Cambridge Checkpoint Mathematics Teacher s Resource 9 Cambridge Education Cambridge University Press Samples

Cambridge Checkpoint Mathematics 9 matches the requirements of stage 9 of the revised Cambridge Secondary 1 curriculum framework. It is endorsed by Cambridge International Examinations for use with their programme. This Teacher’s Resource is intended to be used alongside the Cambridge Checkpoint Mathematics Coursebook 9 and Practice Book 9. The Teacher’s Resource CD-ROM contains: • answers to exercises from the Coursebook • answers to exercises from the Practice Book • teaching notes for each topic • resource sheets • an end-of-year review. Other components of Cambridge Checkpoint Mathematics 9:

To find out more about Cambridge International Examinations visit www.cie.org.uk Visit education.cambridge.org/cie for information on our full range of Cambridge Checkpoint titles including e-book versions and mobile apps.

ISBN 978-1-107-69397-5

9 781107 693975

Mathematics

Teacher’s Resource

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Completely Cambridge – Cambridge resources for Cambridge qualifications Cambridge University Press works closely with Cambridge International Examinations as parts of the University of Cambridge. We enable thousands of students to pass their Cambridge exams by providing comprehensive, high-quality, endorsed resources.

Cambridge Checkpoint

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Coursebook 9 ISBN 978-1-107-66801-0 Practice Book 9 ISBN 978-1-107-69899-4

Greg Byrd, Lynn Byrd and Chris Pearce

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9781107693975 Greg Byrd, Lynn Byrd and Chris Pearce: Cambridge Checkpoint Mathematics Teacher’s Resource 9 Cover. C M Y K

Greg Byrd, Lynn Byrd and Chris Pearce

Cambridge Checkpoint Mathematics Teacher’s Resource 9 Byrd, Byrd and Pearce

Cambridge Checkpoint Mathematics Teacher’s Resource 9

9

© CAMBRIDGE UNIVERSITY PRESS 2013

Introduction

The Cambridge Checkpoint Mathematics course covers the Cambridge Secondary 1 mathematics framework. It is divided into three stages: 7, 8 and 9. This is the Teacher’s Resource for stage 9, to be used in conjunction with Coursebook 9 and Practice Book 9. Similar resources are available for stages 7 and 8. They allow students to build up their knowledge and skills as they work towards the stage 9 Progression test and the Checkpoint examination at the end of stage 9. The curriculum is presented in six areas: number, algebra, geometry, measure, handling data and problem solving. This resource is divided into 19 units, each related to one of the first five content areas. Problem solving is included in all units.

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The structure of the Teacher’s Resource matches that of the Coursebook. Each unit starts by listing the teaching objectives that will be covered in the unit. Problem-solving objectives are indicated by a special icon ( ). Next, there is table outlining a lesson structure and listing references to available resources. Each unit then includes a brief summary of assumed prior knowledge.

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The resource then gives extensive details for each lesson. These include: • key words that students need to know • the main teaching points for the lesson • common misunderstandings and misconceptions • a summary of problem-solving objectives that are addressed, where appropriate • one or more teaching activities to introduce and develop the ideas to be taught and which complement and support the exercises in the Coursebook • comments on some of the questions in the Coursebook • suggestions for suitable homework, usually including a reference to the Practice Book.

The Teacher’s Resource also contains an end-of-year review, in the style of a Progression test, and answers to the Coursebook and Practice Book exercises and the end-of-year review.

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Problem solving

The problem-solving objectives are an important part of the course as they provide a structure for the application of mathematical skills.

Using techniques and skills in solving mathematical problems

A Calculate accurately, choosing operations and mental or written methods appropriate to the numbers and context. B Manipulate numbers, algebraic expressions and equations, and apply routine algorithms.

C Understand everyday systems of measurement and use them to estimate, measure and calculate. E Draw accurate mathematical diagrams, graphs and constructions.

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D Recognise and use spatial relationships in two dimensions and three dimensions.

F Decide how to check results, by: – using rounding to estimate numbers to one significant figure and calculating mentally then comparing with the estimate – considering whether an answer is reasonable in the context of the problem – using inverse operations. G Estimate, approximate and check their working. Solve a range of word problems involving single or multi-step calculations.

