Maths Frameworking Teacher Pack 3.3 by Collins

Teacher Pack 3.3

Rob Ellis, Kevin Evans, Keith Gordon, Chris Pearce, Trevor Senior, Brian Speed, Sandra Wharton

7537891 TEACHER PACK 3.3 title.indd 1

20/05/2014 10:09

Contents Introduction Maths Frameworking and the 2014 Key Stage 3 Programme of study for mathematics Programme of study matching chart

4 Using data

Overview 4.1 Scatter graphs and correlation 4.2 Two-way tables 4.3 Estimation of a mean from grouped data 4.4 Cumulative frequency diagrams 4.5 Statistical investigations Review questions Challenge – Census

viii xv

1 Percentages Overview 1.1 Simple interest 1.2 Percentage increases and decreases 1.3 Calculating the original value 1.4 Repeated percentage changes Review questions Challenge – Exponential growth

1 3

Overview 5.1 Step graphs 5.2 Time graphs 5.3 Exponential growth graphs Review questions Problem solving – Mobile phone tariffs

13 15

Maths Frameworking 3rd edition Teacher Pack 3.3

49 51 53 55 57 57

6 Pythagoras’ theorem Overview 6.1 Introducing Pythagoras’ theorem 6.2 Using Pythagoras’ theorem to solve problems 6.3 The converse of Pythagoras’ theorem Review questions Activity – Practical Pythagoras

17 19 21 23 23

3 Polygons Overview 3.1 Properties of polygons 3.2 Interior and exterior angles of regular polygons 3.3 Tessellations and regular polygons Review questions Mathematical reasoning – Semi-regular tessellations

41 43 45 47 47

5 Applications of graphs

5 7 9 11 11

2 Equations and formulae Overview 2.1 Multiplying out brackets 2.2 Factorising algebraic expressions 2.3 Expressions with several variables 2.4 Equations with fractions Review questions Investigation – Body mass index

35 37 39

25 27

59 61 63 65 67 67

7 Fractions Overview 7.1 Adding and subtracting fractions 7.2 Multiplying fractions and mixed numbers 7.3 Dividing fractions and mixed numbers 7.4 Algebraic fractions Review questions Investigation – Fractions from one to six

29 31 33 33

iii

69 71 73 75 77 79 79

8 Algebra Overview 8.1 Expanding the product of two brackets 8.2 Expanding expressions with more than two brackets 8.3 Factorising quadratic expressions with positive coefficients 8.4 Factorising quadratic expressions with negative coefficients 8.5 The difference of two squares Review questions Challenge – Graphs from expressions

11.3 Solving quadratic equations by drawing graphs 11.4 Solving cubic equations by drawing graphs Review questions Challenge – Maximum packages

81 83 85

127 129 131 131

12 Compound units

Overview 12.1 Speed 12.2 More compound units 12.3 Unit costs Review questions Challenge – Population density

89 91 93

133 135 137 139 141 141

13 Right-angled triangles Overview 13.1 Introduction to trigonometric ratios 13.2 How to find trigonometric ratios of angles 13.3 Using trigonometric ratios to find angles 13.4 Using trigonometric ratios to find lengths Review questions Investigation – Barnes Wallis and the bouncing bomb

9 Decimal numbers Overview 9.1 Powers of 10 9.2 Standard form 9.3 Multiplying with numbers in standard form 9.4 Dividing with numbers in standard form 9.5 Upper and lower bounds Review questions Mathematical reasoning – To the stars and back

95 97 99 101 103 105 107 107

109 111 113 115 117 117

11 Solving equations graphically Overview 11.1 Graphs from equations in the form ay ± bx = c 11.2 Solving simultaneous equations by drawing graphs

Maths Frameworking 3rd edition Teacher Pack 3.3

145 147 151 153 155 155

14 Revision and GCSE preparation

10 Surface area and volume of cylinders Overview 10.1 Volume of a cylinder 10.2 Surface area of a cylinder 10.3 Composite shapes Review questions Problem solving – Packaging soup

143

Overview Practice Revision GCSE preparation: solving quadratic equations GCSE-type questions

157 159 161 162 162

Learning checklists 3-year scheme of work 2-year scheme of work

164 177 182

119 121 125

Percentages

Learning objectives • • • •

How to calculate simple interest How to use a multiplier to calculate percentage increases and decreases How to calculate the original value after a percentage change How to calculate the result of repeated percentage changes

Prior knowledge • •

How to work out a percentage of a given number, with or without a calculator How to write one number as a percentage of another number

Context •

Percentage increase and decrease is probably one of the most common uses of mathematics in real life. Everyone meets it in some form or another even if this is only in terms of financial capability. Here is a website you could visit to find real life applications of percentage: http://www.pfeg.org/

Discussion points • •

Which sets of equivalent fractions, decimals and percentages do you know? From one set that you know (such as 1 = 0.5 = 50%), which others can you work out?

