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Cambridge Lower Secondary
Mathematics TEACHER’S RESOURCE 9
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Lynn Byrd, Greg Byrd & Chris Pearce
Second edition
Digital Access
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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Cambridge Lower Secondary
Mathematics TEACHER’S RESOURCE 9
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Greg Byrd, Lynn Byrd & Chris Pearce
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence.
© Cambridge University Press 2021
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www.cambridge.org Information on this title: www.cambridge.org/9781108783897
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Second edition 2021
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Printed in Great Britain by CPI Group (UK) Ltd, Croydon CR04YY
A catalogue record for this publication is available from the British Library ISBN 978-1-108-78389-7 Paperback with Digital Access
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Cambridge University Press has no responsibility for the persistence or accuracy 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. Cambridge International copyright material in this publication is reproduced under licence and remains the intellectual property of Cambridge Assessment International Education.
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Test-style questions and sample answers have been written by the authors. In Cambridge Checkpoint tests or Cambridge Progression tests, the way marks are awarded may be different. References to assessment and/or assessment preparation are the publisher’s interpretation of the curriculum framework requirements and may not fully reflect the approach of Cambridge Assessment International Education. Third-party websites and resources referred to in this publication have not been endorsed by Cambridge Assessment International Education. NOTICE TO TEACHERS IN THE UK
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.
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.
CONTENTS
Contents Introduction 6 7
How to use this series
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About the authors
How to use this Teacher’s Resource
10
About the curriculum framework
15
About the assessment
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Introduction to thinking and working mathematically
16
Approaches to learning and teaching
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Setting up for success
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Acknowledgements 25
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Teaching notes 1 Number and calculation
26
2 Expressions and formulae
34
3 Decimals, percentages and rounding
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Project Guidance: Cutting tablecloths
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4 Equations and inequalities
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5 Angles 73
Project Guidance: Angle tangle
6 Statistical investigations
86
7 Shapes and measurements
91
8 Fractions 100
Project Guidance: Selling Apples
9 Sequences and functions
113
10 Graphs 121
Project Guidance: Cinema membership
11 Ratio and proportion
132
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
12 Probability 138 13 Position and transformation
148
Project Guidance: Triangle transformations
14 Volume, surface area and symmetry
161
15 Interpreting and discussing results
170
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
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Metacognition Skills for life
Letter for parents
Lesson plan template
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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
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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. Greg Byrd, Lynn Byrd and Chris Pierce
6 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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 onto 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.
7 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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).
9 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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).
10 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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
11 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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.
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 questions section then looks at one of the questions in detail and provides more guidance about developing the skill that it supports
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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Additional teaching notes are provided for the six NRICH projects in the Learner’s Book, to help you make the most of them.
Guidance on selected Thinking and working mathematically questions
12 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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. 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. Answers 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:
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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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 Lower Secondary 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 Cambridge Primary and Lower Secondary Mathematics (0862) curriculum frameworks are available on the Cambridge Assessment International Education website: https://www.cambridgeinternational.org/lowersecondary
15 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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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a st
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lly
Test an idea
ica
Use an example
Characterising and Classifying
at
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Specialising and generalising
Critiquing and Improving
m he at ) m 8 g 01 kin l, 2 or a w tion d an na g ter kin e In hin dg e t bri Th am (C
You can download a list of the Thinking and working mathematically questions set in this stage and their respective characteristics here.
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.
16 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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
17 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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.
26
31 28
34 29
32 27
27 24
25 23
26 24
28 27
29 26
30 29
33 31
27 23
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 44ºC?
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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.
19 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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
D 0.005 × 3
F 5 × 0.03
J 0.005 × 30
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.
Zara’s method.
6x5 ÷ 3x2 6 ÷ 3 = 2 and x5 ÷ x2 = x5 – 2 = x3 So, answer is 2x3.
6x5 5 – 2 3 1 3x2 = 2x = 2x So, answer is 2x3.
2
20 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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
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• 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
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1 100 2.6 ÷ 100 = 2.6 × 1 = 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.
