
2 green + 2 red + 2 yellow, 3 red + 3 yellow,
6 blue + 2 yellow and 3 blue + 3 green + 1 yellow
Differentiation ideas: To extend the activity, again
referring learners to the four coloured rods in Question
6, ask learners to find different ways to make a total
length of 12x + 12. They might include the two ways
used in part c i of Question 6.
Again, learners should show that they have a correct
combination of rods by setting out their working as in
the box shown alongside Question 6.
To make sure that learners are starting part b from the
correct point, first allow them to self-check their answer
to part a. Some learners – especially the more confident
learners who did not read all parts of Question 9
before starting – might have simplified their answer to
part a already. Use this to show that reading the whole
question before starting it is a good idea.
M
Answers: 12 blue, 9 blue + 1 red + 1 green,
6 blue + 2 green + 2 red, 6 blue + 3 red + 1 yellow,
3 blue + 3 green + 3 red, 6 red + 2 yellow,
5 red + 2 green + 1 yellow and 4 red + 4 green.
There are many ways of successfully showing that the
answer from part a simplifies to 2(2x2 + 9). The starting
point for part a depends upon how learners normally
work out the perimeter. They might start by adding
all lengths in turn or by multiplying each of the two
dimensions given by 2 and adding, or by adding the
two dimensions given, then multiplying by 2. The first
two methods will usually lead learners into having to
factorise. With the third method, factorising can be
avoided. Emphasise that with any method, learners
should show full working.
Plenary idea
SA
Perimeter and area (5 minutes)
Resources: Mini white boards
Description: On the board, sketch/display:
1
x
2
4
3
a
x–5
3b
4x
4
3x + 2
2x
Ask learners to write an expression for the area of each
rectangle [4x, 4ax, 3bx − 15b, 6x2 + 4x]. When you have
checked the answers and dealt with any issues, ask
learners to write an expression, in its simplest form,
for the perimeter of each rectangle [8 + 2x, 2a + 8x,
6b + 2x − 10, 10x + 4].
In part c you might need to guide some learners into
realising that the x value only appears when being
squared, so its value of 2 or −2 will always give the same
answer of 4. Make sure that learners give their own
reasons for why Arun is correct.
Homework ideas
Workbook, Section 2.2.
As Section 2.2 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Assessment ideas
Give each learner an exit ticket, cut out from Resource
sheet 2.2: Exit ticket. Learners should complete the
exit ticket just before leaving class. Allow 3–5 minutes
to complete.
38
Original material
© Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
2 EXPRESSIONS AND FORMULAE
There is space for a name on top of the exit ticket.
Ask for learners to put their name only if you want
the accuracy of their answers to count towards a
formal assessment. When learners think that their
feedback is anonymous, the information they give is
often more honest. Reading what learners think they
have learned and how you might help them further will
help you to clarify teaching points for the class for the
next lesson or in revision lessons to come.
LEARNING PLAN
Framework codes
9Ae.02
LANGUAGE SUPPORT
PL
E
2.3 Expressions and indices
Learning objectives
Success criteria
• Understand how to
manipulate algebraic
expressions including:
applying the laws of indices.
• Learners can use the laws
of indices in algebraic
expressions.
For example: to read ‘x2 × x3’ as ‘x to the power
of 2 (or x squared) multiplied by x to the power
4
of 3 (or x cubed)’ and to read ‘( z 3 ) ’ as ‘z to the
power of 3 (or z cubed) all to the power of 4’, etc.
M
There is no new vocabulary in this section.
When working through the Worked example and
the answers to Question 1 it might be useful to ask
different learners to read out the question.
Common misconceptions
How to identify
How to overcome
Forgetting the rules for negative
indices.
Question 9.
Read out the ‘Tip’ box before learners
start Question 9. Check by asking
learners to write 10−5 as a fraction.
SA
Misconception
Starter idea
Indices laws (5–10 minutes)
Resources: Mini white boards or note books
Description: Many learners benefit from a more detailed
explanation of the general rule for multiplication of
numbers expressed as powers: xa × xb = xa + b.
Set this activity before learners start Exercise 2.3.
Work through an explanation of the general rule for
multiplication of numbers expressed as powers.
• On the board, write ‘x2 × x4’.
• Ask learners to suggest another (longer) way of
writing x2. Write ‘x × x’ under the x2 term.
• Ask learners to suggest another (longer) way of
writing x4. Write ‘x × x × x × x’ under the x4.
• Ask learners to suggest another (shorter) way of
writing x × x × x × x × x × x.
• Emphasise that x2 × x4 = x2 + 4 = x6, which is quicker
than writing out all the xs in full and counting.
Now work through a similar explanation of the general
rule for division of numbers expressed as powers.
• On the board, write ‘x5 ÷ x2’.
• Ask learners to suggest another (longer) way of
writing out x5 ÷ x2. You might need to guide them
towards a fraction as an alternative.
39to publication.
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
• Demonstrate how to simplify
x×x×x×x×x
by
x×x
• Ask learners to suggest another (longer) way of
writing out 2x3.
• Ask learners to suggest another (longer) way
of writing out (2x3)2. Be aware that some
learners will suggest 2 × x × x × x × 2 rather than
2 × x × x × x × 2 × x × x × x.
• Ask learners for another (shorter) way of writing
this.
• Emphasise that (2x3)2 = 22x3 × 2 = 4x6, which is quicker
than writing out the 2s and all the xs in full and
counting.
Main teaching idea
Plenary idea
Indices (5–10 minutes)
Resources: Note books
PL
E
cancelling common factors. Cancel an x from the top
and from bottom of the fraction, then do it again.
• Underneath the simplified fraction, write ‘x × x × x’.
• Ask learners for another (shorter) way of
writing this.
• Emphasise that x5 ÷ x2 = x5 – 2 = x3, which is quicker
than writing out all the xs in full, cancelling and
counting.
If you have time, on the board, write ‘(2x3)2’.
Differentiation ideas: For learners who are confused,
ask them to look at the first rule given in the
introduction and to work out the answer of x2 ÷ x2. This
might not give them the answer to the question, but it
can be built on later when discussing that any number/
expression divided by the same number/expression = 1.
1
a2 × a3 =
2
3b2 × 2b3 =
3
a7 ÷ a4 =
4
9b8 ÷ 3b2 =
5
9c 4
=
12c 7
6
8d 7
=
4d
7
e3 ÷ e5 =
8
f 3÷f 2=
9
6g6 ÷ 12g12 =
10 (a4)3 =
11 (4b5)2 =
Assessment ideas: Either peer-mark so learners can get
instant feedback on any types of question they need to
revise further, or take the answers in yourself for a more
formal assessment of the learning objectives.
M
Question 3, Think like a mathematician
(5 minutes)
Description: Ask learners to copy and complete each of
the following:
Learning intention: To understand that x2 ÷ x2 = 1.
Resources: Note books, Learner’s Books
SA
Description: Ask learners to answer part a individually.
Make sure all learners have written an answer and have
given a reason. When completed, ask for a show of
hands for each of the three options: Arun, Sofia and
Zara. Hopefully no learner will have voted for Zara, but
if they have, ask for their reasons first. Do not tell them
that they are incorrect. Then ask for those who chose
Arun (there are usually a few) for their reasons. Finally,
ask for those that chose Sofia’s answer for their reasons.
Return to those learners that chose Arun. They probably
reason (incorrectly) that x2 ÷ x2 = x2–2 = x0 = 0. Suggest
learners substitute in a number for x. Suggest some use 2,
others 3, others 5 and others 10 to substitute. Ask all
learners to work out x2 for their ‘x’ and then to work
out x2 ÷ x2. Obviously, all learners should get the answer
of 1. Learners should now answer parts c and d. When
completed, discuss answers and end the discussion when
all learners understand that just because x2 ÷ x2 = x2–2 = x0
looks like it might = 0, it does not mean it is true! x0 = 1.
In fact, any number or letter to the power zero equals one.
Answers:
1 a5
2 6b5
3
a3
4
3b6
5
6
2d 6
7
e−2 or 12
3
4c3
9
1
1 −6
g or 6
2g
2
10 a12
e
8 f
11 16b10
Guidance on selected Thinking and
working mathematically questions
Conjecturing and convincing
Learner’s Book Exercise 2.3, Question 8
Learners should recognise by now that the way to
answer this type of question is to answer the question
3
set – simplify the expression (3x2 ) – and compare the
answer to those given. Learners can use the difference
between their answer and the incorrect answer to explain
the error made by, in this case, Marcus.
40
Original material
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We are working with Cambridge Assessment International Education towards endorsement of this title.
2 EXPRESSIONS AND FORMULAE
Homework ideas
Assessment ideas
Workbook, Section 2.3.
At various times during Section 2.3, ask individual
learners short, easy-to-answer questions that check
knowledge. Ask questions without warning, and only
ask three or four learners questions. Later in the lesson,
ask three or four different learners. For example, ask
questions such as:
As Section 2.3 might take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
3
x2 × x3, 5x2 × x3, 5x2 × 3x3, x6 ÷ x4, 10x6 ÷ 2x2, 8x 5 ,
PL
E
10 x
10 x 4
, x4 ÷ x6, x5 ÷ x4, 3x3 ÷ 6x6, (x2)3, (2x5)3, etc.
5x
2.4 Expanding the product of two linear expressions
LEARNING PLAN
Framework codes
Success criteria
• Understand how to
manipulate algebraic
expressions including:
expanding the product of
two algebraic expressions.
• Learners can expand two
brackets.
M
9Ae.02
Learning objectives
LANGUAGE SUPPORT
SA
Brackets: used to enclose items that are to be seen
as a single expression
Difference of two squares: an expression of the
form a2 – b2. It can be written as (a + b)(a – b)
Expand: to multiply the terms inside one bracket
by the terms inside the other bracket
Perfect square: a perfect square is a number, or
expression, that can be written as the product of
two equal factors, e.g. 3 × 3 = 9, x × x = x2,
(x + 1)(x + 1) = x2 + 2x + 1
Encourage learners to explain how they are
expanding brackets. This will help you to observe
if they are confident with this skill. It will also help
them to remember the method.
Common misconceptions
Misconception
How to identify
How to overcome
Not completing the product by
not multiplying each pair of terms.
Question 2.
Thorough discussion throughout the
worked example and discussion when
checking answers to Question 1.
Forgetting to simplify the
expression after expanding it.
Question 2.
Thorough discussion throughout the
worked example and discussion when
checking answers to Question 1.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Starter idea
Expanding a bracket (5–10 minutes)
Resources: Mini white boards or note books
Description: Learners to work in pairs or small groups.
Ask learners to expand and, where possible, simplify
these expressions:
2
5(4 − x)
3
6(2 − 3x)
4
x(x − 3)
5
x(3 + 4x)
6
x(2x − 5y)
7
5x(6y − 8x)
8
x(x + 2) + x(x + 5)
9
x(2x + 5) − x(x + 3)
PL
E
4(x + 3)
When completed, ask each pair/group to compare their
answers with another pair/group and to discuss any
differences.
Now display answers on the board and check for any
misunderstandings or gaps in knowledge.
3
6
9
12 − 18x
2x2 − 5xy
x2 + 2x
Answers:
Part A: (n + 4)(n + 1) − n(n + 5) = n2 + 5n + 4 −
(n2 + 5n) = 4
Part B: (n + 6)(n + 1) − n(n + 7) = n2 + 7n + 6 −
(n2 + 7n) = 6
Part C: (n + 10)(n + 1) − n(n + 11) = n2 + 11n + 10 −
(n2 + 11n) = 10
Part D: The difference will be 12. (n + 12)(n + 1) −
n(n + 13) = n2 + 13n + 12 − (n2 + 13n) = 12
Differentiation ideas: A few learners might require you
to repeat the help given to the class for part d in Activity
2.4 on a one-to-one basis.
Only use the extension if learners complete the original
activity without too many problems.
M
2 20 − 5x
5 3x + 4x2
8 2x2 + 7x
If suitable, set this extension to Activity 2.4. Learners
can work individually or in small groups, depending
upon ability. Give each learner or group a copy of
Resource sheet 2.4.
Explain that this is an extension of Activity 2.4 in
Exercise 2.4. Learners should follow the instructions on
the sheet.
1
Answers:
1 4x + 12
4 x2 − 3x
7 30xy − 40x2
get the majority of the class to realise that if they start
with n, the next number is n + 1. They should then be
able to fill in the other two boxes with ‘n + 5’ and ‘n + 6’.
Main teaching idea
Plenary idea
Activity 2.4 (10 minutes (+ 10–15 minutes
for extension activity))
Online brackets (10–20 minutes)
Learning intention: To learn to deal with more
complicated algebra.
SA
Resources: Note books, Learner’s Books
Description: Allowing a number to be represented by
n and the next to be n + 1 is a large conceptual leap
for many learners. Many learners will require some
help (and convincing!) to believe this is true and
actually usable.
Learners must carefully follow the instructions given in
parts a and b to realise that the difference is 5 each time.
Many learners might require help with filling in the
block in part d. On the board, draw a 2 × 2 grid. In the
top left box write the number ‘8’. Ask for the top right
number [9]. Erase the 8 and instead write ‘41’, which is
not on their grid. Again, ask for the number in the top
right box [42] and ask how it was calculated. Insist that
42 is not just the next number, it is 41 + 1. This should
Resources: Mini white boards or suite of computers/
tablets, calculators
Description: Enter ‘transum.org, brackets’ into a search
engine. When directed to the page, learners complete
the level 3 material. This material changes every time it
is clicked on, so no two sessions are the same. Using an
electronic white board will mean that all learners see the
same questions. Using a suite of computers/tablets will
result in learners seeing different questions.
This is a free resource and learners can access this too.
They might want to practise at home, so make sure they
write down the web address of this resource. There are
many other similar resources available.
Remember that the questions increase in difficulty
further down the page, so think about which learner to
ask which question unless learners’ times tables are very
good or they are able to use calculators.
42
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2 EXPRESSIONS AND FORMULAE
When working as a class, ask a learner to show their
answer, discuss as a class if the answer is probably
correct or not. If it is thought that the answer is correct,
type it into the answer box and check. if not, discuss
possible errors and retry.
Discuss part d when completed, asking for how learners
decided the answer would be x2 − 100. It is important
for learners to be able to answer part e and that this
discussion includes that the number (10 in this case) has
been squared.
If each learner has individual access to the site, you
might decide to limit the time taken on the activity. This
will lead to differentiation by outcome, but hopefully all
learners will be successful and practise a valuable skill.
If learners find it difficult to generalise in part e, remind
them of the discussion about the 10 (or −10) being
squared to give the 100 at the end of their answer.
Homework ideas
Workbook, Section 2.4.
PL
E
Assessment ideas: Answers can be discussed and
checked online. Alternatively, at or towards the end of
Exercise 2.4, when directed to the page, learners could
use part of the level 6 material as a test to be answered
in their note books.
Guidance on selected Thinking and
working mathematically questions
Specialising and generalising
Learner’s Book Exercise 2.4, Question 9
After so much practice, learners should have little
difficulty in working out the simplified expansions for
parts a and b. This is the first time that an expansion has
led to just two terms. Learners will probably notice that
there are no x-terms in these expansions, but might not
realise why.
You could ask learners that have difficulty with
questions 1 and especially 2, to repeat Question 2 using
the box method. If learners have used the box method
in Question 4, it is probably not necessary to answer
all questions, just parts d and e. They already have the
answers, but this extra practice will help them to learn
the technique more fully.
Assessment ideas
Use Question 6 as an extended hinge-point question.
By now learners should be able to look at each part of
Question 6 and to decide which is the correct expansion.
Ask any learner with incorrect answers to expand the
brackets, simplify and show you their results. This will
be an instant check for you to decide if you need to
focus on any misconceptions displayed in their answers.
Some learners will always need to expand the brackets
in questions such as Question 6. It is good practice to
encourage this thoroughness.
SA
M
When learners have completed part c it is useful
to discuss this as a class to ensure that all learners
understand what is happening with this type of
question. They should understand that questions such
as (x + 2)(x − 2) are very different in outcome from either
(x + 2)(x + 2) or (x − 2)(x − 2).
As Section 2.4 might take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
2.5 Simplifying algebraic fractions
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ae.02
• Understand how to
manipulate algebraic
expressions including:
simplifying algebraic fractions.
• Learners can simplify
algebraic fractions.
43to publication.
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
LANGUAGE SUPPORT
Algebraic fraction: a fraction that contains an
unknown variable, or letter
You might need to review and remind learners
about previously learnt vocabulary, such as
denominator, numerator, cancel, simplified,
substitute, expression, etc.
Common misconceptions
How to identify
How to overcome
Making similar mistakes with
algebraic fractions as learners
might make with simple fractions.
They might simply multiply the
denominators to find a common
multiple, often resulting in
unnecessarily large numbers. They
might convert the denominator
correctly, but not change
the numerator.
Most of the questions in
Exercise 2.5.
Starter idea activity.
Being confused by the mixture of
fractions and algebra. Forgetting
the rules they have been using
well up until now and ‘simplifying’
terms that cannot be simplified,
Question 6.
Work done and discussions should
stop this from happening. Check
for incorrect answers in Question 6
to ensure this misconception is
not seen.
M
PL
E
Misconception
for example, writing a + 2b = 2ab
8
8
Main teaching idea
Fractions (2–5 minutes)
Resources: Mini white boards or note books
Question 5, Think like a mathematician
(5–10 minutes)
Description: Use this starter idea before working
through the worked example. Set learners a few basic
Learning intention: To understand how to use
substitution to check answers to a simplification.
SA
Starter idea
1
3
1 1
3 2
1 1
4 3
1 3
9 4
5
7
fraction questions, such as + , + , − , + , etc.,
to remind them of the basic skills they need.
When working through the worked example, it is
perfectly acceptable for learners to think of
1
1
2
y − y = y.
3
9
9
y y
− as
3 9
Emphasise that these are just fractions. The same skills
learners use with ordinary fractions apply equally here.
Learners must still ensure that denominators are the
same before adding or subtracting, and if they multiply
the denominator of a fraction, they must multiply its
numerator by the same number.
Resources: Note books, Learner’s Books
Description: Most learners should have little difficulty
answering and discussing parts a to e. When
completed, ask learners to answer part f i. When
completed, ask learners what numbers they substituted
in for x and y. Also ask ‘Why did you choose those
numbers?’. Usually, learners will have chosen x = 1 and
y = 2 because they were the numbers used in the book
for the substitutions in parts a and b. If no learner has
used x = 3 and y = 9, then guide learners to discussing
easier numbers to use. Help them towards the idea
44
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2 EXPRESSIONS AND FORMULAE
that x should be a multiple of three (as
2x
) and that
3
the y should be a multiple of nine (as y ). On the board,
9
write or display:
Assessment ideas: Answers should be checked online
using the ‘Check’ button. Alternatively, at or towards
the end of Exercise 2.5, when directed to the page,
learners could use the first five questions of the level 3
material and the first question of the level 4 material as
a test to be answered in their note books.
Substituting using x = 1
and y = 2
Substituting using x = 3
and y = 9
2x y 2 × 1 2 2 2
+ =
+ = + =
3
9
3
9 3 9
6 2 8
+ =
9 9 9
2x y 2 × 3 9
+ =
+ =
3
9
3
9
6 9
+ = 2 +1 = 3
3 9
Guidance on selected Thinking and
working mathematically questions
6 x + y 6 × 3 + 9 18 + 9 27
=
=
= =3
9
9
9
9
Learner’s Book Exercise 2.5, Question 10
Critiquing and convincing
PL
E
6x + y 6 × 1 + 2 6 + 2 8
=
=
=
9
9
9
9
or
Ask learners ‘Which seems easier to answer?’
Tell learners that they can use any numbers they want
when checking, but they must substitute the same
numbers into both the question and the answer.
Ask learners to complete part f, before self-marking.
After checking answers, discuss which numbers they
used for the substitution and why.
Learners will hopefully start answering by simply
expanding Shania’s answer, which gives Taylor’s answer.
Other learners might show by the reverse method of
factorising Taylor’s answer. Both methods show that
learners can demonstrate this skill.
Differentiation ideas: For learners that do not have the
depth of understanding to decide which numbers to
substitute, allow them just to use 1 and 2.
Explaining which of the two methods a learner prefers
can be difficult for some learners. Writing ‘I prefer
Shania’s method because it is easier’ is acceptable, but you
should follow this by asking the learner to explain to you
why it seems an easier method. You are more likely to get
a proper critique of one or both methods this way.
Plenary idea
Homework ideas
Workbook, Section 2.5.
M
Online brackets (5–10 minutes)
Resources: Mini white boards or suite of computers/
tablets, calculators
SA
Description: Enter ‘transum.org, algebraic fractions’
into a search engine. When directed to the page, learners
complete the level 3 material. This material changes
every time it is clicked on, so no two sessions are the
same. Using an electronic white board will mean that
all learners see the same questions. Using a suite of
computers/tablets will result in learners seeing different
questions.
Ask all learners to answer the first three questions.
Check answers and deal with any issues. Then ask
learners to answer the next two questions. Again, check
answers and deal with any issues. If required, click again
on the level 3 tab and repeat with a new set of questions.
If each learner has individual access to the site, you
might decide to limit the time taken on the activity,
allowing some learners to be able to try several different
sets of questions. This will lead to differentiation by
outcome, but hopefully all learners will be successful
and practise valuable skills.
As Section 2.5 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Assessment ideas
Use peer-marking from Question 4 onwards.
Learners must concentrate on deciding if the work
they are checking is clear, that workings are correct
and easily followed.
Put learners into pairs or small groups who will mark
each other’s work.
This method might take longer than just going over the
answers with learners for them to self-mark and to write
their own notes on improvements to be made, but it is
usually worth the extra time for the greater depth of
understanding it can give.
45to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
After every (or every other) question, learners swap
note books. You go over the answers, learners mark
appropriately, and ask you if they are not sure about
how to mark something unusual. This highlights
misconceptions and possibly correct (but unorthodox)
methods not yet discussed. As learners mature,
learning to mark each other’s work seems to allow for a
greater depth of understanding for many, which, when
combined with class discussions about a point, makes
for real progress in many learners.
LEARNING PLAN
Framework codes
9Ae.04
LANGUAGE SUPPORT
PL
E
2.6 Deriving and using formulae
Learning objectives
Success criteria
• Understand that a situation
can be represented either
in words or as a formula
(including squares and
cubes), and manipulate
using knowledge of inverse
operations to change the
subject of a formula.
• Learners can write, use and
change the subject of a
formula.
Encourage learners to describe their methods
in words. This will help you to assess their
understanding. It will also help them to remember
the method.
M
Changing the subject: rearranging a formula or
equation to get a different letter on its own
Subject of a formula: the letter that is on its own on
one side (usually the left) of a formula
SA
Common misconceptions
Misconception
How to identify
How to overcome
Making mistakes with order of operations
when rearranging formulae. For example,
learners might try to make a the subject of
v = u + at by actually using the rules of the
order of operations, dividing by t first and
then subtracting u.
Question 5.
Worked example and
discussions when checking
Question 5.
Starter idea
Inverse operations (2–5 minutes)
Resources: None
Description: Hold a brief discussion about the four
rules and their inverse operations. For example, ask ‘If
x + 7 = 20, how do you make x the subject?’ If learners
just try to give the answer of 13, agree that 13 is the
value of x, but ask how to show the method (i.e. inverse
of + 7 is − 7). Similarly, use x − 7 = 20, 4x = 20, x = 20.
3
Next, discuss how to make x the subject when x2 = 25.
Learners might, after working through the four
previous rules, give the answer of square rooting
46
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2 EXPRESSIONS AND FORMULAE
Main teaching idea
Differentiation ideas: If learners cannot think clearly
enough to visualise the shapes, on the board,
draw/display:
Plenary idea
Legs on goats (3–5 minutes)
PL
E
straight away. It is, however, more likely that learners
will understand that x = 5 (ignore the other possible
value of −5; if it is mentioned, tell learners that you
will ignore the negative square roots today to simplify
their discussions), but might need reminding/guiding
to remembering about square roots. When you have
discussed this and learners remember that squares and
square roots are inverse operations, ask ‘How could
you make x the subject when x2 = a?’ Then ask ‘How
could you make x the subject when x3 = a?’ You might
need to guide learners by asking ‘If the inverse of
x squared is the square root of x, what is the inverse of
x cubed?’
Resources: Mini white boards or note books
Description: On the board, copy/display the following:
1
Write a formula for the number of legs, L, on any
number of goats, G.
Learning intention: To work out Euler’s formula and to
rearrange it.
2
Use your formula from Question 1 to work out L
when G = 8.
Resources: Note books, Learner’s Books
3
Description: Learners should have little difficulty in
completing the table in part a, even if they need to make
their own sketches to help.
Rearrange your formula in Question 1 to make G
the subject.
4
Use your formula in Question 3 to work out G
when L = 80.
Question 3, Think like a mathematician
(10 minutes)
Assessment ideas: Once learners have finished, ask to
see answers to Question 1. Some learners might have
written G = 4L rather than L = 4G. If they have made
this mistake, explain why their formula is incorrect.
With this incorrect formula, their next answers would
be 2, G = 4L and 320. These answers are wrong, but they
would show that these learners have correctly used their
formula for the other three questions.
M
Working out the formula will be difficult for many
learners – even though they have already used a similar
table and worked out Euler’s formula in Stage 8. A little
guidance from you might be required, but hopefully
most help will come from other learners in the class.
SA
It is not important which rearrangement of Euler’s
formula learners write down, as long as it is based on
the correct formula and that it only has one letter or
the number 2 as the subject (E = F + V − 2, F = E − V + 2,
V = E − F + 2 or 2 = F + V − E). Insist that learners check
that their formula works for all of the shapes in the table.
Even with the Tip, part d can be confusing for some
learners. If required, tell learners that both of the shapes
are types of pyramid. They must work out what type of
base these pyramids have.
Learners’ answers to part e indicate depth of
understanding. If a learner says that, as F = 0, the
shape has no faces, they need some help understanding
what they have just suggested. Other learners will
say that the shape is impossible, but give no other
information. Only learners with good understanding
will not only say the shape cannot be drawn, but will
extend that to say that a shape with 5 edges and 7
vertices cannot be drawn.
For those learners with the correct answer to Question 1
[L = 4G], allow self-marking for the rest of the answers:
32, G = L and 20.
4
Guidance on selected Thinking and
working mathematically questions
Conjecturing and convincing
Learner’s Book Exercise 2.6, Question 13
Some learners will try to convert both the 30 °C to °F
and the 82 °F to °C. While this might be good practice in
using formulae, it shows that the learner is not thinking
logically or planning ahead before attempting this and
therefore probably most other questions.
47to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
All the learner needs to do is to convert one of the
temperatures – probably the 30 °C, as it is an easy
number to use in the formula given – and compare with
the other temperature. It is not sufficient just to say that
·
86 > 82 (or 30 > 27.7), the learner must write that 30 °C
is higher than 82 °F. Tell learners that they will often
have to check that their maths working is ‘translated’
back into language.
Workbook, Section 2.6.
Assessment ideas
Use Question 9 as an extended hinge-point question.
While the last three questions require concentration and
knowledge, it would be a mistake to move on from this
question without all learners being able to answer these
types of problems. Some learners will probably make
some mistakes. Ask successful learners to explain their
methods. You could also set other similar questions,
for example:
PL
E
Homework ideas
Learners could then improve/update their individual
posters if necessary. Learners could store their posters at
home as a possible revision tool towards mid-term/endof-year tests.
As Section 2.6 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
a
d = 5t − 2
b
q= t +r
c
y=
3+t
e
d
e=
2t − h
f
4
SA
M
You could ask learners to make a poster containing
everything they think they need to remember for the
end-of-unit test. The following lesson, it is important
to share the posters in class (e.g. spread the posters out
over a few desks for everyone to look at), rather than
marking them. Discuss the different posters as a class.
When the class agree that a point is important, that
key point could be copied onto the board (by you or
a learner). Agree on as many key points as possible.
Make t the subject of each of these formulae:
48
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3 DECIMALS, PERCENTAGES AND ROUNDING
Unit plan
PL
E
3 Decimals,
percentages and
rounding
Approximate Outline of learning content
number of
learning hours
Resources
3.1 Multiplying
and dividing by
powers of 10
1–1.5
Multiply and divide integers
and decimals by 10 to the
power of any positive or
negative number.
Learner’s Book Section 3.1
Workbook Section 3.1
Resource sheet 3.1A
Resource sheet 3.1B
Additional teaching ideas Section 3.1
3.2 Multiplying
and dividing
decimals
1–2
Estimate, multiply and divide
decimals by integers and
decimals.
Learner’s Book Section 3.2
Workbook Section 3.2
Resource sheet 3.2
Additional teaching ideas Section 3.2
3.3
Understanding
compound
percentages
1
Understand compound
percentages.
Learner’s Book Section 3.3
Workbook Section 3.3
Additional teaching ideas Section 3.3
SA
M
Topic
3.4
1
Understanding
upper and lower
bounds
Understand that when a
number is rounded there are
upper and lower limits for the
original number.
Learner’s Book Section 3.4
Workbook Section 3.4
Additional teaching ideas Section 3.4
Cross-unit resources
Language worksheet: 3.1–3.4
End of unit 3 test
49to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
BACKGROUND KNOWLEDGE
• Understand percentage increase and decrease
(Stage 8).
• Round numbers to a given number of decimal
places (Stage 7). Round numbers to a given
number of significant figures (Stage 8).
• Understand the meaning of the symbols
< and > (Stages 7 and 8).
In this unit learners will deepen their knowledge
and extend their use of decimals, percentages and
rounding, often using smaller or larger numbers
than they have used before in more complicated
situations.
PL
E
For this unit, learners will need this background
knowledge:
• Use knowledge of place value to multiply and
divide whole numbers and decimals by any
positive power of 10 (Stage 7).
• Use knowledge of place value to multiply and
divide integers and decimals by 0.1 and 0.01
(Stage 8).
• Estimate, multiply and divide decimals by
whole numbers (Stage 7). Estimate and multiply
decimals by integers and decimals (Stage 8).
TEACHING SKILLS FOCUS
Reflection
At the end of Unit 3, ask yourself:
• Are learners able to explain what they are
thinking? If the answer is ‘No, not really’, is
that just because they are not used to giving
explanations and so need much more practice?
• Are learners getting better at explaining their
reasoning?
• Are learners getting better at explaining what
mistakes have been made and what to do next in
a problem?
• Are learners more confident explaining when
in pairs or small groups rather than as a
whole class?
• With the more complicated problems, can
learners tell you what they will do, i.e. can they
make a plan?
Remember, if you are the first teacher to use this
very powerful learning tool, your learners might find
it difficult to explain what they are thinking. They will
need more practice.
SA
M
Metacognition
This is a complicated area of learning that can be
simplified to ‘thinking about thinking’.
Throughout this unit, ask learners, whenever
possible, to say out loud what they are thinking.
Usually try to ask at the start or a short way through
answering a problem.
If a question has already been answered, ask
learners what they were thinking while they were
attempting a problem. Also ask learners if they
would now do the problem a different way.
If done regularly, this questioning leads to a process
that can be used throughout learners’ schooling:
‘think about a problem, plan what to do, do the
plan, look back and decide if you could have done
anything better’.
This process teaches learners to understand how to
solve problems effectively, not just get the answer to
a particular question.
50
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3 DECIMALS, PERCENTAGES AND ROUNDING
3.1 Multiplying and dividing by powers of 10
LEARNING PLAN
Learning objectives
Success criteria
9Np.01
• Multiply and divide integers
and decimals by 10 to the
power of any positive or
negative number.
• Learners can multiply and
divide numbers by 10 to
the power of any positive or
negative number.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary in this section.
1
10
You read ‘ ’ as ‘one tenth’. You can also say
‘one over ten’.
1
’ as ‘one hundredth’. You can also say
100
You read ‘
‘one over a hundred’.
You read ‘10−3’ as ‘ten to the power of minus three’.
Common misconceptions
Misconception
How to overcome
Questions 6 and 10.
Encourage learners to convert the
powers to integers or fractions. This
usually clears the confusion. It is
essential that learners understand
M
Being confused when tackling
questions such as Question 6b
(450 ÷ 103) and, more so,
Question 10a (0.25 ÷ 10−1).
How to identify
SA
Starter idea
Getting started (5–10 minutes)
Resources: Note books, Learner’s Book Getting started
exercise, calculators
that ‘÷
1
’ is the same as ‘× 10’,
10
Question 8 will help clarify this.
This exercise is a quick reminder of previous work
that will help learners be more effective with this unit.
It is not a test. After each question it might be useful
to allow self- or peer-marking, allowing learners to
rectify any mistakes after a brief discussion.
Description: Before learners attempt the questions,
discuss what they remember about multiplying by 0.1.
On the board, write ‘5 × 0.1’. Ask ‘How can you do
this mentally?’ Learners might suggest several different
methods [e.g. 5 ÷ 10 or move the decimal place to the left
by one place]. Then ask the same question for ‘5 ÷ 0.1’
[e.g. 5 × 10 or move the decimal place to the right by
one place].
Main teaching idea
You might need to help a few learners with Question 5.
Ask the class for the formula of the circumference of a
circle and then, on the board, write ‘C = 2πr’.
Resources: Note books, Learner’s Books
Question 12, Think like a mathematician
(5 minutes)
Learning intention: To understand the mathematical
process of multiplying and dividing by a negative
power of 10.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Description: Learners should be able to answer
confidently part a. Discuss with learners the two key
points given in the introduction text in Section 3.1 of
the Learner’s Book. Ask if learners used these facts, or
the pattern formed from the answers. Ask which method
seemed easier for this question.
Part b is about spotting the pattern in part a of
decreasing numbers in the answers with decreasing
indices. When all learners have completed part c, have a
brief discussion on their answers.
Differentiation ideas: Some learners might be confused
with parts b and e. If this happens, suggest that they
look at the pattern of the questions and answers
in parts a and d or, if necessary, point out the links
between the indices and the answers.
Plenary idea
I use (5 minutes)
Specialising and generalising
Exercise 3.1, Question 3
Learners should notice that in Question 2 all four
questions with a negative index have an answer less
than 3.2. Importantly, they should also show that they
have noticed that as the indices get smaller, the answer
gets smaller, so any similar question with an index of
less than −4 will have an answer smaller than that of
part h, 0.00032. Make sure all learners can explain why
Arun is correct before moving on. Allow other learners
to explain if necessary.
PL
E
Part d might still cause some confusion. Ask learners
to tell you the six answers, and again, discuss briefly the
two key points given in the introduction text in Section
3.1 of the Learner’s Book. Ask if learners used these
facts, or the pattern formed from the answers.
Guidance on selected Thinking and
working mathematically questions
The three generalisations can all be understood
by looking at the answers to Question 2. These
generalisations are important, and will help learners
when checking their answers in the future. To highlight
the importance of these three statements, perhaps
learners could write them in a brightly coloured box in
their note books, to reinforce their usefulness.
Homework ideas
Workbook, Section 3.1.
Description: Ask learners to copy and complete the
following:
As Section 3.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
M
Resources: Note books or mini white boards
To make a question look easier, instead of:
× 103 I use
SA
× 10 −3 I use
÷ 103 I use
÷ 10 −3 I use
Assessment ideas: Ask learners to compare answers,
discussing any differences. Write the different options on
the board. Discuss as a class the different options and
decide which is best and why.
Answers: Answers will usually be × 1000, ÷ 1000, ÷ 1000,
× 1000.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
You could use Resource sheet 3.1B to provide further
practice. Remember to give learners suitable numbers for
the four central shapes and to add any other constraints
you feel necessary. For example, giving certain powers or
other values as starting points, similar to Question 10 in
Exercise 3.1 in the Learner’s Book.
Assessment ideas
At various times during Section 3.1, ask individual
learners a short, easy to answer question that checks
knowledge. Ask questions without warning, and only
ask three or four learners questions. Later in the lesson,
52
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3 DECIMALS, PERCENTAGES AND ROUNDING
ask three or four different learners, etc.
Use questions such as:
‘What is 102 as an ordinary number?’ [100]
‘What is 103 as an ordinary number?’ [1000]
‘What is 100 as an ordinary number?’ [1]
‘What is 10−2 as an ordinary number?’ [0.01]
‘What is 10−3 as an ordinary number?’ [0.001]
‘What is 10−1 as an ordinary number?’ [0.1]
‘What is the equivalent calculation to ÷
1
?’ [× 100]
100
‘What is the equivalent calculation to ÷
1
?’ [× 1000]
1000
‘What is the equivalent calculation to ÷
1
?’ [× 10]
10
PL
E
‘What is the equivalent calculation to ÷ 10−2?’ [× 100]
‘What is the equivalent calculation to ÷ 10−3?’ [× 1000]
‘What is the equivalent calculation to ÷ 10−1?’ [× 10]
3.2 Multiplying and dividing decimals
LEARNING PLAN
Framework codes
9Nf.06
Learning objectives
Success criteria
• Estimate, multiply and
divide decimals by integers
and decimals.
• Learners can multiply and
divide decimals by integers
and decimals.
M
• Learners understand how to
estimate when multiplying
and dividing decimals by
integers and decimals.
LANGUAGE SUPPORT
SA
Equivalent calculation: a different calculation than
the one you have to do but which gives exactly the
same answer
Encouraging learners to talk through their working
not only helps learners to remember methods, but
it also helps you to check their understanding.
Common misconceptions
Misconception
How to identify
How to overcome
Giving incorrect answers to
questions such as 0.3 × 0.05
(Question 3, card C). Learners
might have worked out the 3 × 5
correctly but have not understood
the issues of the decimal points.
Question 3.
Worked example 3.2 part b.
Discussion after checking
questions 1 and 2.
53to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Starter idea
Deci.mals (5 minutes)
Resources: Mini white boards or note books
Description: On the board, write/display:
Work out:
5 × 15
ii
0.5 × 15
iii
5 × 1.5
iv
5 × 0.15
v
0.5 × 1.5
vi
0.5 × 0.15
PL
E
i
Tell learners that i is easy, 5 × 15 = 75, and all the others
are also based on 5 × 15. Tell learners that from ii
onwards, they need to work out where the decimal point
should be placed.
When completed, ask a learner for the answer to ii.
Ask another learner if they think the answer given is
correct. Whether the learner answers ‘yes’ or ‘no’, ask
why and check with the rest of the class for agreement/
disagreement. Repeat for all questions.
Answers:
i 75
iv 0.75
ii 7.5
v 0.75
Working as a class, ask learners in turn [or get them
into pairs or small groups first and ask each pair/
group in turn] for an answer. Ask the rest of the class
if they agree. If learners disagree, ask them to explain
what the mistake is, not just say the correct answer.
Check using the ‘check’ button below the questions.
Then either get learners to retry [if incorrect] or
congratulate and move on. An alternative is to ask
each learner in turn to type in their answer and then
click the ‘check’ button. They can then get help from
the class if required. Other learners can explain what
mistake they might have made. They should not give
the answer.
iii 7.5
vi 0.075
If each learner has individual access to the site you
might decide to limit the time taken on the activity. This
will lead to differentiation by outcome, but hopefully all
learners will be successful and practise a valuable skill. It
is best for learners to do this activity in pairs rather than
individually.
Differentiation ideas: There are only 12 questions per
page, but you can call up as many pages as you want.
The questions start easy and get more difficult, so ask
learners [or pairs/groups] in an order that will maximise
their chance of success.
Online decimal times (5–15 minutes)
Assessment ideas: Work through a page, click on the
resource again and use some of the questions on the
new page as a test to be answered in note books. When
finished, the tests can be self- or peer-marked when the
check button has been clicked.
Learning intention: To practise mental and written
multiplication involving decimals.
Plenary idea
Resources: Mini white boards or note books
Powten (5–15 minutes)
Description: Do this activity after Question 3.
Resources: Mini white boards or note books
Enter ‘transum.org, decimal times’ into a search engine.
When directed to the page, learners complete the level 1
material. This material changes every time it is clicked
onto, so no two sessions are the same.
Description: Enter ‘transum.org Powten’ into a search
engine. When directed to the page, learners complete the
level 4 material. This material changes every time it is
clicked onto, so no two sessions are the same. This page
gives excellent practice for a common question, similar
to Question 13 in Exercise 3.2 in the Learner’s Book.
SA
M
Main teaching idea
This is a free resource and learners can access this too.
They might want to practise at home, so make sure they
write down the web address of this resource. There are
many other similar resources available.