•

How would you go about finding the percentage equivalents of any fraction?

Associated Collins ICT resources • • •

Chapter 1 interactive activities on Collins Connect online platform Calculating reverse percentages video on Collins Connect online platform Cars and phones Wonder of Maths on Collins Connect online platform

Curriculum references •

•

• •

Develop fluency Select and use appropriate calculation strategies to solve increasingly complex problems Solve problems Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics Number Define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100% Interpret fractions and percentages as operators Recognise and use relationships between operations including inverse operations

Maths Frameworking 3rd edition Teacher Pack 3.3

Fast-track for classes following a 2-year scheme of work •

Although pupils have met percentages before, there are some important and quite challenging concepts in this chapter. The ideas of percentages as a multiplier and the use of multiplicative reasoning are very important to pupils’ confidence and fluency when working with percentages. So, while you may be able to leave out some of the earlier questions in each exercise in the Pupil Book, be careful not to leave out too much or move on too fast.

Maths Frameworking 3rd edition Teacher Pack 3.3

Lesson 1.1 Simple interest Learning objectives

Resources and homework

•

• • • • •

•

To understand what is meant by simple interest To solve problems involving simple interest

Pupil Book 3.3, pages 7–9 Intervention Workbook 2, pages 40–42 Intervention Workbook 3, pages 16–17 Homework Book 3, section 1.1 Online homework 1.1, questions 1–10

Links to other subjects

Key words

•

• •

History – to calculate the impact of interest over time to compare incomes or standard of living

lender simple interest

Problem solving and reasoning help •

The FS and PS questions in Exercise 1A of the Pupil Book require pupils to apply their understanding to a range of increasingly complex financial situations. The challenge activity at the end of the exercise requires pupils to start thinking about how they can use formulae to help them to generalise percentage change questions. This is a powerful idea, which you should develop carefully.

Common misconceptions and remediation •

Pupils often struggle when they start using percentages that are greater than 100. Use reallife examples to help them overcome this. Start with percentages that pupils are comfortable working with, for example: a pair of jeans costs a shop £30 to make. A 50% profit would mean selling the jeans for £30 plus £15 = £45; 100% profit would mean selling the jeans for £30 plus £30 = £60; 150% profit would mean selling the jeans for £30 plus £45 = £75.

Probing questions • •

Talk me through how you would increase, and then decrease £22 by, for example, 25%. Can you do the calculations in a different way?

Part 1 • • •

Ask pupils to work in pairs to discuss what they know about taking out a loan, and to suggest some examples of when they might take out a loan. Then ask pupils to discuss what they know about how they might pay back the loan. Take feedback but do not comment on pupils’ feedback yet.

Part 2 • • • •

• •

Tell pupils that at some point most people will take out a loan, for example, to buy a house. Interest is usually paid to the lender – the person or company that has provided the loan. This may be a good point at which to comment on any suggestions that pupils have made. One type of interest is called simple interest. This is calculated as a percentage. As long as the loan exists, you will pay the lender a percentage of the loan at regular intervals. Ask pupils to work through Example 1 on page 7 of the Pupil Book in pairs, discussing their understanding as they go and recording any questions. Answer any questions. If necessary, pupils can work through examples 2 and 3 before moving on to Exercise 1A. Pupils can now do Exercise 1A from Pupil Book 3.3. Encourage pupils to use the internet to find out about compound interest. Maths Frameworking 3rd edition Teacher Pack 3.3

