Maths Frameworking Teacher Pack 3.2 by Collins

Teacher Pack 3.2

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

7537884 TEACHER PACK 3.2 title.indd 1

20/05/2014 10:07

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

4.3 Two-way tables 4.4 Comparing two or more sets of data 4.5 Statistical investigations Review questions Challenge – Rainforest deforestation

viii xv

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

47 49 51 51

5 Applications of graphs 1 3

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

5 7 9 11 11

53 55 57 59 61 61

6 Pythagoras’ theorem

2 Equations and formulae Overview 2.1 Multiplying out brackets 2.2 Factorising algebraic expressions 2.3 Equations with brackets 2.4 Equations with fractions 2.5 Rearranging formulae Review questions Investigation – Body mass index

Overview 6.1 Introducing Pythagoras’ theorem 6.2 Calculating the length of the hypotenuse 6.3 Calculating the length of a shorter side 6.4 Using Pythagoras’ theorem to solve problems Review questions Activity – Practical Pythagoras

13 15 17 19 21 23 25 25

63 65 67 69 71 73 73

3 Polygons Overview 3.1 Angles in polygons 3.2 Constructions 3.3 Angles in regular polygons 3.4 Regular polygons and tessellations Review questions Activity – Garden design

7 Fractions

27 29 31 33

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

35 37 37

4 Using data Overview 4.1 Scatter graphs and correlation 4.2 Time-series graphs

Maths Frameworking 3rd edition Teacher Pack 3.2

75 77 79 81 83 85 85

39 41 43

iii

8 Algebra Overview 8.1 More about brackets 8.2 Factorising expressions containing powers 8.3 Expanding the product of two brackets Review questions Challenge – Graphs from expressions

12 Compound units 87 89

Overview 12.1 Speed 12.2 More about proportion 12.3 Unit costs Review questions Challenge – Population density

91 93 95

13 Right-angled triangles

Overview 13.1 Introducing 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 Rounding appropriately 9.4 Mental calculations 9.5 Solving problems Review questions Mathematical reasoning – Paper

97 99 101 103 105 107 109 109

10 Prisms and cylinders Overview 10.1 Metric units for area and volume 10.2 Volume of a prism 10.3 Surface area of a prism 10.4 Volume of a cylinder 10.5 Surface area of a cylinder Review questions Problem solving – Packaging cartons of fruit juice

137 139 141 143 145 145

147 149 151 153 155 157 157

14 Revision and GCSE preparation

111 113 115 117 119 121 123

Overview Answers

159 161

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

165 178 182

123

11 Solving equations graphically Overview 11.1 Graphs from equations in the form ay ± bx = c 11.2 Graphs from quadratic equations 11.3 Solving quadratic equations by drawing graphs 11.4 Solving simultaneous equations by graphs Review questions Challenge – Linear programming

Maths Frameworking 3rd edition Teacher Pack 3.2

125 127 129 131 133 135 135

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

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 other, even if it is only in terms of financial capability. Use the following link 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, for example, 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 2 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 3.2

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, be careful not to leave out too much or move on too fast.

Maths Frameworking 3rd edition Teacher Pack 3.2

Lesson 1.1 Simple interest Learning objectives

Resources and homework

• •

• • • • •

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

Pupil Book 3.2, 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 standards 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 question at the end of the exercise requires pupils to start thinking about how they can use formulae to help them to generalise questions about percentage change. This is a powerful idea that you will need to develop carefully.

Common misconceptions and remediation •

Pupils often struggle when they start using percentages greater than 100. Using a real-life example will help pupils to overcome this. Start with percentages that pupils are already comfortable working with, for example, an explanation based on a shop selling a pair of jeans that cost £30 to make. A 50% profit would mean that the shop sold the jeans for £30 plus £15, which is £45; 100% profit would mean selling the jeans for £30 plus £30, which is £60; 150% profit would mean selling the jeans for £30 plus £45, which is £75.

Probing questions • •

Talk me through how you would increase or decrease £22 by, for example, 25%. Can you work it out in a different way?