Remember that the questions get more difficult as you
go down the page, so think about which learner to ask
which question.
Using an electronic white board will mean that all learners
see the same questions. Using a suite of computers/tablets
will result in learners seeing different questions.
Remember that the questions get more difficult as you
go down the page, so think about which learner to ask
which question.
Using an electronic white board will mean that all learners
see the same questions. Using a suite of computers/tablets
will result in learners seeing different questions.
54
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3 DECIMALS, PERCENTAGES AND ROUNDING
Working as a class, ask learners in turn (or get them
into pairs or small groups first and ask each pair/group
in turn) for an answer. Ask the rest of the class if they
agree. If learners disagree, they should explain what the
mistake is, not just say the correct answer. Check using
the ‘check’ button below the questions. Then either
get learners to retry (if incorrect) or congratulate and
move on.
Homework ideas
If each learner has individual access to the site you
might decide to limit the time taken on the activity. This
will lead to differentiation by outcome, but hopefully all
learners will be successful and practise a valuable skill.
Rather than using Resource sheet 3.2 as an Assessment
idea, it could be used as a possible homework, but only
for more confident classes.
Guidance on selected Thinking and
working mathematically questions
Exercise 3.2, Question 14
As Section 3.2 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
PL
E
Assessment ideas: Work through a page using the
‘check’ button, click on the resource again and use
some of the questions from the new page as a test to be
answered in note books. When finished, the tests can be
self- or peer-marked when the ‘check’ button has been
clicked.
Critiquing and improving
Workbook, Section 3.2.
Use this Assessment idea (set aside 10–15 minutes) after
learners have completed at least questions 8 and 9 in
Exercise 3.2. They can work individually or in small
groups, depending upon ability and your intentions –
you might wish to keep the completed resource sheets as
evidence of learning. Each learner or group will need a
copy of Resource sheet 3.2.
All of the mathematical statements require either
< or > to make them true. A reason is required for each
decision. Question 1 has been done, as an example.
Many learners might initially need a little extra help.
If so, ask them for the answer to Question 3 [>] and
help them with the reason that: 0.3 < 0.7, but you are
dividing 20 by each number. The smaller the divisor,
the larger the answer, so the first answer will be the
larger. If this help is given, it is advisable to write it on
the board.
M
All learners should realise that Hugo’s estimation is a
good one. He has rounded the numbers correctly to one
significant figure and has multiplied accurately, so the
actual answer should be around 8.
Assessment ideas
SA
Many learners will try to improve Hugo’s solution
method. It is probably easier for most learners to use
long multiplication or another method (e.g. using a
grid, Napier’s bones, etc.) to get the answer of 8694 and
then to deal with the decimals in a fairly similar way to
Hugo’s method. However, many learners might just say
there are three decimal places in the question, so there
should be three decimal places in the answer, without
referring to 1000.
Answers: 1 <, 2 <, 3 >, 4 >, 5 <, 6 <, 7 >, 8 >, 9 <.
Discuss learners’ reasons for their choices, especially if
they are different from those of other learners.
It is important that learners try to improve Hugo’s
method and not just say that they cannot think of a
better way.
55to publication.
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
3.3 Understanding compound percentages
LEARNING PLAN
Learning objectives
Success criteria
9Nf.05
• Understand compound
percentages.
• Learners can understand and
use compound percentages.
LANGUAGE SUPPORT
PL
E
Framework codes
Compound percentage: when a percentage
increase or decrease is followed by another
percentage increase or decrease
Often questions about percentages are given in
context. Support learners with the language used
in the questions, and make sure that learners
understand what each question is asking before
they attempt to answer it.
Common misconceptions
How to identify
How to overcome
Correctly multiplying by 1.2, but
then dividing by 1.2 instead of
multiplying by 0.8 in questions
such as part a i of Question 3 ‘60
increased by 20%, then decreased
by 20%’.
Question 3 part a i.
Discussion during the introduction
and worked example. Thorough
checking/discussion with
Question 1 part c.
Using, for example, 0.5 for a
decrease of 5% instead of 0.95.
Question 1 part c.
SA
M
Misconception
Starter idea
Percentages (5–10 minutes)
Resources: Mini white boards or note books
Description: Ask learners to write 20% of 200 [40].
Ask learners to increase 200 by 20% [200 + 40 = 240].
Ask learners to decrease 200 by 20% [200 − 40 = 160].
Ask learners to write the multiplier when increasing an
amount by 20% [1.2].
Ask learners to write the multiplier when decreasing an
amount by 20% [0.8].
After each question, ask learners to show their answers
and ask a learner with the correct answer to explain
their method/reasoning.
Check that 0.95 has been used.
Check again in Question 5
parts b i and ii.
If you have time, repeat questions using 5% [10, 210,
190, 1.05, 0.95].
If all learners seem confident, move on. If not, repeat all
questions using 10% [20, 220, 180, 1.1, 0.9].
Main teaching idea
Question 2, Think like a mathematician
(5 minutes)
Learning intention: To understand the effect of
increasing and decreasing an amount by the same
percentage.
Resources: Note books, Learner’s Books, calculators
56
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3 DECIMALS, PERCENTAGES AND ROUNDING
Description: Learners can discuss part a with a partner,
but ask that they try to think what will happen rather
than actually work out the answer before deciding who
is correct. When learners have written who they think is
correct, they can work out the value of the coin [$792]
and write down what mistake Marcus (and perhaps
themselves) made.
Specialising and generalising
Exercise 3.3, Question 10
This is an important question, as this is the basis for
almost all compound interest questions. It is essential
that all learners understand that × 1.04 × 1.04 is the same
as × (1.04)2.
Learners could specialise with parts b and c, by working
out both the long and the short version on their
calculators, to ensure that their answer is correct. When
learners have answered parts b and c they should check
them by self-marking. When all learners have understood
parts a to c, parts d and e i and ii should be straight
forward, leaving learners the now, hopefully, easy task of
generalising for part e iii. If all learners have not got the
correct answer for part e iii, make sure that you ask other
learners to explain why the answer is 5000 × (1.04)n.
PL
E
Part b is interesting. Learners will understand that 2 × 5
is 10 and that 5 × 2 is also 10. They often think, however,
that by swapping the increase and decrease value, that
the answer will somehow become different (usually
greater) than the original answer.
Guidance on selected Thinking and
working mathematically questions
Differentiation ideas: If learners are confused with the
mechanics of how a 10% increase and a 10% decrease
actually gives an overall decrease, you might have to work
through the question with them. Show that 10% of $800
is $80, so now there is $880. 10% of $880 is $88 – which is
more than $80. As you will subtract the $88, you will end
up with less than $800 ($792). Next show learners that
an increase of 10% followed by a decrease of 10% can be
found by × 1.1 × 0.9 or × 0.99. 0.99 is less than 1, so the
answer will always be less than the starting number.
Homework ideas
Workbook, Section 3.3.
Plenary idea
80 (5 minutes)
M
Resources: Mini white boards or note books, calculator
As Section 3.3 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Description: Ask learners to use any method they prefer
to work out:
80 increased by 20%, then decreased by 20%
2
80 increased by 20%, then increased by 20%
3
80 decreased by 20%, then decreased by 20%
SA
1
4
80 increased by 20%, five times in a row
5
80 decreased by 20%, five times in a row
Answers:
1 76.8
4 199.0656
2
5
115.2
26.2144
3
51.2
Assessment ideas: Depending on the ability of the class,
allow peer-marking after each question, or after the
third question or when all five questions have been
completed. Whenever you decide to check answers have
a short discussion about methods, trying to find several
different methods from learners, and then discussing
which method might be the easiest.
Assessment ideas
At various times during Section 3.3, ask individual
learners short, easy to answer questions that check
knowledge. Ask questions without warning, and only ask
three or four learners questions. Later in the lesson, ask
three or four different learners, etc. Use questions such as:
‘What do you multiply 50 by to work out 10%
of 50?’ [0.1]
‘What do you multiply 50 by to increase it
by 10%?’ [1.1]
‘What do you multiply 50 by to decrease it
by 10%?’ [0.9]
‘What do you multiply 50 by to increase it by 10%
and then to increase it by 20%?’ [1.1, then 1.2]
‘What do you multiply 50 by to increase it by 10%
and then to decrease it by 70%?’ [1.1, then 0.3]
‘What do you multiply 50 by to decrease it by 10%
and then to increase it by 7%?’ [0.9, then 1.07]
‘What do you multiply 50 by to decrease it by 1%
and then to decrease it by 99%?’ [0.99, then 0.01]
57to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
3.4 Understanding upper and lower bounds
LEARNING PLAN
Learning objectives
Success criteria
9Np.02
• Understand that when a
number is rounded there are
upper and lower limits for the
original number.
• Learners can work out upper
and lower bounds.
LANGUAGE SUPPORT
PL
E
Framework codes
Lower bound: the smallest value that a rounded
number could have been before it was rounded
Upper bound: the largest value that a rounded
number could have been before it was rounded
Encourage learners to use the terms lower bound
and upper bound. Make sure that learners listen
carefully to phrases such as ‘to the nearest 10’,
‘to the nearest 100’, etc. Remind learners of the
inequality symbols and what each symbol means.
‘<’ is ‘less than’, ‘⩽’ is ‘less than or equal to’, ‘>’ is
‘greater than’ and ‘⩾’ is ‘greater than or equal to’.
Common misconceptions
How to identify
How to overcome
M
Misconception
Thinking of the upper bound
in an inequality as a number
that rounds down.
Question 5, 7.5 ⩽ x < 8.5,
where a learner might think
that 8.5 rounds to 8.
Whenever using an inequality to show the
range of values, remind learners that the
upper bound (e.g. 8.5) rounds up to the next
number. The range is from the first number in
the inequality to just less than the last number.
1
800 ± half of 100
± half of (5–7 minutes)
2
8000 ± half of 100
Resources: Mini white boards or note books
3
80 ± half of 10
Description: Remind learners what the ‘±’ sign means.
4
800 ± half of 10
5
8 ± half of 1
6
80 ± half of 1
7
8 ± half of 0.1
8
80 ± half of 0.1
SA
Starter idea
Ask learners to write the two numbers that are equal
to ‘400 ± half of 100’ [350, 450]. Give learners about
20 seconds to answer and then ask them to show their
answers. Ask a successful learner to explain how they
got their answers. Make sure all learners understand this
question. The same skills will be required for the next
eight questions.
Repeat (ask question, 20 seconds, show, ask to explain)
for the following:
Write the two numbers that are equal to:
Answers:
1 750, 850
4 795, 805
7 7.95, 8.05
2 7950, 8050 3 75, 85
5 7.5, 8.5 6 79.5, 80.5
8 79.95, 80.05
58
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3 DECIMALS, PERCENTAGES AND ROUNDING
Main teaching idea
Question 5, Think like a mathematician
(5 minutes)
Learning intention: To understand that
7.5 ⩽ x ⩽ 8.49999999… and 7.5 ⩽ x < 8.5 are the same,
but 7.5 ⩽ x < 8.5 is easier to write and much easier to use.
Resources: Note books, Learner’s Books
Learners will notice how much easier it is to use Marcus’
method than Sophia’s.
Differentiation ideas: Look out for learners trying
to answer part b using 7.5–8.5 as a shorter method of
writing 7.5 ⩽ x < 8.5. Ask other learners to explain why
this is not correct.
Plenary idea
To the nearest … (5 minutes)
Resources: Mini white boards
Assessment ideas: Learners can compare answers and
discuss any differences before giving the answers for
self- or peer-checking.
PL
E
Description: When completed, discuss as a class.
Ask ‘Is Sofia’s method correct or not?’ Allow learners
to explain that it is correct. It is best for learners to
understand that what she has said is exactly the same
as Marcus’ comment – that 8.49999999… is < 8.5,
but only just!
If learners have difficulty with the last two questions,
read back the answers from the last few questions.
Emphasise the fact that if rounded to the nearest 100,
you add and subtract half of that, i.e. 50, from the
rounded number. If a number is rounded to the nearest
10, you add and subtract half of that, i.e. 5. If rounded
to the nearest 1, you add and subtract half of that,
i.e. 0.5. Now ask ‘How do you work out the bounds of
a number rounded to the nearest 0.1?’ [add and subtract
half of that, i.e. 0.05].
Conjecturing and critiquing
Exercise 3.4, Question 4
You should expect all learners to be able to explain that
7.5 is the first/lowest number that rounds to 8. Ask other
learners to explain if anyone needs this to be clarified.
Many learners will agree with Arun’s statement, but then
should change their minds and agree with Zara.
Part b is a large conceptual leap for most learners.
A conjecture of this nature is difficult, so do not be
surprised if only a few learners – or none at all – give a
good answer at this stage.
M
Description: Any time after completing Question 9.
Guidance on selected Thinking and
working mathematically questions
Learners should be fairly confident with the first four
suggested questions here. Ask learners to use any
patterns or rules they can see to work out the last two
questions. If learners find this difficult, allow them to
work in pairs or small groups to discuss the problem.
SA
One at a time, on the board, write the numbers and
what they have been rounded to. Ask learners to write
the upper and lower bounds of the numbers and what
number they have added and subtracted to work out the
upper and lower bounds.
Start with 900 to the nearest 100 [850 ⩽ x < 950,
900 ± 50].
950 to the nearest 10 [945 ⩽ x < 955, 950 ± 5].
90 to the nearest 10 [85 ⩽ x < 95, 90 ± 5].
9 to the nearest whole number [8.5 ⩽ x < 9.5, 9 ± 0.5].
9.5 to the nearest 0.1 [9.45 ⩽ x < 9.55, 9.5 ± 0.05].
• If a learner suggests 8.5, ask them (with help from
other learners if necessary) to round 7.5 to the
nearest whole number [8] and ask them to round 8.5
to the nearest whole number [9].
• If any learner suggests 8.49, encourage their thinking
and ask if they can think of a number even bigger
than 8.49, but still less than 8.5. If this discussion
works, lead them towards suggesting 8.4999…
The main focus of this question is to make learners
understand that the bounds of 8, when rounded to the
nearest whole number, is between 7.5 and a number less
than 8.5. All learners must understand that 8.5 cannot
be used as 8.5 rounds up to 9. Question 5 will help
learners as using inequalities to show the range of values
removes the confusing issue of ‘up to 8.49999999…’,
showing the use of ‘< 8.5’ instead.
9.55 to the nearest 0.01 [9.545 ⩽ x < 9.555, 9.55 ± 0.005].
59to publication.
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Homework ideas
Workbook, Section 3.4.
As Section 3.4 might take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Tell learners you will give them some facts about
memorable times/distances, all of which have been
rounded. Learners must rewrite the times/distances as
ranges of values using inequalities.
Eliud Kipchoge ran a marathon in less than
two hours. His time was 1:59:40, rounded to the
nearest second.
You could ask learners to make a mind-map containing
everything they think they need to remember for the
end-of-unit test. In the following lesson, it is important
to share the mind-maps in class (e.g. spread out over a
few desks for everyone to look at), rather than marking
them. Discuss the different mind-maps as a class. When
the class agree that a point is important, that key point
could be copied onto the board (by you or a learner).
Agree on as many key points as possible. Learners
could then improve/update their individual mind-maps
if necessary. Learners could store their mind-maps at
home as a possible revision tool towards mid-term/endof-year tests.
2
The current women’s 100 metres record is 10.49
seconds (rounded to 0.01 of a second) set by
Florence Griffith-Joyner in 1988.
Assessment ideas
Answers:
1 1:59:39.5 ⩽ x < 1:59:40.5
2 10.485 seconds ⩽ x < 10.495 seconds
3 762.95 mph ⩽ x < 763.05 mph
4 8847.5 m ⩽ x < 8848.5 m
5 657 015 tonnes ⩽ x < 657 025 tonnes
PL
E
1
The land speed record is held by Andy Green,
driving Thrust SSC at 763.0 mph, rounded to the
nearest tenth of a mph.
4
Mount Everest (or Chomolungma) is 8848 m high,
to the nearest metre.
5
The heaviest ship ever built was the Seawise Giant,
which had a mass of 657 020 tonnes to the nearest
10 tonnes.
SA
M
Towards the end of Exercise 3.4 use this assessment to
check the level of understanding of the exercise. This
can be informal, with learners discussing the range of
values, or as a test to be marked in class or as an exit
ticket set of questions.
3
60
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PROJECT GUIDANCE: CUTTING TABLECLOTHS
PROJECT GUIDANCE: CUTTING TABLECLOTHS
Why do this problem?
This problem offers an interesting context
in which learners can practise working with
percentages and critique the different approaches
and representations used. It also offers learners
an opportunity to use algebra to express
the relationships they discover and to make
generalisations.
Finally, challenge learners to work backwards and
to find out what percentage strip needs to be cut
off to create a tablecloth that uses 75%, and then
50%, of the original cloth.
Key questions
What is the length of the new tablecloth? What is
the width of the new tablecloth?
PL
E
Possible approach
Introduce the context of making a rectangular
tablecloth from a piece of cloth using one cut and
one join, as shown in the diagram.
percentage strip that is cut off and the percentage
of the cloth that is used for the new tablecloth.
Invite learners to work out the percentage of cloth
that has been used to make the new rectangle
when 20% is cut off as shown, and then to share
their methods for working the percentage out.
Some learners might work out the area of the blue
rectangle and then express it as a percentage of
the area of the purple square. Others might work
out the area of the red square as a percentage of
the purple square, and subtract this from 100. This
is a good opportunity to invite learners to critique
the different methods.
You cut off n% of the original square. What will the
dimensions of the new tablecloth be?
Possible support
Start by focussing on cuts which are multiples
of 10%.
Encourage learners to make a table so they can
look for patterns in their results. It might be easier
to work with the area of the leftover cloth rather
than the area of the new rectangle.
Possible extension
Invite learners to consider what might happen
if they started with a 2 m by 1 m piece of cloth
instead of a square.
SA
M
Next, invite learners to explore cutting off strips
which are other percentages of the original cloth.
Challenge them to find a relationship between the
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Unit plan
PL
E
4 Equations and
inequalities
Approximate
number of
learning hours
Outline of learning content
Resources
4.1 Constructing
and solving
equations
1–2
Represent situations in
words or as an equation.
Move between the two
representations and solve
the equation, including
when the unknown is in the
denominator.
Learner’s Book Section 4.1
Workbook Section 4.1
Additional teaching ideas Section 4.1
4.2 Simultaneous
equations
2–2.5
Solve simultaneous linear
equations algebraically and
graphically.
Learner’s Book Section 4.2
Workbook Section 4.2
Additional teaching ideas Section 4.2
4.3 Inequalities
1–1.5
Represent situations in
words or as an inequality.
Move between the two
representations and solve
linear inequalities.
Learner’s Book Section 4.3
Workbook Section 4.3
Additional teaching ideas Section 4.3
M
Topic
SA
Cross-unit resources
Language worksheet: 4.1–4.3
End of unit 4 test
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Understand that a situation can be represented
either in words or as an equation. Move
between the two representations and solve
the equation (integer or fractional coefficients,
unknown on either or both sides) (Stage 8).
• Use knowledge of coordinate pairs to construct
tables of values and plot the graphs of linear
functions (Stage 8).
• Understand that letters can represent open and
closed intervals (Stage 8).
Learners will extend their knowledge of
representing situations in words and algebra
to where the unknown is the, or part of the,
denominator and to using their knowledge to
solve inequalities. Simultaneous equations are
introduced, with learners solving them both
graphically and algebraically.
62
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4 EQUATIONS AND INEQUALITIES
TEACHING SKILLS FOCUS
Now give one question for all learners to attempt,
without help. Is there evidence of learning? Have
the ‘teachers’ done a good job? Did the ‘teachers’
understand what they were teaching? Are there any
aspects that you need to clarify?
Reflection
At the end of Unit 4, ask yourself:
• Do you know what the learners know/knew about
this topic?
• Have you asked questions to look for evidence
of learning, of a depth of understanding of the
topic that shows learners understand how the
maths works, not just that they can get an answer
to a question?
• Are learners confident that if they can suggest
half-formed ideas about a problem, then they
can share it and receive guidance from yourself
or another learner?
• Are you making sure that learners understand
that learning from their mistakes is an excellent
and invaluable process that is encouraged within
the classroom?
M
PL
E
Assessment for learning
A key aspect for assessment for learning is assessing
prior knowledge. While the Getting started exercise
will help find weaknesses, much of this unit is built
on previously learned skills. If any of those skills are
weak or missing it is important to revisit that area of
the Stage 8 work.
You might need to adapt or stop the planned lesson
if the required previous knowledge is missing. If
only part of the class lacks a skill, then this is a great
opportunity for you to get learners to help you
to teach.
Show the skill required to all learners, set three
or four basic questions, put learners in groups
with one or two, ‘learners’ with as many ‘teachers’
as possible.
Listen to the groups. Ask that only one ‘teacher’ is
speaking at any time. Regularly check with ‘learners’
that they understand and that the ‘teacher’ is giving
good feedback to any questions they are asking.
Let learners self-mark their answers to the questions.
Now give slightly harder questions to all learners,
working in pairs – one ‘learner’ and one ‘teacher’
per pair if possible. Allow self-marking.
4.1 Constructing and solving equations
SA
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ae.05
• Understand that a situation
can be represented either
in words or as an equation.
Move between the two
representations and solve
the equation (including
those with an unknown in the
denominator).
• Learners can understand and
write equations, then solve
them.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
LANGUAGE SUPPORT
using algebra. Learners should realise that, unless
instructed to use a particular letter, they can use
any letter. Encourage learners to always write which
letters they are using for which quantities.
When solving equations, encourage learners to talk
through the method, using phrases such as ‘Divide
both sides by 3’ and ‘Subtract 3 from both sides’.
PL
E
Construct: use given information to write
an equation
Solve: calculate the value of any unknown letter(s)
in an equation
Support learners with the language when situations
are represented in words. Make sure that learners
understand the situation and how to rewrite it
Common misconceptions
Misconception
How to identify
How to overcome
Forgetting the brackets in a
situation such as ‘Anders thinks
of a number. He subtracts 5, then
multiplies the result by 12.’, writing
‘n − 5 × 12’ rather than ‘12(n − 5)’.
Activity 4.1.
Worked example 4.1a. If learners
have problems understanding
Worked example 4.1a, discuss
again during Question 3.
Being confused by ‘rules’ about
solving equations.
Any question involving solving.
Many learners find it helpful to
think of an equation as a balance.
So, for example, if they add a term
to one side they must add the
same term to the other side, to
maintain the balance.
practice to allow learners to attempt the questions as
individuals, but discuss answers/problems in pairs/small
groups. Allow self-marking and allow learners to rectify
any mistakes after the discussion.
M
Starter idea
Getting started (5–10 minutes)
Resources: Note books, Learner’s Book Getting
started exercise
SA
Description: Learners should have little difficulty with
the Getting started questions, but before learners
attempt these questions discuss what they remember
about the order in which it is best to solve an equation
y
4
Main teaching idea
Question 3 (2–5 minutes)
Learning intention: To check understanding of the two
methods of solving the equation given.
such as + 10 = 13. It is probably useful to allow learners
Resources: Note books, Learner’s Books
to answer Question 1a, then to check methods, before
allowing them to attempt parts b and c, checking again,
then attempting part d.
Description: Ask learners to answer part a. When
completed, work through the answer on the board:
You might need to remind a few learners what the
inequality signs mean during Question 2.
2(x + 12) = 4x − 6
Once learners have answered Question 3a, ask them to
compare their answers with a partner. Follow this with a
brief class discussion before asking learners to finish the
rest of the questions.
Remember that this is not a test. It is designed to help
learners prepare more effectively for Unit 4. It is good
2x + 24 = 4x − 6
24 = 2x − 6
Subtracting 2x from both sides
30 = 2x
Adding 6 to both sides
15 = x
Dividing both sides by 2
x = 15
64
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4 EQUATIONS AND INEQUALITIES
Ask learners to compare their answer with the answer
shown. Discuss as a class any differences, including
short cuts, longer versions or mistakes. Decide as a class
if the working on the board is clear and easy to follow.
Ask learners to answer part b. When completed, work
through the answer on the board:
2(x + 12) = 4x − 6
Dividing both sides by 2
12 = x − 3
Subtracting x from both sides
15 = x
Adding 3 to both sides
x = 15
If each learner has individual access to the site you
might decide to limit the time taken on the activity. This
will lead to differentiation by outcome, but hopefully all
learners will be successful and practise a valuable skill.
Try to ensure that all learners attempt at least the first
question from each level 1 to 5.
Assessment ideas: Work through the chosen questions
on a page, click on the resource again and use one
or two of the new questions as a test for learners to
answer in their note books. When finished, the tests can
be self- or peer-marked when the ‘check’ button has
been clicked.
PL
E
x + 12 = 2x − 3
Check using the ‘check’ button below the questions.
Then either get learners to retry (if incorrect) or
congratulate and move on.
Again, ask learners to compare their answer with
the answer shown. Discuss as a class any differences,
including short cuts, longer versions or mistakes. Decide
as a class if the working on the board is clear and easy
to follow.
Ask learners to write their favourite method and why
they prefer that method [i.e. answer part c]. When
completed, ask for a vote for each method. Tell
learners that having a favourite method is sensible, but
it is obviously better to be able to use both methods,
depending on the type of question they are given.
Critiquing
Exercise 4.1, Question 5
Although both methods require a similar number of
lines of working to get x = 13, it is important for learners
to think which method they prefer. By thinking about
the different methods, learners should understand both
methods a little better. Most learners will prefer the
‘multiplying out the brackets’ method. Make sure they
explain why they prefer not to use the other method.
They will probably mention that they still need to
expand brackets, so it is best to do that first to stop the
possibility of making a mistake with a method that
many will see as more difficult.
M
Differentiation ideas: Learners should be able to solve
this type of equation using both methods. If this is
very challenging for a learner, concentrate on them
being able to answer using part a’s method, expanding
brackets first.
Guidance on selected Thinking and
working mathematically questions
Plenary idea
Online solving equations (5–20 minutes)
SA
Resources: Mini white boards or note books, calculators
Description: Enter ‘transum.org, equations’ into a search
engine. When directed to the page, learners work on
levels 1 to 5.
With such a range of questions available, this is an
excellent resource. It is suggested that learners should
start at level 1 and attempt the first question. Check
their answer and discuss any issues. They can then
attempt another question at or near the end of the page.
When you think learners are ready to move on, move up
a level. Repeat, stopping at level 5.
Working as a class, ask learners in turn (or get them
into pairs or small groups first and ask each pair/
group in turn) for an answer. Ask the rest of the class
if they agree. If learners disagree, ask them to explain
what the mistake is, not just to say the correct answer.
Homework ideas
Workbook, Section 4.1.
As Section 4.1 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
65to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Assessment ideas
At various times during Section 4.1, ask individual
learners to give the next line of working of one of
the questions, which you point to, on the board. Ask
questions without warning, and only ask two or three
learners questions. Later in the lesson, ask two or three
different learners, etc. Timing is required to get all
questions answered before the end of the lesson.
Suggested questions:
2x + 3 = − 1
2
3(x + 8) = 8x − 1
3
10 − 3x = 6
4
28 = 16 − 4x
30
=5
x
6
4x + 4 = 20 − 4x
7
8x − 12 = 4(x + 12)
8
2x + 6 = 5x − 3
9
4x
−5=3
2
10
24
=6
x+2
Answers:
1 −2
4 −3
7 15
10 2
2 5
3 1
5 6
8 3
6 2
9 4
1
3
PL
E
1
5
4.2 Simultaneous equations
LEARNING PLAN
Framework codes
9Ae.06
Learning objectives
Success criteria
• Understand that the solution
of simultaneous linear
equations:
• Learners can solve
simultaneous linear equations
both algebraically and
graphically.
M
• is the pair of values that
satisfy both equations
• can be found algebraically
(eliminating one variable)
SA
• can be found graphically
(point of intersection).
LANGUAGE SUPPORT
Method of elimination: a method for solving
simultaneous equations when the number of xs or
ys are the same, so you add or subtract the two
equations to eliminate the xs or the ys
Method of substitution: a method for solving
simultaneous equations where you write one of the
equations in the form ‘y = …’ or ‘x = …’ and then
substitute this into the other equation
Simultaneous equations: two or more equations,
each containing several variables
Some of the words used in this unit have other
meanings in everyday life. For example:
‘Simultaneous’ is often used as a describing word
when two or more things happen at the same time.
‘Rearrange’ can be used as a describing word when
objects have been moved, e.g. ‘I rearranged the
furniture’ meaning that, perhaps, the sofa has been
moved to a different place in a room.
‘Substitute’ is a commonly used term in team
sports when one player takes the place of another.
66
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4 EQUATIONS AND INEQUALITIES
CONTINUED
Discuss the different meanings of the words with
learners to help them appreciate how the same
words can have different meanings and can be
used in different ways.
Common misconceptions
Misconception
How to identify
How to overcome
Working out one variable, but
forgetting to substitute in to work
out the second variable.
Starter idea
Questions 2, 3 and 4.
Worked example 4.2 (5 minutes)
Resources: Learner’s Books
Discourage guessing as it is rarely
an efficient strategy. Taking an
algebraic approach, learners can
solve even the most complicated
problems.
PL
E
Trying to use a mental trial
Questions 2, 3 and 4.
and improvement strategy (i.e.
guessing) to try to work out the two
variables.
Resources: Note books, Learner’s Books
Description: With no worked examples explaining what
to do when faced with an equation which does not start
‘y = ’, many learners might initially be confused. Allow
learners time to try to think through the problem. If you
wish to give the whole class a tip (rather than using the
Differentiation ideas), suggest they think about making
this question look more like questions 2, 3 and 4. Some
learners will soon realise that rearranging the second
equation is a good starting point, but they might still
require some reassurance that the method is a useful
one.
M
Description: From Worked example 4.2, read the first
line ‘Solve these simultaneous equations. y = 3x + 1 and
y = x + 9’. Then ask learners to look at the equation as
3x + 1 = x + 9.
Checking answers, ensuring
learners have written values for
both x and y.
Put learners into pairs or small groups. Ask learners to
explain to each other why they know that 3x + 1 = x + 9.
Allow a maximum of two minutes.
Ask a learner to explain to you how they know that
3x + 1 = x + 9.
SA
Ask the class if the explanation was similar to their
explanation.
If the explanation did not include a mention of the
fact that these are simultaneous equations, and that
the y values in both equations will be the same, you
will need to discuss this point further. This is the key
to understanding. In simultaneous equations, because
y = y, learners can rewrite y = 3x + 1 and y = x + 9 as
3x + 1 = x + 9 (i.e. the y values are the same so the 3x + 1
and the x + 9 must also be the same).
Main teaching idea
Differentiation ideas: This can be a big step for
some learners. Even working with a partner does not
guarantee that all learners will work out how to solve
the first pair of equations. When helping, alternate
between suggesting two different methods. The first
pair you help, suggest they rearrange the second
equation making y the subject [y = 5x − 3], so making the
question look like questions 2, 3 and 4. The next pair,
suggest that they substitute for y in the second equation
[5x = 3x + 1 + 3] and work out the value of x. The next
pair, go back to suggesting making y the subject, etc.
Plenary idea
How? (5–10 minutes)
Question 7, Think like a mathematician
(10 minutes)
Resources: Mini white boards
Learning intention: To use their knowledge of algebra to
help solve simultaneous equations.
3x + 2y = 14
Description: On the board, write these two equations:
y = 2x
67to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Ask learners to explain how to solve the simultaneous
equations. They should not solve them, just explain how
to solve them.
Assessment ideas: Learners swap explanations with a
partner. Learners then follow the explanation to check
the explanation allows them to solve the simultaneous
equations correctly. Discuss good/clear explanations as
a class.
Guidance on selected Thinking and
working mathematically questions
Assessment ideas
Many of the questions in Exercise 4.2 could be used as
a test, answered on a separate piece of paper, for you to
mark and keep for evidence of success at this learning
objective. It is recommended that if you use this method,
use Questions 13c and 14a as the class ‘test’. If you
would prefer to use these questions in class as normal,
perhaps when learners have completed Question 14,
you could give them two other questions under test
conditions. Suggestions for questions:
1
Exercise 4.2, Question 10
2x + 2y = 24
PL
E
Specialising and convincing
Solve these simultaneous equations. Use
any method.
If learners have been checking their answers by
substituting in their x and y values, this question should
be straightforward. They will notice that Sofia’s answer
works for both equations, but Zara’s answer only works
for the second equation, the first showing −6 = 24.
If learners solve the simultaneous equations rather than
check the answers, then they need some guidance on
the best way of answering this type of question. You
should also ask them why they have not been checking
their answers.
Homework ideas
2
Solve these simultaneous equations using a
graphical method.
y=x−1
2y = 4x − 10
Answers: 1 x = 4, y = 8
2
x = 4, y = 3
M
Workbook, Section 4.2.
y = 2x
SA
As Section 4.2 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
4.3 Inequalities
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ae.07
• Understand that a situation
can be represented either
in words or as an inequality.
Move between the two
representations and solve
linear inequalities.
• Learners can solve a variety
of inequalities starting from
an inequality or from words.
68
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We are working with Cambridge Assessment International Education towards endorsement of this title.
4 EQUATIONS AND INEQUALITIES
LANGUAGE SUPPORT
Encourage learners to read their solutions out
in words, for example, for ‘x < 3.5’, say ‘x is less
than 3.5’.
Make sure that learners understand that, for
example, ‘6 is greater than or equal to x’ is the
same as ‘x is less than or equal to 6’. Ask them for
other examples of similar equivalent sentences.
PL
E
Inequality: a relationship between two expressions
that are not equal
Solution set: the set of numbers that form a
solution to a problem
Support learners with the language when situations
are represented in words. Make sure that learners
understand the situation and how to rewrite it as an
inequality, choosing the correct inequality sign(s).
Common misconceptions
Misconception
Still getting confused with
inequalities such as x < −3, thinking
that −2 is less than −3.
How to identify
How to overcome
Question 2b.
Discussion with other learners when
checking.
Thinking that an inequality is solved Question 4.
differently to a normal equation.
Regularly saying that inequalities and
equations are solved in the same way.
Question 11.
Discussions during and after
completing Question 11 and
Differentiation ideas in the main
teaching activity idea for Question 11
in the Additional teaching ideas.
M
Getting confused when changing
−x < 6 to x > 6.
Starter idea
Inequalities (10 minutes)
Resources: Mini white boards or note books
SA
Description: Use this starter idea before working
through the introduction to Section 4.3.
On the board, write the equation ‘2x + 11 = 4’.
Ask learners to solve the equation.
The solution should look something like this:
2x + 11 = 4
2x = −7
x = −3.5
Now display ‘2x + 11 ⩾ 4’. Tell learners that this is an
inequality. Remind learners that ⩾ means ‘is greater
than or equal to’ and discuss how it differs from >.
The solution, in words, is ‘x is greater than or equal
to −3.5.’ Emphasise that this is not a single value but
a set of numbers. The numbers −3.5, −3, 0, 4.2, 100,
are all in the solution set. The numbers −3.6, −4, −8,
−20.5 are not in the solution set. Draw a number line to
remind learners how to illustrate the solution set on a
number line:
– 4 –3.5
–3
–2
–1
0
Remind learners that the solid black circle indicates that
−3.5 is included in the solution set.
Learners should know that there are four inequality
signs; <, >, ⩽ and ⩾. Check that learners can remember
the meaning of each inequality sign. For example, the
solution set of 2x + 11 < 4 is x < −3.5 as shown here:
Explain that learners can solve an inequality
algebraically in a similar way to an equation.
In this example: 2x + 11 ⩾ 4
2x ⩾ −7
x ⩾ −3.5
– 4 –3.5
–3
–2
–1
0
The open circle indicates that −3.5 is not included in the
solution set.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Now give learners this inequality to solve:
24 − 2x ⩾ 6 + x
The solution could look like this: 24 − 2x ⩾ 6 + x
24 ⩾ 6 + 3x
18 ⩾ 3x
6⩾x
x⩽6
Plenary idea
x + 20 ⩽ 10 − x (5 minutes)
Resources: Mini white boards
PL
E
The last step might cause problems for learners. Make
sure they understand why the sign has changed. It is
because ‘6 is greater than or equal to x’ is the same
as ‘x is less than or equal to 6’. If learners are not
convinced, insert some possible numerical values for x
such as 5, 4 or 3, to see that both forms mean the same.
For less confident learners, show that the ‘sharp end’
of the inequality points to the x in the last two lines of
the solution.
Differentiation ideas: All learners should be able
to solve 4x + 5 = 17. It would be a good idea for less
confident learners to start by solving this equation
before reminding them that solving 4x + 5 = 17 and
solving 4x + 5 < 17 are done using the same algebraic
techniques. Check that these learners get the correct
solution of x < 3.
It is possible that a learner will start by subtracting x
from both sides and end up with the inequality
−3x ⩾ −18. If they then divide by −3 they might get
x ⩾ 6, which is incorrect. There are two ways to resolve
this error:
Assessment ideas: Ask learners in pairs to compare
answers. Learners should compare their algebra first,
checking they both have the same answer of x ⩽ −5,
and discuss any differences in their methods. Next, they
should check that their number lines are the same with
the same endings to the line, i.e. a solid black circle
above the −5 and an arrow pointing towards the left.
Guidance on selected Thinking and
working mathematically questions
Specialising and convincing
Exercise 4.3, Question 7
M
• One way is to tell learners that if they divide by a
negative number, they must change the direction of
the inequality. This is usually seen as a puzzling rule
to learners.
• Or, you can avoid the rule by telling learners always
to arrange their working with a positive number
of xs on one side of the inequality. So, −3x ⩾ −18
becomes 0 ⩾ 3x − 18 (adding 3x to both sides) which
becomes 18 ⩾ 3x. The result then follows.
Description: Ask learners to show the solution to the
inequality x + 20 ⩽ 10 − x on a number line.
SA
Main teaching idea
Question 6, Think like a mathematician
(3–5 minutes)
Learning intention: To understand methods of checking.
Resources: Note books, Learner’s Books
Description: The focus of this question is for learners
not only to try to answer the question, but to discuss
different learners’ methods of checking. During this
discussion, ask learners for their reasoning as to why
their method works as a check.
Make sure learners understand why, if checking using
x = 3, they get 17 < 17, which is an incorrect statement.
As with many of this type of question, learners often
find it simpler to solve the problem themselves, then
compare to find the mistake. Learners should easily see
that Franco did not multiply out the brackets correctly,
and their answer should show that the solution
is x ⩽ −11.
When learners have substituted in for the three x
values (an integer above, an integer below and the
actual solution) they will hopefully be able to give
an explanation as to why this is a good method of
checking.
Some learners might not realise the significance of the
answers [−30 ⩽ −29, −27 ⩽ −27 and −24 ⩽ −25]. Ask
a successful learner to explain. Check that all learners
understand that −30 is less than −29, so −30 ⩽ −29 is
true, that −27 is equal to −27, so −27 ⩽ −27 is true and
−24 is greater than −25, so −24 ⩽ −25 is not true. This
means that x cannot be −10, but it can be −11 and −12
showing that a solution of x ⩽ −11 must be true.
70
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4 EQUATIONS AND INEQUALITIES
Homework ideas
Assessment ideas
Workbook, Section 4.3.
With so many inequalities to solve, this is an excellent
opportunity for peer-marking. Having learners regularly
swap books (in pairs or groups) for checking/marking
helps learners focus on the important aspects of
the work.
As Section 4.3 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
PL
E
You could ask learners to make a summary containing
everything they think they need to remember for the
end-of-unit test. The following lesson, it is important
to share the summaries in class (e.g. spread out over a
few desks for everyone to look at), rather than marking
them. Discuss the different summaries as a class. When
the class agree that a point is important, copy that point
onto the board (you or a learner). Agree on as many key
points as possible. Learners could then improve/update
their individual summary if necessary. Learners could
store their summaries at home as a possible revision tool
towards mid-term/end-of-year tests.
When a learner makes a mistake, make sure that they
know what mistake they made and how to answer that
question correctly next time. Acknowledge learners who
are able to explain this with a ‘well done’ (as this is what
you need in the class, active learners).
SA
M
If a learner makes a mistake, but does not understand
the mistake they have made or how to correct it, it might
be a good idea for you to help that learner separately.
You could ask other learners to explain or you could
have a class discussion on that question/skill.
