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Cambridge Lower Secondary
Mathematics TEACHER’S RESOURCE 7
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Lynn Byrd, Greg Byrd & Chris Pearce
Digital access Original material © Cambridge University Press 2020. 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 7
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Greg Byrd, Lynn Byrd & Chris Pearce
Original material © Cambridge University Press 2020. 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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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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Additional resources for this publication at www.cambridge.org/delange
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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. 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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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 Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication. been endorsed by Cambridge Assessment International Education.
CONTENTS
Contents Introduction 6 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 Cambridge International assessment
15
Introduction to Thinking and working mathematically
16
Approaches to teaching and learning
23
Setting up for success
25
Teaching notes
1 Integers 26 Project 1: Mixed-up properties
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2 Expressions, formulae and equations
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3 Place value and rounding
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4 Decimals 66
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5 Angles and constructions
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Project 2: Clock rectangles
6 Collecting data
91
7 Fractions 98
Project 3: Fraction averages
8 Shapes and symmetry
114
9 Sequences and functions
126
Project 4: Mole and goose
10 Percentages 137 11 Graphs 143
Project 5: Four steps
12 Ratio and proportion
155
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
13 Probability 164 14 Position and transformation
172
15 Shapes, area and volume
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Project 6: Removing cubes
16 Interpreting results
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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
Diagnostic check and answers Mid-year 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:
Differentiated worksheets and answers Language worksheets and answers Resource sheets
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End-of unit tests and answers
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: 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 learners 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 speaking 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 of this research, 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. 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
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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, focusing on examining work and writing. She has written or co-authored a number of textbooks, homework books, workbooks 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. 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 textbooks, 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 textbooks 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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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 learners to use in class with guidance from the teacher. It contains six units which offer complete coverage of the curriculum framework. A variety of investigations, activities, questions and images motivate learners 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.
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. Teachers can assign learners questions from one or more tiers for each exercise, or learners can progress through each of the tiers in the exercise.
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HOW TO USE THIS SERIES
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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
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Digital Access with all the material from the book in digital form plus editable planning documents, extra guidance, worksheets and Digital Access with all the material from the book in digital form plus editable planning documents, extra guidance, worksheets and more provided as downloadable resources on Cambridge GO.
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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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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 Primary 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 number of learning hours
Outline of learning content
Resources
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Understand how to add and subtract positive and negative integers
Learner’s Book Section 1.1 Workbook Section 1.1 Additional teaching ideas Section1.1
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Topic
1.1 Adding and subtracting integers
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Cross-unit resources: Vocabulary worksheet 1: 1.1–1.3 Vocabulary worksheet 2: 1.4–1.6
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 Before teaching Unit 1, you may want to use the diagnostic check activity to assess whether the learners are ready to begin Stage 7.
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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 Assessment for learning When you ask questions to the whole class or to individual learners, it is better to ask open questions rather than closed questions.
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 Curriculum objective 7Ni.03
Learning intentions
Success criteria
Estimate, multiply and divide integers, including where one integer is negative.
Learners can correctly answer questions such as −3 × 5, 4 × −6 and −18 ÷ 3.
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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
Making mistakes with subtraction
Ask questions such as ‘What is 3 – −4?’ Possible incorrect answers are 1 or −1.
Encourage learners to always change a subtraction to an addition of the inverse.
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Misconception
So 3 – −4 = 3 + 4 = 7.
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
Checking understanding (10 minutes)
Using an inverse operation (5 minutes)
Resources: Getting started exercise at the start of Unit 1 of the Learner’s Book.
Learning intention: To emphasise the fact that addition and subtraction are inverse operations. To encourage reflection on methods.
Description: Give learners 5 minutes to look at the questions. Then on the board write the statement ‘___ is a multiple of 3.’
Resources: Learner’s Book Exercise 1.1, Question 4
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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.
Integers: the whole numbers: . . ., −3, −2, −1, 0, 1, 2, 3, . . . Inverse: the operation that has the opposite effect; the inverse of ‘add 5’ is ‘subtract 5’
CROSS-CURRICULAR LINKS 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.
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The Cross-curricular links feature provides suggestions for linking to other subject areas.
LANGUAGE SUPPORT
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 critiquing
Learner’s Book Exercise 1.1, Question 14
This question has more than one answer. The number of answers is not specified. Learners must look for all possible answers and convince themselves that they have found all of the possible answers.
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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
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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 Primary 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 Primary Mathematics resources map to the Cambridge Primary Mathematics curriculum framework. Scheme of work: a suggested scheme of work that you can use to plan teaching throughout the year.
Each unit includes:
• • •
Differentiated worksheets: these worksheets are provided in variations that cater for different abilities. Worksheets labelled ‘A’ are intended to support less confident learners, while worksheets labelled ‘B’ are designed to challenge more confident learners. Answer sheets are provided. 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.
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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-year 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. • Answers to Learner’s Book questions • Answers to Workbook questions • Glossary
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CAMBRIDGE LOWER SECONDARY MATHEMAT ICS 7: MID-YEAR TEST
Name ______________________________ _____
Date _____________
Stage 7 Mid-year test: Units 1 to 8
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: PROJECT – MIXED-UP PROPERTIES
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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 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/programmes-and-qualifications/cambridge-lower-secondary to find out more.
The Cambridge Primary and Lower Secondary Mathematics curriculum frameworks have been updated from September 2020. They have been designed to encourage the development of mathematical fluency and ensure a deep understanding of key mathematical concepts that are essential to learners as they develop their mathematical knowledge. There is an emphasis on key skills that embed strategies for solving mathematical problems and encourage the communication of mathematical knowledge in written form and through discussion. At the Primary level, it is divided into three major strands: • Number •
Geometry and Measure
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Statistics and Probability
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Algebra is introduced as a further strand in the Lower Secondary framework.
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 Lower Secondary Mathematics curriculum frameworks are available on the Cambridge Assessment International Education website: https://www.cambridgeassessment.org.uk/
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Introduction to Thinking and working mathematically Thinking and working mathematically is an important part of the Cambridge Mathematics 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 7. This section provides examples of questions that require learners to demonstrate the characteristics, along with sentence starters to help learners formulate their thoughts. Use an example
Test an idea
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Specialising and generalising
Give an example
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Say what would happen to a number if . . .
Specialising and generalising Specialising
Specialising involves choosing and testing an example to see if it satisfies or does not satisfy specific maths criteria. Learners look at specific examples and check to see if they do or do not satisfy specific criteria.
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INTRODUCTION TO THINKING AND WORKING MATHEMATICALLY
Example: Sofia reflects pentagon ABCDE in the y-axis.
a
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The diagram shows the object, ABCDE, and its image, A′B′C′D′E′. y A9 A B3 B9 2 E E9 1 C C9 −4 −3 −2 −1 0 1 2 3 4 x −1 −2 D9 D
The table shows the coordinates of the vertices of the object and its image. Copy and complete the table. Object
A(−4, 3)
B(−1, 3)
C(__, __)
D(__, __)
E(__, __)
Image
A′(4, 3)
B′(__, __)
C′(__, __)
D′(__, __)
E′(__, __)
b What can you say about the x-coordinates of the vertices of the object and its image? c What can you say about the y-coordinates of the vertices of the object and its image? Learners show they are specialising when they compare the x and y coordinates for the object and its image, and see the connection in this particular case.
SENTENCE STARTERS • I could try . . .
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• . . . is the only one that . . .
• . . . is the only one that does not . . .
Generalising
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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:
This follows on from parts b and c in the specialising example. Will your answers to parts b and c always be true for whatever shape you reflect in the y-axis? Explain your answers. e In general, when you reflect a shape in the x-axis or the y-axis, will the object and the image always be congruent? Explain your answer. Learners will show they are generalising when they notice the effect on the coordinates of a shape when it is reflected in the x-axis or y-axis, and realise that this happens whatever the shape. They will also understand that a shape and its reflected image are always congruent. d
TIP
Try to reflect some shapes of your own in the y-axis.
SENTENCE STARTERS • I found the pattern . . . so . . .
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Conjecturing and convincing Talk maths
Make a statement
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Conjecturing and convincing
Persuade someone
Conjecturing
Share an idea
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:
a In the function machines shown, the functions are missing. output i input ii input __
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output
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4 3 1
To test your function, put in the input numbers and see if you get the correct output numbers.