Part 1 • • •

Working in pairs, ask pupils 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 to comment on any suggestions raised here. One type of interest is called simple interest. This is calculated as a percentage. As long as the loan exists, the recipient will pay the lender a percentage of the loan at regular intervals. Work through Example 1 on page 7 of the Pupil Book as a class. Pupils can then work through examples 2 and 3. Check pupils’ understanding before they do Exercise 1A. Pupils can now do Exercise 1A from Pupil Book 3.2. You could ask more able pupils to use the internet to find out about compound interest. Maths Frameworking 3rd edition Teacher Pack 3.2

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 b £4.50 c £400 d £450 e £600 f £39 g £164 h £120 2 a £11.05 b £29.75 c £1495 d £8.05 e £8.68 f £212.80 g £50.16 h £1.63 3 a £16 b £144 4 a £95 b £1140 5 a £172.80 b £4147.20 6 £40.88 7 £1792 8 £6 9 Mary pays the most interest, £907.20, compared to only £453.60 for Cameron. 10 1.8% per month 11 1.4% per month 12 1.8% per month 13 a £153.60 b 19.2% Challenge: Using a formula A £60 B a £75 b £225 c £1350

Maths Frameworking 3rd edition Teacher Pack 3.2

Lesson 1.2 Percentage increases and decreases Learning objectives

Resources and homework

•

• • • • •

•

To calculate the result of a percentage increase or decrease To choose the most appropriate method to calculate a percentage change

Pupil Book 3.2, 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

Problem solving and reasoning help •

This lesson reinforces the concept of using percentage as an operator. This is an important step to ensure confidence and fluency in pupils, so make sure that you spend time on this lesson. It often helps to make links to fractions as operators. In the challenge 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 such as: 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, but let them 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 is 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 there are several methods that one can use 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 between methods. Maths Frameworking 3rd edition Teacher Pack 3.2

• • • •

Tell pupils that the multiplier method is often the most efficient, and the focus of this lesson. Say that the multiplier method involves multiplying the original value by an appropriate number to calculate the result of the percentage change. Work through Example 4 on page 10 of the Pupil Book as a class; then have the class work through examples 5 to 7 in pairs. Check pupils’ understanding before they do Exercise 1B. Pupils can now do Exercise 1B from Pupil Book 3.2.

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 £500 b £300 c £456 d £344 2 a £136 b £76.50 c £63.75 d £40.80 3 a £354.69 b £236.02 c £518.30 d £851.51 4 a £602.55 b £508.95 c £719.55 d £571.55 5 a increase of 7% b decrease of 30% 6 a £6240 b £9880 c £10 920 d £12 220 7 5% 8 a 8.9% b 15.4% c 37.9% d 135% 9 a 25% b 11.1% c 3.1% d 69.8% 10 39.3% 11 145% 12 28% 13 Peter is wrong, the width has been reduced by 20%, but the length has been reduced by 25%. Challenge: Population change A 36 000 B 39 600 C 37 620 D 25.4%

Maths Frameworking 3rd edition Teacher Pack 3.2

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 3.2, 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 word

•

Food technology – to calculate comparative dietary components Geography – to calculate populations at a given period, given the current size of the population and a rate of change

original value

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 has depreciated by 5% and is valued at £8500. What was its value 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 3.2

Part 2 •

• •

Use examples 8 to 10 on pages 14 and 15 of the Pupil Book to demonstrate how to use multiplicative reasoning and inverse operations to calculate the percentage change between two values. This concept is quite challenging, so make sure that you take time over it in order to give pupils the opportunity to ask questions. Support less able pupils with more worked examples as part of guided group work. Pupils can now do Exercise 1C from Pupil Book 3.2.

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. Extension: What percentage increase would take the sale price back to its starting point? Answers Exercise 1C 1 a 1.1 b 1.45 c 1.87 2 a 0.95 b 0.58 c 0.43 3 a £250 b £450 c £180 4 a £700 b £64 c £653.95 5 500 6 600g 7 £650 8 £65 9 £67.50 10 a £128 b 240 11 a £56.40 b £456.80 c £879 12 a £54.30 b £143.80 c £281.53 13 a €285 b €83.45 c €1860 14 38.1 million (38 064 516) 15 4280 Financial skills: VAT A £587.50 B £540.50 C 18%

Maths Frameworking 3rd edition Teacher Pack 3.2

d 2.35 d 0.15 d £720 d £60

Lesson 1.4 Using percentages Learning objective

Resources

•

• • • • •

To choose the correct calculation to work out a percentage

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

Links to other subjects

Key words

•

Food technology – to calculate dietary requirement for different people Geography – to calculate comparative populations in different areas or over time

No new key words for this topic

Problem solving and reasoning help •

This lesson continues to develop and consolidate pupils’ understanding of percentages by giving them the opportunity to make choices and decisions about the methods they use in a range of contexts.

Common misconceptions and remediation •

Pupils often learn a set of rules without really understanding them. Pupils often meet the different questions over time, so they never have the opportunity to identify the type of question and make independent decisions about which method to use. Therefore, provide pupils with opportunities to check their understanding by making choices and decisions over the approaches they use, in a range of increasingly complex and unfamiliar situations.

Probing questions •

The answer to a percentage increase question is £12. Make up an easy and a difficult question.