Part 3 •

• •

Introduce the idea of compound interest. Then ask pupils to consider these questions: o ‘If a sum is invested and gains 5% each year, how long will it take to double in value?’ o ‘If an object depreciates in value by 5% each year, how long will it take until only half of the original value remains?’ Are these two answers the same? This will prepare pupils for Lesson 1.2 and Lesson 1.3. Ask more able pupils, ‘Why aren't these two answers the same?’ You could challenge pupils further by asking: ‘Is there a rate that is used for both gain and depreciation, for which those two answers would be the same?’ Answers Exercise 1A 1 a £75 e £600 2 a £11.05 e £8.68 3 £144 4 £1814.40 5 £6451.20 6

a £50

b £4.50 f £212.80 b £29.75 f £212.80

c £400 g 3.8% of £50.16 c £1495 g 3.8% of £50.16

d £450 h £1.625 d £8.05 h £1.625

50 × 100 = 4 b 1250

7 a £13 500 b 54% 8 a £7.50 b £12 c 52 9 Carly: £840, James: £840; Carly pays a higher rate of interest but James pays for longer. 10 17% 11 0.225% 12 0.6% 13 a 1.75 b i 5.25% ii 10.5% iii 21% iv 42% Challenge: Using a formula A £560 B £301 C £10 744  42 = 2.008 D 1 + 2.4100

Maths Frameworking 3rd edition Teacher Pack 3.3

Lesson 1.2 Percentage increases and decreases Learning objectives

Resources and homework

•

• • • • •

•

To use the multiplier method to calculate the result of a percentage increase or decrease To calculate the percentage change in a value

Pupil Book 3.3, pages 10–13 Intervention Workbook 2, pages 40–42 Intervention Workbook 3, pages 16–17 Homework Book 3, section 1.2 Online homework 1.2, questions 1–10

Links to other subjects

Key words

•

• •

Food technology – to calculate increases in a certain food type in the diets of different people

decrease multiplier

• •

increase original value

Problem solving and reasoning help •

This lesson reinforces the concept of using percentage as an operator, and is an important step to ensure confidence and fluency in pupils, so make sure that you take time over it. It often helps to make links to fractions as operators. In the reasoning activity at the end of Exercise 1B in the Pupil Book, pupils will need to apply their understanding of percentage change to the slightly less familiar context of population change. Use the opportunity to tackle any outstanding misconceptions.

Common misconceptions and remediation •

Pupils are often confused when they come across percentages that are greater than 100. Using real-life examples could help, starting with percentages that pupils are able to work with comfortably. Pupils need a good understanding of 100% as a whole before tackling percentage increases and decreases successfully.

Probing questions • • •

How would you find the multiplier for different percentage increases and decreases? How would you find a multiplier to increase, then decrease, by a given percentage? Given a multiplier, how could you tell if this would result in an increase or a decrease?

Part 1 • • • •

Write a variety of percentages on the board, for example: 5%, 10%, 20%, 25%. Then write some quantities on the board, for example: £32, 58 kg, 200 km, £150. Apply each percentage value to each quantity and calculate a percentage increase or decrease, as appropriate. Pupils should be able to do most of these without a calculator. Pupils can decide which percentages they can do easily without a calculator. In pairs, encourage pupils to challenge each other to calculate easier percentages faster than their partner, who should be using a calculator.

Part 2 •

• •

Tell pupils that a percentage change may be: o an increase if the new value is larger than the original value o a decrease if the new value is smaller than the original value. Say that several methods can be used to calculate the result of a percentage change. Ask pupils if they used other methods of calculation in Part 1. If so, ask these pupils to explain the differences among the methods. Maths Frameworking 3rd edition Teacher Pack 3.3

• • •

•

Say that the multiplier method is often the most efficient, and is the focus of this lesson. Tell pupils that the multiplier method involves multiplying the original value by an appropriate number to calculate the result of the percentage change. Ask pupils to work through Example 4 on page 10 of the Pupil Book in pairs, discussing their understanding as they go, and recording any questions. Answer these questions. If necessary, pupils should work through examples 5 and 6 before doing Exercise 1B. Pupils can now do Exercise 1B from Pupil Book 3.3