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What do you think are the missing functions? Test your functions to see if they are correct. If they are not correct, try a different function and test again. Learners will show they are conjecturing when they ask themselves ‘What are the missing functions?’ They can then try different functions and test them with the input numbers they are given.
TIP
SENTENCE STARTERS • I think that . . .
• I wonder if . . .
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.
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INTRODUCTION TO THINKING AND WORKING MATHEMATICALLY
Example: A builder buys a rectangular piece of land. The dimensions of the land are shown in the diagram. a Work out the area of the land, in hectares. 80 m 350 m
SENTENCE STARTERS • This is because . . . • You can see that . . .
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The cost of the land is $12 400 per hectare. The builder says, ‘This land will cost me more than $34 000.’ b Is the builder correct? Explain your answer. Show your working. Learners will show they are convincing when they do calculations to decide if the builder is correct.
• I agree with . . . because . . .
• I disagree with . . . because . . .
Characterising and classifying
Organise into groups
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Spot a pattern
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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:
a In pairs or groups, discuss the best method to use to draw three chords inside a circle that will make a right-angled triangle. b Individually, draw the diagram from part a. c When all learners in your group have drawn a diagram, compare your diagrams. d What do you notice about the longest chord of the circle?
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Learners will show they are characterising when they identify that the longest chord in the circle that forms a right-angled triangle is in fact the diameter of the circle.
SENTENCE STARTERS • This is similar to . . . so . . . • The properties of . . . include . . .
Classifying
Example:
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Classifying involves organising mathematical objects into groups according to their properties. Learners organise objects or numbers into groups according to their mathematical properties. They use Venn and Carroll diagrams. Yasiru has these cards. The cards have different shapes on them. B
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a Classify the cards into groups. You must have at least two groups. You can choose how you organise them, but you must explain why you have put the shapes in these groups. b Re-classify the cards into different groups. You must have at least two groups. Explain why you have put the shapes into their new groups. Learners will show that they are classifying when they sort the cards into groups. It is up to the learners how they classify the cards; they could use symmetry properties, number of equal angles, number of sides, etc.
SENTENCE STARTERS
• . . . go together because . . .
• I can organise the . . . into groups according to . . .
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INTRODUCTION TO THINKING AND WORKING MATHEMATICALLY
Critiquing and improving Consider the advantages and disadvantages and correct if required
Evaluate the method used
Critiquing
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Critiquing and improving
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:
Aika and Hinata use different methods to answer Question 10a. This is what they write: Aika
Hinata
S 1 : 25 000, o 1 m 25 000 m. 25 000 m ÷ 100 = 250 m 250 m ÷ 1000 = 0.25 m o 1 m 0.25 m. 12 × 0.25 = 3 m
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S 1 : 25 000, o 1 m 25 000 m. So 12 m n p d : 12 × 25 000 = 300 000 m 300 000 m ÷ 100 = 3000 m 3000 m ÷ 1000 = 3 m a
Critique Aika’s and Hinata’s methods. What are the advantages and disadvantages of each method? b Whose method do you prefer? Explain why. This question provides an opportunity for learners to practise critiquing when they are shown two different ways to answer a question. They need to be able to follow the working shown, and choose the method that they think is the best. It also gives them the opportunity to explain why they prefer one method over another.
SENTENCE STARTERS
• The advantages of . . . are . . . / The disadvantages of . . . are . . .
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Improving Improving involves refining mathematical ideas to develop a more effective approach or solution. Learners find a better solution. Example: This is part of Arun’s homework.
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Za d M y a g r $120. Za ys $80 d M ys $40. Ty r l g r $630. Hw h d h f m gt?
Ro r Za : M 80 : 40. Tl r f : 80 + 40 = 120 V f t: $630 ÷ 120 = 5.25 Za g: $80 × 5.25 = $420 M g: $40 × 5.25 = $210 Ck: $420 + $210 = $630 ✓
M
Arun has got the answer correct.
However, some of her calculations were quite difficult and she had to use a calculator.
SA
a How can she make the calculations easier? b Rewrite the solution for her. Do not use a calculator. This question provides an opportunity for learners to look at the method that Rafina uses to answer a question. They can then offer suggestions for improving her method in order to make the calculations simpler to do.
TIP
What extra step could Arun add to simplify her solution?
SENTENCE STARTERS
• It would be easier to . . .
• . . . would be clearer and easier to follow.
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APPROACHES TO TEACHING AND LEARNING
Approaches to teaching and learning 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 approach that places student learning at its centre. It focuses on how Learners 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 that 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 Learner’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, learners 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.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
Metacognition Metacognition describes the processes involved when Learners 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:
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. Creativity – finding new ways of doing things, and solutions to problems
•
Collaboration – the ability to work well with
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Communication – speaking and presenting confidently and participating effectively in meetings
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Critical thinking – evaluating what is heard or read, and linking ideas constructively
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Learning to learn – developing the skills to learn more effectively
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Social responsibilities – contributing to social groups, and being able to talk to and work with people from other cultures.
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•
Cambridge learner and teacher attributes
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This course helps develop the following Cambridge Learner and Teacher attributes. Cambridge learners
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 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 teaching and learning is available to download from Cambridge GO (as part of this Teacher’s Resource).
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication. 24
SETTING UP FOR SUCCESS
Setting up for success Our aim is to support better learning in the classroom with resources that allow for increased student autonomy, whilst supporting teachers to facilitate student 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 problemsolving skills.
PL E
Some ideas to consider to encourage an active learning environment are as follows: •
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.
•
Have a display area with student work and vocab 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.
SA
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
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication. 25
CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
1 Integers Unit plan Approximate number of learning hours
1.1 Adding and subtracting integers
2
1.2 Multiplying and dividing integers
2
1.3 Lowest common multiples
2
1.4 Highest common 2 factors 2
Learner’s Book Section 1.1 Workbook Section 1.1 Additional teaching ideas Section1.1
Understand how to multiply and divide positive and negative integers
Learner’s Book Section 1.2 Workbook Section 1.2 Additional teaching ideas Section 1.2
Find the lowest common Learner’s Book Section 1.3 multiple (LCM) of two Workbook Section 1.3 positive integers Additional teaching ideas Section 1.3 Find the highest common factor (HCF) of two positive integers
Learner’s Book Section 1.4 Workbook Section 1.4 Additional teaching ideas Section 1.4
Test for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, 11
Learner’s Book Section 1.5 Workbook Section 1.5 Additional teaching ideas Section 1.5
Understand the relationship between powers and roots for square numbers and cube numbers
Learner’s Book Section 1.6 Workbook Section 1.6 Additional teaching ideas Section 1.6
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1.6 Square roots and 2 cube roots
Resources
Understand how to add and subtract positive and negative integers
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1.5 Tests for divisibility
Outline of learning content
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Topic
Cross-unit resources: Vocabulary worksheet 1: 1.1–1.3 Vocabulary worksheet 2: 1.4–1.6 Resource sheet 1: Mixed-up properties Diagnostic check End of unit test
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1 INTEGERS
BACKGROUND KNOWLEDGE
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TEACHING SKILLS FOCUS
This unit focuses on subtracting integers by adding the (additive) inverse. Learners will be familiar with subtraction related to the position of two points on a number line. Emphasise that changing the subtraction to an addition is a more efficient method. Multiplication and division of integers will be limited to examples where one of the integers is negative. The calculations will be of the form −6 × 2 or 6 × −2 or −6 ÷ 2. Emphasise that multiplication and division are inverse operations. Learners’ knowledge of common multiples is extended to the idea of a lowest common multiple. Learners’ knowledge of common factors is extended to the idea of a highest common factor. This is useful for simplifying fractions efficiently. Learners will learn that the inverse of squaring is finding a square root. The inverse of cubing is finding a cube root. Appropriate notation is introduced. Examples are limited to square numbers and cube numbers.
PL E
Before teaching Unit 1, you may want to use the diagnostic check activity to assess whether the learners are ready to begin Stage 7. This diagnostic check can assist you as the teacher to identify gaps in the learners’ understanding which you can address before teaching this unit. For this unit, learners will need this background knowledge: • Understand how to display integers on a number line (Stage 5) • Be able to add and subtract integers (Stage 6) • Be able to find the multiples of a positive integer (Stage 5) • Be able to find the factors of any positive integer up to 100 (Stage 5) • Understand common factors and common multiples (Stage 6) • Recognise the first ten square numbers (Stage 5) • Recognise the first five cube numbers (Stage 6).