71to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
5 Angles
Unit plan
Topic
Approximate
Outline of learning
number of
content
learning hours
5.1 Calculating
angles
2
2
5.4 Constructions
2
5.5 Pythagoras’
theorem
2
Interior angles of polygons, Learner’s Book Section 5.2
including regular polygons. Workbook Section 5.2
Additional teaching ideas Section 5.2
Exterior angles of
polygons, including regular
polygons.
Learner’s Book Section 5.3
Workbook Section 5.3
Additional teaching ideas Section 5.3
Constructing specific
angles and regular
polygons.
Learner’s Book Section 5.4
Workbook Section 5.4
Additional teaching ideas Section 5.4
Pythagoras’ theorem.
Learner’s Book Section 5.5
Workbook Section 5.5
Additional teaching ideas Section 5.5
Resource sheet 5.5
M
5.3 Exterior
angles of
polygons
Angles in intersecting lines, Learner’s Book Section 5.1
parallel lines, triangles and Workbook Section 5.1
quadrilaterals.
Additional teaching ideas Section 5.1
PL
E
5.2 Interior angles 2
of polygons
Resources
SA
Cross-unit resources
Language worksheet: 5.1–5.5
End of unit 5 test
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• The sum of the angles of a quadrilateral is 360 °
(Stage 7).
• The exterior angle of a triangle is equal to
the sum of the two interior opposite angles
(Stage 8).
• Vertically opposite angles; corresponding and
alternate angles on parallel lines (Stage 8).
• How to construct triangles, perpendicular
bisectors and angle bisectors (Stage 8).
In this unit learners will move on to look at the
sum of the angles in any polygon and the interior
and exterior angles in any regular polygon.
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5 ANGLES
CONTINUED
They will use their knowledge of lines and angles
to solve more complicated problems where they
need to decide which rules to apply. Learners will
then extend their construction skills before being
introduced to Pythagoras’ theorem and using this
to solve problems and to find missing lengths in
triangles. Learners will be using algebra as they
work out missing lengths and angles. If needed,
refer back to the work on algebra in Unit 2.
PL
E
TEACHING SKILLS FOCUS
Active learning
How much should learners be told and how much
should they be given the opportunity to discover
things for themselves? In this unit several formulae
concerning the angles of polygons are developed.
Some of these formulae are not included in the
introductory material but are included in the exercises.
Learners are asked to look at particular examples,
to identify a pattern and then to use that to write a
general formula. This formula can then be tested and
used to generate more results. This process of looking
for patterns and generalising and testing is something
you should encourage in learners, to make them
confident mathematicians. By Stage 9 learners should
be able to carry out this process.
As you are introducing topics or helping learners
to answer questions, reflect on whether you are
giving them the opportunity to come up with
generalisations for themselves. Can they put these
generalisations into an algebraic form? Do not
immediately give learners the formula. Instead, give
them the opportunity to derive it for themselves.
Learners will become more confident and they will
be more likely to remember the formula.
M
5.1 Calculating angles
LEARNING PLAN
Learning objectives
Success criteria
9Gg.09
• Use properties of angles,
parallel and intersecting lines,
triangles and quadrilaterals to
calculate missing angles.
• Learners can answer questions
such as those in Exercise 5.1
that involve several properties.
They can also give reasons for
their answers.
SA
Framework codes
LANGUAGE SUPPORT
There is no new vocabulary in this section.
Knowledge of the terms quadrilateral, parallel
lines, vertically opposite angles, corresponding
angles and alternate angles are assumed from
earlier stages.
Some learners might need support to explain their
reasons. Practice will also help with this.
73to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Common misconceptions
Misconception
How to identify
How to overcome
Making errors with finding
corresponding or alternate angles.
Ask learners to give a reason for
each set in a solution to a problem.
Always ask for justification when
learners give an answer.
Starter idea
Getting ready (15 minutes)
Resources: Learner’s Book
• Question 1: angle sum of a quadrilateral.
• Question 2: exterior angle of a triangle
• Question 3: angles on parallel lines
• Question 4: proof of the angle sum of a triangle
• Question 5: bisecting an angle
Learners will need to use all of these ideas in this unit.
Main teaching idea
• angle sum of a triangle
• exterior angle of a triangle = sum of opposite
interior angles
• angle sum of a quadrilateral
• angles on a line
• angles round a point
• vertically opposite angles
• corresponding angles
• alternate angles.
For example, a learner might point to two angles
and say ‘these angles are equal because they are
corresponding angles’, or they might say ‘this angle is
the exterior angle of a triangle and it is equal to the sum
of these two interior angles’.
PL
E
Description: Ask learners to answer the Getting started
questions in Unit 5. When they have finished, discuss
each question in turn. Make sure that their knowledge
of the subject matter is secure. Here is a list of the
subject of each question:
Try not to prompt learners about what you want. See if
they can identify examples to show all of the following:
Angle knowledge (15 minutes)
M
Learning intention: To encourage learners to apply all
their knowledge of angle properties in a new situation.
If any of the rules are omitted, you will have to
prompt the learners, but do not do this until absolutely
necessary.
Resources: A copy of the diagram shown to display.
Description: Show learners a copy of this diagram:
Differentiation ideas: If learners cannot remember any
of these rules, you might need to draw another diagram
to reinforce that particular point.
SA
Plenary idea
Tell learners that they know a number of properties
of angles in triangles, in quadrilaterals and on parallel
lines. Ask them to show examples of any of these
properties in the diagram. Ask learners to come to the
diagram in turn and to point out an example.
Reflection (5 minutes)
Resources: None
Description: Draw a triangle as shown:
A
a°
B
b°
c°
C
Ask learners to imagine point C moving away to the
right. A and B do not move.
Ask ‘As point C moves, what happens to angle c?’ [it gets
closer and closer to 0].
Ask ‘What happens to a + b?’ [it gets closer to 180 °].
Now show this diagram where the two lines are parallel:
74
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5 ANGLES
more practice of this by asking them to justify answers
in other questions in this exercise where they are not
explicitly asked to do so, such as Question 6.
A
a°
B
Homework ideas
b°
Workbook, Section 5.1.
Ask ‘How do you know that a + b = 180?’ Learners
should be able to use alternate angles to explain this.
PL
E
Finally, ask how the two examples are related. Learners
should see that the first diagram gets more and more
similar to the second diagram as C moves to the right.
Assessment ideas: Replies from learners will indicate
whether they can see how to use properties of angles on
parallel lines in this example.
Guidance on selected Thinking and
working mathematically questions
Convincing
Exercise 5.1, Question 7
As Section 5.1 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
The plenary activity idea in the Additional teaching
ideas asks learners to produce a summary diagram.
Learners could complete this for homework.
Assessment ideas
This unit is about pulling together ideas that were
previously learned separately and being able to choose
the appropriate concept to apply in a given situation.
You can see from learners’ answers to questions,
from their comments during discussion and from any
questions they ask whether they have learnt these skills.
M
This question asks learners to give reasons for their
answers. You want learners to be able to state the
properties of angles as evidence to justify a solution.
This work on angles is particularly useful for developing
this skill. There are only a small number of concepts
that learners might need to use and they should become
skilful at choosing the correct rule. Learners don’t
need to write a lot as long as they clearly state the
particular idea they are using. You could give learners
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
SA
5.2 Interior angles of polygons
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Gg.07
• Derive and use the formula
for the sum of the interior
angles of any polygon.
• Learners can explain how
to calculate the sum of
the angles of a polygon
with a given number of
sides, and then use that
to calculate the interior
angle of the corresponding
regular polygon.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
LANGUAGE SUPPORT
Regular polygon: a polygon where all the sides are
the same length and all the interior angles are the
same size
The word polygon is familiar from earlier stages.
The angles inside a polygon are referred to
as interior angles in this section to distinguish
them from the exterior angles covered in the
next section.
Make sure that learners are confident in using the
names of different polygons, especially polygons
with larger numbers of sides.
Misconception
PL
E
Common misconceptions
Automatically thinking of regular
polygons when they imagine
shapes with more than four sides.
Starter idea
How to identify
How to overcome
Ask a learner to draw a hexagon.
Do they try to draw a regular shape?
The exercises and activities have
examples of polygons that are
not regular.
Making polygons (10 minutes)
Resources: None
Now ask learners to add equilateral triangles to the
other two sides of the square, as shown.
M
Description: Ask each learner to draw a square.
Then ask them to draw an equilateral triangle with
sides the same length attached to one side of the
square. They should draw this, but it might be in a
different orientation.
SA
Ask ‘What is the name of this shape?’ [pentagon].
Ask learners to work out the size of each angle
[60, 90, 90, 150, 150 degrees].
Now ask learners to add an equilateral triangle to the
opposite side to the first triangle, as shown.
Ask ‘What is the name of this shape?’ [octagon].
Ask learners to work out the size of each angle
[60, 60, 60, 60, 210, 210, 210, 210 degrees].
A possible error is to say 150 degrees instead of 210
degrees. If this happens, discuss why it is incorrect [it is
because they have found the exterior angle instead of
the interior angle].
Emphasise that all these angles are inside the shape. For
that reason they are called interior angles.
Ask ‘What is the name of this shape?’ [hexagon].
Ask learners to work out the size of each angle [60, 60,
150, 150, 150, 150 degrees].
76
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5 ANGLES
Main teaching idea
Angles of a polygon (10 minutes)
Learning intention: To learn a way to work out the sum
of the interior angles of a pentagon and the size of each
interior angle of a regular pentagon. The method can be
applied to other polygons.
Resources: Rulers, protractors
PL
E
Description: This activity is described in the introduction
text in Section 5.2 of the Learner’s Book. Ask each
learner to draw a pentagon. They should use a ruler.
The pentagon does not need to be regular, but it must
have five sides.
sides, but it should be at least five. The first polygon
should be irregular, the second polygon should
be regular.
• On the first diagram, show how to divide it into
triangles and hence find the sum of the angles.
• On the second diagram, the regular polygon, show
how to calculate the angles.
Assessment ideas: Ask learners to show their work to a
partner. Their partner could suggest how to improve it.
Looking at learners’ work will show whether they have
understood the ideas in this section.
Ask learners to join any two vertices with a straight line.
Ask ‘what shapes do you have now?’ [a triangle and a
quadrilateral]. Now tell them to divide the quadrilateral
into two triangles.
Ask them to make sure they agree that the angles of
the three triangles combined make the angles of the
pentagon.
Ask ‘What is the sum of the angles of the pentagon?’
[since there are three triangles, this is 3 × 180 ° = 540 °].
Now say that in a regular pentagon, all the sides are the
same length and all the angles are the same size. Ask
‘What is the size of each angle?’ [540 ÷ 5 = 108 °].
Specialising and generalising
Exercise 5.2, Question 9
Learners have the opportunity to generalise from a set
of specific examples. Putting the results for different
polygons in order in a table is a useful technique. In this
case it shows that the angle sum increases by 180 for
each increase in the number of sides. This gives a useful
hint about the form of the equation. Learners need
practice in recognising a pattern and then writing this
in an algebraic form. This second step is something that
many learners find difficult and they need to practise.
M
Finally, ask learners to try to use a ruler and a
protractor to draw a regular pentagon with each side
5 cm long. When finished, learners ask a partner to
check the accuracy of the measurements. Ask ‘How
accurately could you do this?’. Note that it can be
difficult to do the drawing accurately.
Guidance on selected Thinking and
working mathematically questions
SA
Differentiation ideas: Some learners will find the
drawing difficult to do. Alternatively, you could give
them a photocopy of a regular pentagon with a side
of 5 cm and ask them to check by measuring that it is
indeed regular.
Plenary idea
What have you learned? (5 minutes)
Resources: None
Description: There are two distinct ideas here. The first
idea is calculating the sum of the angles of any polygon.
The second idea is using that result to calculate the angle
of a regular polygon.
Ask learners to make summary notes as follows:
• Draw side-by-side two polygons with the same
number of sides. They can choose the number of
Homework ideas
Workbook, Section 5.2.
As Section 5.2 will take more than one lesson, set suitable
parts of the Workbook at the end of each lesson. Only set
questions that can be answered using skills and knowledge
gained from that lesson. Workbooks are aimed at fluency
and consolidation through practice, not as a method to
learn new skills that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
The second main teaching activity idea ‘Regular
polygons’ in the Additional teaching ideas gives another
homework suggestion.
Assessment ideas
The most important idea in this section is dividing a
polygon into triangles and using these to find the sum
of the interior angles. There are a number of questions
that practise this in Exercise 5.2. You can use learners’
answers to these questions to assess their understanding
and competence.
77to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
5.3 Exterior angles of polygons
LEARNING PLAN
Learning objectives
Success criteria
9Gg.08
• Know that the sum of the
exterior angles of any
polygon is 360 °.
• Learners are able to explain
how to calculate the exterior
angle of any regular polygon.
LANGUAGE SUPPORT
PL
E
Framework codes
Exterior angle of a polygon: the angle outside a
polygon between an extension of one side and an
adjacent side
The term exterior angle is familiar from Stage 8 for
triangles and quadrilaterals. It will be applied to
any polygon in this section.
Common misconceptions
Misconception
How to overcome
Give learners an interior angle,
such as 150 °, and ask them to
calculate the external angle.
Use diagrams to illustrate the
exterior angles as much as
possible.
M
Confusing the exterior angle
with the entire angle outside the
polygon, rather than 180 minus the
interior angle.
How to identify
Starter idea
Main teaching idea
Exterior angle of a triangle (5 minutes)
The external angles of a pentagon
(10 minutes)
Resources: None
SA
Description: Ask learners to draw a triangle. Then ask
them to draw an exterior angle at one point. Make sure
they all do this by extending one side of the triangle.
The exterior angle is the sum of two of the interior
angles of the triangle. Ask learners to mark these two
angles. Check that they mark the opposite angles.
Now ask them to repeat this for the exterior angle at a
second point. And then for the third point.
Illustrating on the board what you want learners to do
will help them to follow the instructions.
They should now see that they have marked every
interior angle twice. Ask ‘What does this tell you about
the sum of the exterior angles of a triangle?’ Learners
should see that it is twice the sum of the interior angles
of the triangle. It is 360 °.
Learning intention: To give a practical demonstration of
the sum of the external angles of a pentagon.
Resources: None
Description: This is a way of showing that the sum of the
external angles of a pentagon is 360 °. It is similar to the
method shown in the introduction to Section 5.3 in
the Learner’s Book where you can see this diagram:
c° C
D
d°
E
e°
P
A
a°
b°
B
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5 ANGLES
Learners can carry out a practical version of this. Ask
learners to draw a large pentagon. Ask them to put
a pen (or pencil) on one side (EA on the diagram).
Ask them to move the pen around the pentagon in an
anticlockwise direction, noting how much they turn the
pen at each vertex. You can illustrate this on the board.
Learners should see that, when it gets back to the start,
the pen has made one complete turn.
Check by adding angles each time.
A possible homework is to ask learners to show different
ways of arranging regular hexagons and equilateral
triangles around a point.
Assessment ideas: You can check that learners know the
angles of regular polygons with a small number of sides.
Ask ‘Why is it not possible to use pentagons in any of the
arrangements of regular polygons around a point?’
PL
E
Ask them to mark on the external angles that the pen
has turned through. They have shown that the sum of
the external angles is 360 °.
and a square; two dodecagons and a triangle; a
dodecagon, a square and a hexagon.
An alternative way to do this, if you have enough space,
is to draw the pentagon on the ground and to ask a
learner to walk round the shape. Point out the amount
of turn at each vertex and confirm that overall there is
one whole turn.
Now ask learners to repeat the exercise but moving
the pen clockwise. Again, the pen turns through 360 °.
Marking the angles, they will see that they are vertically
opposite the angles in the first version. This means they
are the same size.
Guidance on selected Thinking and
working mathematically questions
Specialising and generalising
Exercise 5.3, Question 6
This question gives learners the opportunity to recognise
a pattern and express it in algebraic form. They can
then use this formula to generate further results.
Moving from the specific to the general is an important
mathematical skill that learners need to practise.
Finally, ask ‘What happens if the pentagon is regular?’
Make sure learners give the reply that the external
angles are all equal. Each angle is 360 ° ÷ 5 = 72 °. This
means that the internal angle is 180 ° − 72 ° = 108 °. This
confirms a result from Section 5.2.
M
Learners can now start Exercise 5.3 in the Learner’s Book.
CROSS-CURRICULAR LINKS
Differentiation ideas: A class demonstration is
important if some learners cannot understand how to
carry out the activity correctly.
Plenary idea
SA
Review of regular polygons (5 minutes)
Resources: None
Description: Show this table about the interior angles of
regular polygons with the second row empty. Ask learners
to tell you the numbers to put into the second row.
Sides
3
Angle
60 ° 90 ° 108 ° 120 ° 135 ° 144 ° 150 °
4
5
6
8
10
12
Say that this shows that it is possible to arrange two
squares and three equilateral triangles around one
point because 2 × 90 ° + 3 × 60 ° = 360 °. Ask ‘What other
possibilities are there using regular shapes?’
Some possible answers are: six equilateral triangles;
four squares; three hexagons; two hexagons and two
triangles; one hexagon and four triangles; two octagons
Examples of the use of regular polygons are
often found in design and art, particularly in art
from the Islamic world.
Homework ideas
Workbook, Section 5.3.
As Section 5.3 will take more than one lesson, set suitable
parts of the Workbook at the end of each lesson. Only set
questions that can be answered using skills and knowledge
gained from that lesson. Workbooks are aimed at fluency
and consolidation through practice, not as a method to
learn new skills that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Another homework suggestion is given in the plenary
activity idea in these notes.
Assessment ideas
There are two main points to look for. Do learners know
what the sum of the external angles of a polygon is? Can
they use this fact to find the exterior angles of a regular
polygon? Use their answers to questions in the exercises
and during class discussion to assess this.
79to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
5.4 Constructions
LEARNING PLAN
Learning objectives
Success criteria
9Gg.11
• Construct 60 °, 45 ° and 30 °
angles and regular polygons.
• Learners can construct the
angles with an accuracy of
plus or minus 2 degrees.
LANGUAGE SUPPORT
PL
E
Framework codes
Inscribe: when you inscribe a polygon in a circle,
every vertex is on the circle
Learners should be familiar from Stage 8 with
the use of the word ‘construct’ in this context to
mean to draw with a ruler and compasses only
and without using a protractor. Of course, learners
can use protractors to check accuracy after a
construction has been completed.
‘Draw an angle of 30 degrees’ normally means use
a protractor. ‘Construct an angle of 30 degrees’
normally means with just a ruler and compasses.
Common misconceptions
Misconception
How to overcome
Ask learners to bisect a line or
an angle. Check the accuracy by
measurement.
Make sure to carry out the
peer- and self-assessment
requested in Exercise 5.4.
M
Not keeping the compasses
unchanged for successive steps in
constructions that require this.
How to identify
Starter idea
Bisecting a line (5 minutes)
SA
Resources: Ruler and a pair of compasses
Description: Ask each learner to draw a line segment. It
must not be parallel or perpendicular to the sides of the
paper. Now ask each learner to draw the perpendicular
bisector of their line. They can use a ruler and
compasses but not a protractor. They should know how
to do this by putting the compass point on each end of
the line in turn and drawing arcs on each side of the line.
If some learners cannot remember, ask a learner who
can do it to explain.
The construction should look like this. AB is the original
line segment.
A
B
Ask each learner to exchange their construction with a
partner and to check each other’s work. Ask ‘Is it a right
angle?’ They can check with a protractor. Ask ‘Are the
two parts of the original line the same length?’ They can
check by measuring with a ruler. If it is inaccurate, the
learner should draw the diagram again. The most likely
error is changing the length of the compasses when
moving from one end of the line to the other.
80
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5 ANGLES
Main teaching idea
Inscribing a square in a circle (10 minutes)
Guidance on selected Thinking and
working mathematically questions
Learning intention: To practise doing a construction
accurately, with an opportunity for peer assessment.
Critiquing
Resources: Learner’s Book, rulers and compasses
Learners are asked to draw two patterns and then to
think about an alternative method. For example, for
the first pattern they could start with a construction
of a regular hexagon in a circle and join vertices to
make two triangles; or they could draw the internal
hexagon first and add triangles on the edges; or they
could construct first one equilateral triangle and then
another. The question gives an opportunity to evaluate
different approaches.
PL
E
Description: Ask learners to draw a circle with
compasses. Check quickly that this is done correctly.
Then work through Example 1 in the introduction to
Section 5.4 in the Learner’s Book. Give the learners
clear instructions for each step. It is better if the
diameter is drawn so that it is not parallel to the sides of
the paper. This will test construction skills more.
Use peer assessment to check accuracy. Learners should
measure the sides of the square to ensure they are all
equal and measure the angles with a protractor to
ensure they are right angles.
Differentiation ideas: Some learners might need more
support with a demonstration of each step of the
construction. Alternatively, you can let learners who
have successfully drawn the square to give instructions
to learners who cannot complete the task successfully.
Plenary idea
Constructing other angles (10 minutes)
Homework ideas
Workbook, Section 5.4.
As Section 5.4 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
M
Resources: Compasses and ruler
Exercise 5.4, Question 11
Description: Ask learners to remind you of the angles
they have learned to construct. They should say 30, 45,
60 and 90 degrees.
Ask them to describe a method to construct an angle
of 15 °. Two possible ways are:
SA
• Construct 30 ° then bisect it.
• Construct 45 °. Use one side to construct an angle
of 60 °. The angle between is 15 °.
If a learner gives one method, ask for an alternative
suggestion.
In a similar way, ask how to construct 75 °.
Answers could use 90 − 15 or 45 + 30.
You could also use the plenary activity idea in the
Additional teaching ideas as homework instead of
during the lesson. In that case, ask learners to measure
the lengths of the lines themselves. They could also write
a brief description of how they carried out the task.
Assessment ideas
Throughout the exercise questions ask learners to
monitor their progress through peer- or self-assessment.
This gives them immediate feedback on the accuracy
of their construction technique. Where learners make
mistakes they should redraw the diagrams.
Assessment ideas: If you want to assess learners’
performance, you can ask them to actually construct
either of these angles and then use peer assessment to
check accuracy.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
5.5 Pythagoras’ theorem
LEARNING PLAN
Learning objectives
Success criteria
9Gg.10
• Know and use Pythagoras’
theorem.
• Given two sides of a rightangled triangle, the learner
can find the third side,
deciding whether they are
calculating a hypotenuse or
another side.
LANGUAGE SUPPORT
PL
E
Framework codes
Hypotenuse: the longest side of a right-angled
triangle, opposite the right angle
Pythagoras’ theorem: a relationship between the
three sides of a right-angled triangle
Make sure that learners read questions and look at
diagrams carefully to understand if they are being
asked to work out the hypotenuse or another side.
Common misconceptions
Misconception
How to overcome
Check answers to questions in
Exercise 5.5 where the hypotenuse
is given and another side is
required.
Ask questions such as ‘Two sides of
a right-angled triangle are 6 and 9.
What could the third side be?’ This
emphasises the two possible cases.
M
Assuming that they always add
the squares of the two given sides
when asked to find the third side of
a right-angled triangle.
How to identify
Main teaching idea
Revising squares of integers (10 minutes)
Introduction to Pythagoras’ theorem
(15 minutes)
SA
Starter idea
Resources: Calculator (optional)
Description: Ask learners to tell you the first 20 square
numbers and to write them as they do so. They should
know most of these by heart. Then ask learners to find
pairs of square numbers whose sum is also a square
number. They can work in pairs to do this. The easiest
ones to find are the squares of 3, 4 and 5. Others
possible sets of 3 are: 6, 8, 10; 9, 12, 15; 12, 16, 20; 5, 12,
13; 8, 15, 17.
These are the only possibilities where the numbers are 20
or less. Learners might find other sets of three where the
numbers are greater than 20.
Learning intention: To use an example to illustrate a
general result.
Resources: None
Description: Ask learners to draw a rectangle with
sides 6 cm and 8 cm. Draw a diagonal and measure it.
They should agree that the length of the diagonal is
10 cm. Answers of 10.1 or 9.9 would be acceptable from
a drawing, but try to get agreement that it seems to
be 10 cm.
Emphasise that 62 + 82 = 102 and explain that this is
an example of a result that is true for all right-angled
triangles.
82
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5 ANGLES
c
b
Guidance on selected Thinking and
working mathematically questions
a +b =c
2
2
2
Specialising and generalising
a
Exercise 5.5, questions 13 and 14
Both questions 13 and 14 support learners in moving
from particular numerical examples to a generalisation.
At this stage you want learners to think about algebra
when they are generalising. The ability to move from
using numbers to algebra is an important skill that
takes time to develop. Algebra is a challenge for many
learners. The more experience you can give learners
of generalising and expressing a generalisation
algebraically, the more confident they will become. You
want learners to feel that algebra is a natural way to
express generalisations.
PL
E
Say ‘The square of the hypotenuse is equal to the sum
of the squares of the other two sides’. Tell learners
that this is called Pythagoras’ theorem. (You might
like to tell learners that it is named after a man from
the Mediterranean island of Samos who wrote a book
including a proof of this result over 2500 years ago).
Now ask learners to draw a rectangle with sides 7 cm
and 10 cm. Once again draw and measure the diagonal.
Write values that learners suggest.
Suppose the diagonal is d cm. Then by Pythagoras’
theorem, d 2 = 72 + 102
Ask ‘How can you find d ?’ They should see that d 2 = 149
and so d = 149. A calculator gives the square root
as 12.2065… because the square root is irrational.
Ask learners to round this to one decimal place. That
gives d = 12.2 cm. Ask ‘Did you get that answer by
measurement?’
Homework ideas
Differentiation ideas: Some learners will need more
practice. You can ask them to draw other rectangles and
repeat the exercise. For more confident students you can
go straight on to Questions 1 and 2 in Exercise 5.5.
As Section 5.5 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Workbook, Section 5.5.
M
Exercise 5.5 in the Workbook has more questions on
Pythagoras’ theorem. This is an opportunity to give
learners more practice.
Plenary idea
Check your progress (15 minutes)
Resources: ‘Check your progress’ exercise, Learner’s
Book
SA
Description: Ask the learners to do the ‘Check your
progress’ questions. When they have finished, go through
each question. Ask learners to give their answers and to
explain how they obtained them.
Assessment ideas: As learners work on the questions
you can check they are doing them correctly. When
going through the answers, asking learners to explain
their methods is a good way of assessing understanding.
You can also use ‘traffic lights’ to get quick feedback.
(green = very confident, red = very uncertain).
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Assessment ideas
The answers to Exercise 5.5 give the best opportunity to
decide whether learners are using Pythagoras’ theorem
correctly. Are they deciding first whether they are
finding the hypotenuse or one of the other sides? Are
they writing down Pythagoras’ theorem initially to give
an equation, and then rearranging it if necessary before
solving it?
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
PROJECT GUIDANCE: ANGLE TANGLE
Ask learners to think about how they might prove
that angle a is always 135 °. If they give no ideas,
suggest labelling the two equal angles at P with
b, and the two angles at R with c, creating the two
equations a + b + c = 180 ° and 2b + 2c + 90 ° = 180 °,
and deducing the value of b + c and hence a.
Possible approach
Learners will require plain paper, rulers, pencils,
compasses and protractors.
Next, invite learners to explore what happens with
triangles with an angle of 60 °, with triangles with
an angle of 120 °, and with triangles with other
angles. Can they predict what angle a will be for
their chosen angles?
PL
E
Why do this problem?
This problem gives learners an opportunity to
practise construction skills in a context that allows
them to specialise, by investigating carefully
chosen angles of 90 °, 60 ° and 120 °, and then
generalise their results to any triangle.
Invite learners to construct a right-angled triangle
PQR and to bisect the other two angles, as shown
in the diagram. Then ask them to measure the
angles in their triangle.
Key questions
Can you use algebra to explain what is happening?
R
Could you use dynamic geometry to create a
diagram that helps you to understand?
a
P
Q
20 °
Angle Q
Angle R
Angle a
90 °
70 °
135 °
90 °
35 °
135 °
SA
55 °
Possible support
Encourage learners to construct triangles with
angles which are multiples of 10 ° to begin with,
and to write each angle on their diagrams. Make
the distinction between angles that they need to
measure, and angles that they can calculate from
others that they know.
M
On the board collect together results, in a table
as shown:
Angle P
Finally, bring the class together to discuss their
findings and the convincing arguments or proofs
they have constructed.
Possible extension
Invite learners to construct regular polygons using
the angle constructions they have discovered.
For example, how might they construct a regular
octagon inside a square? Or a regular dodecagon
inside a regular hexagon?
Invite learners to comment on anything they
notice. They might be surprised that regardless
of what they choose for angle P or R, angle a is
always 135 °.
84
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6 STATISTICAL INVESTIGATIONS
Unit plan
PL
E
6 Statistical
investigations
Topic
Approximate
Outline of learning content
number of
learning hours
6.1 Data
collection and
sampling
2
6.2 Bias
2
Resources
Understand how to choose
Learner’s Book Section 6.1
the most appropriate data
Workbook Section 6.1
to collect when investigating
Additional teaching ideas Section 6.1
predictions.
Understand the problem of Learner’s Book Section 6.2
bias and how to deal with it. Workbook Section 6.2
Additional teaching ideas Section 6.2
M
Cross-unit resources
Language worksheet: 6.1–6.2
End of unit 6 test
BACKGROUND KNOWLEDGE
SA
For this unit, learners will need this background
knowledge:
• Selecting and justifying data collection and
sampling methods (Stage 8).
• The advantages and disadvantages of different
sampling methods (Stage 8).
In this unit learners will extend the data collection
and sampling skills developed in Stage 8. They
will also look at sources of bias when using a
sample and use data to make inferences and
generalisations.
TEACHING SKILLS FOCUS
Active learning
The statistical objectives are different from the
mathematical objectives in other parts of the
programme of study. They are designed to develop
learners’ ability to carry out increasingly more
sophisticated investigations. Answers to questions
are not just a simple number, as they would be
to numerical calculations. There will often be a
range of justifiable answers. In this unit the idea
of bias is introduced and examined. Questions
encourage learners to think about planning and
writing descriptions. Teaching needs to give learners
opportunities to reflect on what they have written.
In your teaching, use whole-class discussion to
give learners a chance to think about their own
answers and whether they can improve them. In
the exercises, tell learners to work in pairs. This will
encourage discussion and encourage learning.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
CONTINUED
At the end of the unit, reflect on how successful it
was to ask learners to work in pairs. Did it encourage
discussion? Also consider whether whole-class
discussion gave the opportunity to discuss all the
questions, including the questions set as homework.
LEARNING PLAN
Framework codes
9Ss.01
Learning objectives
Success criteria
• Select, trial and justify data
collection and sampling methods
to investigate predictions for a
set of related statistical questions,
considering what data to
collect, and the appropriateness
of each type (qualitative or
quantitative; categorical, discrete
or continuous).
• Learners can complete
the planning for the data
collection in the questions
in Exercise 6.1 of the
Learner’s Book.
• Make informal inferences and
generalisations.
• Learners can make inferences
and generalisations,
supported by data collected.
M
9Ss.05
PL
E
6.1 Data collection and sampling
LANGUAGE SUPPORT
SA
There is no new vocabulary in this section.
Learners are already familiar with the words
‘prediction’ and ‘sample’ in the context of a
statistical investigation.
Support learners during discussions with a partner
or with the whole class.
Common misconceptions
Misconception
How to identify
How to overcome
Not always being clear about the
way to write a prediction to test.
In discussion of the example in the
text in the Learner’s Book and the
questions in Exercise 6.1.
Use the Workbook questions to
give extra practice if necessary.
Starter idea
Ready to start (10 minutes)
Resources: Getting started questions in the
Learner’s Book
Description: Ask learners to complete the Getting started
questions. They can do this in pairs. When they have
completed the exercise, take answers from the learners.
There are many possible answers that can be given.
86
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6 STATISTICAL INVESTIGATIONS
Main teaching idea
Planning a statistical investigation
(20 minutes)
consists of planning a few statistical investigations,
rather than questions with correct or incorrect answers.
Each question can take quite a bit of time. It is best if
learners work in pairs.
Learning intention: To look at all the steps needed to
plan a statistical investigation.
This unit concentrates on planning. Analysing results
comes in Unit 15.
Resources: None
Differentiation ideas: Adapt the amount of support
and guidance you give to the ability of the learners.
Some learners might need more help with thinking of
questions, forming predictions, and so on.
Plenary idea
1
Guidelines (5 minutes)
3
Forming predictions. Again, there are suggestions in
the text. Ask learners to suggest other predictions.
Emphasise the difference between questions and
predictions here. A question can give rise to a
prediction to be tested. The prediction might be
something you think is correct or incorrect or you
might not know. That does not matter. Predictions
do not have to be things you think are true.
Data to collect. Choose one or two predictions and
discuss the data you need to collect. Is the data
continuous, discrete or categorical? Data can be
collected in different ways. For example, for foot
size you could measure the length of the foot or
you could use shoe size. One factor to consider is
which data is easier to find.
Choosing a sample. Learners will know different
ways to choose a sample. Here they should consider
different ways and then decide on what they think is
the best method in the circumstances.
SA
4
Asking questions. There are five example questions
in the text. Ask learners to suggest more questions.
There are no correct and incorrect answers here, you
just want learners to think of different possibilities.
Resources: Exercise 6.1, Reflection, Learner’s Book
Description: Use the reflection question in Exercise 6.1
as a basis for a brief discussion of the key points of this
section. Learners will have different suggestions for the
three pieces of advice. Write these suggestions and try to
get learners to agree on the best three.
Assessment ideas: You could start by asking pairs
of learners to compare their answers to the reflection
question in Exercise 6.1 and to agree between themselves
which is the better set of three. Or they might choose a
mixture of the two sets.
Guidance on selected Thinking and
working mathematically questions
M
2
PL
E
Description: The introductory text in Section 6.1 in
the Learner’s Book discusses the relationship between
height and other body measurements. Read through this
text in detail, adding to the outline in the text. Establish
the procedure for planning a statistical investigation:
When carrying out each step it is a good idea for learners
to work in pairs. Give them a minute or two to discuss
each point in turn. So for point 1, give learners two
minutes to think of some questions in pairs, and then
take feedback. You can do this with the other points too.
A final point to make is that it is a good idea to do a
trial run. Collect the data you have decided on from a
few people. Check that collecting the data in this way is
feasible, and think about any changes you wish to make
to your plan before you start your investigation.
Learners can now start Exercise 6.1. This is different
to most exercises in the Learner’s Book because it just
Convincing
Exercise 6.1, all questions
All the questions in this exercise involve planning
the collection of data to test predictions. Testing a
prediction using statistics is similar to justifying a
conjecture, but the outcome will not be so definite.
Data can support a prediction or cast doubt on it but
it is not a definite proof. The data must be appropriate
for the conclusion to be valid. That is why there is an
emphasis on choosing a sample in the best way possible
and making sure that the sample is representative.
Throughout the exercise, learners are asked to give
reasons for their answers, giving them the opportunity
to practise their convincing skills.
Homework ideas
Workbook, Section 6.1.
Exercise 6.1 in the Workbook has more suggestions
for statistical investigations to plan. As Section 6.1 will
take more than one lesson, set suitable parts of the
87to publication.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Workbook at the end of each lesson. Only set questions
that can be answered using skills and knowledge gained
from that lesson. Workbooks are aimed at fluency and
consolidation through practice, not as a method to learn
new skills that should be taught in class.
6.2 Bias
LEARNING PLAN
Framework codes
9Ss.02
Learning objectives
Success criteria
• Explain potential issues and
sources of bias with data
collection and sampling
methods, identifying further
questions to ask.
• Given a survey result, learners
can ask relevant questions to
check on the validity.
• Identifying wrong or
misleading information.
• Learners can identify the
problems with the diagrams
in questions 4 and 10.
M
9Ss.05
Ask learners to work in pairs or small groups. This
will allow them to discuss ideas, to clarify their
understanding, and to become more confident. This
topic does not have simple numerical answers where it is
obvious whether the answer is correct or not. Your aim
as a teacher is to develop an ability to think critically
and evaluate strategies. This is aided by class discussion
which will enable you both to develop good solutions
and to assess each learner’s understanding.
PL
E
As learners will be working on their own when they do
the homework, you could use a starter in a subsequent
lesson to allow learners to compare, discuss and possibly
improve their answers in pairs.
Assessment ideas
LANGUAGE SUPPORT
SA
Bias: selectivity when choosing a sample that
makes the results unrepresentative
Misleading: information that leads you to an
incorrect conclusion
Use the words ‘bias’ and ‘misleading’ in
discussions, and encourage learners to use these
new words too.
Support learners during discussions with a partner
or with the whole class.
Common misconceptions
Misconception
How to identify
Thinking that bias can be
Ask learners to describe possible
avoided by ensuring that
sources of bias.
different groups of people by
age and gender are represented.
How to overcome
The questions in Exercise 6.2 look at
various sources of bias. Use the main
activity ideas in these notes and in the
Additional teaching ideas to discuss all
possible sources.
88
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6 STATISTICAL INVESTIGATIONS
Starter idea
Plenary idea
A representative sample (5 minutes)
Review of the unit (10 minutes)
Resources: None
Resources: ‘Check your progress’ exercise in the
Learner’s Book
Main teaching idea
Bias (15 minutes)
Description: Ask learners to complete the questions in
the ‘Check your progress’ exercise. They could do this
in pairs.
Assessment ideas: Working in pairs gives an
opportunity for peer- and self-assessment to take place.
PL
E
Description: Say that you want to select a representative
sample of 20 people from the audience in a cinema
or a theatre. Ask ‘How could you do this?’ Focus on
what it means to say that the sample is representative
rather than the mechanics of choosing the sample.
Factors suggested are likely to be age and gender, but
there could be others. Finish by saying that these are
important things to think about when you are doing a
statistical investigation.
Learning intention: To become familiar with the concept
of bias and sources of bias.
Resources: Worked Example 6.2 in the Learner’s Book
Description: On the board, display the scenario in
Worked example 6.2:
Convincing
Exercise 6.2, Question 9
When learners reach Question 9, they will be aware of
the problems involved in choosing an unbiased sample.
This question gives learners an opportunity to apply
those ideas to devise a more sophisticated plan than
simply asking the first people they see.
Homework ideas
Workbook, Section 6.2.
As Section 6.2 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
M
An investigation is carried out to test the prediction that
people in a town are in favour of building a new library.
A survey is carried out on people using a supermarket
between 09:00 and 12:00 one Wednesday and Thursday.
Guidance on selected Thinking and
working mathematically questions
If this scenario is not appropriate to your setting,
change it to something similar. The important point is
that the sample will not be representative.
Ask ‘Why will this sample not be representative?’ Say
that it is a biased sample. Use this new word during the
discussion to reinforce it.
SA
Take suggestions of ways to get a more representative
sample. This could include doing the survey at a library,
doing the survey on different days and at different times
of day, looking for a mixture of males and females and
a range of ages.
Differentiation ideas: Ask learners to discuss ways
to get a representative sample in pairs and then take
suggestions. Check whether some pairs need more
guidance about the type of areas to consider.
You could ask learners to find an example of a survey
on the internet and to look for possible sources of
bias. This could be followed up in a starter activity in
a subsequent lesson, asking individuals to describe any
particularly interesting examples found.
Assessment ideas
Self-assessment is important in this section. When
looking at answers to particular questions, learners
can judge their own responses and learn from them.
Ask learners to work in pairs on the questions in
Exercise 6.2. This will encourage discussion and
the sharing of ideas. This will help to ensure a clear
understanding of the idea of what bias is and possible
sources of bias.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Unit plan
Topic
PL
E
7 Shapes and
measurements
Approximate
Outline of learning content
number of
learning hours
7.1 Circumference 1–1.5
and area of a
circle
7.2 Areas of
1–1.5
compound shapes
1–2
Learner’s Book Section 7.1
Workbook Section 7.1
Additional teaching ideas Section 7.1
Estimate and calculate areas Learner’s Book Section 7.2
of compound 2D shapes.
Workbook Section 7.2
Additional teaching ideas Section 7.2
Use very small or very large
units of length, capacity
and mass.
M
7.3 Large and
small units
Know and use the
formulae for the area and
circumference of a circle.
Resources
Learner’s Book Section 7.3
Workbook Section 7.3
Resource sheet 7.3
Additional teaching ideas Section 7.3
Cross-unit resources
Language worksheet: 7.1–7.3
End of unit 7 test
SA
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Understand π as the ratio between a
circumference and a diameter. Know and
use the formula for the circumference of a
circle (Stage 8).