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Introduction Using understanding and strategies in solving problems H Identify, organise, represent and interpret information accurately in written, tabular, graphical and diagrammatic forms. I Explore the effect of varying values in order to generalise. J Find a counter-example to show that a conjecture is not true. K Present concise, reasoned arguments to justify solutions or generalisations using symbols, diagrams or graphs and related explanations. L Recognise the impact of constraints or assumptions. M Recognise connections with similar situations and outcomes.

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N Consider and evaluate the efficiency of alternative strategies and approaches and refine solutions in the light of these.

Problem solving in stage 9

Each unit in stage 9 focuses on some of the framework statements. The Teaching notes for each unit give specific tips on how to teach problem-solving skills, with suggestions for activities that can be introduced in the classroom. The questions marked with the problem-solving icon in the Coursebook and Practice Book are particularly suitable for exercising the skills described in the selected framework statements for each unit. The table below maps the framework statements from the Problem-solving section of the Cambridge Secondary 1 Stage 9 Mathematics Curriculum Framework against the units in the Coursebook and Practice Book. It indicates problem-solving objectives that provide a particular focus in each unit.

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Many of the teaching activities are designed with these objectives in mind. This is indicated in the lesson plans. The problem-solving icon is used in both the Coursebook and the Practice Book to show questions that address one of the objectives.

Unit 1

✓

Unit 2

✓

Unit 3

✓

B

C

D

E

✓

G

H

N

✓

✓

✓

Unit 5

✓

✓

Unit 6

I

J

K

✓

Unit 8

✓

Unit 9

✓

L

✓

✓

✓

✓

✓

✓

Unit 7

✓

✓

✓

✓

✓

✓

✓

✓

✓

Unit 10

✓

✓

✓

E

✓

Unit 12

✓

Unit 13

✓

✓

✓

Unit 14

✓

Unit 15

✓

✓

✓

✓

Unit 16 Unit 17

F

✓

Unit 4

Unit 11

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A

✓

✓

✓

✓

✓

✓ ✓

Unit 18

✓ ✓

✓

✓

✓

Unit 19

✓

✓ A

B

C

D

E

✓ F

G

H

I

J

K

L

M

N

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Teaching notes

Teaching notes

Unit 2

2 Sequences and functions

Objectives ★★ Know the origins of the word algebra and its links to the work of the Persian mathematician Al’Khwarizmi. ★★ Generate terms of a sequence using term-to-term and position-to-term rules. ★★ Derive an expression to describe the nth term of an arithmetic sequence. ★★ Find the inverse of a linear function. ★★ Calculate accurately, choosing operations and mental or written methods appropriate to the numbers and context

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★★ Manipulate numbers, algebraic expressions and equations, and apply routine algorithms

★★ Present concise, reasoned arguments to justify solutions or generalisations using symbols, diagrams or graphs and related explanations

Possible lessons

Number of 40-minute periods

Resources in Practice Book

Resources in Teacher’s Resource

Pages 16–17

Page 11

Resource sheet 2.1

2

Pages 18–19

Page 12

Resource sheet 2.2

1 or 2

Pages 20–21

Page 13

Resource sheet 2.3

Topic

1

Generating sequences

1 or 2

2

Finding the nth term

3

Finding the inverse of a function

Resources in Coursebook

Assumed prior knowledge

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Lesson

• Students should be competent in basic addition, subtraction, multiplication, squaring and division skills. • They should understand and be able to use term-to-term rules, position-to-term rules and the nth term. • Students should understand and be able to use function machines.

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Unit 2

Teaching notes

Lesson 1  2.1 Generating sequences Coursebook pages 16–17

Key words linear sequence

non-linear sequence

term-to-term rule

position-to-term rule

Main teaching points • Students need clearly to understand the difference between a linear and a non-linear sequence. Encourage them always to look at the difference between several successive terms. • Many students prefer to use the nth term rather than the position-to-term rule when generating or working with a sequence. This is a natural progression for the more able and should be encouraged.