SA
Assessment for learning When you ask questions to the whole class or to individual learners, it is better to ask open questions rather than closed questions. For closed questions there is just one correct answer expected. The answer will be correct or incorrect. Examples of closed questions are ‘What is 3 × −6?’ or ‘How many factors does 18 have?’ Open questions can have a variety of answers. Examples of open questions are ‘How do you work out 3 × −6?’ or ‘Tell me why 4 is not a factor of 18.’ You can use open questions to assess the understanding of learners. The answers to open questions give you important information about the learner’s progress. Open questions often also lead to further questions and create a dialogue.
If a learner gives an incorrect answer or an unclear explanation, don’t immediately say it is incorrect. Ask for comments or suggestions from other learners. Ask other learners to explain and correct any errors. If the explanation is unclear, ask another learner to improve on it. Learners need to understand that getting incorrect answers is an important part of the learning process. To complete investigations successfully, learners need to feel confident about trying out ideas and testing hypotheses. You can encourage these attitudes in the classroom with suitable questioning. Think about some questions you have asked recently. Were the questions closed or open? Could you rephrase any closed questions to make them open questions?
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
1.1 Adding and subtracting integers LEARNING PLAN Curriculum objective
Learning intentions
Success criteria
7Ni.01
Estimate, add and subtract integers, recognising generalisations.
• Learners can successfully work out additions; e.g. 6 + −8, −5 + 3 and −5 + −3.
PL E
LANGUAGE SUPPORT
• Learners can successfully work out subtractions by adding the inverse; e.g. −2 − 4 = −2 + −4 = −6 or 3 − −5 = 3 + 5 = 8.
Positive integers: the whole numbers greater than zero: 1, 2, 3, 4, . . . An integer is a whole number that can be positive, negative or zero (e.g. 3, −5 and 0). You often show integers on a number line. An inverse operation has the opposite effect of a given operation. The inverse of adding 2 is subtracting 2. The inverse of multiplying by 3 is dividing by 3. The inverse of squaring is finding the square root. The inverse of cubing is finding the cube root.
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Integers: the whole numbers: . . ., −3, −2, −1, 0, 1, 2, 3, . . . Inverse: the operation that has the opposite effect; the inverse of ‘add 5’ is ‘subtract 5’ Inverse operation: the operation that reverses the effect of another operation Negative integers: the whole numbers less than zero: −1, −2, −3, −4, . . . Number line: a line used to show numbers in their correct position
Common misconceptions
How to identify
SA
Misconception
Making mistakes with subtraction
Ask questions such as ‘What is 3 − −4?’ Possible incorrect answers are 1 or −1.
Starter idea
Checking understanding (10 minutes) Resources: Getting started exercise at the start of Unit 1 of the Learner’s Book. Description: Give learners 5 minutes to look at the questions. Then on the board write the statement ‘___ is a multiple of 3.’
How to overcome Encourage learners to always change a subtraction to an addition of the inverse. So 3 − −4 = 3 + 4 = 7.
Ask ‘What numbers between 1 and 20 could you write in the space?’ Then on the board write ‘___ is a factor of 20.’ Ask ‘What numbers between 1 and 20 could you write in the space?’ Then on the board write ‘___ is a square number.’ Ask ‘What numbers between 1 and 20 could you write in the space?’
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1 INTEGERS
During this questioning, listen for uncertainties about the meaning of ‘multiple’, ‘factor’ or ‘square number’. Now ask learners to do the Getting started exercise at the start of Unit 1 of the Learner’s Book.
Main teaching idea Using an inverse operation (5 minutes)
Guidance on selected Thinking and working mathematically questions Characterising and critiquing Learner’s Book Exercise 1.1, Question 14 This question has more than one answer. The number of answers is not specified. Learners must look for all possible answers and convince themselves that they have found all of the possible answers. Giving learners questions with more than one possible answer or even no answer at all discourages them from thinking that a question can have just one answer. This question uses the characterising strategy as learners identify the mathematical characteristics of integers on the number line. What can they identify about integers here? Learners can develop critiquing skills when they compare their thinking with that of their partner’s.
PL E
Learning intention: To emphasise the fact that addition and subtraction are inverse operations. To encourage reflection on methods.
Assessment ideas: Learners can compare their answers with those of a partner and check if they have the same answer. Learners can then work together to correct any incorrect answers.
Resources: Learner’s Book Exercise 1.1, Question 4
Description: Learners should start Exercise 1.1. After they have done Question 4, ask a learner ‘How did you do part a?’ Ask other learners ‘Did you use the same method or a different method?’ or ‘What method did you use?’ Repeat with the other parts of Question 4. Explain how you can change each question to a subtraction. You may have already talked about this in the previous discussion.
For example: You can write 6 + __ = 10 as __ = 10 − 6 = 4. Make sure learners can do the other parts of Question 4 in a similar way. Answers: a 4 b −2 c −10 d −6
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. You may want to look at some of the questions in a subsequent lesson.
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Differentiation ideas: Learners who are unsure should check their answers by doing the appropriate addition.
Homework idea
Plenary idea
Using subtraction (5 minutes)
SA
Resources: Learner’s Book Exercise 1.1, Question 14
Description: Let learners look at this question. Ask ‘Did you use a number line to answer this question?’ Ask ‘What is the difference between 7 and −3?’ [7 − −3 = 10]
Ask ‘How can you use this fact to solve the problem without drawing a number line?’ Take suggestions from learners. They are looking for three numbers. One number is −3 + half of 10 = −3 + 5 = 2. Another number is 7 + 10 = 17. The third number is −3 − 10 = −13. Ask learners to use this method with the numbers −4 and 2. For reflection, you could ask ‘What happens when you reverse the subtraction and work out −3 − 7 = −10? Does the method still work?’
Assessment ideas By asking individual learners questions and listening carefully to their comments you can check their understanding. Some learners will be able to make a general statement. Other learners will prefer to make statements using particular examples. This gives an idea of the learner’s depth of understanding. Where appropriate, learners can compare their answers with those of a partner and check if they have the same answer. Learners can then work together to correct any mistakes and to improve their answers. Several questions in Exercise 1.1 include the opportunity for peer assessment.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
1.2 Multiplying and dividing integers LEARNING PLAN Learning intentions
Success criteria
7Ni.03
Estimate, multiply and divide integers, including where one integer is negative.
Learners can correctly answer questions such as −3 × 5, 4 × −6 and −18 ÷ 3.
LANGUAGE SUPPORT
PL E
Curriculum objective
Product: the result of multiplying two numbers; the product of 9 and 7 is 9 × 7 = 63
Common misconceptions Misconception
Not knowing if an answer should be positive or negative
Starter idea
How to identify
How to overcome
Oral questioning
Emphasise the concept of multiplication as repeated addition.
Similarly, −3 × 4 = −3 + −3 + −3 + −3 = −12.
Multiplication practice (10 minutes)
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Resources: A multiplication square
Ask ‘What other pairs of numbers multiply to make −12?’ [1 and −12, −1 and 12, 2 and −6, −2 and 6]
Description: Show this multiplication square: × 4
9 5
3
SA
8 6
Ask learners to complete the multiplication square as quickly as possible. Check answers by asking learners to read out their answers.
Say ‘The first square shows that 3 × 4 = 12.’ (‘3 multiplied by 4 is equal to 12’). Ask ‘What does this mean?’ Elicit the idea that multiplication is repeated addition. 3 × 4 = 3 + 3 + 3 + 3 or 3 × 4 = 4 + 4 + 4 Both give the same answer. Ask ‘What is 3 × −4?’ (‘What is three multiplied by negative four?’) Look for the suggestion that it is −4 + −4 + −4 = −12.
Use the word product. Say ‘The product of −3 and 4 is −12.’ Learners need to know this mathematical meaning of the word product. It is different from the ‘everyday’ meaning. Use Worked example 1.3 if more practice is necessary.
Main teaching idea Discussing methods (10 minutes) Learning intention: To check learners’ understanding. To show that different methods are possible. Resources: Learner’s Book Description: Learners start working on Exercise 1.2. After a few minutes, look at Question 3. Ask an individual learner ‘How did you get your answer?’ Ask other learners ‘Did you use a different method?’ Two possible methods for Question 3a are: 1 9 × 2 is 18, so the answer must be −2. 2 Change it to a division. The answer is −18 ÷ 9 = −2. Either method is acceptable.