This unit reinforces and extends learners’
knowledge of manipulating basic formulae using
knowledge of inverse operations to change
the subject of a formula, especially relating to
the circle. Remind learners of the methods for
calculating the perimeter of a semicircle and the
common mistake of not including any straight lines
in their calculation.
Learners’ knowledge of calculating the area of
compound shapes is extended to estimation and to
calculating areas of more complicated compound
shapes, some involving circles and semicircles.
This unit also extends learners’ knowledge of the
prefixes of the metric system (e.g. milli, centi, kilo,
etc.) to much smaller and larger prefixes.
90
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7 SHAPES AND MEASUREMENTS
TEACHING SKILLS FOCUS
Remind learners that the key to being successfully
involved in this type of learning is that there is
no judgement. The learner asking for help and
the learner giving help are both learning and
improving.
At the end of Unit 7, ask yourself:
• Did learners have useful discussions that solved
issues one of them was having? How do you
know?
• Did a variety of learners do the explaining – or
did you rely on just one or two learners?
• Did the learners that helped other learners
understand the work better themselves because
of the help they gave? Are you sure?
• Did learners that received help from other
learners benefit from it, or did they then need
help/advice from you?
• Are all learners that require help getting it?
• What other ways could you get learners to
explain more to other learners?
M
PL
E
Active learning
Throughout the three sections of Unit 7, if learners
do not understand, or they continue to get the same
type of question incorrect, ask another learner to
explain/help. It is important that you also listen to
the explanation/help given by another learner. You
need to be able to confirm that the help is of good
quality or to ask if another learner would/could
explain the problem in a different way.
Active learning helps to establish good learning
patterns and practice. When a learner can explain
well, it shows that they thoroughly understand
what they are doing and know how to improve.
Also, learners often feel more confident speaking
to other learners, asking more targeted questions,
so becoming more active learners themselves. As
learners get more used to explaining concepts
or asking for specific, targeted help from other
learners, these discussions can happen without
you being present. Hopefully, the practice learners
have had during stages 7 and 8 will mean they are
already confident in this very effective learning skill.
7.1 Circumference and area of a circle
LEARNING PLAN
Learning objectives
SA
Framework codes
9Gg.01
• Know and use the
formulae for the area and
circumference of a circle.
Success criteria
• Learners can calculate the
area and circumference of
a circle given its radius or
diameter, and vice versa.
• Learners can calculate the
area of semicircles.
LANGUAGE SUPPORT
There is no new vocabulary in this section.
Make sure that learners use the terms ‘radius’,
‘diameter’, ‘circumference’ and ‘perimeter’
correctly in any discussions. When a learner
uses any of these terms correctly or incorrectly
(e.g. using ‘radius’ instead of ‘diameter’ or
‘circumference’ instead of ‘perimeter’) ask other
learners if the word used is correct.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
CONTINUED
If the word used is not correct, ask ‘What is the
correct word to use?’
Encourage learners to tell you the formulae they
are using. This will help you to check that they are
using the correct formulae, and it will also help
them to remember the correct formulae.
Common misconceptions
Confusing the two formulae,
writing, for example, C = πr 2 or
A = πd 2.
Being confused about what π is
and how to use it.
Starter idea
How to identify
How to overcome
Questions 3, 4 and especially 7.
Repeatedly asking learners to
tell you the formulae for C and A
throughout the lesson.
Questions 3 and 4.
Discussions during Worked
example 7.1 and during checking
with questions 1, 2 and 3.
PL
E
Misconception
Getting started (15–20 minutes)
Resources: Note books, Learner’s Book Getting
started exercise
When learners have completed part b, it might be useful
to allow learners to compare answers, ensuring that all
learners have the correct answers before working out the
three percentage differences.
M
Description: Before starting the Getting started
questions, discuss what learners remember about
calculating the circumference of a circle, the link
between radius and diameter and how to use the π
button on a calculator.
Description: Check that all learners get the correct answer
to part a before moving on to part b. Learners should
write the full answer first [153.938 040 02…] before writing
their rounded answer of 153.938.
You might need to help some learners with splitting the
‘L’ shape into two rectangles in part a of Question 3.
SA
Ask learners not to use calculators during Question 5
and, if necessary, guide them towards discussing moving
the decimal point as a quick way to work out the answer
to this type of question.
When learners have completed part e, discuss how to
decide when it is best to use π [whenever a calculator is
available!] and when it is best to use an approximation.
Differentiation ideas: If learners struggle with part c,
demonstrate using the rounded answer from part a:
Percentage difference =
153.938 04 − 153.938
0.000 04
× 100 =
× 100 = 0.000 025 98%
153.938 04
153.938 04
This exercise is a quick reminder of previous work that
will help learners to be more effective with this unit.
It is not a test. After each question, it might be useful to
allow self- or peer-marking, allowing learners to rectify
any mistakes after a brief discussion.
Although this answer is much smaller than any of the
others, it shows the method quite clearly for learners to
follow.
Main teaching idea
Area and perimeter (5–10 minutes)
Question 3, Think like a mathematician
(5–10 minutes)
Description: On the board, write/display this question:
Learning intention: To understand the effects of using
different approximations to pi.
Resources: Note books, Learner’s Books, calculator
Plenary idea
Resources: Note books, calculators
For each shape, work out:
i
the area, correct to two significant figures
ii
the perimeter, correct to two decimal places.
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7 SHAPES AND MEASUREMENTS
1
2
Perimeter = 123 cm
10 cm
10 cm
Answer:
πd
2
+ d = 123
or
π 
d  + 1 = 123
2 
3
10 cm
Guidance on selected Thinking and
working mathematically questions
+ d = 123
π d + 2d = 246
d ( π + 2 ) = 246
d=
+1
or
πd
2
+ d = 123
π 2r
2
246
π 2r
π+2
2
+ 2r = 123
+ 2r = 123
π r + 2r = 123
d = 47.8...
r = 23.9...
d = 47.8...
r = 23.9...
r ( π + 2 ) = 123
123
π+2
r = 23.9...
r=
Assessment ideas
Use Question 12 as an opportunity for learners to
explain their working and self- or peer-mark.
Ask learners to complete Question 12, part a. Asking
six different learners in turn, get them to write on the
board their method of calculation for cards A to F. Allow
learners to copy from their note books. After the working
has been written for each card, ask who has written:
exactly the same, a simpler version, a longer version.
Discuss the merits of other suggestions and decide which
method they would probably use in the future.
M
Critiquing and improving
π
2
Assessment ideas: This could be simply self-marked
or it could be used as peer-marking with learners
concentrating on how well/clear the working is set
out and where either you discuss methods or learners
compare methods with each other.
Answers:
1 i 310 cm2 ii 62.83 cm
2 i 39 cm2 ii 25.71 cm
3 i 79 cm2 ii 35.71 cm
123
2
PL
E
d=
πd
Exercise 7.1, Question 5
SA
This question asks learners to critique the two answers.
Both answers have errors that are too often seen in
common questions. Ellie has made a mistake with the
order of operations and has multiplied before squaring.
Hans has confused squaring with doubling. It is
important that learners notice the mistakes and, in class
discussions, why these mistakes might have been made.
Along with notes regarding diameter and radius
conversions, here are probable options chosen by
learners:
Card A: C = πd Card B: A = πr2 Card C: C = πd
C = 3 × 15
A = 3 × 62
C = 3 × 20
Homework ideas
C = 45
A = 3 × 36
C = 60
Workbook, Section 7.1.
so, card v
A = 108
so, card vi
As Section 7.1 will take more than one lesson, set suitable
parts of the Workbook at the end of each lesson. Only set
questions that can be answered using skills and knowledge
gained from that lesson. Workbooks are aimed at fluency
and consolidation through practice, not as a method to
learn new skills that should be taught in class.
so, card i
Card D: A = πr2
Card E: C = πd
Card F: A = πr2
A = 3 × 72
90 = 3 × d
300 = 3 × r2
A = 3 × 49
3 × d = 90
3 × r2 = 300
A = 147
d = 30
r2 = 100
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
so, card iii
so, card iv
r = 10
For your most confident learners only, ask them to work
out the radius of this semicircle:
so, card ii
Once all cards have been discussed/marked, allow
learners to answer part b.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
7.2 Areas of compound shapes
LEARNING PLAN
Learning objectives
Success criteria
9Gg.03
• Estimate and calculate areas
of compound 2D shapes
made from rectangles,
triangles and circles.
• Learners can confidently break
down a 2D shape into simpler
shapes before estimating and
calculating the area.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary in this section.
Encourage learners to describe their methods
in words. This will allow you to check their
understanding and also help them to gain
confidence. Encourage learners to describe
the compound shapes in terms of their simpler
components. Getting one learner to explain how to
draw a specific compound shape to another learner
is a good test of explanation skills.
Common misconceptions
Misconception
How to identify
How to overcome
Questions 5 and 6.
Discussion during part b of Worked
example 7.2 and with part d of
Question 1.
M
Adding instead of subtracting the
area of the hole when a compound
shape has a hole.
Starter idea
b
Compound areas (5–10 minutes)
SA
Resources: Note books
1
Work out the areas of these compound shapes made
from rectangles and triangles.
a
5 cm
15 cm
9 cm
Description: On the board, copy/display these questions
and diagrams:
18 cm
20 cm
2
Work out the length marked x and calculate the
area of this compound shape.
7 cm
x cm
4 cm
5 cm
6 cm
16 cm
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7 SHAPES AND MEASUREMENTS
Depending on the class, you might decide to allow
learners to work in pairs or small groups to speed up the
practising of these skills.
2
8 cm
When completed, ask learners to compare answers with
other learners and decide if one method seems easier/
clearer than another. When learners have compared
answers/methods, give the answers.
Main teaching idea
10 cm
PL
E
Answers:
1 a 39 cm2 b 219 cm2
2 9 cm, 62.5 cm2
6 cm
Question 4, Think like a mathematician
(3–5 minutes)
Answers:
= 10 × 2 +
1
2
= 10 × 6 + × 10 × 8
1
× π × 52
2
= 60 + 40
= 100 cm2
= 20 + 12.5π
= 59.269 908…
= 59.3 m2
Resources: Note books, Learner’s Books
Assessment ideas: Peer-marking is suitable. Learners
need to show full, clear workings and to make sure that
they don’t omit any major steps. The shown solutions
are the only possible ways of working out the areas.
Guidance on selected Thinking and
working mathematically questions
SA
M
• Putting the two ends [A and C] together to make a
circle is easier and quicker than Kira’s method.
• Kira should not have rounded off until she found the
total area. The actual answer should be 257.1 to one
decimal place.
Differentiation ideas: Some learners will be happier
doing the extra work, i.e. using Kira’s method, rather
than putting A and C together to make a circle. As long
as learners understand they have a choice and make a
considered decision then allow them to use whichever
method they feel most confident with.
1
2
2 b×h+ ×b×h
2
Learning intention: To understand and improve a
common solution to a potential exam problem.
Description: There are two key points that must be
discussed here:
1
1 b × h + πr 2
Plenary idea
Compound shapes (5 minutes)
Resources: Mini white boards
Description: On the board, draw/display these
compound shapes. Ask learners to work out the area
of both compound shapes, correct to three significant
figures.
1
Conjecturing and convincing
Exercise 7.2, Question 8
Learners need to explain that the two shapes have the
same area. Explaining that the two semicircles in Shape
B have the same area as the circle in Shape A is a good
start, but learners should explain more clearly, using
either the diameter or radius to show they are the same.
Also, learners need to say that the squares are identical,
both having a side length of 10 cm.
No calculations are required, but writing that
1
1
10 2 − π × 52 = 10 2 − π × 52 − π × 52 might be useful
2
2
for some learners. Other learners will probably
work out that both shapes have an area of
21.460 18… cm2.
2m
10 m
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Homework ideas
Workbook, Section 7.2.
are happy with that). Help learners to focus on the
important aspects of their work, especially showing
working that leads logically to the answer.
As Section 7.2 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Regularly ask if learners have made a mistake. Check
that they know what they did wrong and how to get that
question correct next time. Acknowledge learners who
tell you with a ‘well done’ (as this is what you need in the
class, active learners).
Assessment ideas
If learners make a mistake but do not know what they
have done wrong or how to correct it, depending on
the question (and the learner), either help the learner
separately, asking learners near them to explain, or hold
a class discussion on the question/skill.
PL
E
With so many diagrams, this is an excellent opportunity
for peer-marking. Ask learners to regularly swap books
(in pairs or groups) for checking (and marking if you
7.3 Large and small units
LEARNING PLAN
Framework codes
Success criteria
• Know and recognise very
small or very large units of
length, capacity and mass.
• Learners can identify and use
very small or very large units
of length, capacity and mass.
M
9Gg.02
Learning objectives
LANGUAGE SUPPORT
SA
Prefix: a set of letters that you put in front
of a word
Tonne: a unit of mass such that
1 tonne = 1000 kilograms
Encourage learners to say aloud the prefixes in
the table in the introduction. Support learners with
the pronunciation of the Greek letter in this table,
for example you read the Greek letter μ as ‘mew’,
which rhymes with the word ‘new’.
Also encourage them say the powers of ten in
words, for example ‘ten to the power of twelve’
for ‘1012’.
Make sure that learners understand and recognise
the difference between, for example, ‘one billion’
and ‘one billionth’ and between ‘one million’ and
‘one millionth’, etc. Reading the sentences in
questions 1 and 2 gives good practice at this.
Common misconceptions
Misconception
How to identify
How to overcome
Forgetting the letter used for
a prefix.
Most questions, especially
questions 3 and 8.
Discussion about, and repeated referral to,
the table of prefixes, etc. in the introduction.
Forgetting the multiple for
each prefix.
Question 8.
Discussion about, and repeated referral to,
the table of prefixes, etc. in the introduction.
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7 SHAPES AND MEASUREMENTS
Starter idea
‘Length’, ‘Mass’ and ‘Capacity’
(5–10 minutes)
Resources: Mini white boards or note books
Description: Before starting to work through the
introduction or Worked example 7.3, ask learners to
write all of the metric measurements they can think of.
Give some examples, such as mm and mL.
When completed, discuss the prefixes of the suggestions,
e.g. mm, mg, mL, etc. are all ‘milli’ and explain/elicit
that ‘milli’ means ‘a thousandth’. Note that if any
learner suggests megabyte, gigabyte, etc. you should
write this under the heading ‘Capacity’.
Main teaching idea
Differentiation ideas: For part a, see the help suggested
in the description.
For part c, if learners find it difficult to think of their
own conjecture, suggest that they think about the units
for length, such as the link between mm and cm and
then possibly mm and km or cm and km.
Plenary idea
Prefixes (5–10 minutes)
PL
E
When completed, ask learners for their answers.
Write all new suggestions onto the board, under the
appropriate headings of ‘Length’, ‘Mass’ and ‘Capacity’.
So, 100 millilitres = 10 centilitres – not the 1 cL that
Arun thinks. Arun’s conjecture is incorrect.
Question 4, Think like a mathematician
(5–10 minutes)
Learning intention: To understand some of the
connections between different prefixes.
Resources: Note books, Learner’s Books
Resources: Note books
Description: Ask learners, without referring to their
books, to list all unit prefixes and their letters. Also ask
them to give an example for each. Give a time limit
(depending on the ability of the class) of between 3
and 8 minutes.
[For example: Prefix = tera, letter = T,
e.g. 1 terabyte = 1 000 000 000 000 bytes].
Learners could set their examples out in a table,
as shown:
Prefix letter
example which is the same as
tera
terabyte
T
1 000 000 000 000 bytes
M
Description: Learners must think logically.
Marcus: Learners must decide whether to convert one
tonne to grams or one megagram to tonnes. Most will
decide to convert one tonne to grams:
1 t = 1 × 1000 = 1000 kg (kilograms) = 1000 × 1000 =
1 000 000 g (grams).
SA
Looking at the table in the introduction, 1 000 000 has
a prefix of mega, so 1 000 000 grams is 1 megagram.
Marcus’ conjecture is correct.
Arun: Learners must decide whether to convert
millilitres to centilitres or vice versa. Neither
measurement is very commonly used, so learners will
probably convert 100 millilitres directly to centilitres.
Some learners might be successful, but for many learners
this will be too confusing. If/when help is needed,
suggest that learners look at how many millilitres make
a litre and how many centilitres make a litre and to put
these two values equal to each other.
Learners should write down that 1 litre = 1000 mL and
1 litre = 100 cL.
So, 1000 millilitres = 100 centilitres (divide both by 10).
You may wish to remind learners that there are nine
prefixes in the table in the Learner’s Book.
Assessment ideas: Put learners into small groups of
three or four. Learners compare answers, checking
which prefixes, if any, a learner has missed out and
checking that the size of the examples are accurate.
Allowing learners to use the table in the introduction
will help their checking.
Guidance on selected Thinking and
working mathematically questions
Conjecturing and convincing
Exercise 7.3, Question 11
Explanations might vary, but learners need to show that
they understand that in this case the smallest number
is the shortest time which means it is the fastest, while
the largest number is the longest time which means it is
the slowest.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Homework ideas
Assessment ideas
Workbook, Section 7.3.
At various times during Section 7.3, ask individual
learners short, easy to answer questions that check
knowledge. Ask questions without warning, and only
ask three or four learners questions. Later in the lesson,
ask three or four different learners, etc. Use questions
such as:
As Section 7.3 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
What letter represents nano [n], milli [m], centi [c],
hecto [h], kilo [k], mega [M], giga [G], tera [T] and micro
[μ, pronounced ‘mew’].
PL
E
You could ask learners to make a summary containing
everything they think they need to remember for the
end-of-unit test. The following lesson, it is important to
share the summary sheets in class (e.g. spread the sheets
out over a few desks for everyone to look at), rather than
marking them. Discuss the different summaries as a class.
When the class agree that a point is important, copy that
key point onto the board. Agree on as many key points
as possible. Learners could then improve/update their
individual summary sheets if necessary. Learners could
store their summary sheets at home as a possible revision
tool towards mid-term/end-of-year tests.
What is the prefix for the letter T [tera], G [giga], M
[mega], k [kilo], h [hecto], c [centi], m [milli], μ [micro]
and n [nano].
SA
M
What number do you multiply by when using the prefix
micro [0.000 001], milli [0.001], centi [0.01], hecto [100],
kilo [1000], mega [1 000 000] and giga [1 000 000 000].
98
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8 FRACTIONS
8 Fractions
Unit plan
Approximate number Outline of learning
of learning hours
content
Resources
8.1 Fractions
and recurring
decimals
1–1.5
Decide whether the
decimal equivalent to a
fraction is recurring or
terminating.
Learner’s Book Section 8.1
Workbook Section 8.1
Resource sheet 8.1
Additional teaching ideas Section 8.1
Estimate and calculate
with fractions using
the correct order of
operations.
Learner’s Book Section 8.2
Workbook Section 8.2
Additional teaching ideas Section 8.2
Multiply with fractions.
Learner’s Book Section 8.3
Workbook Section 8.3
Resource sheet 8.3
Additional teaching ideas Section 8.3
Divide with fractions.
Learner’s Book Section 8.4
Workbook Section 8.4
Additional teaching ideas Section 8.4
8.2 Fractions and 1–1.5
the correct order
of operations
1–1.5
8.4 Dividing
fractions
1–1.5
8.5 Making
calculations
easier
M
8.3 Multiplying
fractions
PL
E
Topic
1–1.5
Simplify calculations
containing fractions.
Learner’s Book Section 8.5
Workbook Section 8.5
Additional teaching ideas Section 8.5
SA
Cross-unit resources
Language worksheet: 8.1–8.5
End of unit 8 test
Mid-point test
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Recognise fractions that are equivalent to
recurring decimals (Stage 8).
• Estimate, add and subtract mixed numbers,
and write the answer as a mixed number in its
simplest form (Stage 7, Stage 8).
• Estimate and multiply an integer by a mixed
number, and divide an integer by a proper
fraction (Stage 8).
• Use knowledge of the laws of arithmetic and
order of operations (including brackets) to
simplify calculations containing decimals or
fractions (Stage 8).
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
CONTINUED
The focus of this unit is to develop knowledge
of fractions and their use in more complicated
questions, involving the order of operations to
answer and simplify calculations. Stage 8 work
on decimal equivalents is extended to include
terminating as well as recurring decimals. Learners
will start to use division as a multiplicative inverse
and to learn to cancel common factors to simplify
multiplication and division problems.
PL
E
TEACHING SKILLS FOCUS
At the end of Unit 8, ask yourself:
• Are learners able to explain what they are
thinking? If the answer is ‘No, not really’, is
that just because they are not used to giving
explanations and so need much more practice?
• Are learners getting better at explaining their
reasoning?
• Are learners getting better at explaining what
mistakes have been made?
• Are learners getting better at knowing what to
do next in a problem?
• Are learners more confident explaining when in
pairs or small groups rather than as a whole class?
• With the more complicated problems, can learners
tell you what they will do, i.e. make a plan?
This is a very powerful learning tool, but your
learners might find it difficult to explain what they
are thinking. They will need practice.
SA
M
Metacognition
A complicated area of learning that can be
simplified to ‘thinking about thinking’.
Throughout this unit, ask learners, whenever
possible, to say out loud what they are thinking.
Usually, try to ask at the start or a short way through
answering a problem.
If a question has already been answered, ask what
they were thinking while they were attempting a
problem and if they would now do the problem a
different way.
If done regularly, this questioning leads to a process
that can be used throughout their schooling: ‘think
about a problem, plan what to do, do the plan, look
back and decide if you could have done anything
better’.
This process teaches learners to understand how to
solve problems effectively, not just get the answer to
a particular question.
8.1 Fractions and recurring decimals
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Nf.01
• Deduce whether fractions will
have recurring or terminating
decimal equivalents.
• Learners can deduce whether
a fraction has a recurring
or terminating decimal
equivalent.
100
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8 FRACTIONS
LANGUAGE SUPPORT
Make sure that learners are confident with the
definitions of the different types of decimals.
Throughout this section and the exercises, ask
individual learners to explain/define a meaning
of one of the three key terms shown here. When
a learner has given a correct answer, ask a nearby
learner for an example of that type of decimal.
PL
E
Equivalent decimal: a decimal number that has the
same value as a fraction
Recurring decimal: in a recurring decimal, a digit or
group of digits is repeated forever
Terminating decimal: a decimal number that does
not go on forever
Common misconceptions
Misconception
Assuming that as, e.g. 1 is a recurring
6
How to identify
How to overcome
Question 5a.
Remind learners to always cancel down
a fraction to its simplest form before
deciding if its decimal equivalent is
recurring or terminating.
decimal, all fractions with 6 as a
denominator are recurring – including 3.
6
Starter idea
Getting started (10–15 minutes)
Resources: Note books, Learner’s Book
Getting started exercise
9
9
You might also need to help learners explain that the two
statements made by Marcus and Zara are not different.
M
Description: Learners should have little difficulty with
most of the Getting started questions. Before learners
attempt the questions, discuss what they remember
about fractions. Discuss how to convert a fraction into
a decimal, what is important when adding/subtracting
fractions, how to multiply a fraction by an integer and
what seems strange about dividing by a fraction.
Learners also know that 9 ÷ 9 = 1. You might need to
guide learners to writing:
.
9
9
= 0.9 and = 1
9
9
.
9 9
So, as = then 0.9 = 1
SA
Remember that this exercise is not a test. It is designed
to be a quick reminder of previous work that will
help learners be more effective with this unit. After
each question it might be useful to allow self- or peermarking, allowing learners to rectify any mistakes after
a brief discussion.
Differentiation ideas: If learners don’t understand part
a, suggest they try to cancel down and then answer the
question.
Some learners will not understand the simple proof
shown. This is not surprising really, as it seems counter
intuitive that two numbers that look so different are, in
fact, the same. Try a different approach. Ask learners to
copy and complete each part as you write it:
1
= [0.3 recurring]
3
2
= [0.6 recurring]
3
Main teaching idea
.
.  .
0. 3 + 0. 6 = 0. 9
 
.
Then guide learners into writing 1 = 0.9 .
Question 3, Think like a mathematician
(5–10 minutes)
Plenary idea
.
9
Learning intention: To understand that = 0.9 = 1.
9
1 2
+ = [1]
3 3
5 terminators! (2–5 minutes)
Resources: Note books, Learner’s Books
Resources: Mini white boards
Description: Most learners will answer part a
confidently. Part b might confuse some learners.
Description: On their boards, learners write five
fractions, each with a different denominator, which give
a terminating decimal equivalent.
Answering part b is effectively a simple proof. Learners
. .
.
.
can follow the pattern of 0.1, 0. 2, 0.3, … to get 0.9 .
101
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Assessment ideas: Learners swap boards, and peermark their partner’s fractions using a calculator.
When checked, you could have a class discussion on
which denominators were chosen.
Guidance on selected Thinking and
working mathematically questions
Specialising and generalising
Exercise 8.1, Question 8
Homework ideas
Workbook, Section 8.1.
As Section 8.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practise, not as a method to learn new skills
that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Guide, if necessary, learners to realising that, just
At various times during Section 8.1, ask individual
learners short, easy-to-answer questions that check
knowledge. Ask questions without warning, and only ask
three or four learners questions. Later in the lesson, ask
three or four other learners, etc. Ask questions such as:
PL
E
The work done in Question 5 might help here, but
sometimes it will be necessary for learners to calculate
the decimal equivalents to the fractions. Allow learners
to use a calculator to speed up the checking process for
part a.
1 2
3
like 3 , 6 and 9 are all recurring because they have
denominators with multiples of three (which gives a
recurring decimal), then all of the cards have
fractions which are multiples of seven (which gives
a recurring decimal) – even card E which can be
cancelled down.
M
Part b can be answered in many ways. Hopefully
learners will notice that card E is the only fraction
which can be cancelled down. Other, less useful (but
still correct), options include that card D has the
only non-prime numerator and, more impressively,
that card C is the only fraction which does not have
714285 as part of the recurring decimal (it has 523809
recurring instead).
Assessment ideas
SA
The important thing for learners to appreciate is that
they should only decide if a fraction has an equivalent
recurring decimal by looking at the denominator after
any potential cancelling has taken place.
Does 1 have a recurring decimal equivalent?
3
Does 3 have a recurring decimal equivalent?
4
Does 3 have a recurring decimal equivalent?
5
Does 2 have a recurring decimal equivalent?
7
Does 4 have a recurring decimal equivalent?
9
Does 2 have a terminating decimal equivalent?
3
2
Does have a terminating decimal equivalent?
9
Does 2 have a terminating decimal equivalent?
20
8.2 Fractions and the correct order of operations
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Nf.02
• Estimate, add and subtract
proper and improper fractions,
and mixed numbers, using the
order of operations.
• Learners can estimate and
calculate using fractions and
mixed numbers using the
correct order of operations.
102
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8 FRACTIONS
CONTINUED
Learning objectives
Success criteria
9Nf.03
• Estimate, multiply and divide
fractions, interpret division as
a multiplicative inverse, and
cancel common factors before
multiplying or dividing.
• Learners can estimate and
cancel common factors
before accurately dividing
fractions.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary in this section.
Encourage learners to explain their working. This
will help you to understand if they are using the
correct order of operations. It will also help learners
to remember the correct order of operations.
Common misconceptions
Misconception
How to identify
How to overcome
Incorrectly understanding the order Most questions in Exercise 8.2.
of operations.
Description: Part d is the most important aspect of
this question. Learners need to keep developing their
discussion skills. Explaining how/why they estimated
in a particular way is good practice. It is important
that if learners used different methods, they discuss the
reasons for the different use, and which method (for this
question) is probably more efficient.
M
Starter idea
Practice and discussion.
Fractions (5–10 minutes)
Resources: Mini white boards or note books
SA
Description: Give learners a variety of fraction questions
to check that they can calculate accurately. On the
board, write one question at a time, checking (selfor peer-marking) after each question, dealing with
misconceptions as they arise. Use questions such as:
2 1 13
+ [ ]
3 5 15
3 3 6
− [ ]
5 7 35
2 6
9
+ [1 ]
5 7 35
3 1 3
× [ ]
7 2 14
3 2 1
× [ ]
8 9 12
 1 1
  [ ]
5 25
2
3 3 1
+ [1 ]
8 4 8
6÷
3
[8] 4
7÷
3
1
[9 ]
4
3
Differentiation ideas: If learners find it difficult to start
this question, ask them if they think rounding to the
nearest half or rounding to the nearest whole number
would be easiest. Whatever their answer, suggest they
use that level of rounding to work out an estimate for
the question.
Main teaching idea
Plenary idea
Question 3, Think like a mathematician
(5 minutes)
The answer is 19 (5 minutes)
Learning intention: To understand different methods
of estimating.
Resources: Note books, Learner’s Books
32
Resources: Note books or mini white boards
Description: After a brief discussion about the order of
operations, ask learners to write full workings to show
the following (write/display on the board):
 1 3   3 1  1 1 3 19
 ×  ×  −  − × + =
2 4
4 2
2 2 4 32
103
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Assessment ideas: When completed, ask learners
in pairs or small groups to compare their workings.
Ask learners to look for any differences between their
workings and to decide which method gives the clearest
method. If learners cannot decide which is the best/
clearest method, they should ask you to help decide.
In your discussion after learners have completed the
question, make sure that learners understand that
whether a question asks for mixed numbers to be added,
subtracted, multiplied or divided, converting mixed
numbers into improper fractions will always be a good
step towards calculating the answer.
From your confident learners, the minimum working
expected could be:
Sometimes there might be an easier method, but
converting to improper fractions always works.
 1 3  3 1 1 1 3
×
−
− × +
×
 2 4   4 2  2 2 4
3 1 1 3
= × − +
8 4 4 4
3 1
=
+
32 2
3 16
=
+
32 32
19
=
32
Homework ideas
PL
E
Workbook, Section 8.2.
As Section 8.2 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Most learners should give more steps to show
understanding and clarity.
Guidance on selected Thinking and
working mathematically questions
Critiquing and improving
Exercise 8.2, Question 5
At various times during Section 8.2, ask learners to write
an explanation of how to answer the question they have
just answered. When completed, ask learners to compare
explanations with a partner for them to decide on the
best method for that question. Do not put the same
learners into pairs each time. Working with different
partners will give learners the opportunity to get ideas
from other learners in the class. Questions that you could
use in this way are questions 2c, 4, 6, 7 and 11b.
M
When completed, ask learners to work in pairs or small
groups to compare answers. This will help some learners
to see a different way to explain their own thoughts.
Hopefully, learners will suggest (or see the suggestion) that
working out 1 + 2 and 3 + 5 separately is probably easier
Assessment ideas
5
3
5
6
Throughout Section 8.2 ask learners to estimate all or
part of a question. Ask them to explain why they are
choosing/using the numbers they use.
5
6
SA
than working out 1 + 2 .
8.3 Multiplying fractions
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Nf.03
• Estimate and multiply
fractions and cancel common
factors before multiplying.
• Learners can estimate and
cancel common factors
before accurately multiplying
fractions.
104
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8 FRACTIONS
LANGUAGE SUPPORT
Some of the questions in the exercise are set
in context. If needed, support learners with the
language used, and make sure they understand
what is being asked before they attempt to answer
the question.
Asking learners to explain how they are estimating
answers will encourage learners to practise their
estimation skills and to use appropriate language.
PL
E
Cancelling common factors: dividing the numerator
and denominator of a fraction by a common factor
Encouraging learners to talk through their methods
will help them to remember how to correctly
multiply fractions. Asking learners to ‘think out
loud’ when they are cancelling fractions will allow
you to check understanding and help speed
calculation by using appropriate common factors.
Common misconceptions
How to identify
How to overcome
Cancelling by smaller numbers and
so failing to see the most obvious
of factors, e.g. cancelling top and
bottom of 20 by 2 instead of 10.
Misconception
Question 4.
Working through and discussing
answers to questions 1, 2 and 3.
Cancelling once but failing to
check if further cancelling is
20 10
= .
possible, e.g.
Question 4.
Question 3.
70
70
35
Starter idea
M
Multiplying by fractions (3–5 minutes)
11
should have 2 × as part of the workings. The ‘highest
3
common factor’ aspect can be discussed once part d is
completed.
Resources: Mini white boards or note books
SA
Description: Learners need to be proficient in
multiplying with fractions before they tackle the extra
complications of cancelling common factors. Check that
learners can remember the methods required with a few
straightforward questions such as:
1
1
3
3
× 22, × 42, × 20 and 40 × [11, 14, 15 and 15]
2
3
4
8
1 3, 2 4 and 6 1 [ 3 , 8 and 6 ]
×
×
×
35
7 5 10 15
2 5 3 5
Main teaching idea
Question 3, Think like a mathematician
(5 minutes)
Differentiation ideas: For learners that have difficulty
in answering part a, point to the part of the solution
4 × 11 and get learners to tell you that Sofia should have
6
divided here by 2. When that has been discussed, point
4
to the part of the solution 16 ×
11
24 6
and ask ‘What
common factor should Sofia have used, rather than
4?’ Learners need to understand that 8 is the highest
common factor and that 8 should have been used, not 4
which is just a common factor.
Plenary idea
How? (5–10 minutes)
Learning intention: To understand that using the highest
common factor to cancel by gives the answer in the
simplest terms.
Resources: Mini white boards
Resources: Note books, Learner’s Books
Ask learners to write the steps that are needed to work
out the answer, not just work out the answer. Tell
learners that they will swap notes afterwards, and follow
someone’s instructions to work out the answer.
Description: You could hold a brief classroom
discussion when learners have completed part b, to
make sure that all learners understand that the solution
1
8
2
3
Description: On the board, write the question ‘1 × 6 ’.
105
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Assessment ideas: Learners swap notes with a
partner and use their instructions only to try to work
out the answer. If a learner cannot complete the
answer because the instructions are not sufficient,
they should add notes to the instructions saying how
they can be improved. Discuss as a class the
best/clearest instructions.
Guidance on selected Thinking and
working mathematically questions
Exercise 8.3, Question 11
Workbook, Section 8.3.
As Section 8.3 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Assessment ideas
At various times during Section 8.3, ask individual
learners short, easy-to-answer questions that check
knowledge. Learners can use mental or written methods,
but no calculators. Ask questions without warning,
and only ask three or four learners questions. Later in
the lesson, ask three or four other learners, etc. Ask
questions such as:
PL
E
Characterising and classifying
Homework ideas
The focus here is not the answers to the questions
on the cards, although checking all learners have
the correct answers before moving on to parts b and
c would be useful. The focus is to think about the
different ways to characterise these answers to be able
to place them in sensible groups. When parts b and
c are completed, put learners into groups of about
four and ask them to compare their classifications/
reasoning, noting any unusual groupings to discuss
with the rest of the class.
3
4
2
2
Integer answers: × 16 , × 15, × 30, × 27 , etc.
9
5
3
4
[12, 10, 24, 6]
3
2
3
Mixed number answers: × 18, × 30, 5 × 20, × 55,
9
10
4
6
etc.[13 1 , 6 2 , 16 2 , 16 1 ]
2
3
3
2
1 2 1 4
2 1 2 3 2 3 6 14
Fraction answers: × , × , × , × , etc. [ , , , ]
18 15 15 9
9 4 9 5 9 10 7 27
M
8.4 Dividing fractions
LEARNING PLAN
Learning objectives
Success criteria
9Nf.03
• Estimate and divide
fractions, interpret division
as a multiplicative inverse,
and cancel common factors
before dividing.
• Learners can estimate and
cancel common factors
before accurately dividing
fractions.
SA
Framework codes
LANGUAGE SUPPORT
There is no new vocabulary for this section.
Encouraging learners to explain their working when
dividing fractions will help them to remember the
correct methods.
Asking learners to explain how they are estimating
answers will encourage learners to practise their
estimation skills and to use appropriate language.
In Exercise 8.4, there are several opportunities for
conjecturing and convincing. Support learners with
any language they find difficult.
106
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8 FRACTIONS
Common misconceptions
Misconception
How to identify
How to overcome
Inverting the first or both fractions.
Question 4.
Discussions during Worked example 8.4 and
questions 1, 2 and 3.
Starter idea
1
Division reminder (3–10 minutes)
4
Resources: Mini white boards or note books
1
3
1
4
1
5
1
2
1
10
1
6
2 ÷ , 4 ÷ , 3 ÷ , 20 ÷ , 5 ÷ , 6 ÷ , etc.
[6, 16, 15, 40, 50, 36]. To help to make sure that all
learners can remember, ask learners to explain how they
got their answer after each question.
When you are sure that all learners can remember how
to use the reciprocal method, give non-unit fractions as
divisors, but make sure the answers are integers, such as:
2
2
3
2
7
3
2 ÷ , 6 ÷ , 6 ÷ , 6 ÷ , 7 ÷ , 9 ÷ , etc.
3
3
4
5
10
[3, 9, 8, 15, 10, 12]
5
3
4
8 4
÷
9 27
20 ÷
3
6
1 5
÷
6 18
2 11
4 ÷
5 3
Assessment ideas: Allow peer-marking. Learners
should check that the working is easily followed, not just
that the answer is correct. As usual, discuss any errors
and how to avoid them.
4
Answers:
Question 6, Think like a mathematician
(5–10 minutes)
Learning intention: To understand the generalisations
of what happens when dividing by proper fractions,
improper fractions and mixed numbers.
SA
Resources: Note books, Learner’s Books
Description: Suggest that learners give at least
four examples of divisions before completing the
generalisations. When completed, check the answers
(self-marking) and discuss the numbers used by learners
and why they chose those numbers.
Differentiation ideas: To extend, ask learners how
removing the word ‘positive’ from the questions would
affect the answers.
Plenary idea
Division check (10 minutes)
Resources: Mini white boards or note books
Description: On the board, write/display the following
questions:
5 5
5
= 20 ×
= 25
4
41
1 20 ×
4
3
80
2
= 26
3
3
2 20 × =
3
3
1 18
1
18
3
× =
×
=
6 5 16
5
5
1
2
2
3
4
5 16
5 16
2
× =
×
=
8 15 1 8 15 3 3
5
8 27
8 27
6
×
= 1 × 1 = =6
9 4
1
9
4
M
Main teaching idea
2
PL
E
Description: Remind learners of the fraction divisions
they did in Stage 8 using simple questions, integer
answers only, such as:
4
5
5 15
÷
8 16
20 ÷
2
11
22
11
22
3
6 4 5 ÷ 3 = 5 ÷ 3 = 5 × 11 =
2
22
3
6
1
× 1 = =1
5
5
5
11
Guidance on selected Thinking and
working mathematically questions
Conjecturing and convincing
Exercise 8.4, Question 10
There are many similar ways for learners to make a
conjecture as to Sofia’s method. The most common
would probably be something like:
C =πd
C
=d
π
14
=d
3
d = 4.66… cm
When learners have answered part
1
7
b [14 ÷
22 99
7
9 1 9
1
= × = × = = 4 cm], discuss fully the
7
7
22 1 2 2
2
different methods/numbers used to explain Sofia’s value.
Try to get the class to determine the best option.
107
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Homework ideas
Workbook, Section 8.4.
As Section 8.4 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Assessment ideas
4
5
Estimate:
Answer:
2 ÷ 2 =1
Answer:
PL
E
1
2
Ask a learner to write on the board their answer to ‘1
Change to improper fractions:’. Discuss this answer as a
class, noting learners who need to change their answer.
Repeat for each of the other four steps. All learners
should end by having:
Estimate:
Ask learners to answer Question 5a. While they are
working, on the board, write:
1 ÷1
When learners have completed Question 5a ask learners
to tell you their estimate. It is likely that someone will
just give ‘1’ as an answer – explain that you need the
whole answer: 2 ÷ 2 = 1.
3
2
1
Change to improper fractions: ÷
2
Invert and multiply: 3 × 5
3
Cancel common factors:
2
1 5 5
× =
2 3 6
91
9
5
3
5
×
2
93
1
Change to improper fractions:
4
Multiply:
2
Invert and multiply:
5
Check with estimate: ≈ 1✓
3
Cancel common factors:
4
Multiply:
5
Check with estimate:
5
6
This can be used to aid self- or peer-marking. Encourage
learners to use exactly this method. If a question tells
the learner to do a question in a certain way in a test,
even if the learner knows a better way, the suggested
method must be used.
M
8.5 Making calculations easier
LEARNING PLAN
Learning objectives
Success criteria
9Nf.04
• Use knowledge of the
laws of arithmetic, inverse
operations, equivalence and
order of operations (brackets
and indices) to simplify
calculations containing
decimals and fractions.
• Learners can simplify
calculations containing
decimals and fractions.