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Common misunderstandings and misconceptions

Problem solving

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• Students may think that if a sequence increases (or decreases) according to a simple term-to-term rule such as ‘multiply by 2’ (× 2) or ‘divide by 2’ (÷ 2) it must be a linear sequence. Make sure that students can recall all the information and skills listed under ‘Assumed prior knowledge’. * In a linear sequence of numbers the terms in the sequence increase (or decrease) by the same amount each time. This means that a sequence is only linear if consecutive terms in the sequence change by repeated addition of the same number, or repeated subtraction of the same number. • Students may think that if a sequence increases (or decreases) by a more complex position-to-term rule such as ‘multiply by 5 and add 3’ (× 5 + 3) it is a non-linear sequence. • BIDMAS plays a more important role as the algebra increases in complexity. Students must be clear as to the BIDMAS order. • The problem-solving objective ‘manipulate numbers, algebraic expressions and equations, and apply routine algorithms’ is addressed in the whole of Exercise 2.1, but specifically questions 3, 4, 7 and 8, and in the activity below. A valuable element of this exercise is practice in working out terms of sequences, both linear and nonlinear. The problem-solving questions and activities require students to apply this skill in more unusual settings, which requires some flexibility of thought and progression of logic.

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Activity

• Set students this activity after they have completed Exercise 2.1. They can work individually or in small groups. Each student will need a copy of Resource sheet 2.1. • Explain that students will use the grid on the sheet to model the spread of a forest fire. The fire spreads to any square that shares an edge with a square that is already on fire. They should start in the centre and work outwards. The first few ‘terms’ have been provided on the sheet. Students can use different colours for successive ‘terms’. • If this is done as a class activity, draw the following table on the board and fill it in as students progress through the task. The table is also provided on the resource sheet. 1

Squares on fire

1

2

3

4

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Term

5

• Ask students to fill in the numbers for terms 2 and 3 [5, 13]. • Ask students to predict the fourth term [25], and then draw to check. Give them the hint, if required, to look at the difference between the terms. • Ask students to check that they understand by predicting and drawing the fifth term [41]. • Assuming that students have successfully completed the table, ask them to describe the sequence with a term-toterm rule [+ 4, + 8, + 12, …]. • Ask students to repeat the forest fire model, but this time with an equilateral triangle as the starting point. A triangular grid is provided on the resource sheet. Term

1

2

3

4

5

Squares on fire

1

[4]

[10]

[19]

[31]

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Teaching notes

Unit 2

• They should find that the term-to-term rule is +3n, or + 3, + 6, + 9, … • If there is time, develop the activity with a discussion on how fast a fire might spread if it were modelled using hexagons. If you have suitable paper (for example, isometric dot paper), this could be set as homework.

Comments on questions

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• Q3  Students could start by drawing nine dashes, writing the number 18 in place of the 10th dash and working back towards the fourth term. They will need to have derived the inverse operation ‘add 4’. This is a fairly straightforward problem-solving question, manipulating numbers by applying routine algorithms. • Q4  A similar method to Q3 is appropriate here, too. Some students may require a pen and paper or even a calculator. • Q7  Students will need to substitute for the fifth term to find that answer C is correct, as the third term works for all four possible rules. • Q8  There is only one possible answer for each of the missing numbers in the position-to-term rule in questions 1 and 2 of Shen’s homework. A simple trial-and-improvement method will yield the correct ‘missing numbers’ in the rules, of 7 and 5. This will allow students to work out the missing terms.

Homework

• Students can repeat the activity, using hexagons. • Practice Book page 11

Lesson 2  2.2 Finding the nth term Coursebook pages 18–19

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Key words

arithmetic sequence

nth term

Main teaching points

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• Encourage students to set the work out in a table, as shown in the introduction to this topic. Using this method will help with other, more complex sequences in future work. It is not, however, the only method available. Another common method for finding the nth term of a simple sequence such as 8, 11, 14, 17, … is very similar, but does not involve the use of a table. The difference between the terms is always 3, so the rule must include 3n. Looking at the first term (8), the difference between 3 × 1 and 8 is +5, so the rule is 3n + 5. • Students must become confident with substituting into simple expressions for nth terms. Any student who has difficulty in giving the first three terms in questions 1–4 of Exercise 2.2 needs immediate help, supported by further practice with straightforward nth terms such as 2n, 2n + 1, 3n + 1, 3n − 1. • In question 6 of Exercise 2.2, make sure that students work out the difference between all successive pairs of terms and do not just find the difference between the first two terms. This is a very bad habit to fall into. Students who do this may still get linear nth term questions right, but will get non-linear questions wrong.