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1 INTEGERS
Now on the board write ‘23 × __ = −391’. Ask ‘How can you find the missing number?’ Here, the easiest method is division (with a calculator): 391 ÷ 23 = 17, so the answer is −17. Do a similar exercise with Question 4. Look at how you can write the divisions as multiplications. Answers: 3 a −2 b −6 c 7 d 5
Plenary idea Review (10 minutes) Resources: Board to write on
• When you subtract a negative number from a positive number, the answer is always positive. • When you subtract a positive number from a negative number, the answer is always negative. For reflection, you could ask ‘Why are general statements such as these statements useful?’ Assessment ideas: Check that learners can use examples to test the general rules.
Guidance on selected Thinking and working mathematically questions
PL E
Differentiation ideas: Some learners might find it difficult to consider different methods. Encourage learners to learn one method and always use that.
Two possibilities are:
Description: Display these statements one at a time:
Learner’s Book Exercise 1.2, Questions 11 and 12
These two questions are an interesting contrast. Question 9 asks straightforward multiplications. Question 10 reverses the situation by giving the answers and asks what the question might be. This is a much better question for encouraging mathematical thinking. It is not clear whether there is more than one solution or none at all. Instead of ‘Here is a question, what is the answer?’ try asking ‘Here is the answer, what is the question?’
Homework idea
Set suitable parts of Workbook Section 1.2 as homework. Workbook Section 1.2 also includes another worked example. The worked example gives learners support when they are working alone. Some of the questions address Thinking and Working Mathematically skills. You might wish to discuss these questions in class.
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1 When you multiply a negative number by a positive number, the answer is always negative. 2 When you divide a negative number by a positive number, the answer is always negative. 3 When you add a negative number and a positive number, the answer is always negative. 4 When you subtract a negative number from a positive number, the answer is always negative. Ask ‘Are these statements correct or incorrect?’
Specialising and generalising
Ask learners to discuss each of these statements in groups for a minute. Then discuss as a whole class to agree on a conclusion. Statements 1 and 2 are correct.
SA
Statement 3 is sometimes correct. Ask ‘When is statement 3 correct?’ It is correct when the negative number is larger in magnitude than the positive number; e.g. −7 and 4, 7 is larger than 4.
Statement 4 is never correct. Ask ‘Can you give a correct general statement about subtraction?’
Assessment idea Use questions in discussions that require learners to explain how they reached an answer. Asking learners to explain is a good way of finding out what they understand. Peer discussion also provides selfassessment opportunities.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
1.3 Lowest common multiples LEARNING PLAN Learning intentions
Success criteria
7Ni.04
Understand lowest common multiple.
Learners can find lowest common multiples and explain their method.
LANGUAGE SUPPORT
PL E
Curriculum objective
Common multiple: a number that is a multiple of two (or more) different numbers; 24 is a common multiple of 2 and 3 Lowest common multiple: the smallest possible common multiple of two (or more) numbers; 24 is
the lowest common multiple of 6 and 8. You can abbreviate lowest common multiple to LCM. Multiple: the result of multiplying a number by a positive integer; the first four multiples of 3 are 3, 6, 9 and 12
Common misconceptions Misconception
How to overcome
A learner might think that because 6 × 4 = 24, then 24 is the lowest common multiple of 6 and 4.
Encourage learners to write the multiples of each number first.
M
Learners might not appreciate the significance of the word lowest in this context.
How to identify
Starter idea
Multiples (15 minutes)
Resources: Each pair of learners needs a dice.
SA
Description: The aim of this activity is to make sure that learners understand the idea of common multiples. Put learners in pairs. Each pair writes the numbers 1 to 20 on a sheet of paper. Learners throw the dice alternately. They cross off any multiple of the number thrown. The game ends when a learner cannot find a number to cross off. Encourage learners to correct their partner if an error is made. Discuss the game afterwards. Ask questions such as ‘Which is the best number to throw?’, ‘Which is the worst number to throw?’ or ‘What number do you need to throw to cross off 10 or 11 or 12?’
Use the word multiple. Say ‘If you throw 3 you can cross off multiples of 3.’
Use the term common multiple. Say ‘You can cross off 15 if you throw 3 or 5. 15 is a common multiple of 3 and 5.’
Main teaching idea An example (5 minutes) Learning intention: To find the LCM by listing multiples. To solve problems systematically. Resources: Learner’s Book, Worked example 1.5 Description: Look at the method used in Worked example 1.5 in Section 1.3. Ask ‘Why is it not necessary to write out the multiples of 10?’, ‘Could you start with the multiples of 10 and then the multiples of 6 afterwards?’ Learners can then start Exercise 1.3. Check on learners’ recall of multiples by asking for multiples of a number chosen at random. Differentiation ideas: If needed, spend more time on this activity or give some learners a second similar example.
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1 INTEGERS
Plenary idea Review of progress (10 minutes) Resources: Learners work in pairs. Each pair needs two dice.
Homework idea 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 of the problems should take place immediately.
PL E
Description: Ask learners to work in pairs. Learners take it in turns to throw two dice twice, adding the scores each time. The learner throwing then finds the lowest common multiple of the two totals. The other learner in the pair decides whether the thrower is correct or not. The learners swap roles and repeat.
another example of this earlier in this unit. A simple question would be ‘Find the LCM of 4 and 9.’ [36] The question ‘The LCM of two numbers is 36, what are the numbers?’ is more challenging, as it has several answers. This type of question also develops understanding, as well as the skills of conjecturing and convincing.
For reflection, ask ‘Can you identify an easy/difficult pair of numbers to answer?’ Assessment ideas: Watch learners as they play. Listen to their comments and be prepared to discuss disagreements.
Guidance on selected Thinking and working mathematically questions
Assessment idea
Opportunities to check learner understanding will occur during discussion. Some activities include peer assessment.
Conjecturing and convincing
Learner’s Book Exercise 1.3, Questions 10 and 11
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Questions 10 and 11 are examples of generating a question by reversing a simple question. There is
1.4 Highest common factors LEARNING PLAN
Learning intentions
SA
Curriculum objective 7Ni.04
Understand highest common factor (numbers less than 100).
Success criteria Learners have strategies to find the highest common factor of two numbers.
LANGUAGE SUPPORT
Common factor: a number that is a factor of two (or more) numbers Factor: a factor of an integer will divide into that integer without a remainder; 6 and 8 are factors of 24
Highest common factor: the largest factor of two (or more) other numbers. You can abbreviate highest common factor to HCF. 7 × 8 = 56, so 7 and 8 are factors of 56. 1 is a factor of any positive integer. 5 is a common factor of 15 and 40.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
Common misconceptions Misconception
How to identify
How to overcome
Learners do not think of 1 and the number itself as factors of a given number.
Check when learners list the factors.
Remind learners of these two factors as much as possible.
Starter idea
For example,
Reminder of factors (10 minutes)
24
or
18 45
or
30 40
.
Show learners that you could divide the numbers 16 and 16
PL E
Description: On the board write ‘20 is a multiple of 5.’
16
Then, under the first sentence, write ‘5 is a ______ of 20.’
24 in the fraction
Ask ‘What is the missing word?’ Learners should know the word factor.
you use the HCF, then you need to do only one division.
Ask learners in turn to give more pairs of similar sentences. The first should be ‘20 is a multiple of __.’
Answers: 7 a 7
The factors of 20 are 1, 2, 4, 5, 10 and 20. Make sure learners do not forget 1 and 20.
b
Then collect their answers. Ask ‘Can you see any patterns?’
SA
• ‘It is not true that larger numbers always have more factors.’ • ‘Prime numbers have just two factors.’ (Learners might not know about prime numbers.) • ‘Square numbers have an odd number of factors.’ If learners do not give the answers listed, then elicit them. The act of looking for factors and writing all the factors of a number is the focus here. Look at learners’ work as they are finding factors. Make sure they always include 1 and the number itself. Make sure learners use the words factor and multiple correctly.
Main teaching idea
Simplifying fractions (5 minutes)
Learning intention: To make connections between different areas of mathematics.
5
8
Reflection For example: When you divide the numerator and the denominator by the highest common factor, you have the fraction in its simplest form. 8 a 1
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Possible answers include:
repeatedly by 2 or by 4. However, if
Question 7 is an example of this. Learners should consider this when they answer Question 8.