Part 3 • • •

Ask pupils to work in pairs on the following problem: ‘The answer to a percentage increase question is £10. Make up an easy question and one that is difficult.’ Pairs should share their questions with another pair. As a class, summarise by discussing what makes the questions easy or difficult. Answers Exercise 1B 1 a £5136 b £4464 c £7056 d £1440 2 a £135 b £92.40 c £71.10 d £12.75 3 a £351.43 b £190.55 c £542.07 d £769.48 4 a £564.53 b 14.7 c 3.42 5 a 7.5% b 0.925 6 a £43 200 b £67 200 c £91 200 d £103 200 7 131.89% 8 a £672 b 29.5% 9 The price of a car can increase by any percentage, but the most a price can decrease by is 100%, making it worth nothing. 10 a +28.1% b −21.9% c +116.8% d −53.9% 11 A change from 870 to 1240 is an increase of 42.5%, but a change from 1240 to 870 is a decrease of 29.8%. 12 a increase of 38.5% b decrease of 13.0% c increase of 20.5% 13 a 25% reduction b 22.2% reduction c 40% reduction 14 a 624 bolivars b 973.44 bolivars c 1518.57 bolivars Reasoning: Population change A Derby +23.7%, Liverpool +12.4%, Plymouth −20.8% So Derby has the largest percentage change. B Derby +51.8%, Liverpool −42.7%, Plymouth +12.9% So Derby again has the largest percentage change.

Maths Frameworking 3rd edition Teacher Pack 3.3

Lesson 1.3 Calculating the original value Learning objective

Resources and homework

•

• • • • •

Given the result of a percentage change, to calculate the original value

Pupil Book 3.3, pages 14–16 Intervention Workbook 2, pages 40–42 Intervention Workbook 3, pages 16–17 Homework Book 3, section 1.3 Online homework 1.3, questions 1–10

Links to other subjects

Key words

•

History – to calculate the impact of interest over time to compare incomes or standard of living

No new key words for this topic

Problem solving and reasoning help •

This lesson continues to develop the concept of using percentage as an operator by looking at the inverse to calculate the percentage change or to calculate an initial value. Pupils will need to be fluent with this concept so that they are confident in applying their understanding of percentages to real-life problems. Encourage discussion to challenge any misconceptions.

Common misconceptions and remediation •

Pupils are often confused by percentages that are greater than 100. Pupils also make the mistake of pairing an increase with an equivalent decrease. For example, they expect a 50% increase that is followed by a 50% decrease to return them to the starting value. This misconception results from using additive instead of multiplicative reasoning. This lesson tackles this misconception, and in Part 3 pupils should draw on the work they did on inverse relationships during the lesson to check and consolidate their understanding.

Probing questions • • • • • • •

The answer to a percentage increase question is £12. Make up an easy and a difficult question. After the addition of 8% interest, Anna has savings of £850. What was the original amount? After one year a car depreciates by 5%. It is valued at £8500. What was the value of the car at the beginning of the year? After being given a 2% increase, a shop assistant earns £7.50 per hour. What was his hourly rate before the increase? At a 25% discount sale, a boy pays £30 for a pair of jeans. What was the original price? Explain how to find a multiplier to calculate an original value after a proportional increase or decrease. True or false: ‘The inverse of an increase by a percentage is not a decrease by the same percentage.’ Justify your answer.

Part 1 •

•

Give pupils five minutes to tackle this question on their own; then to pair-share without discussion (five minutes): ‘At a sale, prices were reduced by 33%. After the sale, prices were increased by 50%. What was the overall effect on the prices? Explain how you know.’ Say that you will revisit this at the end of the lesson to see who has changed their mind.

Maths Frameworking 3rd edition Teacher Pack 3.3

Part 2 •

• •

Use examples 7 to 9 on pages 14 and 15 in the Pupil Book to demonstrate how to use multiplicative reasoning and inverse operations to calculate the percentage change between two values. In pairs, ask pupils to discuss their understanding as they go, and to record any questions. Answer these questions. This is quite a challenging concept so make sure that you take time over it. Pupils can now do Exercise 1C from Pupil Book 3.3.

Part 3 • •

Revisit the question in Part 1. Encourage pupils to use what they have done on percentages as a multiplier and the use of inverse relationships to explain their response to this question. Ask pupils what percentage increase would take the sale price back to its starting point. Answers Exercise 1C 1 a £450 b 66.7 kg c 3.5 m d £6760 2 a £3200 b £480 c 176 kg d 180 3 a 480 b 635 4 a 450 g b 544.5 g 5 a 280 b 717 6 £52.02 7 a £84.60 b £385.20 c £750 (749.58) 8 a £118.66 b £236.48 c £300.15 9 £141.50 10 a 23% b €392.50 11 a 30 072 827 (approx. 30 073 000) b 7 754 726 (approx. 7 755 000) Challenge: At the gym The original membership fee was £128. The old membership was 240 members. The amount raised from 240 members each paying £128 = £30 720. The amount raised from 180 members each paying £160 = £28 800. The gym lost money by raising the fees.