Common misunderstandings and misconceptions

Problem solving

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• Students may not use the rules summarised as BIDMAS correctly with sequences such as 6 – 2n. • Some students will fail to check at least three differences between successive terms.

• The problem-solving objectives ‘manipulate numbers, algebraic expressions and equations, and apply routine algorithms’ and ‘present concise, reasoned arguments to justify solutions or generalisations, using symbols or diagrams and related explanations’ are addressed in questions 8, 9 and 10 of Exercise 2.2. Finding the nth term of a sequence is such a fundamental part of algebra and its eventual uses that students must be very confident at the end of this exercise. Some students may benefit from repeating the majority of this exercise in two or three weeks’ time. • The problem-solving objective ‘manipulate numbers, algebraic expressions and equations, and apply routine algorithms’ is addressed in the activities below.

Activity 1 • More-able students can do this activity after working through the worked examples. Less-able students should do it after they have completed the exercise. © CAMBRIDGE UNIVERSITY PRESS 2013

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Unit 2

Teaching notes

0,

11, 15, 19 0 1, 2

,1 1, 4

–

4, –

, 13

3, 8

2, 5

30

0, 2, 2

, 8,

15,

24

0

3, 4

,1

0, 3

2, 6

1

9 2, 1

33

,

, 10

6, 8

4,

7, 1

n2 + 10

30,

3, 7 ,

15 , 23

, 26

17 10,

, 14

, 12

3, 0, 1

16

,3 , 26

+n

27,

8n – 5

3n – 2

2

n

–1

24,

1, 4, 7, 10, 13

PL

+5

n2

7n

8

+1

21,

–5

, 18

n2 + 4

3n –1

, 24

6, –1, , 6 – –9,

8, 1 2, 1 6

0 5, 2 1 , , 10 0, 5

11, 14, 17

5, 8 ,

1,

0, 4 ,

10n – 8 2

, 19

8, 1 2, 1 6, 2 0, 2 4

4 5, 2 1 , 3, 8

13, 17, 21

6n – 3

n2 + 2

4n + 4 n

3, 11, 19, 27, 35

–2

+1

+4

3n

Activity 2

1

2

5n

n

2n

+1



8, 12, 16, 20, 24

2, 12, 22, 32, 42

M

3n

–1

11, 14, 19, 26, 35

4n –

0 –1

5 6, 2 1 , 4, 9

n2 + 3 2

n

4

5

4n –

– 5n

2

2, 5, 8, 11, 14

5n

, 14

4, 7, 12, 19, 28

3, 6, 11, 18, 27

3n +

2

n

5, 8, 13, 20, 29

4, 9

4

5, 9 ,

–1

3, 9, 15, 21, 27

4n +

2

n

SA

1

1

4n +

+ 2n

11 , 9, 7 , 3, 5

• Split the class into groups of two to six, depending upon ability. Each group will need a copy of Resource sheet 2.2 and a pair of scissors. The task is to cut out the individual triangles and reposition them so that each nth term has the appropriate first five terms next to it. The answer is in the shape of a regular hexagon; see below. • Remind students of the link between the position-to-term rule and the nth term. For example, from Q5 of Exercise 2.1: * term = position number + 5 can be written as n + 5, 2 2 * term = position number + 4 can be written as n + 4. • Answer

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• Set students this activity after they have completed Exercise 2.2. They can work in teams of four to six. Each student will need two sheets of paper. * On one sheet, they will write their own nth terms. The number of nth terms they are asked to write depends upon the time available, but 12 to 15 is usually a sensible number. * On the other sheet they will write the first five terms for each of their nth terms. • Explain that this is a competition, so they should make the nth terms as tricky as possible. • If necessary, impose a few extra rules at this point, such as on the use of decimals and/or fractions, large numbers, positive or negative. If calculators are allowed, most rules will not be necessary. • Depending on what you wish students to practise, decide on which piece of paper they swap with another team. Give the students a fixed, but realistic, time to give the answers to the sheet they have (i.e. either the first five terms or the nth term). At the end of the allotted time, sheets are swapped back and marked. • The team with the most correct answers wins.