Ask ‘Have we got all the factors?’ Learners should be able to agree that there are six factors.
Now, ask learners to work in pairs to find numbers with exactly: four factors, two factors, eight factors, three factors and five factors.
24
b You simplify
25 36
by dividing 25 and 36 by a
common factor. Since 1 is the only common factor, the fraction cannot be simplified.
Differentiation ideas: Using the HCF is not essential. Learners who find it difficult can use repeated division, using any common factor each time. The activity can be extended to include vulgar fractions.
Plenary idea The relation between factors and multiples (10 minutes) Resources: Squared paper Description: Ask learners to number a grid, as shown here. Instruct them to use a pencil so that they can correct any mistakes.
Resources: Learner’s Book Exercise 1.4, Questions 7 and 8 Description: Write some fractions and ask learners to simplify them.
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1 INTEGERS
Multiples 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 Number 6 7 8 9 10
Ask ‘What do you notice about the columns?’ The columns contain the factors of the number. Learners can extend the grid if desired. This shows the link between factors and multiples. Ask ‘Can you describe the link between factors and multiples to a partner?’ Assessment ideas: Check that learners draw the grid correctly.
PL E
Guidance on selected Thinking and working mathematically questions
Specialising and generalising
Learner’s Book Exercise 1.4, Question 8
On each row in turn, mark a cross on the multiples of 1, 2, and so on. The first two rows are as shown. Multiples 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 Number 6 7 8 9 10
This question links simplifying fractions to highest common factors. Often, mathematics is taught as separate topics. It is important to emphasise the connection between different topics whenever you can. When making a conjecture, it is helpful to have this broader view and not to be confined within one particular topic. Often, discoveries in mathematics are made by seeing connections between apparently different topics.
Homework idea
M
At the end of each lesson, set suitable parts of Workbook Section 1.4. Set only those questions that can be answered using the 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 or discussions with any of the problems should take place immediately.
Shown here is the completed grid.
SA
Multiples 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 Number 6 7 8 9 10
Assessment idea Assessment can be from learners’ comments made during discussions or by observing them when working in pairs. Listen to learners’ explanations. Are they clear? Do they use appropriate mathematical vocabulary? Are they convincing? There are several opportunities to use peer assessment.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
1.5 Tests for divisibility LEARNING PLAN Learning intentions
Success criteria
7Ni.05
Use knowledge of tests of divisibility to find factors of numbers greater than 100.
Learners can show that 46 128 is divisible by 6 but is not divisible by 9.
LANGUAGE SUPPORT
PL E
Curriculum objective
Divisible: one whole number is divisible by another whole number when it is a multiple of that other whole number Tests for divisibility: tests you can use to decide if one number is divisible by another number
24 is divisible by 3 because 24 ÷ 3 is exactly 8 with no remainder. This is equivalent to saying 3 is a factor of 24. Similarly, 24 is not divisible by 5 and 5 is not a factor of 24.
Common misconceptions Misconception
How to overcome
Carefully check that learners can answer the questions in Exercise 1.5 correctly.
Use examples as part of whole class discussions. Ask learners to explain their method.
M
Adding the wrong digits when using the rule for divisibility by 11.
How to identify
Starter idea
Divisibility by 3 (20 minutes)
SA
Description: Ask ‘What are the multiples of 3?’ As learners give you multiples of 3, write the numbers on the board.
Ask learners to work in pairs. Learners add the digits of the numbers. Ask ‘Can you make a conjecture?’ Give them a few minutes. If learners make a conjecture, they should check it for other multiples of 3. After a few minutes, ask for their answers. Learners might present their results in different ways. The answer you want is that the sum of the digits of any multiple of 3 is itself a multiple of 3. They might, for example, say that the sum is always 3 or 6 or 9 or . . . Ask ‘Is this true for all multiples of 3?’ Ask learners to give you a large multiple of 3. (They could use a calculator to find a large multiple by working out 3 × a large number.) As a class, agree that this is always the case.
Say ‘69 is a multiple of 3. This means that 69 is divisible by 3. 69 ÷ 3 does not have a remainder.’ Ask learners to say sentences using the word divisible. Tell learners the test for divisibility by 3: ‘A number is divisible by 3 when the sum of the number’s digits is divisible by 3.’ Give more examples to make sure learners understand this test. Now ask ‘Can you find a test for divisibility by 9?’ Learners work in pairs again. Check the results after a few minutes. The answer you want is that ‘a number is divisible by 9 when the sum of the number’s digits is divisible by 9’. To check learners’ understanding of the vocabulary, ask them to rephrase a sentence with the word factor as a sentence with the word divisible, and vice versa. For example, ‘6 is a factor of 42’ and ‘42 is divisible by 6’. You could say ‘42 is a multiple of 6’ is the same as ‘42 is divisible by 6’.
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1 INTEGERS
Main teaching idea
Listen to the discussion between pairs of learners as they work on the problem. Do their comments show that they know and can use the tests?
Missing digits (15 minutes) Learning intention: To use tests for divisibility Description: On the board write the number ‘472*’. Explain that the final digit is missing. Ask ‘This number is divisible by 2. What is the missing digit?’ This should not be difficult for learners to answer.
Divisible by Final digit 2 0, 2, 4, 6, 8 3 4 5 6 7 8 9 10 11
Plenary idea
Review (10 minutes)
Description: On the board write the digits 1, 3, 5, 6, 9. Say that a number has these five digits.
Ask ‘Is this number divisible by 2?’ [Only if 6 is the final digit.]
Ask ‘Is this number divisible by 3?’ [Yes, because the sum of the digits is 24.]
Continue to ask about divisibility by 4, 5, 6, 8, 9, 10, 11. Learners could discuss this in pairs before you check the answers as a class discussion. For reflection, you could ask ‘Why are the tests for 3 and 9 similar?’ Assessment ideas: This is an opportunity to check that learners have learnt the divisibility tests and can use them.
M
Ask learners to work in pairs to complete the table. Listen to learners’ discussions as they complete this task, to identify misconceptions. This is the completed table: Divisible by Final digit
0, 2, 4, 6, 8 2, 5, 8 0, 4, 8 0, 5 2, 8 5 0, 8 5 0
SA
2 3 4 5 6 7 8 9 10
Differentiation ideas: As an extension activity, change the number to 381*2 and ask learners to repeat the exercise. For this number, some of the rows in the table are impossible to complete.
PL E
Write down this table:
Answers: See table on previous page.
When learners have finished, check their tables and discuss the methods they used. Ask ‘Which row in the table was the easiest to complete?’, ‘Which row in the table was the most difficult to complete?’ Ask ‘How did you find the final digit when the number is divisible by 7?’ 7 has no test, so doing a division is the only way to find the missing digit.
Guidance on selected Thinking and working mathematically questions Conjecturing and convincing Learner’s Book Exercise 1.5, Question 13 To prove that a statement is false just requires a counterexample. To prove that a statement is true requires reasoning and can be more difficult. The first statement in Question 13 is: ‘A number is divisible by 8 when it is divisible by 2 and by 4.’ A simple counterexample is 12, which is divisible by 2 and 4 but is not divisible by 8. This is enough to show that the statement is false. There are lots of other examples but you only need to give one example. The second statement is: ‘A number is divisible by 10 when it is divisible by 2 and by 5.’ This is true. Here is a possible proof:
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication. 37
CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
A number divisible by 2 ends in 0, 2, 4, 6 or 8. A number divisible by 5 ends in 0 or 5. A number that is divisible by both 2 and 5 must therefore end in 0. This means that the number is divisible by 10. The third statement is: ‘A number is divisible by 15 when it is divisible by 3 and by 5.’ This is true because 3 and 5 are prime numbers, but a proof is beyond the scope of this book.
At the end of each lesson, set suitable parts of Workbook Section 1.5. Set only those questions that can be answered using the 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 or discussions with any of the problems should take place immediately.
PL E
Learners might convince themselves that it is true by looking at the form of numbers that are divisible by 3 and 5, using the tests to find them.
Homework idea
Numbers ending in 0 are 30, 60, 90, 120, 150, . . . Numbers ending in 5 are 15, 45, 75, 105, 135, . . .
It is apparent that these numbers are the multiples of 15, but this is not the same as a proof.