Maths Frameworking 3rd edition Teacher Pack 3.3

Lesson 1.4 Repeated percentage changes Learning objective

Resources and homework

•

• • • •

To calculate the result of repeated percentage changes

Pupil Book 3.2, pages 17–19 Intervention Workbook 2, pages 40–42 Intervention Workbook 3, pages 16–17 Online homework 1.4, questions 1–10

Links to other subjects

Key words

•

History – to calculate the impact of interest over time to compare incomes or standard of living

No new key words for this topic

Problem solving and reasoning help •

This lesson builds on the work of the previous lessons to consolidate pupils’ understanding of percentages by giving them multi-step problems, which require them to apply repeated percentage changes. Make sure that all pupils are confident with using multiplicative reasoning for percentage change before tackling this lesson.

Common misconceptions and remediation •

Pupils often learn rules without really understanding them. This means that pupils may meet the different questions over time, and so may never have the opportunity to identify the type of question and make independent decisions about which method to use. Provide pupils with opportunities, in a range of increasingly complex and unfamiliar situations, to check their understanding by making choices and decisions over the approaches they use. Pupils need to be confident and fluent with the ideas developed so far, particularly that of a multiplier and using multiplicative reasoning when completing percentage increase or decrease questions.

Probing questions •

The answer to a percentage change question is £15. Make up an easy multi-step question with this answer, and one that is more difficult.

Part 1 • • • • • •

This activity recaps earlier learning. Write four quantities on the left-hand side of the board, for example: 45, 60, 56, 12. On the right-hand side, write, for example: 120, 300, 200, 160. Match the quantities; one from the left-hand side and one from the right-hand side. Calculate, with or without a calculator, the percentage that the left value is of the right value. Some pairs are obvious (12 and 120); some pairs are clearly non-calculator (45 out of 300). Some pairs have whole-number answers but are not obvious, such as 56 out of 160 (35%). Discuss the appropriate methods to use for the answers that are not obvious.

Part 2 •

Tell pupils that now that they know how to answer a range of percentage questions, they need to be able to decide what type of question they have been given and what method to use to solve it. Pupils will gain practice by tackling more complex multi-step problems. Point out that pupils will need to draw on all their learning from this chapter plus what they already know about percentages, as well as making links to fractions, decimals and ratio.

Maths Frameworking 3rd edition Teacher Pack 3.3

• •

Ask pupils, in pairs, to work through Example 10 on page 17 in the Pupil Book. They should discuss their understanding as they go, and record any questions. Answer the questions. Pupils can now do Exercise 1D from Pupil Book 3.3.

Part 3 •

• • • • •

Write a question from Exercise 1D in the Pupil Book on the board, for example, FS question 4: ‘Gabrielle put £2500 in a savings account. At the end of one year she was paid interest of 2% and she left this in her account. At the end of a second year she was paid interest of 2.5%. How much did she have at the end of the second year?’ Discuss with pupils how they could write a formula for this problem. Then ask: ‘What if the original amount is changed? What if the interest rate is changed?’ Now ask pupils to write a similar formula, for example, where the same interest rate of 2% is paid on the original deposit for five years. Encourage pupils to see if they can write the formula more efficiently using powers. (This will move pupils towards the idea of compound interest.) As homework, pupils could research the ways in which different types of interest are used. Answers Exercise 1D 1 9.504 kg 2 546 3 £13 024 4 £2613.75 5 86 bpm 6 a 259 b −4% 7 a 1.9152 m (1.92 m) b 28% (27.68%) 8 The multipliers are 1.16, then 1.25. The product of 1.16 and 1.25 is 1.45, which is the multiplier for an increase of 45%. 9 21% 10 56% 11 58% 12 1.4 × 1.4 = 1.96, so the percentage increase is 96%. 13 a no change b no change c 10% reduction Investigation: Up and down A 1.1 × 0.9 = 0.99, a reduction of 1% B a 0.96, a reduction of 4% b 0.91, a reduction of 9% c 0.84, a reduction of 16% C a reduction of N 2 % D a reduction of 0.25%; 1.05 × 095 = 0.9975 = 1 – 0.0025; 0.0025 = 0.25%

Maths Frameworking 3rd edition Teacher Pack 3.3

Review questions • •

(Pupil Book pages 20–21)

The review questions will help to determine pupils’ abilities with regard to the material within Chapter 1. The answers are on the next page of this Teacher Pack.