SA
Framework codes
LANGUAGE SUPPORT
Strategies: methods
Explaining their methods to each other will help
learners to express their ideas more clearly, and
to understand appropriate methods. Encourage
discussion among learners as they work through
Exercise 8.5.
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8 FRACTIONS
Common misconceptions
Misconception
How to identify
How to overcome
Trying to do too many
simplifications at a time, causing
confusion and mistakes.
Question 4, 7, 10, etc.
Insist on full working.
Starter idea
Plenary idea
Decimals ⇆ fractions (5–10 minutes)
Fractions or decimals? (5 minutes)
Resources: Mini white boards
Description: This starter idea is to refresh learner’s
memories of equivalent decimals and fractions. Ask
one question at a time, checking for accuracy and
understanding by self-marking and discussions.
Description: On the board, write ‘4.2 × ’. Ask learners
PL
E
Resources: Mini white boards or note books
Ask learners to give the fraction equivalent of some
straight forward decimals, e.g:
1
3
0.5 [ 1 ], 0.75 [ 3 ], 0.3 [ 3 ], 0.11 [ 11 ], 0.333… [ ], etc.
2
4
100
10
Then ask learners to give the decimal equivalents to
some simple fractions, e.g:
1
2
1
21
1
[0.25], [0.666…], [0.2], [0.1],
[0.21], etc.
5
3
10
100
4
Main teaching idea
to write workings and the answer. When completed
[2 5 or 2.625], ask learners to explain why they chose the
8
method they used. This can be done orally in a small
group, or in writing. When finished, discuss as a class
what seemed to be the best method.
Now, on the board, write ‘7.2 × 1 ’. Ask learners to write
4
workings and the answer. When completed
4
[1 or 1.8], ask learners to explain why they chose the
5
method they used. This can be done orally in a small
group, or in writing. When finished, discuss as a class
what seemed to be the best method and why it might
be a different method than the best method for the first
question.
M
Question 9, Think like a mathematician
(5–10 minutes)
5
8
Learning intention: To develop and decide upon different
strategies.
Resources: Note books, Learner’s Books
Description: Hopefully, no learners would even consider
4
SA
converting the 1 2 or the 2 to a decimal as they are both
9
3
recurring decimals.
Strategies might be to convert the decimals to
fractions and then to square, or to square the decimals
and then to convert to a fraction before multiplying by
the mixed number, after converting it to an improper
fraction. It is important that learners discuss their
methods and the reasons for their decision, e.g. ‘it’s
easy to square 0.2, but not so easy to square 0.75’. It is
these discussions which might help learners to improve
a strategy and to use those thoughts to develop
plans later.
Differentiation ideas: Some learners will find it difficult
to decide what strategies to use. Ask these learners to
answer each question using any method they want (no
calculators!) and to use their method in discussions.
Assessment ideas: Checking answers can be done by
self- or peer-marking, however, the main assessment of
learners reasoning is best carried out by discussion.
Guidance on selected Thinking and
working mathematically questions
Critiquing and convincing
Exercise 8.5, Question 5
Learners should be able to explain why they think
a method is good (and not so good) based on their
experience and personal preferences. Most learners will
prefer Akeno’s method as it is shorter. Some learners
might prefer Dae’s method because they prefer to work
on part of a question at a time or because they don’t like
having to cancel within fractions. The reasons for the
preference is not important. It is the process of looking,
deciding and explaining that is important.
It is preferable that learners understand that when the
12 cm is replaced by 15 cm, then Akeno’s method might
need to be adapted as 15 is not easily divided repeatedly
by 2.
109
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Homework ideas
Assessment ideas
Workbook, Section 8.5.
With so many changes in methods throughout the
section, this is an excellent opportunity for learners to
assess each other by discussing the method chosen for
each individual question – where there is a choice. If
learners are in pairs, a brief discussion on their method
can take place (30 seconds maximum) followed by a
brief class discussion when any pair has a different
approach to a question.
As Section 8.5 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
SA
M
PL
E
You could ask learners to make a worked example list
containing everything they think they need to remember
for the end-of-unit test. The following lesson, it is
important to share the worked example lists in class (e.g.
spread out over a few desks for everyone to look at),
rather than marking them. Discuss the different worked
example lists as a class. When the class agree that a
point is important, that key point could be copied onto
the board. Agree on as many key points as possible.
Learners could then improve/update their individual
worked example lists if necessary. Learners could store
their worked example lists at home as a possible revision
tool towards mid-term/end-of-year tests.
110
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PROJECT GUIDANCE: SELLING APPLES
PROJECT GUIDANCE: SELLING APPLES
Why do this problem?
This problem offers an engaging context in which
learners can practise calculating with fractions. The
three parts of the problem offer increasing levels
of challenge. The final part challenges learners to
adapt and improve on their initial attempts.
third part can be an ongoing challenge in which
learners try to improve on the largest number of
days achieved by the class so far.
Possible approach
The first part of the problem is designed to
introduce the idea of eating three apples at the
start of each day and then selling a fraction of what
is left. This could be used as a starter activity at the
beginning of the lesson.
How many apples would Arun have left after he has
eaten his next three apples?
PL
E
Key questions
Arun eats three apples and has 15 apples left.
What fraction of the apples could he then sell?
Suggest that learners work in pairs on the second
part of the problem. As they are working, share
any useful strategies that learners use, which might
include observations about the denominators of
the fractions, or useful ways of recording what they
have tried.
Possible support
If learners are struggling to record their thinking
effectively, suggest that they could use a tree
diagram: at each stage, branching off the fractions
it would be possible to sell next, so that all
possibilities are checked.
Possible extension
Challenge learners to prove that they have found
the largest number of days that the apples can
last for.
SA
M
Before starting on the third part of the problem,
encourage learners to share any insights that they
found while they worked on the second part. The
Are there any numbers of apples that Arun
wouldn’t want to be left with?
111
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Unit plan
PL
E
9 Sequences and
functions
Approximate Outline of learning content
number of
learning hours
Resources
9.1 Generating
sequences
1–2
Generate linear and non-linear
sequences.
Learner’s Book Section 9.1
Workbook Section 9.1
Resource sheet 9.1
Additional teaching ideas Section 9.1
9.2 Using the
nth term
1–1.5
Understand and describe nth
term rules algebraically.
Learner’s Book Section 9.2
Workbook Section 9.2
Resource sheet 9.2
Additional teaching ideas Section 9.2
9.3
Representing
functions
1–1.5
Generate outputs and inputs
from a given function involving
indices.
Learner’s Book Section 9.3
Workbook Section 9.3
Resource sheet 9.3A
Resource sheet 9.3B
Additional teaching ideas Section 9.3
M
Topic
SA
Cross-unit resources
Language worksheet: 9.1–9.3
End of unit 9 test
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Understand term-to-term rules, and generate
sequences from numerical and spatial patterns
(Stage 8).
• Understand and describe basic nth term rules
algebraically (Stage 8).
• Understand that a linear function is a
relationship where each input has a single
output. Generate outputs from a given function
and identify inputs from a given output by
considering inverse operations (Stage 8).
The focus of this unit is to extend knowledge
gained from Stage 8. This will expand the variety
of mathematical situations where learners can
effectively use sequences and functions. These
situations will involve non-linear sequences and
functions with indices.
112
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9 SEQUENCES AND FUNCTIONS
TEACHING SKILLS FOCUS
per pair if possible. Allow self-marking.
Now give one question for all learners to attempt,
without help. Is there evidence of learning? Have
the ‘teachers’ done a good job? Did the ‘teachers’
understand what they were teaching? Are there any
aspects that you need to clarify?
At the end of Unit 9, ask yourself:
• Do you know what the learners know/knew about
this topic?
• Have you asked questions to look for evidence
of learning, of a depth of understanding of the
topic that shows learners understand how the
maths works, not just that they can get an answer
to a question?
• Are learners confident that if they can suggest
half-formed ideas about a problem, then they
can share it and receive guidance from yourself
or another learner?
• Did you tell learners that learning from their
mistakes is an excellent and invaluable process
that is encouraged within the classroom?
M
PL
E
Assessment for learning
A key aspect for assessment for learning is assessing
prior knowledge. The Getting started questions will
help find weaknesses. However, much of this unit is
built on previously learned skills, and as such, if any
of those skills are weak or missing it is important
to revisit that area of the Stage 8, or even Stage 7,
work.
You might need to adapt or stop the planned lesson
if the required previous knowledge is missing. If
only part of the class lacks a skill, then this is a great
opportunity for you to get learners to help teach.
Show the skill required to all learners, set three or
four basic questions, put learners in groups with one
or two ‘learners’ with as many ‘teachers’ as possible.
Listen to the groups, ask that only one ‘teacher’ is
speaking at any time. Regularly check with ‘learners’
that they understand and that the ‘teacher’ is giving
good feedback to any questions they are asking.
Let learners self-mark their answers to the questions.
Now give slightly harder questions to all learners,
working in pairs – one ‘learner’ and one ‘teacher’
9.1 Generating sequences
LEARNING PLAN
Learning objectives
Success criteria
9As.01
• Generate linear and
quadratic sequences from
numerical patterns and from
a given term-to-term rule
(any indices).
• Learners can generate
increasing and decreasing
linear and non-linear
sequences.
SA
Framework codes
LANGUAGE SUPPORT
Linear sequence: a sequence of numbers in which
the difference between consecutive terms is
the same
Non-linear sequence: a sequence of numbers in
which the difference between consecutive terms is
not the same
113
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
CONTINUED
Encourage learners to read aloud the rules of
each sequence. Emphasise, for example, that ‘the
term-to-term rule is square and subtract 5’ means
the same as ‘the term-to-term rule is square, then
subtract 5’. Make sure also that learners notice
when the term is being squared and when it is
being cubed.
PL
E
It might be useful to review the sequence key
terms and vocabulary from stages 7 and 8 (e.g.
sequence, term, consecutive terms, term-to-term
rule, sequence of patterns, generate).
Make sure that learners understand the difference
between linear and non-linear sequences by using
the example of two sequences such as 2, 4, 6,
8, 10, … and 1, 2, 4, 7, 11, …
Common misconceptions
Misconception
Incorrectly dealing with negative
numbers.
Starter idea
How to identify
How to overcome
Question 10 part b.
Discuss the results of squaring and
cubing negative numbers.
Getting started (10 minutes)
Resources: Note books, Learner’s Book
Getting started exercise
Inevitably, some learners will show one or more
increasing sequences [i.e. * < 24] and one or more
decreasing sequences [i.e. * > 24] and say that Zara is
correct. Ask these learners to use a logical argument to
justify their answer, not just to show a few sequences
where her statement is true.
M
Description: Learners who were confident with Stage 8
Unit 9 should have little difficulty with much of the
Getting started questions. It might, however, be useful
to have a brief discussion about the difference between a
term-to-term rule and a position-to-term rule.
Logically, for learners to understand why Zara’s
statement uses 24, they should let * = 24 first. This gives
a sequence of 3, 3, 3, …
You might need to help a few learners with Question 4,
where you might need to remind them of the process of
working out the nth term.
SA
This exercise is a quick reminder of previous work that
will help learners to be more effective with this unit.
It is not a test. After each question it might be useful to
allow self- or peer-marking, allowing learners to rectify
any mistakes after a brief discussion.
Differentiation ideas: If learners find it difficult to
start, suggest that they decide on a number for * and
work out the first four terms of the sequence. Ask ‘What
number did you use for *?’ and ‘Why did you choose
this number?’. If required, suggest that they substitute
23, then 24, then 25 for *, then try a couple of decimal
numbers. Hopefully with this amount of data, learners
will be able to form an opinion.
Plenary idea
Main teaching idea
The first three terms (5–10 minutes)
Question 9, Think like a mathematician
(5 minutes)
Resources: Mini white boards
Learning intention: To practise justifying using a
logical argument.
Description: Ask learners to write the first three
terms for each of the following term-to-term rules.
Tell learners that the first term in each sequence is 2.
Resources: Note books, Learner’s Books, calculators
1
add 2
Description: Encourage learners to write any working,
especially any sequences they make.
2
subtract 3
3
multiply by 10
4
divide by 2
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9 SEQUENCES AND FUNCTIONS
5
add 1 , then multiply by 4
Homework ideas
6
multiply by 4, then add 1
Workbook, Section 9.1.
7
add 2, then divide by 2
8
add 2, add 4, add 6, …
9
cube, then subtract 5
2
2
As Section 9.1 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
10 subtract 3, then square
Answers:
1 2, 4, 6
4
2, 1, 1
2
7 2, 2, 2
10 2, 1, 4
2 2, −1, −4
3 2, 20, 200
5 2, 10, 42
6 2, 8 , 34
8 2, 4, 8
9 2, 3, 22
1
2
1
2
When checked, discuss any problems. Any learner with
only six or fewer correct answers should receive
extra help.
Guidance on selected Thinking and
working mathematically questions
Critiquing and convincing
You could extend the ‘Forest fire’ main teaching idea
in the Additional teaching ideas (Resource sheet 1) by
asking learners to repeat the activity using hexagonal
grid paper.
Assessment ideas
Ask learners to close their Learner’s Books. Give
them mini white boards (or similar) to show you their
answers.
Read Question 8 part a from Exercise 9.1. Allow
learners time to answer, then discuss answers as a class
with learners self-checking. Now read Question 8 part b.
Give learners 20 seconds to write their answer. Learners
swap boards with a partner and peer-mark. If a learner
makes a mistake, ask other learners to help to explain
the correct method and answer.
M
Exercise 9.1, Question 11
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
PL
E
Assessment ideas: Peer-mark.
This mistake, or versions of it, are regularly seen in
answers to questions. This type of mistake shows a total
lack of understanding of how to move from one term to
another term in a sequence.
Repeat this for Question 8 parts c and d.
To check understanding, after learners have completed
and checked Question 10, give them one more question.
On the board, write:
Ideally, learners will show an understanding that to
move from the 6th term to the 5th term you need to
divide the term by three, the inverse of ‘× 3’. Confident
learners might show a much deeper understanding,
saying that from the 6th term to the 5th term is ÷ 3,
from the 5th term to the 4th term is ÷ 3 and from the
4th term to the 3rd term is ÷ 3, so ÷ 3 ÷ 3 ÷ 3 is the same
as ÷ 27. Then show that 486 ÷ 27 = 18, which is the 3rd
term. Hopefully learners will check this by showing
that 18 × 3 = 54 [4th term], 54 × 3 = 162 [5th term] and
162 × 3 = 486.
Write the first 3 terms.’
SA
Learners must be able to point out that Tania’s method
is incorrect. The most straightforward method is usually
to show that if the 3rd term is 243, then the next term
would be 243 × 3 = 729, which is already much larger
than the 486 value of the 6th term.
‘First term is 6
term-to-term rule is subtract 4 and
square
Now either give learners a set time (e.g. 20 or 25
seconds) to answer then to show. Alternatively, you
could set this question as a race, depending on what you
think will work best for your class. When complete, ask
learners to either show you their answers [6, 4, 0] or to
compare their answers with a partner, discussing any
differences.
115
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
9.2 Using the nth term
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9As.02
• Understand and describe nth term rules
algebraically (in the form an ± b, where a
and b are positive or negative integers or
• Learners can work out
and use the nth term rule
for linear and quadratic
sequences.
n
PL
E
fractions, and in the form a , n2, n3 or n2 ± a,
where a is a whole number).
LANGUAGE SUPPORT
Quadratic sequence: a sequence of numbers in
which the second difference between consecutive
terms is the same; the highest power in the nth
term rule is 2 (squared)
It might be useful to review the sequence key
terms and vocabulary from stages 7 and 8 (e.g.
position number, nth term).
Encourage learners to read aloud the nth terms.
Several of the questions in Exercise 9.2 ask learners
to explain their answers. Support learners with any
vocabulary as needed.
M
Common misconceptions
How to identify
How to overcome
Not understanding how to use the
nth term to work out terms in a
simple sequence.
Question 2.
Any learner who has difficulty in giving the first
three terms in Question 2 needs immediate
help. Support them with further practice with
straightforward nth terms such as 2n, 2n + 1, 3n + 1,
3n − 1 before using n2, n2 + 1, etc.
SA
Misconception
Starter idea
Ask learners to compare their answers, self-checking
and discussing any differences, working out the correct
answers themselves.
First four terms (2–5 minutes)
Resources: Note books
Description: Ask learners to write the first four terms of
these sequences:
1
first term is −1
2
first term is 2term-to-term rule is add 3,
add 5, add 7, …
3
first term is 1
3
term-to-term rule is add 4.
term-to-term rule is add 1 .
Answers: 1 −1, 3, 7, 11 2
3
2, 5, 10, 17 3
1 2 3 4
, , ,
3 3 3 3
It might be useful for more confident learners to refer
back to the questions/answers of this starter when they
have answered Question 1 parts a, b and c in Exercise 9.2.
• Point out that in Question 1 here, the ‘add four’
refers to the ‘4n’ in the nth term in part a.
• The n2 in part b is shown here in Question 2 as the
differences between the terms is increasing each time
by 2.
n
•
means that from one term to the next, the
3
1
3
increase is .
116
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9 SEQUENCES AND FUNCTIONS
Main teaching idea
Question 5, Think like a mathematician
(5–10 minutes)
Learning intention: To be introduced to quadratic
sequences
Resources: Note books, Learner’s Books
This method is very important. Not only does it help
learners to recognise the difference between linear,
quadratic and other sequences, but it is part of an
important method for working out nth terms of more
complicated quadratic (and cubic) sequences.
Differentiation ideas: Check working for part b. Make
sure that learners write the differences between the
above numbers, not directly underneath.
Resources: Mini white boards
3
n2 + 10
4
n3 + 10
5
n
10
Answers:
1 11, 12, 13, 14
2 20, 30, 40, 50
3 11, 14, 19, 26
4 11, 18, 37, 74
5 1 , 2 or 1 , 3 , 4 or 2
10 10
As Section 9.2 may take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Assessment ideas
SA
10n + 10
A possible improvement for some learners would be
using trial and improvement. When discussed as a class,
point out that this method is generally much faster than
Taki’s method, but possibly still slower than Miyo’s
method.
M
Description: On the board, write/display these nth term
rules. Ask learners to write the first four terms of the
sequences.
2
Some learners might prefer to work out the entire
sequence up to the appropriate point. While this is good
practice for sequence generation, it can be very time
consuming, and with so many steps involved, the risk
of making an error is increased. Hopefully, learners
will note the speed at which Miyo’s method reaches
the correct answer. Miyo’s method does require better
algebra manipulation skills than Taki’s method, but
none of the skills required are new.
Workbook, Section 9.2.
10 (5–10 minutes)
n + 10
Exercise 9.2, Question 10
Homework ideas
Plenary idea
1
Critiquing
PL
E
Description: This is a very straightforward task. Most
learners should be able to complete this question with
confidence. This does not mean that this question is not
important.
Guidance on selected Thinking and
working mathematically questions
5 10 10
5
Assessment ideas: Allow peer-marking. When checked,
discuss any types of nth term rule (especially involving n3)
that learners might need further practice with.
If you want to concentrate on working out the nth term
of quadratic sequences, enter ‘transum.org, quadratic
sequences’ into a search engine. When on the site,
learners answer the Level 1 material. As usual with this
site, each visit gets new questions, so make sure your
learners write the web address of this useful revision
tool. If learners have individual access to the site, allow
a set time to answer the five questions. If learners are
working from an electronic white board, they could
write full working and answers in their note books, with
learners discussing answers before they are entered into
the page for checking. If required, learners could work
in pairs or small groups for the last two questions which
usually start with negative numbers, although the nth
term will still be in the form n2 + c.
117
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
9.3 Representing functions
LEARNING PLAN
Learning objectives
Success criteria
9As.03
• Understand that a function
is a relationship where each
input has a single output.
Generate outputs from
a given function and identify
inputs from a given output
by considering inverse
operations (including indices).
• Learners can use functions to
calculate inputs and outputs.
Learners can work out the
reverse equation for functions
involving indices.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary in this section.
Encourage learners to say aloud their methods for
completing tables of values. Encourage learners to
use the specific vocabulary, especially to use the
term ‘inverse operation’ correctly.
Common misconceptions
How to identify
How to overcome
Question 8.
Discussion about Lara’s method, making sure
learners understand why the inverse of x2 is x.
Introduce x3 and its inverse at the same time.
M
Misconception
Not understanding that the inverse
of x2 is x and that the inverse of
x3 is 3 x .
Starter idea
SA
Squares and cubes, roots too! (5 minutes)
Resources: Mini white boards or note books, calculators
Description: Ask learners to write the first ten square
numbers, without the use of a calculator. Ask a learner
(or several learners) to tell you the answers. On the
board, write the numbers vertically [1, 4, 9, 16, 25, 36,
49, 64, 81, 100]. Ask all learners to check that they have
the correct answers. If not, can they see a mistake?
Next, on the board, put a square root sign ( ) over
each of the answers given. Now ask learners to give the
answers to the questions (i.e. 1, 4 , 9 , 16 , etc.). Ask
a learner (or several learners) to tell you the answers and
to write in the answers [1, 2, 3, 4, …].
Ask learners ‘What does the work on the board show?’
Guide learners, if necessary, saying that the inverse of
square is square root.
Repeat, using the numbers 1, 2, 3, 4 and 5 with cubing
[1, 8, 27, 64, 125] and then cube rooting [1, 2, 3, 4, 5].
Discuss squaring fractions and negative numbers.
2
2
Use examples such as  1  ,  1  , (−3)2 and (−4)2
 2  5
1 1
, 9, 16]. Also discuss cubing fractions and negative
4 25
3
3
numbers. Use examples such as  1  ,  1  , (−2)3 and
 2  5
1 1
, − 8, − 125].
(−5)3 [ ,
8 125
[ ,
Main teaching idea
Question 3, Think like a mathematician
(5 minutes)
Learning intention: To use indices in functions with more
complicated numbers.
Resources: Note books, Learner’s Books
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9 SEQUENCES AND FUNCTIONS
Description: Encourage learners to show working
whenever they are using fractions. This will not only
lead to more correct answers, but it will also help
learners to see why an answer is incorrect. Any learner
using decimals will probably only do so if they are using
a calculator.
When discussing their equations in part d, look out for
learners who have written y = 2x2 instead of y = (2x)2
for function ii or y = x + 23 instead of y = (x + 2)3 for
function iii.
Plenary idea
Exit ticket (5 minutes)
 1
 
5
2
1
5
1
5
= × . Now an answer of
1
will be suggested.
25
Agree, but remind learners that (add this to the board):
 1
 
5
2
1
5
= ×
1
12
1
or 2 = .
25
5
5
Learners should now be more able to show and explain
their answers.
Homework ideas
Workbook, Section 9.3.
PL
E
Differentiation ideas: For learners who have difficulty
with this question, you could review the work in the
starter idea activity which looks at how to square and
cube fractions and negative numbers.
answer, remind learners that (add this to the board):
Resources: Resource sheet 9.3B: Exit ticket
Description: Give each learner an exit ticket, cut out
from Resource sheet 9.3B. Learners should complete the
exit ticket just before leaving class. Allow 3–5 minutes to
complete.
Assessment ideas: When you have taken in the exit
1
4
tickets, you can quickly check for accuracy [9, 10, ,
You could ask learners to make a poster containing
everything they think they need to remember for the
end-of-unit test. The following lesson, it is important
to share the posters in class (e.g. spread out over a few
desks for everyone to look at), rather than to mark
them. Discuss the different posters as a class. When
the class agree that a point is important, that key point
could be copied onto the board (by you or a learner).
Agree on as many key points as possible. Learners
could then improve/update their individual posters if
necessary. Learners could store their posters at home
as a possible revision tool towards mid-term/end-ofyear tests.
SA
M
100]. Reading what learners thought they have learned
and how to improve their understanding might take
longer, but has obvious benefits of you understanding
what learners think they have learned and what you have
been trying to teach them. It will also help you to clarify
teaching points for that learner/class for the next lesson
or in future lessons.
As Section 9.3 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Guidance on selected Thinking and
working mathematically questions
Specialising and convincing
Exercise 9.3, Question 11
Squaring fractions can cause problems for some
learners. It might be useful to give an example/
reminder before learners start Question 11. On the
2
board, write ‘ 1  ’. Ask learners to ‘give an answer’
 5
Assessment ideas
At various times during Section 9.3, ask individual
learners short, easy-to-answer questions that check
knowledge. Ask questions without warning, and only
ask three or four learners questions. Later in the lesson,
ask three or four other learners, etc.
When asking a question, point to a function machine
from Exercise 9.3 (e.g. Question 2 a i) and ask for the
output given an input.
Later in the lesson ask for the input given an output.
You will need to be more careful when asking these
questions. Make sure that the numbers you choose give
a suitable answer.
or ‘simplify’. Regardless of a correct or incorrect
119
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
10 Graphs
Unit plan
Approximate
number of
learning hours
Outline of learning content
Resources
10.1 Functions
2
Writing linear functions in
different ways.
Learner’s Book Section 10.1
Workbook Section 10.1
Additional teaching ideas Section 10.1
10.2 Plotting
graphs
2
Moving from function to
table of values to graph.
Learner’s Book Section 10.2
Workbook Section 10.2
Resource sheet 10.2
Additional teaching ideas Section 10.2
10.3 Gradient
and intercept
2
Using the equation of a
straight line to find the
gradient and the y-intercept.
Learner’s Book Section 10.3
Workbook Section 10.3
Additional teaching ideas Section 10.3
10.4
Interpreting
graphs
2
Real life examples of graphs.
Learner’s Book Section 10.4
Workbook Section 10.4
Additional teaching ideas Section 10.4
PL
E
Topic
M
Cross-unit resources:
Language worksheet: 10.1–10.4
End of unit 10 test
BACKGROUND KNOWLEDGE
SA
For this unit, learners will need this background
knowledge:
• Understand that a situation can be described in
words or as a linear function in the form
y = mx + c (Stage 8).
• Be able to construct a table of values and draw
a graph of a linear function in the form
y = mx + c (Stage 8).
• An equation of the form y = mx + c, where m is
an integer, corresponds to a straight-line graph
with gradient m and y-intercept c (Stage 8).
• Read and interpret graphs with more than one
component (Stage 8).
In this unit learners will also look at linear functions
given in the form ax + by = c and move between
the two forms of a linear equation. They will move
on to plotting graphs of linear functions when y is
given implicitly and of quadratic functions of the
form y = x2 ± a.
Learners will also look at straight-line graphs with
fractional, positive and negative gradients and find
the gradient and y-intercept of these straight lines.
Learners will then apply their skills to looking at
real-life graphs and to using compound measures
to compare graphs.
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10 GRAPHS
TEACHING SKILLS FOCUS
the value of c. This technique can be used in
many situations and can be applied to questions
in many mathematical contexts. Think about this
when you are teaching this unit, but also when you
are teaching other topics. Given a question, think
about how it might be changed to reveal a general
pattern or structure. Moving from the particular to
the general is one of the overarching ideas that
defines mathematics.
PL
E
Active learning
A number of the questions in the exercises in this
unit encourage learners to make generalisations.
Starting with the equation of a line they can look at
the effect of changing one number while keeping
other numbers constant. For example, suppose a
line has the equation 2x + 3y = 12. Changing the
12 to different values will give a series of parallel
lines. This leads to the generalisation that 2x + 3y = c
2
3
is a straight line with a gradient of − whatever
10.1 Functions
LEARNING PLAN
Framework codes
Success criteria
• Understand that a situation
can be represented either in
words or as a linear function
in two variables (of the form
y = mx + c or ax + by = c),
and move between the two
representations.
• Learners can move between
representations in words and
symbols, answering questions
of the type in Exercise 10.1.
M
9As.04
Learning objectives
SA
LANGUAGE SUPPORT
There is no new vocabulary in this section.
The questions in this section are set in context.
Support learners with the required language
and make sure that they understand what each
question is asking them to do before they attempt
to answer it.
Encourage learners to read out their functions in
words.
Common misconceptions
Misconception
How to identify
Thinking that a letter, not a word, always Listen to what learners
represents a number. For example, not
say in discussion.
realising that ‘b + g’ could represent ‘the
number of boys plus the number of girls’
and not just ‘boys + girls’.
How to overcome
Always use the correct terminology. For
example, b is 'number of boys' and not
just 'boys'.
121
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Starter idea
Plenary idea
Ready to start (10 minutes)
Interpreting functions (5 minutes)
Resources: Getting started exercise, Learner’s Book
Resources: None
Description: Tell learners to complete the Getting
started questions. If they finish quickly, they can check
each other’s answers in pairs. When all learners have
finished, mark the questions together, taking answers
from different learners. Look out for any areas of
misunderstanding that need to be addressed before
continuing with this unit.
Description: Describe the following situation:
‘A warehouse has large and small boxes.
The mass of a small box is 3 kg.
The mass of a large box is 5 kg.
There are x small boxes and y large boxes.’
PL
E
Ask learners to work in pairs and discuss the
interpretation of each of the following equations. Write
the equations one at a time.
Main teaching idea
Implicit functions (10 minutes)
Learning intention: To understand that functions can be
written in an implicit form.
Resources: Learner’s Book
Description: This activity is about functions written in
an implicit form, but learners do not need to know the
term ‘implicit’.
Start by looking at the situation at the beginning of
Section 10.1. Ali and Bella have a total of $37. If Ali
has a dollars and Bella has b dollars, you can write
a + b = 37.
Finally, ask pairs to use the information to find the
number of boxes of each size. They will probably do this
by trying out some values. They should find that x = 12
and y = 3. They should be able to show that these values
satisfy all three equations.
Assessment ideas: Asking learners to work in pairs will
help learners who are still having difficulty matching
algebraic and verbal descriptions of a situation. Look
for pairs who might be finding this activity difficult and
support them.
M
Suggest values for one variable and ask learners to work
out the value of the other variable. Make sure learners
realise that they can do this by subtraction.
• x + y = 15
(total number of boxes is 15)
• 3x + 5y = 51
(total mass is 51 kg)
• x = 4y(there are 4 times as many small
boxes as large boxes)
Give pairs time to discuss this before taking feedback.
Move on to the situation in Worked example 10.1.
Fatima buys c pencils at $2 each and k pens at $6 each
and she spends a total of $30.
SA
Ensure that learners understand that they can write this
as 2c + 6k = 30.
Here is a table of possible values.
Guidance on selected Thinking and
working mathematically questions
Generalising
Exercise 10.1, Question 9
c
0
3
6
9
12
15
k
5
4
3
2
1
0
Do not give this table to the learners, but use it to ask
questions such as ‘Could Fatima buy three pens and
four pencils?’ Learners need to check that c = 3 and k = 4
would satisfy the equation.
Generalising highlights the fact that any function can
have a number of different interpretations. It is an
example of the power of mathematics that algebraic
formulations can show the similarities between different
situations. This will not be seen so clearly in verbal
descriptions.
They can now start Exercise 10.1.
Homework ideas
Learners will be asked to produce tables of values in
Section 10.2.
Workbook, Section 10.1.
Differentiation ideas: Concentrate on the formation
of the equation and give more practice with this if
necessary. For example, what if she spends $40 rather
than $30?
As Section 10.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
122
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10 GRAPHS
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Assessment ideas
During this topic, it is useful to ask learners to work
in pairs on some of the questions and activities. When
learners are trying to move from a verbal situation to
an algebraic representation or vice versa, discussion
with a partner will help to clarify ideas and to support
self-assessment.
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
• Use knowledge of coordinate
pairs to construct tables of
values and plot the graphs
of linear functions, including
where y is given implicitly in
terms of x (ax + by = c), and
quadratic functions of the
form y = x2 ± a.
• Given a function, for example,
2x + 3y = 24, learners can
complete a table of values
and use it to draw a graph.
M
9As.05
PL
E
10.2 Plotting graphs
LANGUAGE SUPPORT
SA
Linear function: a function with a
straight-line graph
If needed, remind learners of the vocabulary
associated with graphs (e.g. intercept). Remind
learners that ‘axes’ is the plural of ‘axis’.
When a line is written as, for example, y = 2x + 5
or 3x + 4y = 36, this can be called a function or the
equation of the line. Either term can be used and
learners should recognise both of them.
Common misconceptions
Misconception
How to identify
How to overcome
Not realising that the graph
of a linear function must be a
straight line.
Make a quick check of learners’
graphs in Exercise 10.2; errors will
be obvious.
Emphasise this idea. If points
are not in a straight line, learners
should realise that they have made
an error and correct it.
Drawing the graph of a quadratic
function such as y = x2 − 2 with a
point at the bottom instead of a
smoothly changing curve.
Exercise 10.2 has questions asking
learners to draw quadratic graphs.
Check learners’ graphs. Point out
this common error in discussion.
Graphical software can be used to
indicate the correct shape.
123
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Starter idea
x
y
Functions and graphs (5 minutes)
Resources: None
Description: Display the function y = 3x + 12. Ask
learners to suggest a context for this function. Any
sensible suggestion is fine. If you get no response, you
could suggest something about ages or money or masses.
Then ask ‘Where does the line cross the y-axis?’ They
should know that this is when x = 0, so it must be at
(0, 12). They might mention the y-intercept, since this
term was introduced in Stage 8. Finally ask ‘Where does
the line cross the x-axis?’ If a learner suggests an answer,
ask them to explain how they found it. The answer will
probably imply solving the equation 0 = 3x + 12 which
has the solution x = −4 and so the point required
is (−4, 0).
Main teaching idea
1
2
3
4
5
6
12
10
8
6
4
2
0
Ask ‘What will the graph look like?’ If learners are
unconvinced it is a straight line, ask them to draw it.
It crosses the axes at (0, 12) and (6, 0).
For a third example, look at 2x + 3y = 12. Ask learners in
pairs to try to find pairs of values. This is more difficult
and will often involve fractions. Give them a couple of
minutes to do this. Then say that it is useful to start by
finding where the line will cross the axes.
PL
E
Now ask ‘How would you draw a graph of this
function?’ Learners should suggest a table of values.
Ask for the value of y when x has particular values,
such as 2 or 4 or 1.
0
Tables of values (15 minutes)
When y = 0, the equation becomes 2x = 12, so x = 6 and
this gives the point (6, 0).
Put these in a table of values:
x
y
Resources: None
SA
Description: On the board, show the function x + y = 12.
Ask for pairs of values for x and y. Learners should be
able to suggest pairs of values. As they suggest values,
put them in a table of values, with each pair in the
next column.
x
y
When you have six pairs of values, rewrite the table with
the values of x in increasing order.
Ask learners to say what the graph will look like,
without drawing it. They should be able to describe
a straight line sloping downwards from left to right.
Ask ‘Where will the line cross the axes?’. You are
looking for (12, 0) and (0, 12).
Now repeat with the equation 2x + y = 12. By choosing
particular values of x learners can work out the
corresponding values of y. A table will include some of
these values, and it could include negative values too:
0
6
4
0
This suggests putting in x-values between 0 and 6 to find
the corresponding y-value.
So, when x = 1, then 2 + 3y = 12, so 3y = 10 and so y =
10
3
or 3 1 . Learners should leave the value as a fraction and
3
not approximate with a decimal.
M
Learning intention: To understand how to create a table
of values for a function in the form ax + by = c.
When x = 0, the equation becomes 3y = 12, so y = 4 and
this gives the point (0, 4).
Point out that you could also start with a y-value.
When y = 2, then 2x + 6 = 12, so 2x = 6 and x = 3.
These values could go in the table.
x
y
0
1
3
6
4
31
3
2
0
Ask learners to suggest two more pairs of values.
Put these pairs of values in the table and then ask them
to draw the graph. Check that they get a straight line
passing through (0, 4) and (6, 0).
Note: this example is also in the Learner’s Book as
Worked example 10.2.
If graphical software is available, learners could use it to
check the graphs of the equations in this activity.
Differentiation ideas: If learners require further
practice, ask them to draw the graph of x + y = 12.
You can make this more challenging by asking for
negative values of x and y. A useful extension for more
confident learners is to ask them to compare the graphs
of 2x + 3y = 12 and 3x + 2y = 12.
124
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10 GRAPHS
Plenary idea
Identifying a line (5 minutes)
Resources: Resource sheet 10.2
Description: On the board, draw/display each of these
lines in turn, one at a time, on the same axes. Either use
graph drawing software, or use the graphs supplied in
Resource sheet 10.2. After you draw/display each graph,
ask learners to tell you the equation.
Assessment ideas: Ask learners to discuss in pairs what
the equation is for a minute before you take answers.
This will give them thinking time and it will give you
an opportunity to identify learners who are finding this
difficult. They will need extra support.
Guidance on selected Thinking and
working mathematically questions
Generalising
Homework ideas
Workbook, Section 10.2.
As Section 10.2 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
PL
E
• y=6
• y = 2x
• x+y=6
• x + 2y = 6
Each time, ask learners to give you the coordinates of
one or two points to check.
a generalisation. In Question 11 the steps have been
removed. Learners should use the experience of
the two previous questions to find and describe the
generalisation. This could be by drawing another
example or by considering the general case immediately.
Assessment ideas
It is easy to check learners’ work when they are drawing
graphs. A quick look will show whether they have
drawn the graph correctly or not. You could also ask
learners to compare their graphs with a partner and ask
‘Do your graphs look the same? If not, which graph
is correct?’
M
Exercise 10.2, questions 9, 10 and 11
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Each of these questions forms a sequence. In questions 9
and 10 learners are led through a sequence of steps,
drawing a family of lines and then looking for
SA
10.3 Gradient and intercept
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9As.06
• Understand that straight-line
graphs can be represented
by equations. Find the
equation in the form
y = mx + c or where y is
given implicitly in terms of
x (fractional, positive and
negative gradients).
• Rearrange the equation
2x + 5y = 20 to find the
gradient and the y-intercept
of the line.
125
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
LANGUAGE SUPPORT
you to check their working, and it will also help
them to remember the methods.
Model phrases such as ‘As the x-coordinate
increases by a, the y-coordinate decreases by b’.
Encourage learners to use phrases such as these
when describing straight-line graphs.
PL
E
There are no new words in this section.
The words 'gradient' and 'intercept' are familiar
from Stage 8. Learners were also reminded of these
words in the Getting started exercise at the start of
this unit.
Encourage learners to talk through their methods
when they are rearranging equations. This will allow
Common misconceptions
Misconception
Looking at an equation such
as 2x + 5y = 20 and saying that
the gradient is 2 , omitting the
5
minus sign.
Starter idea
How to identify
How to overcome
Questions in Exercise 10.3.
Tell the learners to rewrite the
equation in the form y = … before
finding the gradient.
Main teaching idea
A reminder of gradient (5 minutes)
More about gradients (10 minutes)
Resources: Graphical software, if possible
Learning intention: To understand that the gradient of a
line can be a fraction.
Resources: None
The coefficient of x in the equation is 2.
2
From the graph, as the value of x increases by 1,
the value of y increases by 2.
SA
1
If learners are unsure about the second point, write
down some coordinates to make it clear: (0, 3), (1, 5),
(2, 7), and so on.
Now, on the board, write the equation y = 3x + 3. Ask
‘What does this graph look like?’ When learners have
described a straight line through (0, 3) with a gradient
of 3, draw the graph to confirm this. Finally, on the
board, draw the line y = x + 3. Do not show the equation.
Ask ‘What is the equation for this line?’ If learners
put a 1 as a coefficient of x, point out that this is
not necessary.
1
2
Description: On the board, write the equation y = x + 4.
M
Description: On the board, show a graph of y = 2x + 3,
using graphical software if possible. Show the equation
as well. Ask ‘What is the gradient of this line?’ Learners
should be able to say that the gradient is 2. Ask ‘How
do you know this?’ They should be able to give two
explanations:
Also show this table of values with only the first row
filled in:
x
y
0
1
2
3
4
5
6
Ask learners to copy the table and to complete the
second row. Show them the completed table to check
their answers, leaving the values as mixed numbers and
not as decimals:
x
y
0
1
4
1
4
2
2
3
5
1
5
2
4
5
6
6
1
6
2
7
Then ask learners to draw the graph. Show them the
graph to check. Ask ‘What is the gradient?’ From
the equation, they can see that the gradient is 1 .
2
Demonstrate how the graph also shows the gradient.
As x increases by 1, y increases by 1 . Point out that this
2
is equivalent to saying that if x increases by 2, then y
increases by 1, as shown in this diagram:
126
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10 GRAPHS
x
y= 2 +4
If learners are not clear about this, substitute numbers
for a and b such as 3x + 5y = 30 and go through the
rearranging with those numbers.