Assessment idea
The aim in this section is to make learners familiar with the tests for divisibility so that they can apply them confidently to any number. You can use peer assessment as a check and reinforcement. Working in pairs, one learner writes a number and asks whether it is divisible by a specified number, such as 3, 5 or 11. The other learner’s answer should include a reason. The first learner must judge whether this is correct.
1.6 Square roots and cube roots
M
LEARNING PLAN
Learning intentions
Success criteria
7Ni.06
Understand the relationship between squares and corresponding square roots, and cubes and corresponding cube roots.
Learners can write the relationship between a square number or cube number and its square or cube root in different ways, using the correct symbols.
SA
Curriculum objective
LANGUAGE SUPPORT
Consecutive: two numbers are consecutive if they are next to each other when written in order Cube number: the result of multiplying three ‘lots’ of the same whole number together; 125 is a cube number equal to 53 = 5 × 5 × 5 = 125 Cube root: the number that produces the given number when three ‘lots’ of the number are multiplied together; the cube root of 125 is 3 125 = 5
Index: a number used to show a power; in 23, 3 is the index Square number: the result of multiplying a whole number by itself; 81 is a square number equal to 92 = 9 × 9 = 81 Square root: the square root of a number, multiplied by itself, gives that number; the square root of 36 is 36 = 6
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1 INTEGERS
CONTINUED The square root of a given number is the number whose square is that number. The square root of 49 is 49 = 7 because 72 = 49. Square numbers have an integer square root.
The cube root of a given number is the number whose cube is that number. The cube root of 64 is 3 64 = 4 because 43 = 64. Cube numbers have an integer cube root.
Common misconceptions How to identify
How to overcome
Misinterpreting index notation for squares and cubes.
Asking learners the value of a square number or cube number.
Reinforce the meaning of squared and cubed whenever they are used.
Using the root symbols incorrectly.
Look at learners’ written work.
Always model correct use.
Starter idea
PL E
Misconception
Square numbers and square roots (10 minutes) Resources: Board to write on
Description: On the board write ‘16, 81, 25’. Ask ‘What do these numbers have in common?’ These numbers are square numbers.
Repeat the activity with 90 and ask for comments again. This time learners can say that it is not an integer and it lies between 9 and 10. However it is not obvious whether it is closer to 9 or 10.
M
Ask learners to write ten more square numbers. Check their answers.
Description: Learners should not use a calculator during this activity. Write down 16 = 4 and 25 = 5. Then write down 17 and ask learners what they can say about it. Possible answers are: it is not a whole number; it is between 4 and 5; it is close to 4 but a little bit larger. Make sure that learners can understand how to find two consecutive integers between which the square root must lie.
Write ‘42 = 16’ on the board and say ‘4 squared equals 16’. Ask learners to say similar statements.
Introduce the term square root: ‘If 4 squared equals 16, then the square root of 16 equals 4.’
SA
Using symbols, on the board write: ‘42 = 16’ and ‘ 16 = 4’.
Ask learners to write similar expressions for other square numbers. Use peer assessment in pairs for learners to check each other’s expressions and to identify any errors. Resolve any problems. Learners can now do Exercise 1.6, Questions 1 and 2.
Main teaching idea
Approximations with square roots (10 minutes)
Learning intention: To understand that roots of any number can be approximated using known results. To be able to find the integer closest to a particular root, and identify two consecutive integers between which a root must lie.
You can do the same thing with a cube root. Ask about 3 25 . Since 23 = 8 and 33 = 27 it is between 2 and 3 and 3 will be the closest integer. Differentiation ideas: Some learners may find it easier to understand if you start with a list of square numbers. A number line can be a useful visual aid. For more confident learners you can introduce the notation 4 < 17 < 5. Assessment ideas: Use this as an opportunity to check that learners can confidently move between a statement about a square and a statement about a square root.
Plenary idea Reinforcing ideas (5 minutes) Description: Ask learners to write the first eight square numbers and the first four cube numbers. Give learners 2 minutes to complete the task. Then use peer assessment in pairs for learners to check that the answers are correct.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
You could ask ‘Which increase in size is faster: square numbers or cube numbers? Could you plot them on a graph to show this?’
Homework idea
Assessment ideas: You can see as learners are working on this whether they know their square and cube numbers.
At the end of each lesson, set suitable parts of Workbook Section 1.6. Set only those questions that can be answered using the 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 or discussions with any problems should take place immediately.
Guidance on selected Thinking and working mathematically questions
PL E
Generalising
Workbook, Section 1.6
Learner’s Book, Exercise 1.6, Question 14
Using the diagram in Question 14d is very helpful to show the pattern. The diagram can also lead to a convincing demonstration of why the results are true. Diagrams can be very helpful to make conjectures and to move from a particular result to a general result. In Question 14d, the diagram illustrates a general result for any number of odd numbers. The diagram can be more effective to convince learners of the result because they can literally ‘see’ it.
Assessment idea
Use peer assessment and class discussion. Learners can assess their own progress in the unit at the end of this section. Some of the questions in the Exercises and some of the lesson ideas give examples of this.
PROJECT GUIDANCE: MIXED-UP PROPERTIES
Set learners the challenge of arranging all nine property cards in the grid. Explain that they can choose how to arrange the number cards. While the learners work on the task in pairs, walk around and listen to their reasoning. Listen for any misconceptions that arise. Once learners have found an arrangement that works, invite them to consider if it is possible to swap any property cards to make new arrangements. Finish off by discussing learners’ insights and discoveries. Key questions Which cards could go in lots of different places? Which cards can go in only a few places? Possible support Learners could start with a smaller grid, using just four of the numbers. Possible extension Challenge students to find arrangements of numbers or property cards for which it is impossible to complete the grid.
SA
M
Why do this problem? This problem challenges learners to apply their knowledge of properties of numbers in an engaging context. This activity helps to develop the skills of characterising and classifying by encouraging learners to identify the properties of a set of numbers and to organise them accordingly. Possible approach Each pair of learners will need a set of cards. Draw a table on the board, like the one below. 4
2
5
7
*
3 6
Download Resource sheet 1: Mixed-up properties. Invite learners to suggest property cards that could go in the cell marked *.
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2 EXPRESSIONS, FORMULAE AND EQUATIONS
Unit plan Topic
PL E
2 Expressions, formulae and equations Approximate Outline of learning content number of learning hours
Introduction and 30 minutes Getting started 2.1 Constructing 1–2 expressions
Resources
Learner’s Book Getting started
Learner’s Book Section 2.1 Understand that you can use letters to represent unknown Workbook Section 2.1 numbers, variables or constants. Additional teaching ideas Section 2.1 Understand that the order of operations applies to algebraic terms and expressions (four operations).
M
Understand that you can represent a situation in words or as an algebraic expression.
1–2
Understand that you can Learner’s Book Section 2.2 represent a situation in words or Workbook Section 2.2 as a formula. Additional teaching ideas Section 2.2
2.3 Collecting like terms
1–2
Collect like terms of algebraic expressions.
Learner’s Book Section 2.3 Workbook Section 2.3 Additional teaching ideas Section 2.3
2.4 Expanding brackets
0.5–2
Understand how to expand brackets using a constant.
Learner’s Book Section 2.4 Workbook Section 2.4 Additional teaching ideas Section 2.4
SA
2.2 Using expressions and formulae
2.5 Constructing 0.5–2 and solving equations
Understand that you can Learner’s Book Section 2.5 represent a situation in words or Workbook Section 2.5 as an equation and then solve Additional teaching ideas Section 2.5 the equation.
2.6 Inequalities
Understand that you can use letters to represent an open interval.
0.5–2
Learner’s Book Section 2.6 Workbook Section 2.6 Additional teaching ideas Section 2.6
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
Topic
Approximate Outline of learning content number of learning hours
Resources
PL E
Cross-unit resources: Resource sheet 2.6: Key words Vocabulary worksheet 1: 2.1–2.3 Vocabulary worksheet 2: 2.4–2.6 End of unit test
BACKGROUND KNOWLEDGE
Here is the main difference between arithmetic and algebra: Arithmetic: 2 + 3 = __ Algebra: 2 + 3 = x where the x represents a number you don’t know the value of yet. This can be very simple but, as it is used almost all the time in science, engineering, economics, computer programming and it also has a huge part to play in the rest of mathematics, it can also be very difficult! Think of it as a game in which a simple answer has been hidden in a more complicated situation and it is your job to get to the simple answer.