Challenge – Exponential growth •

• • •

•

(Pupil Book pages 22–23)

This challenge activity gives pupils the opportunity to extend their learning by exploring the connected but unfamiliar context of exponential growth that links this lesson with other areas of mathematics. This activity will also prepare pupils for some of the work in Chapter 5 (Applications of graphs). This activity makes strong links to drawing graphs. You may want to revisit some of the graphs with which pupils are familiar, and how to use a table to help pupils plot a graph. As a warm-up to this activity, ask pupils to summarise what they have learnt during the lessons in this chapter. Say that this activity requires them to build on this knowledge. Before pupils begin to work on questions 1 to 4, you may also want to check their understanding of the multiplier method for percentage increases and decreases, which pupils met in this chapter, particularly in Lesson 1.2. You could ask more able pupils to develop this further by exploring real-life examples of exponential growth on the internet. Encourage these pupils to explore exponential decay. Answers to Review questions 1 a 67.65 m b 52.09 m c 94.71% 2 a £1123.20 b 31.2% 3 £679.20 4 £263.16 5 £47 6 a 27% (26.97%) b 13.5% (13.48%) Answers to Challenge – Exponential growth 1 a 1000 × 1.2 × 1.2 = 1440 b 1000, 1200, 1440, 1728, 2074, 2488 c

c 35.3% (35.29%) d 67.3% 7 a 30 cm b 15 cm 8 a 229 b 305 9 15 859 10 a 120 b 61 11 £85.55

2 a Number 0 1 2 3 4 5 of years Number 1000 1300 1690 2197 2856 3713 of deer

Maths Frameworking 3rd edition Teacher Pack 3.3

3 Number 0 1 2 3 4 5 of years Number 1000 1400 1960 2744 3842 5378 of deer

5 a

a 1000 × 0.8 × 0.8 = 640 b 1000, 800, 640, 512, 410, 328 c

Maths Frameworking 3rd edition Teacher Pack 3.3

Equations and formulae

Learning objectives • • • •

How to expand more complex expressions containing brackets How to factorise algebraic expressions containing powers of variables How to manipulate expressions containing several variables How to solve equations with the variable in the denominator of a fraction

Prior knowledge • • • •

How to collect like terms in an expression How to use one or two operations to solve equations How to substitute values into a formula What a highest common factor (HCF) is

Context •

This chapter builds on previous work on expanding a term over brackets, and recalls expansion of two such terms and simplification of the results by collecting like terms. This is an introduction to factorisation of terms into a bracket with a numerical and/or algebraic coefficient outside. Pupils are then shown how to factorise a quadratic expression. Finally, pupils will learn how to solve equations involving fractions.

Discussion points • • • •

What steps do you follow when expanding a bracket? What happens if the bracket has a negative coefficient? What is a variable and why do we use them? What strategies can be used to solve for unknowns in algebraic equations? Why strategies can be used to solve equations containing fractions?

Associated Collins ICT resources •

Chapter 2 interactive activities on Collins Connect online platform

Curriculum references •

•

Develop fluency Substitute values in expressions, rearrange and simplify expressions, and solve equations Solve problems Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems

Maths Frameworking 3rd edition Teacher Pack 3.3

• •

Algebra Understand and use standard mathematical formulae; rearrange formulae to change the subject Use and interpret algebraic notation, including: o coefficients written as fractions rather than as decimals o brackets

Fast-track for classes following a 2-year scheme of work •

Much of this chapter will be unfamiliar to pupils. However, some pupils may be familiar with expanding brackets. Check that all pupils can expand brackets with negative coefficients fluently before moving on to the rest of the chapter. If pupils grasp the concepts quickly they can move on to the more challenging questions that are towards the end of each exercise in the Pupil Book.

Maths Frameworking 3rd edition Teacher Pack 3.3