1
1 2
1 2
Assessment ideas: Using letters instead of numbers
focuses on the structure of the problem rather than
particular values. Learners who can describe the process
in general terms will be able to deal with any particular
example. However, if you need to use particular values
learners can still show an understanding of the method
required.
2
0 1 2 3 4 5 6 7
x
1
Now, on the board, write the equation: y = x + 4. Ask
3
learners, ‘Without finding any values, what does this
graph look like? In particular, what is the gradient?’ [1 ]
3
2
Continue in the same way with y = x + 4 [gradient is 2 ].
3
3
1
When this is understood, on the board, write y = 4 − x
2
and a table of values with just the x-values
written in.
x
0
1
2
3
y
4
3
1
2
3
2
1
2
4
5
6
2
1
1
2
1
Specialising and generalising
Exercise 10.3, Question 8
In this question learners are asked to look for patterns in
a number of examples and then to state their findings in
the form of a generalisation. Moving from the particular
to the general is an important mathematical skill that
learners must try to develop.
Homework ideas
Workbook, Section 10.3.
As Section 10.3 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
M
Ask learners to tell you what y-value to write in each case.
Ask ‘What does the line look like?’ Learners should be
able to describe it. In particular, they should see that the
gradient is − 1 . They should be familiar with negative
2
Guidance on selected Thinking and
working mathematically questions
PL
E
y
7
6
5
4
3
2
1
0
gradients. Check understanding by replacing 1 in the
2
equation with other fractions, keeping the minus sign.
SA
Differentiation ideas: You may need to go more slowly
with the equations involving thirds. You can ask learners
to work in pairs and to create a table of values and use
it to draw a graph in each case. Learners need to be sure
what a fractional or negative gradient looks like.
Plenary idea
Generalising (5 minutes)
Resources: None
Description: On the board, write the equation
ax + by = 30. Say that a and b are positive integers.
Ask ‘How would you rearrange this equation to make y
the subject?’ They should see that first they subtract ax
and then divide by b.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Another idea is to ask learners to make a small poster
to demonstrate how to rearrange an equation of the
form ax + by = c, where a, b and c are positive integers,
to make y the subject, and hence to find the gradient
and y-intercept of the graph.
Assessment ideas
Look at learners’ answers to the questions in Exercise
10.3 to ensure that their algebraic manipulation is
correct. The plenary activities in these notes and in the
Additional teaching ideas also give opportunities to
check on understanding.
127
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
10.4 Interpreting graphs
LEARNING PLAN
Learning objectives
Success criteria
9As.07
• Read, draw and interpret
graphs and use compound
measures to compare graphs.
• Learners can calculate
and interpret the gradient
of a linear graph in a
realistic context.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary in this section.
You might need to remind learners that you read
the '/' in compound units as 'per'. So 50 km/h is '50
kilometres per hour', and so on.
If needed, support learners with the necessary
language to describe the real-life situations shown
by the graphs. Also support learners with the
language needed to compare graphs.
Common misconceptions
Misconception
How to overcome
Ask learners to explain a method
when finding the gradient of a line.
Complete appropriate questions in
Exercise 10.4 correctly.
M
Using one point and the origin to
find the gradient when a line does
not pass through the origin.
How to identify
Starter idea
Main teaching idea
Compound units (5 minutes)
Interpreting a linear graph 1 (10 minutes)
Resources: None
Learning intention: To interpret a linear graph that
passes through the origin in a practical situation and to
use the graph to form an equation.
Emphasise that in all cases the answer is a distance
divided by a time and the units are in the form distance/
time (‘distance unit per time unit’). These are called
compound units, but that term is not required by
learners and is not used in the Learner’s Book.
Resources: The graph from the introductory text in
Section 10.4 in the Learner’s Book
Description: On the board, display this graph passing
through (100, 240):
Distance (m)
SA
Description: Say, or on the board, write: ‘A train travels
45 km in 15 minutes’. Ask learners, in pairs, to work
out the speed of the train. Ask them to try to give the
speed in different units. After a couple of minutes take
an answer from one pair. Then ask a different pair to
give a different answer. Then ask if a third pair can give
a different answer again. You can expect answers of
180 km/hour and 3 km/minute to be given first. Other
possible answers are 3000 m/minute or 50 m/s (that is
3000 ÷ 60).
250
200
150
100
50
0
0
50
100
Time (s)
128
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10 GRAPHS
Explain that the graph shows the distance travelled by
a runner. (Note: You can refer to the runner as male
or female. The introduction text in Section 10.4 of the
Learner’s Book does not specify gender.)
Point out the different scales on the two axes and talk
about what they represent.
Ask ‘The graph is a straight line. What does this tell
you?’ [The speed is constant]
Guidance on selected Thinking and
working mathematically questions
PL
E
Ask ‘What distance has the runner travelled after 50 s?
100 s?’ [120 m and 240 m]
Assessment ideas: Asking a learner to explain how
they found an answer gives you an opportunity
to assess their understanding. If you do this with
the whole class listening, then all learners have the
opportunity to assess their own understanding. They
will also have an opportunity to correct any incorrect
solution offered. This is both self-assessment and peerassessment.
Now ask ‘What is the speed?’ Learners could calculate
120
240
the speed as
or
, but in either case the answer is
50
100
2.4 m/s. Make sure that learners include the units.
Now say that the runner runs d m in t s and ask for an
equation for d in terms of t.
Learners should see that d = 2.4t and realise that 2.4 is
the gradient of the line. Emphasise this point.
Use the formula to find the distance for some other
times that could be on or beyond the graph. For
example, when t = 70 then d = 168, that the runner has
travelled 168 m after 70 s; or when t = 200 then d = 480,
that the runner has travelled 480 m after 200 s.
Exercise 10.4, questions 10 and 11
Questions 10 and 11 ask learners to show the method
used to find the answers. This is a way of convincing
others about the accuracy of their answers. Learners
need to show any calculations they do in a way that can
be read and understood by others.
Homework ideas
Workbook, Section 10.4.
As Section 10.4 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
M
Finally, ask ‘What assumption do you make if you use a
value of t beyond the graph?’ You assume that the speed
does not change.
Convincing
Differentiation ideas: For extension, specify a
distance and ask for the time taken. For example,
400
= 167 s to
when d = 400 m, then 400 = 2.4t and t =
2.4
the nearest second.
SA
Plenary idea
Review of progress (10 minutes)
Resources: ‘Check your progress’ exercise in the
Learner’s Book
Description: Ask learners to complete the ‘Check you
progress’ questions at the end of Unit 10 in the Learner’s
Book. When they have completed the questions, go
through the answers. Do this by asking individual
learners to give you their answers and to explain how
they worked them out. If a learner gives an incorrect
answer, then ask another learner to explain the mistake
and to correct the answer.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
You could also ask learners to make brief summary
notes of the important points they need to remember
from this unit.
Assessment ideas
There are a number of questions in Exercise 10.4 where
learners have a graph and an equation for that graph.
Encourage learners to use self-assessment to check their
accuracy. To do this, they:
• Choose a value for the x variable.
• Use the equation to calculate the y variable.
• Check that this corresponds to a point on the graph.
129
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
PROJECT GUIDANCE: CINEMA MEMBERSHIP
Why do this problem?
This problem offers a real-life context in which
to explore simultaneous equations. Learners can
compare and critique the merits of numerical,
graphical and algebraic representations and use
these different approaches to develop convincing
arguments.
Finally, set the challenge of creating a suitable
pricing scheme for the rival cinema, using algebraic
and graphical techniques. Towards the end of
the lesson, you could invite groups of learners
to present their pricing schemes, together with
supporting evidence.
Key questions
• How would you advise someone who watched
just a few films?
PL
E
Possible approach
Introduce the context of a cinema in which you can
choose to buy tickets for $18, or you can pay $20
every three months for Bronze membership, and
then pay just $8.50 for each ticket.
use these methods, and then support them in
understanding how to interpret their solutions.
Invite learners to discuss whether they would
choose to pay the membership fee or not.
When learners have got the idea that the choice
of membership will depend on how many films
they are likely to watch, introduce the Gold and
Silver membership schemes and invite learners to
compare these memberships.
• How can you show clearly and quickly the
different membership schemes?
Possible support
Learners could create a table showing how much it
costs to see 1, 2, 3, … films for each membership
scheme, and then plot this information on a graph.
Possible extension
You could encourage learners to experiment
with different membership schemes using graphplotting software.
SA
M
While they are working, look out for learners who
are approaching the problem algebraically or
graphically. Then bring the class back together
and invite learners to compare their methods. If
nobody has used algebraic or graphical methods,
it might be necessary to prompt learners to
• How would you advise someone who watched a
lot of films?
130
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11 RATIO AND PROPORTION
Unit plan
PL
E
11 Ratio and
proportion
Topic
Approximate
number of
learning hours
Outline of learning
content
Resources
11.1 Using ratios
1–1.5
Use ratios and
equivalence.
Learner’s Book Section 11.1
Workbook Section 11.1
Additional teaching ideas Section 11.1
Understand the
relationship between
two quantities when they
are in direct or inverse
proportion.
Learner’s Book Section 11.2
Workbook Section 11.2
Resource sheet 11.2
Additional teaching ideas Section 11.2
11.2 Direct and
0.5–1
inverse proportion
M
Cross-unit resources
Language worksheet: 11.1–11.2
End of unit 11 test
BACKGROUND KNOWLEDGE
SA
For this unit, learners will need this background
knowledge:
• Understand how ratios are used to compare
quantities to divide an amount into a given ratio
with two or more parts (Stage 8).
• Use knowledge of equivalence to simplify and
compare ratios (different units) (Stage 8).
• Understand and use the relationship between
ratio and direct proportion (Stage 8).
The focus of this unit is to extend and deepen
learners’ knowledge and understanding of ratio
and proportion, learning new skills, including
using inverse proportion, to add to those already
encountered in stages 7 and 8.
TEACHING SKILLS FOCUS
Active learning
Throughout both sections of Unit 11, if learners do
not understand or they continue to get the same
type of question incorrect, ask another learner to
explain/help. It is important that you also listen to
the explanation/help given by another learner. You
need to be able to confirm that the help is of good
quality or to ask if another learner would/could
explain the problem in a different way.
Active learning helps to establish good learning
patterns and practice. When a learner can explain
well, it shows that they really understand what
they are doing and know how to improve. Also,
learners often feel more confident speaking to
131
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
CONTINUED
• Did a variety of learners do the explaining – or
did the class rely on just the most confident
learners in the class?
• Did the learners that helped other learners
understand the work better themselves because
of the help they gave? Are you sure?
• Did learners that received help from other
learners benefit from it or did they then need
help/advice from you?
• Are there any specific learners in your class that
benefit from active learning?
• Are all learners that require help getting it?
Are some learners so lacking in confidence that
they do not ask for help? What are you doing
about that?
• What other ways could you get learners to
explain more to other learners?
PL
E
other learners, asking more targeted questions,
so becoming more active learners themselves. As
learners are now more used to explaining concepts
and asking for specific, targeted help from other
learners, these discussions can happen without you
being present. Hopefully the practice learners have
had during stages 7 and 8 and earlier in this book
will mean they are already confident in this very
effective learning skill.
Remind learners that the key to being successfully
involved in this type of learning is that there is no
judgement. The learner asking for help and the
learner giving help are both learning and
improving.
At the end of Unit 11, ask yourself:
• Did learners have useful discussions that solved
issues one of them was having?
11.1 Using ratios
M
LEARNING PLAN
Learning objectives
Success criteria
9Nf.08
• Use knowledge of ratios
and equivalence for a range
of contexts.
• Learners can work backwards
and use knowledge of the
relative size of the parts
of a ratio.
SA
Framework codes
LANGUAGE SUPPORT
There is no new vocabulary for this section.
The example and questions in this section are all
set in context. Support learners with the language
used, and make sure that they understand what each
question is asking before they start to answer it.
Encourage learners to read aloud their working.
This will help you to check that they are using the
correct method, and will also give them practice at
explaining their method.
132
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11 RATIO AND PROPORTION
Common misconceptions
Misconception
How to identify
How to overcome
Misreading questions such as questions 3
and 4, thinking the amount shown is the
total, not one part.
Questions 3 and 4.
Discussion during questions 1
and 2. Careful checking during
questions 3 and 4.
Starter idea
Getting started (10–15 minutes)
PL
E
Resources: Note books, Learner’s Book
Getting started exercise
Differentiation ideas: Most learners who struggle
with this type of question often just need to discuss it.
A brief discussion with another learner, or yourself,
will clarify their thoughts. This type of working is so
close to other work previously done that learners should
quickly understand.
Description: Learners should have little difficulty
with the majority of the Getting started questions,
but before learners attempt the questions, discuss
what they remember about sharing in a ratio. Guide,
if necessary, learners to discussing that they need to
work out the ‘number of parts’, then work out the
value of ‘each part’, before working out any amounts/
totals.
You might need to help some learners with Question 4,
where they should compare the ratios by either changing
the two ratios to the form n : 1 or n : 15 or by making
equivalent fractions from those in part a.
Sweet! (3–5 minutes)
Resources: Mini white boards
Description: On the board, write/display the following:
Ali and Zac share some sweets in the ratio 4 : 5.
a
Ali gets 24 sweets. How many sweets does Zac get?
b
Zac gets 25 sweets. How many sweets were
shared out?
Assessment ideas: Learners could show their answers
[30, 45]. A quick look at the mini white boards will
quickly show you if any learners require further help.
M
This exercise is a quick reminder of previous work that
will help learners to be more effective with this unit. It
is not a test. After each question it might be useful to
allow self- or peer-marking, allowing learners to rectify
any mistakes after a brief discussion.
Plenary idea
Main teaching idea
SA
Question 7, Think like a mathematician
(5 minutes)
Learning intention: To understand different methods for
ratio questions.
Resources: Note books, Learner’s Books
Description: When discussing the two options, learners
should understand that Nia’s method is quick and that
Rhys’ method would be more useful if the amounts the
others should pay were also required.
The easiest method for this question is just to double
$36.25, as 5 parts is half of the whole [10 parts] –
however, this method won’t always work. This sort of
problem-solving skill is to be commended, as any learner
who suggests this obviously is thinking clearly, however,
Nia’s and Rhys’ methods will always work.
Guidance on selected Thinking and
working mathematically questions
Specialising and convincing
Exercise 11.1, Question 13
Learners have not encountered a question of this type
previously in this exercise. Tell learners to work with a
partner to allow for discussion on how to approach this
problem. Tell learners that there are several different
methods to approach this problem. Allow learners 4 or 5
minutes working before asking for answers [110 g syrup,
220 g butter and 440 g oats]. Discuss the method used to
get the correct answer – it is likely that several learners
will not have the correct solution. They can get extra
practice with this type of problem with the plenary idea
in the Additional teaching ideas. If you require other
practice, you could:
• change the ratio in Question 13 to 1 : 2 : 5. The
process is identical, but the multiplier is slightly more
difficult [88]. The answer is 88 g syrup, 176 g butter
and 440 g oats.
133
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
• change the ratio in Question 13 to 1 : 2 : 3. The
process is similar, but butter becomes the limiting
factor. The multiplier is 125 and the answer is 125 g
syrup, 250 g butter and 375 g oats.
Ask learners to write an explanation of how to answer,
giving clear method and reasons. Insist that learners do
not have to work out the answer, but their explanation
must allow another learner to work out a similar question.
Homework ideas
This can be done individually, but is often more useful
when done in pairs. Learners should discuss their
methods first. Encourage them to talk about the shortest
way to explain the best method.
Workbook, Section 11.1.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Assessment ideas
• When learners read their method to the class, be
unnecessarily critical, and try to find anything at all
that is unclear in any way.
• When learners swap their notes, ask learners to use
the other learner’s answer only to help work out the
following question:
Billie and Caz share some money in the ratio 5 : 3.
Billie gets $75.
On the board, write/display this question:
Billie and Caz share some money in the ratio 3 : 7.
Billie gets $75.
The two best ways of checking this work is for learners
to read out their answers [$175, $250] or to swap notes:
PL
E
As Section 11.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
How much money does Caz get?
b
What is the total amount of money?
How much money does Caz get?
What is the total amount of money?
When answered, ask learners if the notes from the other
learner were well enough explained so they could answer
the new question and work out the correct answers
[$45, $120].
Discuss what learners think are the key parts of the
explanation to help answer any question of this type.
M
a
a
b
11.2 Direct and inverse proportion
SA
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Nf.07
• Understand the relationship
between two quantities when
they are in direct or inverse
proportion.
• Learners can identify and use
direct and inverse proportion.
LANGUAGE SUPPORT
Inverse proportion: two quantities are in inverse
proportion if, when one quantity increases the
other quantity decreases in the same ratio
Remind learners of the definition of direct
proportion. Ensure that they understand the
difference between direct and indirect proportion.
134
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11 RATIO AND PROPORTION
CONTINUED
Do this by asking for and discussing examples
of both.
Again, the examples and questions in this
section are set in context. Support learners in
understanding the language and make sure that
they understand what each question is asking
before they attempt to answer it.
Common misconceptions
How to identify
How to overcome
PL
E
Misconception
Not understanding when two
Any question might cause
quantities are in inverse proportion. confusion, especially questions 1
and 11.
Starter idea
How much? (5 minutes)
Resources: Mini white boards or note books
Description: Ask ‘Five identical toys cost $20.
How much does one toy cost?’
Learners will be able to work out that the cost of one
toy is $4. When you have the answer, on the board,
show/explain this method to show working:
5 toys = $20
÷5
1 toy = $4
Ask ‘4 boxes cost $2. Using this method (point at the
working on the board) work out the cost of 1 box.’
When completed, ask a learner to show their method on
the main board. It should be:
SA
4 boxes = $2
÷4
Description: Depending on your class, you might decide
to have a discussion before starting Question 1 to ensure
that learners have a clear understanding of direct and
inverse proportion as described in the introduction and
in Worked example 11.2. Learners might also need some
guidance about the ‘neither’ category. This can be a little
difficult for learners to understand. Suggest that if the
quantities are not in direct proportion and not in inverse
proportion learners write ‘neither’.
Differentiation ideas: c and f are examples of inverse
proportion. Learners can discuss the fact that with
part f you have to assume that all the people work at the
same rate – if they don’t, it is not inverse proportion as
the ratios are not the same.
M
÷5
Repeated discussions about
why a relationship is in inverse
proportion.
÷4
1 box = $0.50 [or 50 cents]
If necessary, give other questions, such as: ‘10 pizzas
cost $65, what is the cost of 1 pizza?’
Main teaching idea
Question 1, Think like a mathematician
(5 minutes)
Learning intention: To be able to decide if two quantities
are in direct proportion, inverse proportion or neither.
Resources: Note books, Learner’s Books
Plenary idea
Exit ticket (3 minutes)
Resources: Resource sheet 11.2: Exit ticket
Description: Give each learner an exit ticket, cut out
from Resource sheet 11.2. Learners should complete the
exit ticket just before leaving class. Allow 2–3 minutes
to complete.
Assessment ideas: There is no space for a name on top
of the exit ticket. Ask for learners to put their name if
you prefer, or ask that only those learners who would
like some specific help to put their name on their exit
ticket. You will probably recognise the handwriting even
without a name. However, the perceived anonymous
aspect might lead to more honest feedback. This means
that reading what learners think they have learned and
how you might help them further will help you to clarify
teaching points for the class for the next lesson or in
revision lessons to come.
135
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Guidance on selected Thinking and
working mathematically questions
Conjecturing and convincing
Exercise 11.2, Question 10
Learners who do not fully understand when two
quantities are in either direct or inverse proportion
might have difficulty in understanding this question.
Give no help or advice for this question. Ask all learners
to answer part a, even if they are guessing.
Homework ideas
Workbook, Section 11.2.
PL
E
During part b make sure you listen to as many pairs as
possible to check that you know which learners might
think that Arun is correct. Some learners might even
think that neither is correct, and that the roller coaster
will take 8 minutes to complete its ride.
You could ask learners to make a mind map containing
everything they think they need to remember for the
end-of-unit test. The following lesson, it is important
to share the mind-maps in class (e.g. spread out over a
few desks for everyone to look at), rather than marking
them. Discuss the different mind maps as a class. When
the class agree that a point is important, that key point
could be copied onto the board (by you or a learner).
Agree on as many key points as possible. Learners
could then improve/update their individual mind maps
if necessary. Learners could store their mind maps at
home as a possible revision tool towards mid-term/
end-of-year tests.
Use Question 10 as a hinge-point question. As discussed
in the Thinking and working mathematically guidance,
this will help check that learners understand the
different types of proportionality. It will be important
to listen to the discussions in part b and to note learners
that are still unsure of how to decide if a question is
referring to direct, indirect or no proportionality.
SA
M
As Section 11.2 is unlikely to take more than one lesson,
set all or part of the Workbook at the end of the lesson.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Assessment ideas
136
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12 PROBABILITY
12 Probability
Unit plan
Approximate
number of
learning hours
Outline of learning
content
Resources
12.1 Mutually
exclusive events
2
Calculating probabilities
when there are a number
of mutually exclusive
events.
Learner’s Book Section 12.1
Workbook Section 12.1
Additional teaching ideas Section 12.1
12.2 Independent
events
2
Identifying whether events
are independent or not.
Learner’s Book Section 12.2
Workbook Section 12.2
Additional teaching ideas Section 12.2
12.3 Combined
events
2
Calculating the
probabilities of simple
combined events.
Learner’s Book Section 12.3
Workbook Section 12.3
Resource sheet 12.3
Additional teaching ideas Section 12.3
12.4 Chance
experiments
2
PL
E
Topic
M
Carrying out experiments
Learner’s Book Section 12.4
and comparing outcomes Workbook Section 12.4
with expected frequencies.
Additional teaching ideas Section 12.4
Cross-unit resources
Language worksheet: 12.1–12.4
End of unit 12 test
SA
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Understand that complementary events have a
total probability of 1 (Stage 8).
• Be able to use tables, diagrams and lists to
identify outcomes of combined events (Stage 8).
• Be able to find the probabilities of equally likely
combined events (Stage 8).
• Understand how to compare experimental
probabilities with theoretical outcomes
(Stage 8).
In this unit, learners will move on to identify
mutually exclusive events. They will also be
introduced to independent events and will
learn how to identify whether two events are
independent or not. Learners will then extend their
knowledge of theoretical probabilities. They will
calculate probabilities of combined events with
and without a tree diagram. They will calculate
expected frequencies and compare them with
observed outcomes recorded from an experiment
or simulation.
137
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
TEACHING SKILLS FOCUS
their value. It is insufficient to simply write
an answer.
As a teacher you can model the way you want
learners to present their work when you are doing
the starter activities and main activities. Look at your
display board at the end of one of these activities. Is
it easy to read? Are the points set out clearly? Does
it indicate to the learners the way you want them to
present their solutions and arguments? This can be
something to focus on as you teach this unit.
PL
E
Active learning
Mathematics is about communication. One of
the skills you are trying to develop is for learners
to be able to explain what they are doing and to
justify their conclusions. This is something learners
often find difficult. In this unit learners are given
opportunities to practise this skill. For example,
when they are asked to decide whether two events
are independent or not, they need to calculate
probabilities and show how they are calculating
12.1 Mutually exclusive outcomes
LEARNING PLAN
Framework codes
Success criteria
• Understand that the
probability of multiple
mutually exclusive events can
be found by summation and
all mutually exclusive events
have a total probability of 1.
• Work out the probability
of events happening or
not happening when given
the probabilities of a set of
mutually exclusive outcomes.
M
9Sp.01
Learning objectives
LANGUAGE SUPPORT
SA
Mutually exclusive: events are mutually exclusive if
only one of them can happen at one time
The examples and questions in this section are set
in context. Support learners with the language and
make sure that they understand what each question
is asking before they attempt to answer it. Make
sure, in particular, that they notice if in each question
it says that the event does or does not happen.
Use examples to demonstrate the meaning of
‘mutually exclusive’ and encourage learners to use
this term when appropriate.
Model the use of the notation ‘P( )’ to mean
‘the probability of’ and make sure that learners
understand and can use this notation.
Common misconceptions
Misconception
How to identify
How to overcome
Thinking that 'mutually
exclusive' and 'different' mean
the same thing.
Ask learners to identify pairs of events that
are mutually exclusive and pairs that are not
mutually exclusive in a particular context.
Successfully complete the
questions in Exercise 12.1 and
explain individual answers.
138
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12 PROBABILITY
Misconception
How to identify
How to overcome
Thinking that one of a set
of mutually exclusive events
must happen.
Emphasise in discussion that this is not the
case by using appropriate examples.
Ask learners to give examples
of mutually exclusive events.
Starter idea
For example, the probability of getting a counter that is
red or blue is 0.2 + 0.35 = 0.55. Ask individual learners to
suggest further examples.
Ready to start (10 minutes)
Resources: Getting started exercise in the Learner’s Book
Main teaching idea
Finally, remind learners that the probability that an
event does not occur is 1 − the probability that it does.
For example, P(not red) = 1 − P(red) = 1 − 0.2 = 0.8; P(not
(red or blue)) = 1 − P(red or blue) = 1 − 0.55 = 0.45.
PL
E
Description: Ask learners to complete the Getting
started questions. These questions recap ideas that
will be familiar from Stage 8. When they have finished
the questions, go through the answers. Ask individual
learners to say what answer they got and how they
found it. The first three questions are on probability.
The fourth question is a reminder of addition and
multiplication of simple fractions, a skill which learners
will need in this unit. Try to identify any areas of
weakness or misunderstanding that might need some
remediation before continuing with the unit.
Mutually exclusive events (10 minutes)
Ask individual learners to suggest further examples.
Learners can now start Exercise 12.1 in the
Learner’s Book.
Differentiation ideas: If necessary, you can use the
introduction to Section 12.1 and Worked example 12.1
in the Learner’s Book to give further support.
Plenary idea
Summary (5 minutes)
Resources: None
Description: Ask for the main points to remember from
this section.
Resources: A box containing coloured counters. This is
just a prop to start a discussion. Any objects of different
types such as coloured pencils or coins would be
an alternative.
Look for a description of mutually exclusive events and
the fact that if A and B are mutually exclusive events,
then P(A or B) = P(A) + P(B).
M
Learning intention: To understand the meaning of the
phrase ‘mutually exclusive’ and its implications.
SA
Description: Show the box and say that it contains
counters of different colours. Ask learners to imagine
taking out a counter at random. On the board, write
these probabilities:
P(red) = 0.2 P(green) = 0.25
P(yellow) = 0.05
P(blue) = 0.35
Tell learners that ‘P(red)’ is a convenient way to write
‘the probability of red’.
Explain that getting a red counter and getting a green
counter are mutually exclusive events. This means that
they cannot both happen. One or neither might happen
but not both.
Describe other possible events that combine more than
one outcome. For example, getting a counter that is red
or blue. Ask for more examples.
Now say that you can find the probability of these
types of events by adding the individual probabilities.
Give learners this algebraic description and ask them to
give you an example of what A and B could be.
Assessment ideas: By asking individual learners to give
you examples you can check that they understand the
concept of mutually exclusive events and are able to
think of examples of their own.
Guidance on selected Thinking and
working mathematically questions
Convincing
Exercise 12.1, Question 13
Question 13 reverses the normal type of question.
Instead of starting with the number of counters,
the learner is asked to work backwards from the
probabilities and find the number of counters of each
colour. The learner can justify the answer by showing
that the correct probabilities are possible. One feature
of the question is that there is more than one answer.
139
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Any multiple of 12 is correct. Point this out to learners
if they do not notice it. This is a good learning point.
Look for all possible answers, even if you are not
directed to do so. A common example of this is solving
an equation such as x2 = 9 where there are two solutions,
3 and −3. Learners often forget the second solution.
Homework ideas
Workbook, Section 12.1.
Assessment ideas
The questions in the exercise do not include
opportunities for self-assessment. However, you can
look at learners’ answers to the questions to assess
understanding. Use the plenary idea activities in these
notes and in the Additional teaching ideas to ensure that
learners can identify mutually exclusive events in cases
where the probabilities are known, such as rolling a dice,
and in cases where probabilities are not known, such as
weather events.
PL
E
As Section 12.1 will probably take more than one
lesson, set suitable parts of the Workbook at the
end of each lesson. Only set questions that can be
answered using skills and knowledge gained from
that lesson. Workbooks are aimed at fluency and
consolidation through practice, not as a method to
learn new skills that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
12.2 Independent events
LEARNING PLAN
Framework codes
Success criteria
• Identify when successive
and combined events are
independent and when they
are not.
• Learners can explain whether
two successive events are
independent or not in simple
situations such as rolling a
dice or choosing a ball at
random from a bag.
SA
M
9Sp.02
Learning objectives
LANGUAGE SUPPORT
Independent events: if the probability that event B
happens is the same, whether event A happens or
not, then A and B are independent events
Ensure that learners understand the difference
between ‘independent events’ and ‘mutually
exclusive events’. Give examples and ask learners
for examples to support this.
A lot of the questions in Exercise 12.2 require
learners to explain their answer. Support learners
with the language required if needed.
Again, also support learners with the language of
examples and questions in context. It is important
that they understand what a question is asking
before they attempt to answer it.
140
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12 PROBABILITY
Common misconceptions
Misconception
How to identify
How to overcome
Assuming that independent
events are mutually exclusive.
Include discussion of examples where
this is not the case. For example when
rolling a dice, the events roll an even
number and roll a number less than 3.
Complete the questions in
Exercise 12.2 which include relevant
examples, such as questions 6, 7
and 9.
Starter idea
Here is a summary:
• First ball is black? Yes P(second ball is black) = 0.4
• First ball is black? No P(second ball is black) = 0.4
The probability has not changed. P(S) = 0.4 whether
F occurs or not. You say that F and S are independent
events. The first event happening or not does not change
the probability of the second event happening.
PL
E
Recap of mutually exclusive events
(5 minutes)
Resources: None
Description: Ask learners to imagine rolling a fair dice.
Ask learners to write three mutually exclusive outcomes,
one of which must happen. Then ask them to write the
probability of each event.
Take suggestions from individual learners. Check that
they are correct. The probabilities will be fractions.
They will be sixths or equivalent fractions.
A typical answer is: an odd number, 1 ; a 6, 1 ; an even
2
6
number less than 5, 1 .
3
If F happens, the first ball is black. It is not replaced.
The bag now has four balls B, W, W, W.
Now the probability that the second ball is black is
1
= 0.25.
4
On the other hand, if F does not happen and the first
ball is white, the bag now has four balls B, B, W, W.
This time the probability that the second ball is black is
M
Finally, ask why the probabilities of these three
outcomes should add to 1 (because one of them
must happen). Each learner should check that these
probabilities do add to 1.
Now change the experiment. This time the first ball is
not replaced. Work through the consequences with the
learners.
Main teaching idea
Independent events (10 minutes)
Learning intention: To introduce the idea of two events
being independent or not.
SA
Resources: A bag with two black balls and three white
balls, or something similar, is a useful prop, but it is not
essential
Description: Explain that there are two black balls and
three white balls in a bag.
Say that you are going to choose at random two balls,
one at a time, and look at the colour.
Here are two events:
F = first ball is black
S = second ball is black
You choose one ball at random. It is black. Now you
put it back into the bag. Now you choose another ball
at random. Ask ‘What is the probability that the second
ball is black?’ [0.4]
Now suppose that you do the same again, but this time
the first ball is white. Ask ‘What is the probability this
time that the second ball is black?’ [it is still 0.4]
2 1
= = 0.5 .
4 2
This time P(S) is not the same in both cases. It depends
on whether F has happened or not. This means that F
and S are not independent.
When learners understand this they are ready to start
Exercise 12.2.
Differentiation ideas: If learners are uncertain, repeat
the procedure with a different number of balls of each
colour, in a different ratio.
Plenary idea
Review (5 minutes)
Resources: None
Description: Use rolling a fair dice as an example. Ask
for learners to tell you two events that are independent
when you roll a fair dice once. They have already
seen examples of this, so they should be able to recall
one example. Ask if these events are mutually exclusive.
They will not be.
141
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Now ask learners to try to think of two mutually
exclusive events that are independent. They will not be
able to! Bring out the idea that if one event happens
then the probability of the other event happening is
zero. On the other hand, if one event does not happen
then the probability of the other event happening is
increased.
Guidance on selected Thinking and
working mathematically questions
Classifying
Exercise 12.2, Question 7
Homework ideas
Workbook, Section 12.2.
As Section 12.2 will probably take more than one
lesson, set suitable parts of the Workbook at the end of
each lesson.
Workbook Exercise 12.2 has further questions that
can be used as homework to reinforce and strengthen
understanding of the idea of independent events.
PL
E
Assessment ideas: Listen to learners’ comments as they
discuss these examples to make sure they have a clear
understanding of the terms ‘mutually exclusive’ and
‘independent’ and the difference between them.
the course. It is useful for learners to be aware of the
distinction between the two at this stage.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Assessment ideas
When you look at the explanation that learners give to
the questions in Exercise 12.2 you will be able to assess
their understanding of independent events. You will also
see whether they can justify a solution clearly. It might
be the case that some learners find it difficult to calculate
and use probabilities in their arguments. An alternative
is to make a statement such as ‘if the first ball is black,
then the probability that the second ball is black is less’.
This shows an understanding of the underlying concept.
M
Question 7 is about choosing two balls at random from
a bag consecutively. The important point is whether the
first ball is replaced before the second ball is chosen or
not. In the first case the events are independent. In the
second case the events are not independent. This is an
important categorical distinction because it affects how
probabilities are assigned. Subsequent work in this unit
concentrates on independent events, but consideration
of events that are not independent will come later in
12.3 Combined events
SA
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Sp.03
• Understand how to find the
theoretical probabilities of
combined events.
• Learners are able to put
probabilities on a tree diagram
when events are independent
and to use them to calculate
probabilities of different outcomes.
LANGUAGE SUPPORT
There is no new vocabulary in this section.
Learners are familiar with tree diagrams from
Stage 8 and with independent events from
Section 12.2.
Again, support learners with the language needed
to understand questions given in a real-life context.
In particular, make sure that they notice if an event
does or does not happen in each question.
142
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12 PROBABILITY
Common misconceptions
Misconception
How to identify
How to overcome
Not knowing when to add
probabilities and when to
multiply them.
Answers to questions in
Exercise 12.3 or in class discussion.
Emphasise that you always multiply
the probabilities on the branches of
a tree diagram.
Using Resource sheet 12.3 means that learners will not
have to copy the tree diagrams in questions 7 to 10. This
will save a lot of time in the classroom.
Say that when two events are independent you can find
the probability of both events happening by multiplying
individual probabilities.
PL
E
So P(6 and head) = P(6) × P(head) = 1 × 1 = 1 .
Starter idea
6
Fractions and equivalent decimals
(5 minutes)
Resources: None
Description: Say that a digit from 1 to 10 is generated at
random. Ask ‘What is the probability that the number
is a 4?’ Ask for the answer as a decimal [0.1] and as a
fraction  1  .
10 
Repeat this with other events where the probability has
different values. Here are some examples:
1
5
P(even number) = 0.5 or 1 ; P(less than 3) = 0.2 or ;
2
3
P(more than 4) = 0.6 or .
5
12
Ask learners to work out some other probabilities in the
same way. Here are some suggestions:
P(even number and tail) = 1 × 1 = 1 .
2 2 4
2 1 2 1
P(more than 2 and tail) = × = = .
3 2 6 3
You could ask learners to suggest other examples.
Differentiation ideas: You might need to give more
support if learners find the arithmetic with fractions
difficult. You could ask learners to work in pairs to
support each other. You could look at the starter idea in
the Additional teaching ideas to do more work on this.
M
Check that learners can simplify the fractions correctly.
2
You can explain why this seems sensible in the following
way. If you carry out this experiment repeatedly, in onesixth of the trials you get a 6 and in one-half of those
you get a head. So you get both a 6 and a head in onehalf of one-sixth of the trials. That is one-twelfth.
Then say that now there are nine digits, 1 to 9. Ask ‘Why
are fractions better than decimals for your answers this
time?’ Look for the answer that ninths cannot be written
exactly as decimals, so it is better to leave the answers as
exact fractions.
SA
Main teaching idea
Multiplying probabilities (10 minutes)
Learning intention: To understand that you calculate the
probability that two independent events both happen by
multiplying the two individual probabilities.
Resources: A coin and a dice as visual aids (not
essential)
Description: Say that you are rolling a fair dice and
flipping a coin. Ask ‘What is the probability of rolling
a 6?’   Then ask ‘What is the probability of the coin
1
6
landing on a head?’  
1
2
Ask ‘Are these events independent?’ [Yes, whether you
get a 6 or not does not affect the probability of a head]
Plenary idea
Do tree diagrams help? (5 minutes)
Resources: None
Description: Say that you are looking at two independent
events and you want to find the probabilities of both.
Ask ‘How are tree diagrams useful?’ Learners might
give different answers. Try to get the idea that they show
all the possible outcomes and make it easier to answer
supplementary questions.
Assessment ideas: Look for a recognition that you
always multiply the probabilities on the branches and
that the sum of the probabilities of the outcomes is 1. If
you feel that learners are still not confident with using
tree diagrams then remind them of the second main
activity idea in the Additional teaching ideas and repeat
that activity if necessary.
143
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Guidance on selected Thinking and
working mathematically questions
Workbook Exercise 12.3 has further questions that can
be used as homework to reinforce understanding when it
is appropriate to multiply probabilities.
Convincing
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Exercise 12.3, Question 11
Homework ideas
Workbook, Section 12.3.
Assessment ideas
The best way to assess learners’ understanding is
to look at the tree diagrams they have completed in
Exercise 12.3. This can be done quickly. Have they put
the correct probabilities on the branches? Have they
multiplied these probabilities and done so correctly?
Have they used these answers to answer supplementary
questions?
PL
E
Question 11 does not specify the particular diagram
required. Learners can choose, but it is expected
that they will use the experience of earlier questions
and realise that a tree diagram is an efficient way
to do this. Setting out the solution in this way makes
it easy to justify. Encourage learners to always set
out their solutions in a way that can be understood
by others.
As Section 12.3 will probably take more than one
lesson, set suitable parts of the Workbook at the end of
each lesson.
12.4 Chance experiments
M
LEARNING PLAN
Learning objectives
Success criteria
9Sp.04
• Design and conduct chance
experiments or simulations,
using small and large
numbers of trials. Calculate
the expected frequency of
occurrences and compare
with observed outcomes.
• Learners can take a sample
from a set of random
numbers and compare
expected frequencies of
different events’ probabilities,
appreciating the relationship
between the two.
SA
Framework codes
LANGUAGE SUPPORT
Relative frequency: if an action is repeated, the
relative frequency of a particular outcome is the
fraction of times when that outcome occurs
In Stage 8 relative frequency was referred to
as experimental probability in the Framework.
Learners will be familiar with the idea of calculating
this fraction from a sample.
Make sure that learners do not confuse ‘frequency’
and ‘relative frequency’ or ‘relative frequency’ and
‘probability’. Sometimes, to avoid confusion, a
probability calculated without an experiment, such
as the score on a dice or the result of a spinner, is
called ‘theoretical probability’.
Support learners with the language used in
questions set in context.
144
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12 PROBABILITY
Common misconceptions
Misconception
How to identify
How to overcome
Confusing relative frequency and
(theoretical) probability.
Ensure that learners use the terms
relative frequency and probability
correctly and point out errors if
they occur.
There are plenty of opportunities
to recognise the difference in the
questions in Exercise 12.4.
Using a spinner (5 minutes)
Resources: None
learners could do this in pairs so that you do not have
too many results to deal with. When they have done it,
take the results from each learner in turn. Put the results
in a table like this:
PL
E
Starter idea
Description: Say that a spinner has four coloured
sections; red, blue, green and yellow.
Say that all the colours are equally likely. Ask ‘What is
the probability of red?’ [0.25]
Sixes in
ten rolls
Cumulative
frequency
Total rolls
Relative
frequency
3
3
10
0.3
1
4
20
0.2
0
4
30
0.133
3
7
40
0.175
Now say you spin the spinner ten times. Ask ‘How many
reds would you expect?’ They will probably say 2 or 3
(2.5 is not possible!). Ask ‘Why?’ They will say that a
quarter of 10 is 2.5.
50
Ask about other frequencies. For example: 1, 0, 5 or 7.
Allow comments. Establish the fact that any number
could happen.