Comments on questions • Q8  Encourage students to give full explanations why neither student has the correct term, rather than just saying they don’t and giving the nth term of 12  n + 3 12 .

• Q9  Students need to show their understanding that if a sequence is decreasing, then the nth term cannot have a positive n. © CAMBRIDGE UNIVERSITY PRESS 2013

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Teaching notes

Unit 2

• Q10  Again, a full answer is required here. Encourage students to do what the question asks: ‘Explain how you worked out your answer.’ If students write down the steps they take to answer this (and the next) question, this will lead into a useful classroom discussion about the most efficient method(s).

Homework • Practice Book page 12

Lesson 3  2.3 Finding the inverse of a function Coursebook pages 20–21

Key words inverse function

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Main teaching points

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• Many students can confidently work with functions, and possibly equations, but seem less sure about working with mappings. Keep reminding students that these are all essentially the same at this stage and that they can swap them around, mentally, if it helps. • If students find these questions difficult, especially questions 1–4, drawing the function machine usually helps to work out the inverse. • It is important to encourage students to think of the order of operations of the function. Without that information, they cannot find inverse functions. When helping students who are having difficulty, encourage them to verbalise the order of the operations within the function in the question. If they do not get this aspect right, they need to do more work on recognising the BIDMAS aspect of order of operations. Once they do get this right, just saying the order is often enough for them to realise what the inverse is.

Common misunderstandings and misconceptions

• Students are frequently confused more by division questions, especially in identifying the correct order of

(

)

(

)

operations, such as in question 3c y   =   x   +  1 and 3d y   =   x − 4  . This is particularly the case if they are not 2 3 confident with normal BIDMAS rules.

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Problem solving

• The problem-solving objective ‘calculate accurately, choosing operations and mental or written methods appropriate to the numbers and context’ is addressed in question 6 of Exercise 2.3 and in the activity below. Using a function machine to write out a function is valuable practice and helps students to understand the process of that function, and so deal with the inverse. As students deepen their understanding, many will not need to rely upon function machines. • The problem-solving objectives ‘calculate accurately, choosing operations and mental or written methods appropriate to the numbers and context’, ‘manipulate numbers, algebraic expressions and equations, and apply routine algorithms’ and ‘present concise, reasoned arguments to justify solutions using symbols, diagrams or graphs and related explanations’ are addressed in the activity.

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Activity

• Set this activity after students have completed Exercise 2.3. They can work in pairs or small groups of equal size. Each group will need a set of cards cut out from Resource sheet 2.3. • Students should shuffle the cut-out cards and place them into two separate piles, one with the ‘×’ and ‘÷’ cards, another with the ‘+’ and ‘−’ cards. • Each team requires an ‘x’ card. The aim of the game is to use inverse functions. • The game starts with one team taking the top card from each pile, for example, ‘× 2’ and ‘− 3’. • The team can arrange these cards, and their x card, in two ways, either ‘x × 2 − 3’ or ‘x − 3 × 2’,. They decide which will be more difficult to work with and then present the expression they have formed to the opposing team. • The opposing team writes down the inverse function and shows the other team their answer. If they are correct, they get one point; if they are proved wrong, the other team get two points. • The first team to reach five points (for example) wins. © CAMBRIDGE UNIVERSITY PRESS 2013 Copyright Cambridge University Press 2013

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Unit 2

Teaching notes

Comments on questions • Q5  This is conceptually a very difficult question. Many students will be quite challenged, especially with parts a ii and a iii. The first necessary step towards success is in setting out the question carefully. The next is realising that the inverse of ‘× −2’ is ‘÷ −2’. • Q6  Students should know the different terminology by now, but some may still need to look again at the first few lines of the introduction to this exercise, showing the three ways students should be able to represent a function. Again, knowledge of BIDMAS will play a part, if students are to be successful. Encourage students to read the question and to write down the mathematics as they go. This is the best way to achieve the correct method.