M
For this unit, learners will need this background knowledge: • Be able to multiply and divide with positive and negative numbers (Stage 7, Unit 1). This is the first unit in which learners will use algebra. To be successful in this unit, learners will need basic addition and subtraction skills. Learners will also need to be aware of negative numbers and to be able to multiply and divide a negative integer by a positive integer. Algebra is very similar to arithmetic. It uses the same rules, such as +, − , × and ÷. In arithmetic, the only unknown part of anything is the answer. Algebra introduces the use of an unknown value, which you usually show as any letter of the alphabet. Often, you use the letter x.
TEACHING SKILLS FOCUS
SA
Language awareness To help you to highlight and concentrate on language awareness, take time before the lesson to make sure you know the key words learners will meet during a unit. Make sure you are clear in your understanding of the key words/terms. Use the glossary if necessary. Give all learners a copy of Resource sheet 2.6: Key words. You can download this resource from Cambridge GO. Read out each word/term in turn. Afterwards, ask learners ‘Do you know what any of these key words mean?’ Discuss any ideas learners have. Emphasise
that by the end of the unit they will know the meaning of all of these key words. As you work through the unit, refer to Resource sheet 2.6: Key words. Encourage learners to fill in (with an explanation or an example) the meaning of a word/term in the list when they meet each word/ term in the unit. An alternative is to look at the key words at the end of the unit. If you choose to ask learners to complete Resource sheet 2.6: Key words as you work through Unit 2, you can still give another copy of the Resource sheet at the end of the unit to check learners’ understanding.
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2 EXPRESSIONS, FORMULAE AND EQUATIONS
CONTINUED At the end of Unit 2, ask yourself ‘Do the learners understand and feel confident in using the key words language?’ If the answer is yes, then this work has been successful. If the answer is no, then how can you improve how you discuss and use the key words?
PL E
During each section, refer to the key words/terms as often as possible. Encourage learners to use the key words/terms during any classroom discussions. When a learner uses a key word/term, ask another learner what the key word/term means. If you do this throughout the unit, you could give learners Resource sheet 2.6: Key words as a class test at the end of Section 2.6.
2.1 Constructing expressions LEARNING PLAN
Learning intentions
Success criteria
7Ae.01
Understand that letters can be used to represent unknown numbers, variables or constants.
Learners understand Worked example 2.1 and the suggested discussion after in ‘Common misconceptions’ below.
7Ae.02
Understand that the laws of arithmetic and order of operations apply to algebraic terms and expressions (four operations).
Learners understand that the order of operations rules apply to algebra. Learners can use these rules to write algebraic terms and expressions.
7Ae.04
Understand that a situation can be represented either in words or as an algebraic expression, and move between the two representations (linear with integer coefficients).
Learners can read an expression written in words and convert it to an expression written in algebraic terms, and vice versa.
SA
M
Curriculum objective
LANGUAGE SUPPORT
Coefficient: a number in front of a variable in an algebraic expression; the coefficient multiplies the variable Constant: a number on its own (with no variable) Equation: two different mathematical expressions, both having the same value, separated by an equals sign (=) Equivalent expression: an expression that means the same thing as another expression
Expression: a collection of symbols representing numbers and mathematical operations, but not including an equals sign (=) Term: a single number or variable, or numbers and variables multiplied together Unknown: a letter (or letters) in an equation, for which the value (or values) is yet to be found Variable: a symbol, usually a letter, that can represent any one of a set of values
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
CONTINUED Use terms such as ‘twice’ (× 2), ‘three times’ (× 3) and ‘half’ (÷ 2), and check that learners understand and can use these terms. For learners who have difficulty with some of the language, read the questions aloud using emphasis
to aid understanding and therefore learning. For example, after reading ‘I think of a number, x. I multiply the number by 6’ in your usual tone, emphasise ‘then I add 1’.
Misconception
PL E
Common misconceptions If a letter has been used (e.g. x) and a value is suggested (e.g. x could be 6), then x is always 6.
That the order of operations does not apply because ‘they are letters, not numbers’.
How to overcome
In Worked example 2.1, suggest that Mathew is 6 years old, then work out the ages of the other children. Ask ‘Does Mathew have to be 6 years old?’ A ‘yes’ reply shows misunderstanding.
Ask ‘If Mathew is 8 years old, how old are the other people?’ Then work out the ages. Now ask ‘If Mathew is 4 years old, how old are the other people?’, etc.
Repeated errors with Question 6.
Remind learners that the order of operations does apply here because all letters can have a number value. ‘In algebra, a letter represents a number that you don’t know the value of yet.’
Explain that the x is there because you do not know Mathew’s age.
Resources: Learner’s Book Exercise 2.1, Question 5; notebooks or mini whiteboards.
M
Starter idea
How to identify
Getting started (15 minutes)
Resources: Notebooks, Learner’s Book Getting started exercise
SA
Description: Learners should have little difficulty with the Getting started exercise. Before learners attempt the questions, discuss what they remember about the order of operations. Ask learners for any questions they could think of that might be done incorrectly if they didn’t use the correct order of operations. If necessary, guide them towards discussing questions such as 5 + 3 × 2 and (10 − 4) ÷ 2.
Main activity idea
Question 5 Think like a mathematician (15–20 minutes) Learning intention: To understand that the laws of arithmetic and order of operations apply to algebraic terms and expressions (four operations). To understand that a situation can be represented either in words or as an algebraic expression. To gain confidence with using mathematical language.
Description: This activity gives learners practice with using the correct order of operations with algebraic terms. On the board write/display: a I think of a number x. I multiply the number by 5, then subtract 3. b I think of a number x. I multiply the number by 5, then subtract the result from 3. Put learners into suitable pairs or small groups. Learners read parts a and b. Ask ‘What is the main difference between the two sentences?’ Let learners discuss this (maximum of 2 minutes). Then discuss as a class. During this discussion, a learner might say that parts a and b are the same. Ask another learner to explain why this is not true. If learners are still unsure, read parts a and b again, but replace x with a number, for example, 10. This will show that the two statements are not the same. Following the discussion, ask one learner from each pair/group to read out [or write on the board or on a mini whiteboard] an answer. Ask another pair/group if
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2 EXPRESSIONS, FORMULAE AND EQUATIONS
the answer is correct or not. If the answer is incorrect, explain what mistakes have been made. Now click on the DC icon. Allow learners to attempt Question 5. Learners should write parts a to e and their answers in their notebooks.
d
b 4x − 9
x + 7 2
c
x − 1 6
e 25 − 2x
Differentiation ideas: Both for pairs and small groups, putting a mathematically able learner with weaker learners can help all learners develop. However, beware of learners simply copying from or relying on another learner. Putting equally matched learners together might slow some of the discussions, but you will be more confident that all learners understand when they have the correct answers.
Plenary idea
What have you learnt? (5 minutes) Resources: Notebooks
By rewriting cards B, C, D, F, K and L, learners should easily group cards appropriately, [ADGK, BI, CHJL, EF]. Discuss with learners that as x + y and y + x are equivalent expressions (the same, although they look a little different), do they think that x − y and y − x are equivalent expressions too. What evidence or logic can be used to decide if x − y and y − x are equivalent expressions.
PL E
Answers: a 6x + 1
For most classes, the best way to approach this question is for learners to write down each description and then to rewrite it in its usual form; that is, simplified. Part a will allow learners to be reminded that 2n + 3, 2 × n + 3 and 3 + 2 × n are all equivalent, but that 2n + 3 is the usual way that a mathematician would write it.
Workbook, Section 2.1
As Section 2.1 will probably take more than one lesson, at the end of each lesson set suitable parts of the Workbook as homework. Set only those questions that can be answered using the 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 or discussions with any problems should take place immediately.
M
Description: Learners write an explanation of how to decide if brackets are needed when reading a description and writing out its expression. Learners could give some examples to make their explanation clearer. Include examples of division, as well as multiplication.
Homework idea
SA
When learners have completed writing their explanations, ask ‘Will the explanation you have written be useful for revision later in the year?’, ‘Is there anything you now would like to add?’ Assessment ideas: Learners could swap explanations with a partner to check that their explanations are clear and easily understood. When learners get their work back, they can change anything in their explanation to make it more easily understood.