70
80
The numbers shown are just an example. Insert the
numbers from your own class. Explain what relative
frequency means and round the figures to 3 d.p. if
necessary. When you have a set of results, ask for
comments on the values of the relative frequency. They
will probably be going up and down, but the variation
could be decreasing as the number of throws increases.
Ask ‘What is the probability of rolling a 6?’ [ 1 or 0.167
6
to 3d.p.] The value of the relative frequency for a large
number of rolls should be close to this, but there will
always be some variation. Now ask learners to draw a
graph to show the changing relative frequency. Use a
scale of ten rolls to 1 cm on the x-axis and 2 cm to 0.1 on
the y-axis. Join the points with straight line segments.
M
Now say that you spin the spinner 100 times. Ask ‘How
many reds do you expect this time?’. They should agree
that about 25 is reasonable.
60
Say that spinning seven or more reds in ten spins does
not seem to be impossible. It would make you think that
the spinner must be unfair. Then say that 70 or more
reds in 100 spins is the same proportion. Ask ‘Would
you think that the spinner was unfair in this case?’.
SA
You are looking for agreement that 7 or more out of
10 is possible but 70 or more out of 100, which is the
same proportion, is extremely unlikely and suggests
something is biased with the spinner or the person
doing the spinning. That’s all that is required from
this discussion.
Main teaching idea
Relative frequency and dice (15 minutes)
Learning intention: To develop the idea of relative
frequency and link it to sample size and probability.
Resources: A dice for each learner, graph paper
Description: Ask each learner to roll a dice ten times and
to record the frequency of 6. If you have a large class,
1
Draw a line (y = 6 ) for the probability too. Tell learners
to read the scale carefully. Ask them to peer-assess each
other’s graphs. Ask for comments on the shape. These
will be similar to the comments on the values of the
relative frequency. The line showing the changing relative
1
6
frequency will be close to the line y = after a large
number of throws. A lot of variation is possible at first.
145
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Differentiation ideas: Some learners might find it
difficult to draw the graph accurately. If you have
suitable graphical software you could use that to draw
the graph. Spend as much time as required to show
how the relative frequency is calculated and to ensure
that learners can do this. The starter activity idea in
the Additional teaching ideas gives more practice with
writing fractions as decimals, if needed.
Plenary idea
Convincing
Exercise 12.4, Question 8
Question 8 asks learners to plan their own experiments.
They need to decide what to test and what sample size to
take. They are expected to draw conclusions from their
experiment, based on the values calculated.
Homework ideas
PL
E
Check your progress (10 minutes)
Guidance on selected Thinking and
working mathematically questions
Resources: ‘Check your progress’ exercise at the end of
the unit
Description: Ask the learners to answer the ‘Check you
progress’ questions. When they have completed the
questions, go through the answers to each question.
Discuss how they found each answer and what point
the question illustrates. (e.g. this question is about
independent events).
As Section 12.4 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
Assessment ideas
In this unit the emphasis is on peer assessment and
self-assessment. The questions specifically ask for this.
Ensure that learners are doing this as they work through
the exercise and listen for appropriate discussion. Look
for opportunities to join the discussion and support
where necessary.
SA
M
Assessment ideas: This is an opportunity for selfassessment. It gives learners an opportunity to see if
they are confident about answering typical questions.
They can also identify any topic that they still feel
unsure about. Give an opportunity for learners to
ask any questions they might have as you go through
the answers.
Workbook, Section 12.4.
146
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13 POSITION AND TRANSFORMATION
Unit plan
PL
E
13 Position and
transformation
Topic
Approximate
number of
learning hours
Outline of learning
content
Resources
13.1 Bearings and
scale drawings
1
Use bearings and scaling
to interpret position on
maps and plans.
Learner’s Book Section 13.1
Workbook Section 13.1
Resource sheet 13.1
Additional teaching ideas Section 13.1
13.2 Points on a
line segment
1–1.5
13.3
Transformations
1–2
Use coordinates to find
Learner’s Book Section 13.2
points on a line segment. Workbook Section 13.2
Resource sheet 13.2
Additional teaching ideas Section 13.2
M
Transform points and 2D Learner’s Book Section 13.3
shapes by combinations
Workbook Section 13.3
of reflections, translations
Resource sheet 13.3A
and rotations.
Resource sheet 13.3B
Resource sheet 13.3C
Resource sheet 13.3D
Additional teaching ideas Section 13.3
1
Enlarge 2D shapes,
Learner’s Book Section 13.4
determine the scale
Workbook Section 13.4
factor and centre of
Resource sheet 13.4A
enlargement. Understand
Resource sheet 13.4B
changes in perimeter
Additional teaching ideas Section 13.4
and area of squares
and rectangles when
enlarged.
SA
13.4 Enlarging
shapes
Cross-unit resources
Language worksheet: 13.1–13.4
End of unit 13 test
Project resource sheet: Triangle transformations
147
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
BACKGROUND KNOWLEDGE
• Understand that the centre of rotation, direction
of rotation and angle are needed to identify and
perform rotations (Stage 8).
• Enlarge 2D shapes, from a centre of
enlargement (outside or on the shape) with
a positive integer scale factor. Identify an
enlargement and scale factor (Stage 8).
In this unit, learners will extend their previous
work on bearings and using maps and scales,
and on finding the coordinates of points on a line
segment.
They will also deepen their knowledge of
reflections, rotations and translations from Stage 8,
especially with combining transformations.
Learners will extend their previous work on
enlargements by considering a centre of
enlargement inside the original shape. They will
also look at the effect of an enlargement on the
perimeter and area of a square or rectangle.
PL
E
For this unit, learners will need this background
knowledge:
• Use knowledge of scaling to interpret maps and
plans (Stage 7).
• Understand and use bearings as a measure of
direction (Stage 8).
• Use knowledge of coordinates to find the
midpoint of a line segment (Stage 8).
• Translate points and 2D shapes using vectors,
recognising that the image is congruent to the
object after a translation (Stage 8).
• Reflect 2D shapes and points in a given mirror
line on or parallel to the x- or y-axis, or y = ± x
on coordinate grids. Identify a reflection and its
mirror line (Stage 8).
• Rotate shapes 90 ° and 180 ° around a centre
of rotation, recognising that the image is
congruent to the object after a rotation
(Stage 7).
M
TEACHING SKILLS FOCUS
SA
Assessment for learning
A key aspect for assessment for learning is assessing
prior knowledge. While the Getting started
questions will help find weaknesses, much of this
unit is built on previously learned skills. As such, if
any of those skills are weak or missing it is important
to revisit that area of the Stage 8 work.
You might need to adapt or stop the planned lesson
if the required previous knowledge is missing. If
only part of the class lacks a skill, then this is a great
opportunity for you to get learners to help teach.
At the end of Unit 13, ask yourself:
• Do you know what learners know/knew about
this topic?
• Have you asked questions to look for evidence
of learning, of a depth of understanding of the
topic that shows learners understand how the
maths works, not just that they can get an answer
to a question?
• Are learners confident that if they can suggest
half-formed ideas about a problem, then they
can share it and receive guidance from yourself
or another learner?
• Do you tell learners that learning from their
mistakes is an excellent and invaluable process
that is encouraged within the classroom?
148
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13 POSITION AND TRANSFORMATION
13.1 Bearings and scale drawings
LEARNING PLAN
Learning objectives
Success criteria
9Gp.01
• Use knowledge of bearings
and scaling to interpret
position on maps and plans.
• Learners can use bearings
and scales.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary for this section.
Remind learners of the ways of referring to a scale,
for example, ‘1 cm represents 100 cm’, ‘1 to 100’ or
‘1 : 100’, and make sure that they are comfortable
with using all three ways. Encourage them to say
scales in the different ways during discussions.
Worked example 13.1 and questions in
Exercise 13.1 involve a lot of words. Support
learners with the language as needed and make
sure that they understand what each question is
asking before they attempt to answer it.
Common misconceptions
Misconception
How to overcome
Worked example discussions
and almost all questions in
Exercise 13.1.
Remind the class repeatedly that a
bearing is always measured from north
and always measured clockwise.
Question 4.
Checking will confirm understanding:
2 000 000 cm = 20 km.
M
Measuring the acute angle when
inappropriate.
How to identify
Incorrectly converting large
numbers (from cm to km).
SA
Starter idea
Getting started (10 minutes)
Resources: Note books, Learner’s Book Getting started
exercise, Resource sheet 13.1
Description: Learners should have little difficulty with
most of the Getting started questions. Before learners
attempt the questions, discuss what they remember
about scale drawing. Perhaps a simple example would
be useful, such as, ‘A scale drawing of a bus has a scale
of 1 : 20. What does ‘1 : 20’ mean?’ Then ask ‘The scale
drawing of the bus measures 30 cm. How long is the
bus?’ [600 cm or 6 m] and ‘Using the same scale, how
wide is a 1.6 m window on the scale drawing?’ [8 cm]
This exercise is a quick reminder of previous work that
will help learners be more effective with this unit. It is
not a test. After, or occasionally during, each question it
might be useful to allow self- or peer-marking, allowing
learners to rectify any mistakes after a brief discussion.
Main teaching idea
Question 3, Think like a mathematician
(5 minutes)
Learning intention: To solve a problem using a sketch.
Resources: Note books, Learner’s Books
Description: Learners do not need to draw an accurate
diagram to answer this question. A sketch will show
that the yacht and the speedboat could meet because the
bearing lines cross. This means that the boats could meet
at this point.
Differentiation ideas: To extend, ask learners to draw
a scale drawing and to measure the distances from the
intersection to where the yacht and speed boat start, to
find out how far they travel before their paths meet. To
extend further, discuss the speed of the two boats.
149
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Plenary idea
Homework ideas
There and back (3–5 minutes)
Workbook, Section 13.1.
Resources: Mini white boards
As Section 13.1 might take more than one lesson,
set suitable parts of the Workbook at the end of
each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Description: Tell learners that a harbour is 9 km on a
bearing of 070 ° from a lighthouse.
Guidance on selected Thinking and
working mathematically questions
Critiquing and convincing
Exercise 13.1, Question 5
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
PL
E
• Ask learners to draw a scale drawing of the harbour
and lighthouse using a scale of 1 : 150 000.
• Ask learners for the bearing of the lighthouse from
the harbour.
Assessment ideas: You could use peer-marking.
Learners should measure the distance between the
harbour and lighthouse [6 cm] and check/measure the
bearing given [250 °]. Discuss any errors [a 2 mm and
a 2 ° error are acceptable] and how to avoid them.
Use Question 7 as an extended hinge-point question.
All learners must be able to answer this question with
confidence and a fair degree of accuracy before continuing
with the exercise. Ask learners to compare diagrams to
try to spot inaccuracies. Obviously, an inaccurate answer
will suggest there are inaccuracies or misunderstandings.
Learners need to know what those problems are, and
discuss/practise them, before moving on.
M
Teshi has made a careless error. This type of error is
easily made. When learners have realised the mistake,
they must draw an accurate diagram for part b.
Recognising common errors and explaining them will
help learners to avoid these types of errors in their
future work.
Assessment ideas
13.2 Points on a line segment
SA
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Gp.02
• Use knowledge of
coordinates to find points on
a line segment.
• Learners can use coordinates
to find points on a
line segment.
Using Resource sheet 13.2 means that learners will not
have to copy diagrams in Question 1. There are also
pre-drawn axis for questions 10 and 11. This will save
a lot of time in the classroom and also eliminate a
potential source of error.
150
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13 POSITION AND TRANSFORMATION
LANGUAGE SUPPORT
There is no new vocabulary for this section.
Encourage learners to read aloud phrases such as ‘one third of the way along’, ‘three quarters of the way
along’, etc.
Common misconceptions
How to identify
How to overcome
Assuming that one end of the line
segment is at the origin (0, 0).
Question 9.
Detailed discussions with Question 8.
PL
E
Misconception
Starter idea
Main teaching idea
Midpoints on a grid (2–5 minutes)
Resources: Mini white boards or note books
Description: On the board, draw/display this grid and
line segments.
C
I
Learning intention: To understand a method to work
out points that lie on a line segment when neither end is
at (0, 0).
Resources: Note books, Learner’s Books
J
B
Description: All learners should notice that the coordinates
(1, 2) are incorrect as they do not lie on the line segment
AB. Hopefully, the discussions that took place during
Question 3 will help here. It might be useful for all learners
to complete parts a, b and c and then to discuss as a class
the answers and why the method works.
M
D
y
6
5
4
A
3
2
1
Question 8, Think like a mathematician
(5 minutes)
K
1 2 3 4 5 6 x
SA
–6 –5 –4 –3 –2 –1 0
–1
E
F
–2
P
Q
–3
–4
–5
–6
M
G
H
L
N
Ask learners to write the midpoint of the line segment
CD [(−5, 4)]. Discuss how learners got their answer.
Once all learners seem confident with the method
to work out this midpoint, practise more of these
straightforward questions: EF, AB and GH. Discuss
answers, then move on to the midpoints with fractional
coordinates: IJ, PQ, KL and MN.
Differentiation ideas: The main aim for this question is
to help learners to understand how to work out a point
on a line segment when neither end is at (0, 0). You
might need to give some learners guidance for part b,
which can be checked by seeing their answer to part c.
Part d is useful revision, but not essential for this
section. Give as much help as needed to any learner that
requires it for part d i, and then the pair should be able
to answer part d ii.
Plenary idea
O to A to B to … (2–5 minutes)
Resources: Mini white boards or note books
Description: On the board, write/display these questions:
The origin is a word used for the point (0, 0).
The origin O, point A, and point B are equally spaced
along the same line such that the distance OA is equal to
the distance AB.
A is the point (5, 1).
151
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
a
What are the coordinates of point B?
b
C is the next point along the same line so that the
distance BC is equal to distances OA and AB. What
are the coordinates of the point C?
There are several valid methods for answering part c.
Again it might be helpful to allow learners to answer it,
then to discuss as a class.
c
The points continue along the line so that the next
point is labelled with the next letter of the alphabet.
What are the coordinates of point H?
When parts d, e and f have been attempted, allow selfmarking for parts d and e, then discuss answers for part
f. Make sure that all learners understand by getting
several successful learners to explain how they worked
out their answers.
d
What are the coordinates of the point that lies 1 of
4
the way along OH?
Homework ideas
Workbook, Section 13.2.
b (15, 3)
d (10, 2)
PL
E
Answers:
a (10, 2)
c (40, 8)
Assessment ideas: Peer-assessment is useful here.
Learners need to concentrate on the coordinate answers
given, but also should notice any methods they don’t
understand. Discuss any misunderstood methods with
the learner when answers are returned.
Guidance on selected Thinking and
working mathematically questions
Specialising, generalising and convincing
Exercise 13.2, Question 5
Assessment ideas
When Question 5 has been successfully completed, ask
learners to write their own question similar to Question
5 parts a, b, c and d. When written, learners should
write the answers on a separate sheet of paper. Learners
should then swap questions with a partner, answer the
questions, swap back and mark. You could discuss
any incorrect answers briefly as a class and correct
answers found.
M
This is the first question where learners need to project a
line rather than work out a fraction of a line.
As Section 13.2 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
SA
Part a has a tip box suggesting that learners could draw
a diagram to help. Encourage less confident learners to
draw an accurate diagram, with x-axis from 0 to 8 and
y-axis from 0 to 12. For more confident learners who
want a diagram, encourage them to make a quick sketch
of the situation.
Most classes benefit from answering parts a and b,
then discussing answers and methods. The key
aspect for many learners is to visualise that the
line continues, with letters equally spaced along it.
Learners need to understand that if they know the
distance between two consecutive points (e.g. O and A
are two horizontal units and three vertical units apart)
they can work out the next consecutive point. Also,
they should understand that if two points are, for
example, ten places apart, they can find this distance
by multiplying the distances of two consecutive
points by 10.
Next, ask learners to work out the midpoint between
their point at (0, 0) and their point in part d of their
question. This might be a good time to tell/remind
learners that the point (0, 0) is often called the origin,
which is why many of the questions call (0, 0) point O.
On the board, write five or six ‘(0, 0) – ( , )’. In turn,
ask five or six learners for the coordinate in their part d
(write this coordinate on the board in the appropriate
space) and ask them for the midpoint. Ask other
learners if they agree. You can choose a learner to write
working on the board.
If you think this will be useful, ask learners to work out
1
1
the point that lies and/or of the way between each
5
4
of the five or six pairs of coordinates now on the board.
Try to ensure that the answers are integers for the less
confident classes. For more confident classes you can
also use integer or fractional coordinates.
152
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13 POSITION AND TRANSFORMATION
13.3 Transformations
LEARNING PLAN
Learning objectives
Success criteria
9Gp.03
• Transform points and 2D
shapes by combinations
of reflections, translations
and rotations.
• Learners can identify
transformations. They can
transform shapes by a
combination of reflections,
translations and rotations.
PL
E
Framework codes
9Gp.04
9Gp.05
• Identify and describe a
transformation (reflections,
translations, rotations and
combinations of these) given
an object and its image.
• Learners understand what
is needed to give a precise
description of a reflection,
translation or rotation.
• Recognise and explain
that after any combination
of reflections, translations
and rotations the image is
congruent to the object.
• Learners understand that
regardless of the number of
reflections and/or translations
and/or rotations the image
is always congruent to
the object.
M
Using Resource sheet 13.3A means that learners will not have to copy diagrams in questions 2, 3 and 4. This will save a
lot of time in the classroom and also eliminate a potential source of error.
LANGUAGE SUPPORT
SA
There is no new vocabulary for this section.
Encourage learners to say aloud the
transformations. This will allow you to check that
they are including all of the necessary information.
It will also help learners to remember which
information they need to give for a precise
description of each type of transformation.
Make sure that learners read carefully each
question, and that they notice, for example, if a
rotation is clockwise or anticlockwise.
Common misconceptions
Misconception
How to identify
How to overcome
Making errors when not using tracing paper,
especially when working with more complicated
shapes for both reflection and rotation.
Questions 2, 3, 4.
Carefully checking for accuracy and
giving tracing paper to learners.
Repeatedly transforming the original shape,
rather than transforming the new shape, when
consecutive transformations are asked for.
Questions 2, 3, 4.
Worked example 13.3 part a.
Discussions following completion
of Question 2.
153
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Starter idea
Main teaching idea
Transformations (5–10 minutes)
Question 4, Think like a mathematician
(15 minutes)
Resources: Mini white boards or note books, Resource
sheet 13.3B
Learning intention: To understand that using the same
set of transformations, but in a different order, can
result in the final images being in different locations.
Description: On the board, draw/display this grid:
0
–10–9 –8 –7 –6 –5 – 4 –3 –2 –1
–1
Description: This question is much faster when learners
have a copy of Resource sheet 13.3A.
Tell learners to draw each transformation in turn, and
not to try to only draw the final position of the shape.
1 2 3 4 5 6 7 8 9 10
x
In part d, learners should notice that the pairs of
instructions are the same, but in part ii the order is
reversed. It is obvious that the two shapes drawn for
each of parts a, b and c are in different locations on
the grid meaning that the order is important. In the
discussions that follow, discuss why the order is so
important for some pairs of transformations.
Differentiation ideas: When part c has been completed,
it might be useful to ask learners in small groups to
compare their drawings. Their drawings should be
identical. Learners should discuss any differences. You
can join the discussion if they cannot decide on the
correct diagram.
Many learners will find it difficult to answer part d iv
without using two translations. This question might
be simplified by replacing shape Z with a rectangle
sharing three of Z’s vertices. When this has been done
and discussed – i.e. showing that two reflections or two
translations give the same outcome regardless of order –
Z could then be used to check understanding.
M
–2
–3
–4
–5
–6
–7
–8
–9
–10
Resources: Note books, Learner’s Books
PL
E
y
10
9
8
7
6
5
4
3
2
1
SA
Draw a 2-by-2 right-angled triangle at various points
on the grid. Ask learners to discuss the triangle’s
transformation when transformed by reflection, rotation
or translation. You could ask an individual learner to
draw the transformation on the board, or you could ask
all learners to write the coordinates of the vertices of
the triangle after the transformation. Give instructions
such as:
• reflect the triangle in the x-axis, the y-axis, the line
x = −3, the line y = 5, etc.
• rotate the triangle 90 °/180 ° clockwise/anticlockwise
about the point (0, 0)/(2, 3)/(−4, 3), etc. Ensure that,
initially at least, the centre of rotation is close to, or
on, an edge of the triangle.
• translate the triangle two squares left and three
squares up, translate the triangle with the column
5
.
 −1
vector 
This can be adapted to individual/group work by giving
learners copies of Resource sheet 13.3B.
Plenary idea
A to B (5–10 minutes)
Resources: Mini white boards
Description: On the board, draw/display this grid and
question:
y
6
5
4
3
2
1
B
0
1 2 3 4 5 6x
A
154
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13 POSITION AND TRANSFORMATION
Describe a combined transformation that transforms
shape A to shape B.
When checked and returned, ask ‘Will your combination
of transformations still work if the rectangles are
replaced by triangles?’ On the board, draw/display
this grid:
y
6
5
4
3
2
1
0
B
A
1 2 3 4 5 6x
Specialising, generalising, characterising
and convincing
Exercise 13.3, Question 9
Ask learners to read the entire question. Ask them what
they will do first [draw the grid and shapes].
You might decide to only set part a before discussing
learners’ answers and dealing with any issues that arise.
PL
E
Only learners who find this very difficult should be
allowed to copy the diagram. Most learners should
attempt this question by only looking at the diagram on
the board. When complete, use peer-marking to check.
There are many correct combinations. Most correct
combinations will involve a rotation and a translation.
Some combinations may start with a reflection in the
line y = x. You might need to assist some learners in
their checking.
Guidance on selected Thinking and
working mathematically questions
Discuss parts b and c as one question. Learners will
probably suggest several different correct methods
and this will lead naturally onto the assumption that
there are many different combined transformations
that take G to H. Explanations will differ, but by the
end of the discussion, learners should understand that
as long as one of the transformations is a reflection in
any line ‘x = a number’, then there is a translation that
will translate G to H. The realisation that any ‘x = a
number’ will work [e.g. x = 1 000 000, x = −1 000 000 or
even x = 1.125] means that there is an infinite number of
combinations that will transform G to H.
Homework ideas
Workbook, Section 13.3.
As Section 13.3 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
If you prefer learners to work on paper, for your
records, then give out copies of Resource sheet 13.3D.
Assessment ideas
SA
M
When checked, ask learners to describe a combined
transformation that transforms shape A to shape B.
Again, there are numerous correct answers. Most correct
combinations will probably start with a 90 ° clockwise
rotation about some point.
Assessment ideas: Peer-marking/checking.
Use Question 11 as a class ‘test’. If learners can answer
these questions, they obviously understand how to deal
effectively with a combination of transformations.
There are only four pairs of transformations, but the
nature of the question will mean that learners will
usually do many trials before working out each correct
pair of triangles.
155
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
13.4 Enlarging shapes
LEARNING PLAN
Learning objectives
Success criteria
9Gp.06
• Enlarge 2D shapes, from
a centre of enlargement
(outside, on or inside the
shape) with a positive
integer scale factor. Identify
an enlargement, centre of
enlargement and scale factor.
• Learners can enlarge
2D shapes from a centre
of enlargement.
• Analyse and describe changes
in perimeter and area of
squares and rectangles when
side lengths are enlarged by a
positive integer scale factor.
• Learners can identify the
centre of enlargement
and scale factor of an
enlargement and can
understand the changes
in the resulting perimeter
and area of squares
and rectangles.
PL
E
Framework codes
9Gp.07
M
Using Resource sheet 13.4A means that learners will not have to copy diagrams in Questions 1, 2,
3 and 4, and there are pre-drawn axes for questions 10 and 12. This will save a lot of time in the
classroom and also eliminate a potential source of error.
LANGUAGE SUPPORT
SA
Ray lines: lines that start at a fixed point and
continue forever
Where descriptions are required, encourage
learners to read aloud their descriptions. Ask other
learners to decide if all relevant points have been
made and if the language is clear. If learners use
different language for the same description, ask
learners to decide which description is clearer.
You might be required to guide on the most
appropriate mathematical language to use.
Common misconceptions
Misconception
How to identify
Not choosing corresponding points or
not being accurate enough in drawing
the lines when finding a centre of
enlargement by drawing lines through
corresponding points.
Questions 9, 10, 12 and 13. Discussion during part b of Worked
example 13.4 and careful checking
of answers for questions 9, 10, 12
and 13.
Giving the scale factor but not the
Questions 9 and 10.
coordinates of the centre of enlargement
when asked to describe an enlargement.
How to overcome
Discussion during part b of Worked
example 13.4 and careful checking,
and possibly discussion, when
marking Question 9.
156
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13 POSITION AND TRANSFORMATION
Starter idea
Plenary idea
Scale factor (2–5 minutes)
Exit ticket (3–5 minutes)
Resources: Mini white boards or note books
Resources: Resource sheet 13.4B: Exit ticket
Description: Use this starter activity idea before working
through the introduction or Worked example 13.4.
Description: Give each learner an exit ticket, cut out
from Resource sheet 13.4B. Learners should complete
the exit ticket just before leaving class. Allow 3–5
minutes to complete the sheet.
On the board, display this diagram:
0
Ask learners to write their names on the ticket. Learners
should complete the exit ticket before giving it to you at
the end of the lesson.
B
A
PL
E
y
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 x
Ask learners to write the scale factor of the
enlargement [2].
Although locating the centre of enlargement is covered
in part b of Worked example 13.4, discuss how learners
think they could find the centre of enlargement for the
diagram. Getting the correct answer is not important.
Thinking how to get the correct answer is the important
part of this exercise.
The exit tickets can be returned unmarked for self- or
peer-marking and discussion or marked by you as a
more formal record of individual success.
Answers: a scale factor 2 enlargement, centre (−2, −4)
b (−3, −1), (−3, 3) and (5, −1)
Guidance on selected Thinking and
working mathematically questions
M
Main teaching idea
Assessment ideas: Reading through the learners’
comments will help you determine the effectiveness
of the lesson. If you have regularly used exit tickets,
learners will be used to saying what would help them to
achieve at a higher level.
Question 5, Think like a mathematician
(10 minutes)
Specialising, critiquing and convincing
Learning intention: To understand ratios of perimeters
and areas when a square is enlarged.
This question highlights a surprisingly common mistake.
Learners must realise that all of the enlargements they
have drawn using ray lines drawn from the centre of
enlargement through the image vertices are similar to
those where the centre of enlargement is within the
image. The centre of enlargement is not an indicator of
location of one of the vertices of the enlargement.
SA
Resources: Note books, Learner’s Books
Description: Check and correct Question 4 answers
before starting Question 5. Learners must use their
diagrams from Question 4 to work out the perimeter
and area of their rectangles labelled B, C and D.
It might be useful to ask learners to answer parts a and
b and then to check learner’s answers. Any errors or
misunderstandings should be dealt with through class
discussion before learners attempt to write their rules.
Differentiation ideas: You might need to remind some
learners that the perimeter is the distance around the
shape and that the area of a square is base multiplied by
height. Many less confident learners will not, despite the
tip box, realise that the rule linking the ratio of lengths
to the ratio of areas involves squared numbers. You
might need to give some guidance, either when learners
are answering part d or when other learners point it out
during the discussions during part f.
Exercise 13.4, Question 3
Homework ideas
Workbook, Section 13.4.
As Section 13.4 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
You could ask learners to make a worked example list
containing everything they think they need to remember
for the end-of-unit test. The following lesson, it is
important to share the worked example lists in class (e.g.
spread out over a few desks for everyone to look at),
157
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
rather than marking them. Discuss the different worked
examples as a class. When the class agree that a point is
important, copy that key point onto the board. Agree
on as many important key points as possible. Learners
could then improve/update their individual lists if
necessary. Learners could store their worked example
lists at home as a possible revision tool towards midterm/end-of-year tests.
Assessment ideas
Ask learners what was their first step. They should reply
that they made a drawing of a grid and the two shapes:
0
K
L
K
1 2 3 4 5 6 7 8 9 10 11 x
Discuss that it should now be easy to see where the
rays cross and to write the coordinates of the centre of
enlargement. Ask if anyone’s drawing actually had all
four rays crossing exactly at (6,5) – congratulate all those
that did!
Ask a learner how they decided on the scale factor of
enlargement [scale factor 3] and briefly discuss their
method and its accuracy.
1 2 3 4 5 6 7 8 9 10 11 x
SA
0
L
y
11
10
9
8
7
6
5
4
3
2
1
M
y
11
10
9
8
7
6
5
4
3
2
1
Now ask what their next step was. They should reply
that they drew rays (lines) from each object’s vertices
through the corresponding image vertices. Ask here who
needed to make at least one of their rays longer as it
needed to be longer to see where the rays all crossed.
PL
E
As Question 12 is the first question with the object
inside the image, it is often useful to work through a
possible full solution to the question once completed by
learners. This will allow for peer- and self-marking, and
possibly useful discussion.
Discuss any different approaches and errors (and their
consequences) in plotting the shapes.
158
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PROJECT GUIDANCE: TRIANGLE TRANSFORMATIONS
PROJECT GUIDANCE: TRIANGLE TRANSFORMATIONS
Why do this problem?
This problem invites learners to explore the
effects of combining multiple transformations. By
classifying the effects of different transformations,
learners can begin to understand what happens
when you combine multiple transformations, and
to make conjectures about which combinations of
transformation might be equivalent.
Invite learners to suggest ways of getting from one
triangle to the other triangles using fewer or more
transformations. This could lead to discussions of
how some of the transformations are inverses of
each other, and some transformations are selfinverse.
Possible approach
Invite learners to look at the blue, red and pink
triangles. What can they say about how you might
get from one triangle to another triangle? Allow
some time for discussion. Learners might notice
that the red triangle is three times as big as the
blue triangle, and that it is in a different orientation,
and that the pink triangle and the blue triangle are
mirror images of each other.
Key questions
What clues are there about which transformations
might have been used?
PL
E
Finally, learners could create their own chains of
transformations to challenge each other.
If you know how to get from one triangle to
another, how can you work out how to get back?
Possible support
Learners might find it useful to draw the triangles
on tracing paper to help them to perform the
transformations.
You could also relax the restriction of using exactly
three transformations.
Possible extension
Invite learners to explore how the order in which
transformations are performed affects the final
outcome. Which combinations of transformations
can be done in any order?
M
Then share the fifteen transformation cards from
the Project resource sheet: Triangle transformations
so that learners can cut them out and choose
different combinations of the transformations.
Challenge learners to find sets of three
transformations which map each triangle onto the
other triangles.
How does the triangle change if you reflect it? Or
rotate it? Or enlarge it?
SA
Bring the class together and share what learners
have found. Take time to discuss that there are
different ways to achieve the same overall result.
159
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Unit plan
PL
E
14 Volume, surface
area and symmetry
Topic
Approximate
number of
learning hours
Outline of learning content
Resources
14.1 Calculating
the volume of
prisms
1–1.5
Derive and use the formula
for the volume of prisms
and cylinders.
Learner’s Book Section 14.1
Workbook Section 14.1
Resource sheet 14.1
Additional teaching ideas Section 14.1
Calculate the surface
area of cubes, cuboids,
triangular prisms, pyramids
and cylinders.
Learner’s Book Section 14.2
Workbook Section 14.2
Additional teaching ideas Section 14.2
14.2 Calculating
1–1.5
the surface area of
triangular prisms,
pyramids and
cylinders
1
Identify reflective symmetry Learner’s Book Section 14.3
in 3D shapes.
Workbook Section 14.3
Resource sheet 14.3A
Resource sheet 14.3B
Additional teaching ideas Section 14.3
M
14.3 Symmetry in
three-dimensional
shapes
SA
Cross-unit resources
Language worksheet: 14.1–14.3
End of unit 14 test
BACKGROUND KNOWLEDGE
For this unit, learners will need this background
knowledge:
• Derive the formula for the volume of a triangular
prism. Use the formula to calculate the volume of
triangular prisms (Stage 8).
• Calculate the surface area of cubes, cuboids,
triangular prisms and pyramids (Stage 8).
• Identify reflective symmetry and order of
rotational symmetry of 2D shapes and patterns
(Stage 7).
• Understand that the number of sides of a
regular polygon is equal to the number of lines
of symmetry and the order of rotation (Stage 8).
The focus of this unit is to extend learners’
understanding and use of volume and surface area
to more complicated prisms, especially cylinders.
Learners will also extend their knowledge of lines
of symmetry to include planes of symmetry in three
dimensions.
160
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14 VOLUME, SURFACE AREA AND SYMMETRY
TEACHING SKILLS FOCUS
This process teaches learners to understand how to
solve problems effectively, not just how to get the
answer to a particular question.
At the end of Unit 14, ask yourself:
• Are learners able to explain what they are
thinking? If the answer is ‘No, not really’, is
that just because they are not used to giving
explanations and so need much more practice?
• Are learners getting better at explaining their
reasoning?
• Are learners getting better at explaining what
mistakes have been made and what to do next in
a problem?
• With the more complicated problems, can learners
tell you what they will do, i.e. make a plan?
PL
E
Metacognition
This is a complicated area of learning that can be
simplified to ‘thinking about thinking’.
Throughout this unit, ask learners, whenever
possible, to say out loud what they are thinking.
Try to ask questions at the start or a short way
through answering a problem.
If a question has already been answered, ask what
learners were thinking while they were attempting
a problem and if they would now do the problem a
different way.
If done regularly, this questioning leads to a process
that can be used throughout learners’ schooling:
‘think about a problem, plan what to do, do the
plan, look back and decide if you could have done
anything better’.
14.1 Calculating the volume of prisms
M
LEARNING PLAN
Learning objectives
Success criteria
9Gg.04
• Use knowledge of area and
volume to derive the formula
for the volume of prisms and
cylinders. Use the formula
to calculate the volume of
prisms and cylinders.
• Learners can derive and use
the formulae for the volume
of prisms and cylinders.
SA
Framework codes
LANGUAGE SUPPORT
There is no new vocabulary for this section.
If necessary, remind learners of the vocabulary
of 2D shapes to describe the cross-section
of 3D prisms. Also, make sure that learners
understand words such as ‘triangular’ to mean ‘in
the shape of a triangle’, ‘circular’ to mean ‘in the
shape of a circle’, ‘rectangular’ to mean ‘in the
shape of a rectangle’, etc.
When answering questions about cylinders, make
sure that learners read the question carefully, and
in particular that they notice if the question and/
or the diagram are referring to the radius or the
diameter of the cross-section. Similarly, make sure
that learners notice which units are being used, for
example, cm, mm, m, and that they check that all
the units are the same before doing any calculations.
Encourage learners to read aloud their method.
This will allow you to check their work, and it will
also help learners to remember how to calculate
correctly the volume of a prism.
161
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Common misconceptions
How to identify
How to overcome
Not understanding that a cylinder
is a prism.
Question 4.
Make sure that learners understand that a
prism is a shape that has the same crosssection all the way through, perpendicular
to its length, so a cylinder is also a prism.
Forgetting the formula for the
volume of a prism, and forgetting
that they know the formula for the
area of the cross-section.
Questions 3, 4 part b, 8.
Throughout the lesson, emphasise that
the formula for the volume of any prism is
the area of its cross-section (or end area)
multiplied by its length (or depth or height,
depending upon orientation).
Starter idea
PL
E
Misconception
Getting started (10 minutes)
Resources: Note books, Learner’s Book
Getting started exercise
Description: Learners should have little difficulty with
most of the Getting started questions. Before learners
attempt the questions, discuss what they remember
about the formulae for the circumference and area
of circles.
Description: You could ask learners to answer part
a with their partners, then discuss answers with the
class. Welcome any use of ‘a prism is a 3D shape that
has the same cross-section along its length’ from the
introduction as part of learners’ justifications. Some
learners will benefit from imagining that the cylinder is
on its side and is being sliced like a loaf of bread; every
slice will look the same – circular.
It is usually useful for learners to set out working for
their answer to part c, e.g.:
Volume = area of cross-section × length
M
You might need to give some learners a prompt with
Question 4, reminding them that the formula for
the volume of the prism is the end area of the prism
multiplied by the length of the prism. You could also
give learners a brief reminder that a net is similar to
cutting open a box and laying the faces flat and drawing
what you see – it’s a good way to work out the total
surface area, as you can see and work out the area of
each individual face.
Resources: Note books, Learner’s Books
SA
Remember that this is not a test. This exercise is
designed to prepare learners for Unit 14. It is good
practice to allow learners to attempt the questions as
individuals, but discuss answers/problems in pairs/small
groups when required.
Main teaching idea
Question 4, Think like a mathematician
(3–5 minutes)
Learning intention: To understand a cylinder is a
prism and to work out the formula for the volume of
a cylinder.
= πr2 × h
= πr2h
Differentiation ideas: You might need to remind learners
that the volume of any prism is the end area multiplied
by the length (or height in this case). You might need to
give some learners a prompt to recall the area of a circle.
Plenary idea
Odd one out (3–5 minutes)
Resources: Note books
Description: Ask learners to use volume to work out
which of the following prisms is the ‘odd one out’ (i.e.
to find the shape which has a different volume from the
other shapes).
On the board, draw/display these shapes:
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14 VOLUME, SURFACE AREA AND SYMMETRY
Homework ideas
5 cm
10 cm
Workbook, Section 14.1.
5.84 cm
10.88 cm
12.36 cm
4.91 cm
14.33 cm
9.95 cm
14.73 cm
9.95 cm
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
PL
E
7.44 cm
As Section 14.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
10.75 cm
Answer: The ‘odd one out’ is the trapezoidal prism, with
a volume of 775 cm3. The other three shapes have a
volume of 785 cm3 (when all are rounded to the nearest
whole number).
Assessment ideas
At various times during Section 14.1, ask individual
learners questions such as:
‘What is the formula for the volume of a cylinder?’
[V = πr2h]
‘What is the formula for the volume of a triangular
prism?’ [V = 1 × b × h × d]
Assessment ideas: First ask ‘Who thinks the odd
one out is the cylinder because it has integer lengths?’
Learners who agree have not worked out the volumes.
2
Peer-checking is useful here. Ask learners to check/
compare working with a partner. If both have the
same working and answers then they are probably both
correct.
2
M
Guidance on selected Thinking and
working mathematically questions
‘What is the formula for the volume of a cuboid?’
[V = b × h × d]
‘What is the formula for the volume of a trapezoidal
prism?’ [V = 1 (a + b ) × h × d]
Critiquing and improving
Exercise 14.1, Question 5
SA
On first look, this solution seems good: formula written,
correct substitution, square of 5 worked out first and,
although Sara has not written the full answer, she
has written the correct answer after being rounded.
However, Sara has ignored the units. Unfortunately, this
mistake is seen too often with this type of question.
Learners need to check before calculating that the units
are the same. If, as in this case, the units are not the
same, learners need to convert one of the units to be the
same as the other unit. Learners can choose between
5 mm = 0.5 cm or 2 cm = 20 mm before working out the
correct answer of 1.57 cm3 (3 s.f.) or 1570 mm3 (3 s.f.).
‘What is the formula for the volume of any prism?’
[V = area of cross-section × length]
Ask questions without warning, and only ask two or
three learners questions. Later in the lesson, ask two or
three other learners, etc.
As learners are working, regularly ask individuals
questions, for example:
‘What methods are you using?’
‘What are you thinking when you look at this
question?’
‘What is your plan for solving this question?’
‘Can you think of a quicker/better method for
working out the answer?’, etc.
163
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
14.2 Calculating the surface area of triangular prisms,
pyramids and cylinders
LEARNING PLAN
Learning objectives
Success criteria
9Gg.05
• Use knowledge of area, and
properties of cubes, cuboids,
triangular prisms, pyramids
and cylinders to calculate
their surface area.
• Learners can calculate the
surface area of triangular
prisms, pyramids and
cylinders.
LANGUAGE SUPPORT
PL
E
Framework codes
Encourage learners to read aloud their methods.
This will allow you to check their work, and it will
also help learners to remember how to calculate
correctly the surface area of 3D shapes.
Make sure that learners carefully read questions,
and especially that they notice when they are being
asked to work out a volume and when they are
being asked to work out a surface area.
M
There is no new vocabulary for this section.
Ensure that learners are confident in the names of
2D shapes, to describe the faces of a 3D shape.
As in Section 14.1, make sure that learners
understand words such as ‘triangular’ to mean ‘in
the shape of a triangle’, ‘circular’ to mean ‘in the
shape of a circle’, ‘rectangular’ to mean ‘in the
shape of a rectangle’, etc.