Homework • Practice Book page 13

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Resource sheet 2.1

SA Term

1

Squares on fire

1

2

3

4

Teaching notes

Unit 1

5

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M Term

1

Triangles on fire

1

2

3

4

5

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Resource sheet 2.2

Unit 1

Teaching notes

, 9,

–4,

16,

6

–1,

, 10

4, 1

1, 2

0

–

0, 2

4

4

0, 4, 8, 12, 16

6, 2 , 9,

, 24

n

2

+2

–5 4

3, 6, 11, 18, 27

2, 1

14, 19

, 20

10, 15

n+ 4

0, 3

, 8,

3,

4, 7, 12, 19, 28

3, 9, 15, 21, 27

,

0, 5

8, 1

14

n2 + 4

5n

11,

15,

3, 5

, 7,

8n

–5 4

0, 2

, 16 ,2

2

24

8, 1

, 35

19, 27

+4

–1

2

1, 8, 1

+1

2

5

6, 1

9, –

, 5,

3n

E

2n

+1

–2

n

5, 9

, 16

6, – 1,

1

7, 2

, 13 ,1

, 13

4n

5n

25

, 8,

1, 4, 7, 10, 13

, 14

4, 7

n2 + n

n2

1, 4

10, 12

11, 14, 19, 26, 35

n2

, 29

+1

,

3, 7

0, 3, 8, 15, 24

, 20

8

3n –1

12,

32 22,

3n + 2

n 2, 5, 10, 17, 26

2

–5

4n + 4

0

+1

2

n

, 13

– 10n

, 42

2,

–1

–1

–3

5, 8

, 17 3, 8, 13, 18, 23

4n

6n

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+5

5n

4n – 4

7n

1

0

–1

M

#

+ 2n

, 14

, 19

11, 15 2

n

18 , 11

–2

, 30

+ 3n

21, 24, 27, 30, 33

n2 + 3

3n

, 20

5, 8

, 12

12, 19, 26, 33, 40

SA 2, 6

n2 – 1

Cut along the lines to form triangular tiles. Fit the tiles together so that the questions and answers are correctly matched at the edges that meet.

9, 1

1

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Cambridge Checkpoint Mathematics 9 matches the requirements of stage 9 of the revised Cambridge Secondary 1 curriculum framework. It is endorsed by Cambridge International Examinations for use with their programme. This Teacher’s Resource is intended to be used alongside the Cambridge Checkpoint Mathematics Coursebook 9 and Practice Book 9.

M SA The Teacher’s Resource CD-ROM contains: • answers to exercises from the Coursebook • answers to exercises from the Practice Book • teaching notes for each topic • resource sheets • an end-of-year review.

Other components of Cambridge Checkpoint Mathematics 9: Coursebook 9 ISBN 978-1-107-66801-0 Practice Book 9 ISBN 978-1-107-69899-4

Completely Cambridge – Cambridge resources for Cambridge qualifications Cambridge University Press works closely with Cambridge International Examinations as parts of the University of Cambridge. We enable thousands of students to pass their Cambridge exams by providing comprehensive, high-quality, endorsed resources.

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To find out more about Cambridge International Examinations visit www.cie.org.uk Visit education.cambridge.org/cie for information on our full range of Cambridge Checkpoint titles including e-book versions and mobile apps.

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9781107693975 Greg Byrd, Lynn Byrd and Chris Pearce: Cambridge Checkpoint Mathematics Teacher’s Resource 9 Cover. C M Y K

Greg Byrd, Lynn Byrd and Chris Pearce

ISBN 978-1-107-69397-5 © CAMBRIDGE UNIVERSITY PRESS 2013

9 781107 693975

Cambridge Checkpoint Mathematics Teacher’s Resource 9 Byrd, Byrd and Pearce

Cambridge Checkpoint Mathematics Teacher’s Resource 9

Greg Byrd, Lynn Byrd and Chris Pearce

Cambridge Checkpoint

Mathematics

Teacher’s Resource

9