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 must be able to decide on the type of question and thus the method to use to solve it. Make it clear 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. Write the questions from Example 11 on page 17 of the Pupil Book on the board: The ratio of men to women in a keep-fit class is 4 : 3. a Work out the percentage of the keep-fit class that is made up of men. b What is the number of women as a percentage of the number of men? Maths Frameworking 3rd edition Teacher Pack 3.2

•

• • • • •

Working in pairs, ask pupils to describe in their own words what type of questions these are and what mathematics they will have to use. Encourage them to identify key words that will help them to decide what to do. Pupils do not need to work out the answer yet. Take feedback (pupils share key words). As a class, look at the solution in the Pupil Book. This is challenging but very important in terms of pupils being able to apply their learning independently. Ensure that you take the time to give pupils the opportunity to ask questions. You could go through a similar process with pupils for Example 12. Support less able pupils with more worked examples as part of guided group work. You may need to provide some simple examples to help them review what they already know. Pupils can now do Exercise 1D from Pupil Book 3.2.

Part 3 •

• • •

        

Revisit what you did at the end of Lesson 1.2 when pupils decided what made a percentage increase question easy or difficult. Tell pupils that now they will extend this, using all they know about percentages as well as fractions, decimals and ratio to design their own percentage questions. They need to make up an easy question and one that is difficult. Encourage more able pupils to design multi-step questions. Share some examples with the class and discuss what makes them easy or difficult. Use grouping to support or challenge pupils. Answers Exercise 1D 1 a 39.8% b 31 2 a 87.5% b 700% 3 a 75% b 52.9% c 60% 4 39% 5 a 80 b 80% c 125% 6 a 44.4% b 80% c 125% 7 a 28.3% b 38.1% c 76.9% 8 a 60% b 240 c 66.7% 9 a 1742 b 56.5% c 1218 10 a 8.6% b £407.15 c £84 11 a £50.60 b £29.80 c £79.81 d £142.80 12 a 300 b1:4 Challenge: Different representations A 48 dogs : 32 cats can be simplified to 3 : 2. 3 : 2 is the same as 60% : 40%, so 60% of the animals are dogs. 32 simplifies to 2 48

B The ratio of cows to sheep is 5 : 3. 62.5% of the animals are cows. The number of sheep is 3 the number of cows. 5

C The ratio of children to adults is 1 : 14. 93.3% of the passengers are adults. The number of children is 1 of the number of adults. 14

Maths Frameworking 3rd edition Teacher Pack 3.2

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 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 challenge 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 that pupils are familiar with 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. You may want to check pupils’ understanding of the multiplier method for percentage increase and decrease, which they 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.

Maths Frameworking 3rd edition Teacher Pack 3.2

Answers to Review questions 1 a i £225.50 ii £22.55 iii £35.75 iv £310.75 b i 12.7% ii 2.4% iii 95.7% iv 131.3% 2 a 78 m b 42 m 3 a £81 b 18% 4 She paid £45 in interest and also paid back the £300. 5 a £12 b £25 6 Alison 14.1% increase, Wayne 91.1% increase, Jasmine 57.6% decrease 7 a £56.25 b £ 738 c £48 d £58 8 a 64 cm2 b 10 cm c 100 cm2 d 56.25% 9 a 560 b 806 10 a £85.20 b £108.40 11 30 pairs Answers to Challenge – Exponential growth 1 £3000, £4000, £5000, £6000 2 a 100% + 50% = 150% = 1.5 b 1000 × 1.5 = 1500 c £2250 d £3375 f and g

e 2250, 3375, 5063, 7594

h straight line will continue i curve will continue to get steeper a 1.25 b 1250, 1563, 1953, 2441, 3052 c

d same starting point

Maths Frameworking 3rd edition Teacher Pack 3.2

e increases more gradually

Equations and formulae

Learning objectives • • •

How to expand brackets and factorise algebraic expressions How to solve more complex equations How to rearrange formulae

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 previously learned algebraic techniques and moves on to more advanced methods of algebraic manipulation. These include: expanding brackets with negative coefficients, factorising algebraic expressions, solving linear equations involving fractions and rearranging formulae.

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 do we need to be able to rearrange formulae?

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 Algebra Use and interpret algebraic notation, including: z Substitute numerical values into formulae and expressions, including scientific formulae Simplify and manipulate algebraic expressions to maintain equivalence by: o collecting like terms o multiplying a single term over a bracket

Maths Frameworking 3rd edition Teacher Pack 3.2

• •

o taking out common factors o expanding products of two or more binomials Understand and use standard mathematical formulae; rearrange formulae to change the subject Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)

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 3.2