Common misconceptions
How to identify
How to overcome
Not understanding the formula for the
area of the curved surface of a cylinder.
Worked example 14.2.
See Starter idea.
Confusing volume with surface area.
Question 2.
Check answers.
SA
Misconception
Starter idea
Surface area of a cylinder (1–2 minutes)
Resources: Rectangular piece of paper
Description: Show learners a rectangular piece of paper.
Roll the piece of paper into a tube, so that the width of
the paper becomes the circumference of the tube. Ask
learners ‘What is the formula for the circumference of
the tube?’ [2πr or πd] Then slowly unroll the paper to
show again the rectangle. Demonstrate again rolling and
unrolling the paper, to link the circumference of the tube
to the width of the rectangle. The width of the rectangle
is 2πr or πd. Ask ‘What is the area of the rectangle?’
[width × length = 2πrl or πdl] Again, demonstrate rolling
and unrolling the paper so that learners see that the area
of the rectangle is the same as the area of the curved
surface area of the tube.
Main teaching idea
Question 4, Think like a mathematician
(15 minutes)
Learning intention: To use algebra skills with the
formula for the surface area of a cylinder.
Resources: Note books, Learner’s Books
Description: In questions 1 to 3, learners have used
formulae, but separated out, working out areas of circles
and rectangles separately. In part a of Question 4
164
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14 VOLUME, SURFACE AREA AND SYMMETRY
learners are asked to use those formulae to make just
one formula for the whole surface area. Many learners
will have to think hard to make this formula, but will be
successful. Many learners will require some guidance for
part b. Making sure learners start with the correct surface
area formula of SA = 2πr2 + 2πrh will obviously help.
Differentiation ideas: For learners who find it difficult
to start, remind them that they have been using formulae
to work out the surface are of a cylinder already. They
need to put these formulae together. Point out the net of
a cylinder in Question 1 and suggest that this question
should help them make up their formula. When your
less confident learners get to SA = πr2 + πr2 + 2πrh or
SA = 2πr2 + 2πrh, you might decide to miss out part b or
to give direction as to what to do.
Plenary idea
Surface area (5–10 minutes)
Specialising, characterising and convincing
Exercise 14.2, Activity 14.2
A cuboid is obviously the easiest shape to use for this
activity. You could ask all learners to use a cuboid first.
Less confident learners could suggest several different
sized cuboids. You could ask more confident learners
to suggest dimensions for a triangular prism and/or
a cylinder.
For further ideas of shapes to work with, you could
discuss various boxes/containers that learners may have
seen, such as:
M
Resources: Note books
Guidance on selected Thinking and
working mathematically questions
PL
E
You will probably need to give some learners some
guidance with part c. Start by writing h = 2r on the board.
Tell learners to substitute h in their formula with 2r.
Learners usually find working easier if they use the
formula SA = 2πr(r + h) rather than SA = πr2 + πr2 +2πrh or
SA = 2πr2 + 2πrh. If learners are successful with part c, they
will usually be successful with part d too.
Or you could use peer-marking. You could ask learners
to give a mark out of 5 for each set of working – 1 being
unclear and hard to follow working, 5 being very clear
and easily followed working. Also checking the answers
are correct [1 136 cm2, 2 112π cm2 or 351.858… cm2, 3
132 cm2, 4 273.205… cm2].
Description: This plenary is almost thorough enough to
be a class test. Ask learners to work out the surface area
of each shape.
SA
On the board, draw/display these shapes:
2 cm
10 cm
4 cm
2
5 cm
3 cm
10 cm
10 cm
10 cm
4 cm
3
Homework ideas
Workbook, Section 14.2.
10 cm
4 cm
1
When discussing answers, focus on the reasons why a
company might choose one shape instead of another
shape. The shape itself might be seen to be important,
but so is cost. The smaller the surface area the cheaper
the material.
4
Assessment ideas: You could take in learners’ working
and mark it as evidence of learning.
As Section 14.2 will take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Assessment ideas
With so many sets of working, this is an excellent
opportunity for peer-marking. Regularly ask learners
to swap books (in pairs or groups) for checking (and
marking if you are happy with that). This will help
learners to focus on the important aspects of their work.
165
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
This form of marking, with practice, will allow learners
to become more aware of what examiners look for when
marking (i.e. clear and easy-to-follow logical working).
Regularly ask if learners have made a mistake. Also ask
them if they know what they did wrong and how to get
that question correct next time. Acknowledge those who
tell you, with a well done (as this is what you need in the
class, active learners).
help that learner separately, asking learners near them to
explain or having a class discussion on the question skill.
As learners are working, regularly ask individual
learners questions, for example:
‘What methods are you using?’
‘What are you thinking when you look at this
question?’
‘What is your plan for solving this question?’
‘Can you think of a quicker/better method for
working out the answer?’, etc.
PL
E
Also ask if learners have made a mistake and do not
know what they have done wrong or how to correct it.
Depending on the question (and the learner) you could
14.3 Symmetry in three-dimensional shapes
LEARNING PLAN
Framework codes
9Gg.06
Success criteria
• Identify reflective symmetry
in 3D shapes.
• Learners can identify
reflective symmetry in
3D shapes.
M
LANGUAGE SUPPORT
Learning objectives
SA
Isometric paper: paper covered with lines or dots
that form congruent equilateral triangles
Plane: a flat surface
Plane of symmetry: a plane that divides a 3D shape
into two congruent halves that are mirror images of
each other
Make sure that learners realise that a 2D line of
symmetry divides a shape into two congruent
parts. Similarly, a 3D plane of symmetry divides a
solid into two congruent solids.
Make sure that learners consistently use ‘line
of symmetry’ in two dimensions and ‘plane of
symmetry’ in three dimensions.
Common misconceptions
Misconception
How to identify
How to overcome
Thinking that a plane passing through opposite
edges of a cuboid is a plane of symmetry. This
is equivalent to thinking that the diagonal of a
rectangle is a line of symmetry in two dimensions.
Question 2 part b.
See Starter idea.
Having difficulty in drawing 3D objects and their
planes of symmetry.
Questions 1, 2 and 3.
Main teaching idea.
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14 VOLUME, SURFACE AREA AND SYMMETRY
Using Resource sheet 14.3A means that learners will not
have to copy diagrams in questions 1, 2 or 5. This will
save a lot of time in the classroom and also eliminate a
potential source of error.
Description: Drawing planes of symmetry can be a
difficult skill to master. Learners need to practise this.
Use triangle dot or, preferably, isometric grid paper. Use
this suggestion before starting Exercise 14.3.
Resource sheet 14.3B also provides isometric paper
for Question 3.
Ask learners to draw a cube, side length 2 cm, or three
dots and two spaces if the dot/grid paper does not have
a 1 cm spacing.
Starter idea
Most common mistake (2–5 minutes)
PL
E
Resources: None
On the board, draw/display what their diagram should
look like:
Description: Learners often think that a plane passing
through opposite edges of a cuboid is a plane of
symmetry. This is equivalent to thinking that the
diagonal of a rectangle is a line of symmetry in two
dimensions.
On the board, draw/display this diagram:
Discuss any differences between learners’ diagrams and
the diagram on the board. Emphasise that using the
lines on the grid is important.
Now ask learners to draw another cube, again with
a side length of 2 cm. When completed, ask learners
to draw a horizontal plane of symmetry through the
middle of the cube.
M
Not a
plane of
symmetry
SA
Ask learners to explain to a partner why the plane
passing through the cuboid is not a plane of symmetry.
Allow two or three minutes. Ask several learners
to explain their reasons to the class. If none of the
explanations are acceptable, suggest that learners think
about what they could draw to explain why a rectangle
has not got a diagonal line of symmetry.
For example, for
then reflecting.
On the board, draw/display what their diagram should
look like:
to be a line of symmetry,
would mean that
would be a rectangle.
Main teaching idea
Drawing boxes (10–15 minutes)
Learning intention: To be able to draw a cube and a
plane of symmetry on isometric grid paper.
Resources: Resource sheet 14.3B
or
Discuss any differences between learners’ diagrams
and the diagrams on the board. Emphasise that
using the lines on the grid is still important. Many
learners will need to redraw their diagram at this
point. Some learners will take a few attempts to get a
correct diagram.
Now ask learners to draw another cube and ask them to
draw a vertical plane of symmetry through the middle
of the cube.
167
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
On the board, draw/display what their diagram should
look like:
rather than the infinite number of lines of symmetry
in reality.
Homework ideas
Workbook, Section 14.3.
or
or
or
There are six different ways for the diagonal to be drawn
through the cube. To check this diagram, it is best for
learners to compare answers. Put learners into groups
that have drawn the diagonal in the same axis so that
they can compare diagrams.
Differentiation ideas: Some learners will need several
attempts and some guidance to be able to draw the cube
on the grid paper. Practice is the key!
Plenary idea
Two cubes (5 minutes)
You could ask learners to make a poster containing
everything they think they need to remember for the
end-of-unit test. The following lesson, it is important
to share the posters in class (e.g. spread out over a few
desks for everyone to look at), rather than marking
them. Discuss the different posters as a class. When
the class agree that a point is important, that key point
could be copied onto the board (by you or a learner).
Agree on as many key points as possible. Learners
could then improve/update their individual posters if
necessary. Learners could store their posters at home as
a possible revision tool towards mid-term/end-of-year
tests.
PL
E
Finally, ask learners to draw another cube of side
length 2 cm and ask them to draw a diagonal plane of
symmetry through the middle of the cube.
As Section 14.3 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Resources: Isometric grid paper (e.g. Resource
sheet 14.3B)
Use Question 5 as a hinge-point question, especially if
learners are using Resource sheet 14.3A.
M
Description: Ask learners to draw two cubes of side
length 4 cm.
Assessment ideas
When completed, ask learners to draw a different plane
of symmetry on each cube.
SA
Assessment ideas: Allow peer-marking. Learners
should first check that the cubes do have a side length of
4 cm. Next, they should check that the plane is clearly
drawn and passes through the middle of the cube.
Alternatively, learners could hold up their diagrams for
you to move around the class looking at their diagrams.
It is a very quick process to decide if each diagram is
correct or not.
Guidance on selected Thinking and
working mathematically questions
Characterising and convincing
If learners cannot draw on the locations of the three
planes of symmetry, they need additional help.
Note that Resource sheet 14.3A has four copies of the
diagram. Only three copies are required, but learners
might be tempted to draw on an incorrect plane of
symmetry just to draw on each diagram.
As learners are working, regularly ask individual
learners questions, for example:
‘What methods are you using?’
‘What are you thinking when you look at this
question?’
‘What is your plan for solving this question?’
‘Can you think of a quicker/better method for
working out the answer?’, etc.
Exercise 14.3, Question 7
This question checks a common mistake, discussed in
both stages 7 and 8. Less confident learners might still
think that a circle has just one or two lines of symmetry,
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15 INTERPRETING AND DISCUSSING RESULTS
Unit plan
PL
E
15 Interpreting and
discussing results
Topic
Approximate
number of
learning hours
Outline of learning content
Resources
15.1 Interpreting
and drawing
frequency
polygons
1–1.5
Draw and interpret
frequency polygons.
Learner’s Book Section 15.1
Workbook Section 15.1
Additional teaching ideas Section 15.1
15.2 Scatter
graphs
0.5–1
Draw and interpret scatter
graphs.
Learner’s Book Section 15.2
Workbook Section 15.2
Resource sheet 15.2
Additional teaching ideas Section 15.2
Draw and interpret backto-back stem-and-leaf
diagrams.
Learner’s Book Section 15.3
Workbook Section 15.3
Resource sheet 15.3
Additional teaching ideas Section 15.3
Use mode, median, mean
and range to compare two
grouped data distributions.
Learner’s Book Section 15.4
Workbook Section 15.4
Resource sheet 15.4
Additional teaching ideas Section 15.4
M
15.3 Back-to-back 0.5–1
stem-and-leaf
diagrams
1–1.5
SA
15.4 Calculating
statistics for
grouped data
15.5 Representing 0.5–1
data
Choose, explain and use a
useful representation in a
given situation.
Learner’s Book Section 15.5
Workbook Section 15.5
Additional teaching ideas Section 15.5
Cross-unit resources
Language worksheet: 15.1–15.5
End of unit 15 test
End-of-year test
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
BACKGROUND KNOWLEDGE
The focus of this unit is to extend learners’
knowledge of drawing and interpreting various
methods of displaying data and of comparing
distributions.
Learners will move from looking at frequency
diagrams to using frequency polygons. They
will also be introduced to lines of best fit and
correlation on a scatter graph. Learners will use
back-to-back stem-and-leaf diagrams and will
learn how to calculate statistics for grouped data,
including modal class, the interval where the
median lies, and estimates for the range and mean.
PL
E
For this unit, learners will need this background
knowledge:
• Draw and interpret: Venn and Carroll diagrams,
tally charts, frequency tables and two-way
tables, dual and compound bar charts, pie
charts, frequency diagrams for continuous
data, line graphs and time series graphs,
scatter graphs, stem-and-leaf diagrams and
infographics (stages 7 and 8).
• Use knowledge of mode, median, mean and
range to compare two distributions, considering
the interrelationship between centrality and
spread (Stage 8).
TEACHING SKILLS FOCUS
• Do you know what the learners know/knew about
this topic?
• Have you asked questions to look for evidence
of learning, of a depth of understanding of the
topic that shows learners understand how the
maths works, not just that they can get an answer
to a question?
• Are learners confident that if they can suggest
half-formed ideas about a problem, then they
can share their ideas and receive guidance from
yourself or another learner?
• Do you tell learners that learning from their
mistakes is an excellent and invaluable process
that is encouraged within the classroom?
SA
M
Assessment for learning
A key aspect for assessment for learning is assessing
prior knowledge. While the Getting started
questions might help you to find some weaknesses,
much of this unit is built on previously learned skills.
As such, if any of those previously learned skills are
weak or missing, it is important to revisit that area of
the Stage 8 work.
You might need to adapt or stop the planned lesson
if the required previous knowledge is missing. If
only part of the class lacks a skill, then this is a great
opportunity for you to get learners to help teach.
At the end of Unit 15, and the Stage 9 course,
ask yourself:
15.1 Interpreting and drawing frequency polygons
LEARNING PLAN
Framework codes
Learning objectives
Success criteria
9Ss.03
• Record, organise and
represent categorical,
discrete and continuous
data – frequency polygons.
• Learners can draw and
interpret frequency polygons.
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15 INTERPRETING AND DISCUSSING RESULTS
LANGUAGE SUPPORT
etc.) when explaining their work and in group
discussions.
You could also remind learners of the meaning of
the inequality symbols (for example: ‘<’ means
‘less than’, ‘⩽’ means ‘less than or equal to’, etc.)
to make sure that they understand which numbers
lie in which class intervals.
PL
E
Frequency polygon: a chart made up of straightline segments that shows frequencies
Midpoint: the middle value in a class interval
Encourage learners to use the specific vocabulary
(for example: frequency, midpoint, frequency
table, frequency diagram, frequency polygon,
Common misconceptions
Misconception
How to identify
How to overcome
Working out the midpoints correctly but plotting Question 2.
them incorrectly, either at the start or end of the
interval.
Discussion during Worked example 15.1
and when discussing Question 1.
Question 5.
Discussions when checking answers to
Question 5, allowing learners to change
their answer for part d when discussions
have finished.
Not actually comparing, but simply describing
various features of both frequency polygons
when asked to compare two frequency
polygons.
Starter idea
Getting started (15 minutes)
Question 3, Think like a mathematician
(5–10 minutes)
Learning intention: To choose their own class intervals
for a data set.
M
Resources: Note books, Learner’s Book
Getting started exercise
Main teaching idea
Description: Learners would probably benefit from a
short class discussion before attempting each of the
three Getting started questions.
SA
For Question 1, learners should tell each other that the
frequency table will probably need a ‘Tally’ column.
They should also realise that an age of ‘15’ in the table
would lie in the first class of 10 < a ⩽ 15, as the ‘⩽ 15’
means ‘less than or equal to 15’.
For Question 2, learners should tell each other that the
stem-and-leaf diagram must have a key to explain the
numbers and that the ‘stem’ part will have the numbers
0, 1, 2, 3 and 4.
For Question 3, hopefully no learners need to be
reminded of the mean, median, mode or range, but it is
a good idea to check.
Resources: Note books, Learner’s Books
Description: Several sensible options for class widths are
possible; 10, 15 or 20. A class width of 15 gives a good
balance of accuracy and ease of drawing. A class width
of 10 is more accurate, but has more points to plot.
A class width of 20 is less accurate but has fewer points
to plot. It is generally accepted that 5 or 6 is a good
number of points for a frequency polygon, so a class
width of 15 is possibly the best choice, but a class width
of 10 is probably the easiest to work with.
Differentiation ideas: Less confident learners will prefer
to use classes that they have previously used, so they will
probably choose, or be guided towards, a class interval
of 10.
171
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
How? (5–10 minutes)
Guidance on selected Thinking and
working mathematically questions
Resources: Note books
Characterising, critiquing and convincing
Description: On the board, draw/display this table.
Exercise 15.1, Question 8
Tell learners that the table shows the heights of some
bamboo plants.
Many learners will think personally, not mathematically,
when answering part d of the question, even though
they were given a tip in part c. Less confident learners
often try to think of reasons such as:
Height, h (cm)
Frequency
240 ⩽ h < 250
6
250 ⩽ h < 260
14
260 ⩽ h < 270
6
270 ⩽ h < 280
2
Ask learners to write the method they would use to draw
a frequency polygon to display this data.
Assessment ideas: This can be peer-marked in two
main ways:
1
Learners can be given the basic steps and then check
their partner’s work. The steps are:
•
add a midpoint column to the table
•
work out midpoints [245, 255, 265, 275]
Homework ideas
Workbook, Section 15.1.
As Section 15.1 will probably take more than one
lesson, set suitable parts of the Workbook at the end
of each lesson. Only set questions that can be answered
using skills and knowledge gained from that lesson.
Workbooks are aimed at fluency and consolidation
through practice, not as a method to learn new skills
that should be taught in class.
Marking should be done by learners at the start of the
next lesson. Any help/discussions with any problems
should take place immediately.
M
•draw axes – x-axis from 230 to 290 in 10s or 5s
and y-axis from 0 to 15 in 1s
•
plot mid-points against frequency
•
join up the points in order.
Learners are asked to draw the frequency polygon
using the instructions given. Learners must be told
to only use the instructions given, not just to draw
the frequency polygon because they can. Some
learners might have missed out vital information
and the drawing will not be possible.
SA
2
• Liza only asked her friends
• Liza is a girl, so her data is biased against boys
or similar, rather than reflecting on the fact that one
polygon has 25% more data than the other polygon
(as more girls than boys were surveyed).
PL
E
Plenary idea
Assessment ideas
Use Question 4 as an extended hinge-point question.
This question asks learners to display all of the skills
required to draw a frequency polygon from raw data.
Take extra time when checking/marking/discussing this
question when completed. Ensure learners have drawn
a sensible frequency table, with midpoints accurately
calculated. Ensure that the frequency diagram matches
the information from the learner’s table, that axes
are clearly labelled and that lines are neatly and
accurately drawn.
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15 INTERPRETING AND DISCUSSING RESULTS
15.2 Scatter graphs
LEARNING PLAN
Learning objectives
Success criteria
9Ss.03
• Record, organise and represent
categorical, discrete and
continuous data – scatter graphs.
• Learners can draw and
interpret scatter graphs.
9Ss.05
• Interpret data, identifying
patterns, trends and
relationships, within and
between data sets, to answer
statistical questions.
• Learners can interpret
relationships shown by scatter
graphs, and understand
that correlation does not
automatically mean that the
change in one variable is
the cause of the change in the
values of the other variable.
LANGUAGE SUPPORT
PL
E
Framework codes
Encourage learners to use the vocabulary
‘positive correlation’, ‘negative correlation’ and
‘no correlation’ in discussions. Also encourage
learners to describe what the correlation means in
specific examples (for example, ‘This scatter graph
shows negative correlation. As the age of the car
increases, the value of the car decreases’, etc).
M
Correlation: the relationship between two variable
quantities
Line of best fit: a line on a scatter graph that shows
the relationship between the two sets of data
Scatter graph: a graph showing linked values of
two variables, plotted as coordinate points, that
might or might not be related
SA
Common misconceptions
Misconception
How to identify
How to overcome
Joining up the points in a scatter
graph.
Question 1.
Discussion during the introduction.
Tell learners that they must not join the
points in this type of diagram. The only
line drawn is the line of best fit.
Thinking that, just because the
data does not seem to originate
from the origin, there is no
positive correlation.
Question 2.
Discussion during the starter activity
idea and checking after completion of
Question 2.
Thinking that a line of best fit
can be extended below and/or
beyond the data set and be used
to make predictions.
Questions 5 and 6.
Discuss examples to show why this
should not be done, e.g. heating
water in a pan: more heat = higher
water temperature, until it boils,
then more heat does not change the
water temperature.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Starter idea
Learners should be aware that a positive correlation
does not have to ‘start’ at (0, 0). Use the dotted lines in
the two diagrams on the board to discuss this.
Correlation (3 minutes)
Resources: Mini white boards or note books
Description: Use this starter idea after working through
Worked example 15.2.
Another brief class discussion about the strength of
correlation often helps learners to understand correlation
itself more clearly.
Question 2, Think like a mathematician
(5–10 minutes)
Learning intention: To check a conjecture and to
understand the limitations of a line of best fit.
Resources: Note books, Learner’s Books
PL
E
On the board, draw/display two sets of axes, with points
marked with small crosses, as shown:
Main teaching idea
Description: For part f, learners need to understand
that any sort of accurate estimation will only be
possible for a very short distance away from their data.
The further the line is extended away from their data,
the less accurate it is likely to be.
Discuss here why it is not possible to predict from a line
of best fit before or after the data:
Ask learners to identify the type of correlation
(positive). Discuss/explain that the scatter graph on the
left has a stronger correlation than the scatter graph on
the right, as the points are closer to an imaginary line.
• With a temperature of 44 °C the shop might not sell
many drinks as people might not go outside in that
temperature.
Differentiation ideas: Some learners will not
understand part a. Tell these learners that they need
to look at the titles of the data (maximum daytime
temperature and number of cold drinks sold) and decide
what type, if any, correlation they would expect.
SA
M
On the board, draw/display the two identical dashed lines
of best fit, each passing through the middle of the data.
• There is no data to show that the correlation is the
same after or before these points.
Use the term ‘line of best fit’. Tell learners this is a
straight line which is an ‘educated guess’ through the
middle of the data, in the direction of the correlation.
Actually, there are computer programs and calculators
which will calculate the line of best fit as an equation,
but these are beyond the scope of this unit.
Explain that the closer to this line the dots are arranged,
the stronger the correlation. In a perfect correlation,
all the points are on the line.
Repeat that the diagram on the left shows strong
correlation, the diagram on the right shows
weaker correlation.
If required, suggest that the line of best fit is just a line
that should have about half the points on one side of the
line and half of the points on the other side of the line.
Tell learners that the line can pass through points, but it
doesn’t have to.
Plenary idea
Check Question 2 (3 minutes)
Resources: Note books, Learner’s Books and calculators
Description: Ask learners to look back at their scatter
diagram and line of best fit for Question 2. Ask them to
calculate the mean coordinate and to plot this point.
Assessment ideas: When complete, ask learners to
compare their lines of best fit. Whose line went through
the mean point?
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15 INTERPRETING AND DISCUSSING RESULTS
Guidance on selected Thinking and
working mathematically questions
Homework ideas
Specialising, characterising and convincing
As Section 15.2 might take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
Exercise 15.2, Question 4
Assessment ideas
Use Question 6 as a class ‘test’. If learners can answer
this question they obviously understand how to ‘draw
and interpret scatter graphs’.
PL
E
This is a straightforward but important question. When
an outlier is identified (the term outlier is not required
in this section, but it is a useful statistical term) it is
often useful to try to explain why it does not seem to
follow the rest of the data. Learners often leave this
type of question blank, and yet are more than capable
of giving a correct answer. Learners need to think
of the most obvious reason possible. For example, in
Question 4, you could ask ‘Why would a taxi take longer
than anticipated to complete a journey?’ There are
many obvious reasons: the taxi was asked to wait, there
could have been road works or traffic congestion, the
passenger asked the taxi driver to stop to get some food
on the way home, the list is endless – and all correct.
Workbook, Section 15.2.
It is not necessary to tell learners that this is a ‘test’, just
tell learners that you will mark the question.
15.3 Back-to-back stem-and-leaf diagrams
LEARNING PLAN
Success criteria
• Record, organise and
represent categorical,
discrete and continuous
data – back-to-back stemand-leaf diagrams.
• Learners can draw and
interpret back-to-back
stem-and-leaf diagrams.
SA
9Ss.03
Learning objectives
M
Framework codes
LANGUAGE SUPPORT
Back-to-back stem-and-leaf diagram: a way of
displaying two sets of data on one stem-and-leaf
diagram
Tell learners that when a question asks them to
compare back-to-back stem-and-leaf diagrams,
they should check and compare each type of
average they have worked out during the
question.
175
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Common misconceptions
How to identify
How to overcome
Trying (and failing) to draw an
ordered stem-and-leaf diagram
without first drawing a stem-andleaf diagram displaying the data in
its original order.
Questions 2 and 5.
Encourage learners always to start by drawing a
stem-and-leaf diagram displaying the data in its
original order. Then they should redraw the stemand-leaf diagram, putting the numbers in order,
to produce an ordered stem-and-leaf diagram.
This will make it easy to check that they have
included all the numbers.
Forgetting to check that the
number of pieces of data is the
same as the number of ‘leaves’ in
their stem-and-leaf diagram.
Questions 2 and 5.
Ask learners to check during discussions.
PL
E
Misconception
Starter idea
Ordered stem-and-leaf diagram:
Drawing a stem-and-leaf diagram
(5 minutes)
Resources: Mini white boards or note books
Description: Learners can do this activity individually
before working through Worked example 15.3.
Key: 5 | 8 means 58 kg
5
6
7
8
9
8
0
1
0
0
9
1
2
2
2
9
2
3
5
5
4
3
6
4
5
9
4
8
9
9
On the board, copy/display this table and question:
Main teaching idea
Question 2, Think like a mathematician
(5–10 minutes)
M
The masses, in kilograms, of 25 adults are shown in
this table:
73 62 85 71 64 89 80 59 72 69 78 60 64
82 58 92 69 59 75 95 61 90 64 73 86
Draw an ordered stem-and-leaf diagram to show this data.
SA
Remind learners that they should draw an unordered
stem-and-leaf diagram first. They can then use their
unordered stem-and-leaf diagram to draw the ordered
stem-and-leaf diagram.
When completed, peer-marking is useful, primarily to
ensure that the ‘leaves’ are spaced out properly and that
learners have completed an unordered stem-and-leaf
diagram before the ordered stem-and-leaf diagram.
Answer:
Unordered stem-and-leaf diagram:
Key: 5 | 8 means 58 kg
5
6
7
8
9
9
2
3
5
2
8
4
1
9
5
9
9
2
0
0
0
8
2
4
5
6
9
3
1
Learning intention: To draw a back-to-back stem-andleaf diagram. To calculate and make decisions based on
statistical measures.
Resources: Note books, Learner’s Books
Description: Remind learners that they must draw an
unordered stem-and-leaf diagram first. They can then
use the unordered diagram to draw the ordered diagram.
Remind learners to check that there are the same
number of ‘leaves’ in both diagrams to help ensure they
have included all of the data.
Differentiation ideas: You might need to remind some
learners of the method used in Question 1. Suggest they
draw the ‘stem’ first, then put the data for the city car
park on the right-hand side of the stem and then the
data for the beach car park on the left side of the stem.
4
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15 INTERPRETING AND DISCUSSING RESULTS
Plenary idea
Drawing a back-to-back stem-and-leaf
diagram (5 minutes)
Resources: Note books or mini white boards
Part c helps to check that learners understand the use
of the statistics they have just found. The mean and
possibly the median are the important factors here to
determine which group is overall the fastest. The mode
is of little significance for this type of choice.
Description: Ask learners to draw a back-to-back stemand-leaf diagram representing the data of Dieter’s and
Billie’s friends in centimetres.
Homework ideas
Dieter’s friends’ heights: 167, 159, 159, 169, 171, 167,
162, 157, 172, 160, 167, 158
As Section 15.3 might take more than one lesson, set
suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
PL
E
Billie’s friends’ heights: 167, 168, 155, 182, 176, 175, 167,
187, 167, 170, 169, 178
Assessment ideas: You can use peer-marking. On the
board, draw/display this diagram:
Billie’s friends’ heights
5
9 8 7 7 7
8 6 5 0
7 2
Key: 7 | 16 means 167 cm
Dieter’s friends’ heights
15
16
17
18
Workbook, Section 15.3.
7 8 9 9
0 2 7 7 7 9
1 2
Key: 15 | 7 means 157 cm
Ask learners to check that the work they are marking has:
The work on this section involves diagrams that are
easily incorrectly drawn as well as working out various
statistics relating to large data sets. This is an excellent
opportunity for peer-marking. Having learners regularly
swap books for checking/marking will help learners to
focus on the important aspects of their own work.
Learners will focus on the diagrams they are checking,
making sure they have been drawn so that the numbers
are in order of size, smallest nearest the stem, that there
is a key to explain what the numbers mean and that
all the numbers are in line, vertically and horizontally.
Learners will also focus on the statistics they are
checking, making sure they have clear working and that
the comments are true and make sense.
M
• the numbers in order of size from smallest to largest
starting at the stem
• a key to explain the numbers
• all the numbers in line vertically and horizontally.
Assessment ideas
Guidance on selected Thinking and
working mathematically questions
SA
Conjecturing and convincing
Exercise 15.3, Question 3
Learners should be able to work out answers for
part a fairly confidently. You might need to give some
learners some guidance when comparing the times
taken using the statistical measures from part a. As ever
with this type of question, learners should comment
on the obvious points, such as the boys’ mode time is
0.6 seconds slower than the girls’ mode time, the girls’
median time is 0.8 seconds slower than the boys’ median
time, boys’ times are more consistent than the girls’ times
as the boys have the smallest range, the boys’ mean time
is 1.16 seconds faster than the girls’ mean time.
Regularly ask if learners have made a mistake.
For any mistakes:
If learners know what they did wrong and how to
get that question correct next time, acknowledge this
with a well done (as this is what you need in the class,
active learners).
If learners do not know what they have done wrong
or how to correct it, depending on the question
(and the learner), you could either help that learner
separately, asking learners near them to explain,
or have a class discussion on the question/skill.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
15.4 Calculating statistics for grouped data
LEARNING PLAN
Learning objectives
Success criteria
9Ss.04
• Use mode, median, mean
and range to compare two
distributions, including
grouped data.
• Learners can use mode,
median, mean and range to
compare sets of grouped
data, understanding that
these comparisons are made
with estimates as the data
is grouped.
PL
E
Framework codes
LANGUAGE SUPPORT
There is no new vocabulary for this section.
Make sure that learners can use the vocabulary for
grouped data statistics (for example, ‘modal class
interval’ not ‘mode’, ‘the class interval where the
median lies’ not ‘median’, etc.)
Ensure that learners talk about ‘the estimate of the
mean’ and ‘the estimate of the range’ instead of
‘the mean’ and ‘the range’.
M
Common misconceptions
Misconception
How to identify
How to overcome
Not using the midpoint of the class
interval to calculate the mean.
Question 2.
Part b ii of Worked example 15.4
and Question 1.
SA
Starter idea
Mean, median, mode and range (5 minutes)
Resources: Mini white boards or note books
Description: Learners could attempt this activity before
looking at Worked example 15.4. Learners can work
individually or in small groups, depending upon ability.
On the board, display a simple data set such as 2, 2, 2,
3, 3, 7, 9. Ask learners to calculate the mean, median,
mode and range. They should find these values:
• mean = 2 + 2 + 2 + 3 + 3 + 7 + 9 = 28 = 4
7
7
• median = 3, the middle value of the data when put
in order
mode = 2, the most common value
• range = biggest value − smallest value = 9 − 2 = 7.
Change the data set by putting another 2 at the start of
the list.
• Ask ‘Has the mean changed?’ [yes] Ask why.
• Ask ‘Has the median changed?’ [yes] Ask why.
Discuss how to work out the new median. The new
centre is halfway between the last 2 and the first 3,
so the new median is 2 + 3 = 2.5.
2
• Ask ‘Has the mode changed?’ [no] Ask ‘Adding what
number would change the mode?’ [3] Ask why.
• Ask ‘Has the range changed?’ [no] Ask ‘Adding what
number would change the range?’ [any number
<2 or >9]
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15 INTERPRETING AND DISCUSSING RESULTS
Main teaching idea
Question 3, Think like a mathematician
(5 minutes)
Learning intention: To help to understand why the
middle value of a class interval is used when estimating
the mean.
Resources: Note books, Learner’s Books
Differentiation ideas: If learners do not understand,
ask them to look at the table in Question 2, at the row
for 50 ⩽ m < 60. Ask ‘Do you think it is likely that all
12 learners in the class interval of between 50 and 60 kg
would all be 50 kg?’ [no] Then ask ‘Do you think it is
likely that all 12 learners in this class interval would all
be 60 kg?’ [no] Ask ‘Do you think the learners would
have different masses from 50 to 60 kg?’ [yes] Then
ask ‘What guess would be sensible for the mass of the
learners in the 50 ⩽ m < 60 group?’ [55 kg]
Exercise 15.4, Question 4
Focus on the answers to part d, as this part shows
learners’ understanding. There are various factors to
take into consideration. A learner might say that they
would choose the Moorlands, hoping to be lucky with
their waiting time and be in the modal group. Most
learners, however, will choose the Heath as both the
median and mean are lower. Whichever hospital the
learner decides on, make sure they have used all three
measures to explain their choice.
Homework ideas
Workbook, Section 15.4.
As Section 15.4 will probably take more than one lesson,
set suitable parts of the Workbook at the end of each
lesson. Marking should be done by learners at the
start of the next lesson. Any help/discussions with any
problems should take place immediately.
You could ask learners to make a worked example list
containing everything they think they need to remember
for the end-of-unit test. The following lesson, it is
important to share the worked example lists in class
(e.g. spread out over a few desks for everyone to look
at), rather than marking them. Discuss the different
worked examples as a class. When the class agree that a
point is important, that key point could be copied onto
the board (by you or a learner). Agree on as many key
points as possible. Learners could then improve/update
their individual lists if necessary. Learners could store
their worked example lists at home as a possible revision
tool towards mid-term/end-of-year tests.
M
Plenary idea
Characterising and convincing
PL
E
Description: Most learners will use basic logic to work
out that using the middle value is more sensible than
assuming that all masses are either the lowest or the
highest value in the class interval. Some learners might
have some difficulty in explaining why this is true.
A brief group or class discussion to clarify thoughts can
be useful here.
Guidance on selected Thinking and
working mathematically questions
Resource sheet 15.4: Exit ticket (5 minutes)
Resources: Resource sheet 15.4: Exit ticket
SA
Description: Give each learner an exit ticket, cut out
from Resource sheet 15.4. Learners should complete the
exit ticket just before leaving class. Allow 5 minutes to
complete.
Answers:
a 10 < m ⩽ 20
c 19 kg
b 10 < m ⩽ 20
d 40 kg
Assessment ideas: Mark yourself or check answers and
return for self-marking next lesson.
As this question is almost identical to Worked example
15.4, any learner making a mistake here might need
further help with this topic.
Assessment ideas
Use Question 2 as an extended hinge-point question.
Although early in the exercise, any learner making
mistakes here needs some direct assistance to decide on
the misunderstanding.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
15.5 Representing data
LEARNING PLAN
Learning objectives
Success criteria
9Ss.03
• Record, organise and represent
categorical, discrete and
continuous data. Choose and
explain which representation to
use in a given situation.
• Learners can choose
and use an appropriate
diagram, graph or chart to
represent data.
LANGUAGE SUPPORT
PL
E
Framework codes
There is no new vocabulary for this section.
Encourage learners to use as much of the mathematical language as they can that they have learned and
practised so far in Unit 15.
Common misconceptions
Misconception
How to overcome
The activity.
Discussing representations with
learners while they work in class.
M
Using an inappropriate diagram,
graph or chart to represent data.
How to identify
Starter idea
Key features (10–20 minutes)
Resources: Note books
SA
Description: Hold a class discussion asking for the key
features of: Venn and Carroll diagrams, tally charts and
frequency tables, dual and compound bar charts, line
graphs and time series graphs, scatter graphs, stemand-leaf diagrams, frequency polygons, two-way tables,
infographics and pie charts.
You might decide to write on the board key features of
any of the diagrams, graphs or charts which required the
most guiding to get those key features.
Main teaching idea
Project Australia! (60–120 minutes)
Learning intention: To apply the skills and knowledge of
representing data learned since Stage 5.
Resources: Note books, Learner’s Books, scrap paper,
large sheets of paper, calculators
Description: Allow learners to use the Learner’s Book
and any previously made notes in their note books.
This extended activity is aimed at learners applying
their representing data skills by making a poster of
the various information given about Australia. It is
suggested that learners use scrap paper to try different
representations. When learners are satisfied that they
have enough variety of representations, they can then
produce a poster.
Assessment ideas: When completed, display the
posters over desks for learners to look at and compare.
Discuss the best parts of different posters, methods of
representing data which are very effective, and those
methods that are not quite so clearly understood.
Adding to, or altering, posters could be useful for some
learners who have made errors.
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15 INTERPRETING AND DISCUSSING RESULTS
Differentiation ideas: Some learners will find it
difficult to know where to begin. To help them,
you could suggest:
Homework ideas
Workbook, Section 15.5.
PL
E
• using frequency polygons to show the ages of the
population – they could either draw two polygons,
one male and female, or draw one polygon showing
both males’ and females’ ages
• drawing a pie chart to show the percentage of the
main languages spoken in Australia or of where
electricity production comes from
• drawing an infographic giving some facts about the
Great Barrier Reef.
Section 15.5 will take more than one lesson. It is
probably best to set the homework from the Workbook
when the posters are complete and have been discussed.
You might decide, however, to set the homework at
the end of the first lesson and ask learners to use their
homework within their poster.
SA
M
You might decide to ask all learners to display one set of
information, e.g. the percentages of language spoken, as
a homework. This can be checked and discussed in class
before adding to their posters.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
PROJECT GUIDANCE: CYCLE TRAINING
two rides. Draw attention to the fact that the wind
reduces the average speed by 6 km per hour on the
outward journey but increases the average speed
by 6 km per hour on the return journey. Finally,
learners could explore what happens to the total
journey time as the average speeds are affected by
stronger winds.
Key questions
If you know the distance travelled and the time
taken, how can you work out the average speed?
PL
E
Why do this problem?
This problem offers learners an opportunity to
make decisions about how to represent real-life
data, and invites them to characterise the key
points of different journeys. The last part of the
investigation leads to an important generalisation
about the effects on the total journey time when
the average speed for the outward journey and the
return journey are different.
Possible approach
Present learners with the outward and return
journey data for the first day. Invite them to
discuss how they might represent the data. There
are some decisions to be made about how best
to draw a distance–time graph, especially since
the tables present time in terms of distance, but
it is conventional to plot time on the x-axis and
distance on the y-axis. You could also suggest that
learners plot ‘distance from home’ on the y-axis so
that the graph slopes up for the outward journey
and then back down for the return journey, meeting
the x-axis to show the total journey time.
Possible support
Learners could use a spreadsheet or graph-plotting
software to help them to represent the data.
Learners might find it simpler to find the speed
in km per hour by considering the distance that
would be travelled in 60 minutes, rather than using
a formula.
Possible extension
Learners could create an algebraic expression
for the total journey time if the average speed is
(30 − v) on the outward journey and (30 + v) on the
return journey. Learners might go on to use their
expression to explore what happens as v increases,
and then to construct a convincing argument to
explain why the journey will always be completed
more quickly on a calm day than on a windy day.
SA
M
When learners have plotted the first graph, invite
them to draw the graph to represent the second
journey on the same set of axes. Ask learners to
describe the differences between the two journeys
and to suggest plausible explanations for the
differences. Then challenge learners to work out
the average speed for the outward journey and
the average speed for the return journey of the
If Marcus cycles against the wind on the outward
journey, and with the wind on his return journey,
how will his overall journey time compare to the
overall time that would be taken on a calm day?
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