Guidance on selected Thinking and working mathematically questions Characterising and classifying
Ask learners to make up their own question (and to give answers on a separate piece of paper) that is similar to Question 9 of Exercise 2.1 in the Learner’s Book. For marking, learners, working in pairs, can swap homework, answer their partner’s question (in a method similar to how they attempted Question 9) and then discuss while comparing answers. This will take approximately the same amount of time as was taken to complete Question 9, so you will need to plan this time into the lesson structure.
Assessment idea Allow learners to compare answers. This will encourage discussion when learners have different answers. This discussion should lead to deeper understanding. Learners can check their understanding by self-marking when you read out the answers or write the answers on the board.
Learner’s Book, Exercise 2.1, Question 7 In this question, learners must classify expressions into groups of equivalent expressions. They need to know what characterises an equivalent expression.
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CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
2.2 Using expressions and formulae LEARNING PLAN Learning intentions
Success criteria
7Ae.05
Understand that a situation can be represented either in words or as a formula (single operation), and move between the two representations.
Learners can read a formula written in words and convert it to a formula written in algebraic terms, and vice versa.
LANGUAGE SUPPORT
PL E
Curriculum objective
Derive: construct a formula or work out an answer Formula: an equation that shows the relationship between two or more quantities
Formulae: plural of formula Substitute: replace part of an expression, usually a letter, by another value, usually a number
Common misconceptions Misconception Not using the correct order of operations.
How to overcome
Incorrect answers to parts f, g of Question 2
Check answers to part f of Question 2 before allowing learners to continue.
If lots of learners get incorrect answers, write the word on the board and give plenty of time for learners to get familiar with this new process. If that doesn’t work, ask a learner who has the correct value to explain how they worked it out.
M
Starter idea
How to identify
M + A + T + H + S (10 minutes) Resources: Mini whiteboards
SA
Description: This activity encourages fluency with mental substitution. Write the title on your whiteboard. Underneath the title, write/display:
Main activity idea
‘M = 1, A = 2, T = 3, H = 4 and S = 5.’
Biggest and smallest (10–15 minutes)
Say ‘I will say a word. You add together the value of the letters in your head, write the answer on your whiteboard and then hold it up.’
Learning intention: To gain confidence with repeated substitution and gain fluency in simple substitution.
Give an example: ‘What is the value of SAT? SAT is spelt S, A, T. You need to add together 5, 2 and 3, which equals 10.’
Resources: Notebooks or mini whiteboards
Ask learners for the value of any word spelt from the letters MATHS that is suitable for the class. For example: AT = 5, AS = 7, AH = 6 (or HA = 6), HAHA = 12, AM = 3 (or MA = 3), MAT = 6, HAT = 9, MATS = 11, HATS = 14, MAST = 11, SHAM = 12, SMASH = 17, ASTHMA = 17, SHAMMAS = 20, etc.
Description: Ask learners to copy this table. Value of a
a + 5
a − 4
a × 3
a ÷ 2
5
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2 EXPRESSIONS, FORMULAE AND EQUATIONS
You want learners to fill in only the biggest and smallest values. They should leave the others blank. Give an example. In the first row, give the value as 5 (as shown in the table), and do your thinking ‘out loud’. Say ‘Which will give the biggest answer? Probably either the add or the multiply. 5 + 5 = 10, but 5 × 3 = 15, so a × 3 gives the biggest answer.’ Write 15 under the a × 3 heading.
Now give the other values of a, which are 10, 2, −1 [emphasise the negative number. Tell learners to be careful!], −5, −20. Remind learners that you want only the biggest and smallest answers, none of the others. Answers: Value of a
a + 5
5
a − 4 1
10 −2
−1
4
−5
−5
0
Resources: Notebooks or mini whiteboards Description: Tell learners that a builder, Matilda, installs six internal doors into every house she builds. Question 1: How many internal doors does Matilda need for two houses? [Ask for an answer from a learner who rarely offers answers.] Answer = 12 doors. Ask ‘How did you get the answer?’ If they say 6 × 2 = 12 (or 2 × 6 = 12), move on. If the reply is 6 + 6 = 12, agree that it works, but suggest that an easier method might be six doors × number of houses, two, so 6 × 2 = 12.
a ÷ 2
15
5
−15 −60
−10
Question 2: Write a formula, using letters, for the number of doors needed in x houses.
Ask for learners’ answers. D = 6x (where D is the number of doors) will be the most common answer, but any letter = 6x is acceptable. Remind any learners who wrote D = 6h that you said x houses.
Question 3: Matilda also uses the formula W = 7x. Write this formula on the board. If x is the number of houses, discuss what W could be. Write a formula in words for W = 7x (e.g. the number of windows needed = 7 × the number of houses). Hopefully ‘windows’ will be the obvious answer, but accept any answer for which learners might think that seven per house might be needed.
M
7
−20
a × 3 30
2
Matilda the builder (5–10 minutes)
PL E
Next, say ‘Which will give the smallest answer? Probably either the subtraction or the division. 5 − 4 = 1, and 5 ÷ 2 = 2.5, so a − 4 gives the smallest answer.’ Write 1 under the a − 4 heading.
Plenary idea
SA
Ask learners to compare their table with that of a partner and to check that their tables are the same. If their tables are different, they can try to work out the correct answer. Only help if learners cannot agree. Ten out of 12 correct should be the minimum standard. If a learner’s score is below that, either they have not understood the task or they have not understood negative numbers.
Differentiation ideas: If learners find it difficult to answer mentally, let them fill in all cells in the table and ask them to circle the biggest and smallest numbers. Also encourage these learners to do written workings. For the more able learners, as an extension, ask them if there are any integer values for a that give a pair of equal biggest answer [no, there are not] or a pair of equal smallest answer [yes, 8, −2 and −10]. This could be set as a possible homework, but tell learners to draw a new table, starting with the ‘Value of a’ as 10 and then 9, 8, 7, 6, . . . , and to find any patterns that might exist.
Assessment ideas: Learners can self-mark after each question.
Guidance on selected Thinking and working mathematically questions
Critiquing and improving Learner’s Book, Exercise 2.2, Question 10 Part a is the most important part of this question. Learners must understand that until they have been told what the letters represent, they could mean anything. So, x could represent the cost of an adult ticket or the cost of a child ticket (or anything else for that matter; x could represent the number of letters in their uncle’s name!), but until you write the meaning of the letter you just don’t know. If learners understand that x + y could mean ‘cost of an adult ticket + cost of a child ticket’ or ‘cost of a child ticket + cost of an adult ticket’, they have understood the basic concept.
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication. 47
CAMBRIDGE LOWER SECONDARY MATHEMATICS 7: TEACHER’S RESOURCE
Homework idea
Assessment idea
Workbook, Section 2.2
Allow learners to alter their answers after discussion, and then to self-mark.
As Section 2.2 will probably take more than one lesson, at the end of each lesson set suitable parts of the Workbook 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.
LEARNING PLAN
PL E
2.3 Collecting like terms Curriculum objective
Learning intentions
Success criteria
7Ae.03
Understand how to manipulate algebraic expressions by collecting like terms.
Learners can collect like terms, understanding that ab and ba are equivalent.
LANGUAGE SUPPORT
Simplify: gathering, by addition and subtraction, all like terms to give a single term
M
Collecting like terms: gathering, by addition and subtraction, all like terms Like terms: terms containing the same letter(s)
Common misconceptions
How to identify
How to overcome
5a − a = 5
See Starter idea.
See Starter idea and Main activity.
SA
Misconception 3a + 4b = 7ab
See Starter idea.
See Starter idea and Main activity.
5ab + 2ab + 3ba = 7ab + 3ba
See Starter idea in the additional teaching ideas and discussion.
See Starter idea in the additional teaching ideas and discussion.
4xy + 3yx cannot be simplified.
Discussion during Question 7.
Discussion during Question 7 (see Main activity in the additional teaching ideas).
Starter idea
Fruity! (10 minutes) Resources: Notebooks
Description: Do the following activity before going through Section 2.3 introduction or Worked example 2.3.
Write the section title ‘Collecting like terms’ on the board. Explain that this is a very important part of algebra, but that they are probably very good at it already. Ask ‘What is three apples add four apples?’ Hopefully, learners will say ‘seven apples’.
Original material © Cambridge University Press 2020. This material is not final and is subject to further changes prior to publication. 48