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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
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The following are the teaching approaches underpinning our course content and how we understand and define them.
Active learning is a teaching approaches 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
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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
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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 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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.
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• • • • • •
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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.
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Cambridge learners
Reflective as learners, developing their ability Reflective as learners themselves, developing to learn. 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.
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).
23 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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.
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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-whiteboards, 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
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1 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 Introducing the lesson: 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.
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3 Managing 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. 4 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. 5 Plenary and reflection: 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. 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. We offer a range of Professional Development support to help you teach Cambridge Lower Secondary Mathematics with confidence and skill. For details, visit cambridge.org/education
24 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
ACKNOWLEDGEMENTS
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Acknowledgements
25 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
Unit plan Topic
Approximate Outline of learning content number of learning hours
1.1 Irrational 2 numbers
1.2 Standard 2 form 2
Resources
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.
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
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
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1.3 Indices
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1 Number and calculation
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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.
26 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 NUMBER AND CALCULATION
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?
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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.
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9Ni.04
Learning objectives
LANGUAGE SUPPORT
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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.
27 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
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
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• 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.
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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).
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7 18 = 18.4666666... 15 1 3 = 3.14285714... 7
(here the digit 6 repeats).
(here there is a sequence of
6 repeating digits 142 857) 15 17
6 = 6.882352941... (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 = 6.8823529411764705 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 = 2.6457513...
They will find: 2.62 = 6.76; 2.652 = 7.0225; 2.6462 = 7.001 316; 2.64582 = 7.00025764 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?
28 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 NUMBER AND CALCULATION
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.
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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.
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.
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Assessment ideas: Ask the learners to check each other’s diagram and assess whether it is correct.
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.
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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.
29 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
LANGUAGE SUPPORT Standard form: you write a number in standard form as a × 10n where 1 ⩽ a < 10 and n is an integer. Scientific notation: the same as standard form.
Mathematicians use the term standard form. Scientists use the term scientific notation. The meaning is identical.
Common misconceptions How to identify
How to overcome
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.
Check that answers in Exercise 1.2 are written correctly.
Always emphasise this point in discussion, giving examples of correct and incorrect forms.
Starter idea Powers of 10 (5 minutes) Resources: None
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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.
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. 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.
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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!
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.
Ask some reverse questions such as ‘What power of 10 is 100 000?’.
Main teaching idea
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Standard form (10 minutes)
Learning intention: To learn how to write large numbers in standard form. 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.
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.
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. Learners might also think of examples in other subject areas such as science (astronomical masses and distances) or geography (populations).
Explain that numbers written in this way are said to be in standard form.
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1 NUMBER AND CALCULATION
Ask a similar question about small numbers. A science teacher might be able to suggest examples here with which the learners will be familiar. 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
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.
Homework ideas 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.
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Characterising and generalising
Cross curricular links
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.
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
There are opportunities for peer assessment in the activities and in some of the questions in Exercise 1.2.
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1.3 Indices LEARNING PLAN
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
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Framework codes
and that 2−3 =
1 23
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.
31 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE
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
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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
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Negative powers (20 minutes)
Learning intention: To extend the range of definitions of powers to include zero and negative integers. 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: 32
33
34
35
36
9
27
81
243
729
33
34
35
36
9
27
81
243
729
Ask ‘What numbers go in the two empty columns to continue the pattern?’.
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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.
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Learners should see that this is the pattern. 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
1
3
9
33 34 27
81
35
36
243
729
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.
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1 NUMBER AND CALCULATION
3−3
3−2
3−1
30
31
32
33
34
1 27
1 9
1 3
1
3
9
27
81
1 3
1 3
1 9
The table shows that 3−1 = = 1 and 3−2 = = 1 27
3−3 = =
1 33
35
36
243 729 1 32
and
A negative power of 3 is the reciprocal of the corresponding positive power.
1 5
So 5−2 = 2 =
1 25
1 2
and 2−5 = 5 =
1 32
and 80 = 1
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.
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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
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.
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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
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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.
33 Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
