Maths Frameworking Teacher Pack 1.3 by Collins

Teacher Pack 1.3

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

7537839 TEACHER PACK 1.3 title.indd 1

14/01/2014 17:52

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

3.4 Surface area and volume of cubes and cuboids Review questions Problem solving – Design a bedroom Answers

4 Decimal numbers Overview 4.1 Multiplying and dividing by 10, 100, 1000 and 10 000 4.2 Ordering decimals 4.3 Estimates 4.4 Adding and subtracting decimals 4.5 Multiplying decimals 4.6 Dividing decimals Review questions Financial skills – Porridge is so good for you! Answers

1 3 5 7 9 11 13 13 14

2 Sequences Overview 2.1 Function machines 2.2 Sequences and rules 2.3 Finding missing terms 2.4 Working out the nth term 2.5 Other sequences Review questions Mathematical reasoning – Valencia Planetarium Answers

Maths Frameworking 3rd edition Teacher Pack 1.3

43 45 47 49 51 53 55 57 57 58

5 Working with numbers 15 17 19 21 23 25 29

Overview 5.1 Square numbers and square roots 5.2 Rounding 5.3 Order of operations 5.4 Multiplication problems without a calculator 5.5 Division problems without a calculator 5.6 Calculations with measurements Review questions Problem solving – What is your carbon footprint? Answers

29 30

3 Perimeter, area and volume Overview 3.1 Perimeter and area of Rectangles 3.2 Perimeter and area of compound shapes 3.3 Area of some other 2D shapes

41 42

xvi

1 Using numbers Overview 1.1 Charts and financial mathematics 1.2 Positive and negative numbers 1.3 Simple arithmetic with negative numbers 1.4 Subtracting negative numbers 1.5 Multiplying negative numbers Review questions Problem solving – Where in the world? Answers

39 41

59 61 63 65 67 69 71 73 73 74

6 Statistics

35 37

Overview 6.1 Mode, median and range

iii

75 77

6.2 The mean 6.3 Statistical diagrams 6.4 Collecting and using discrete data 6.5 Collecting and using continuous data 6.6 Data collection Review questions Challenge – Maths tournament Answers

9.6 Properties of triangles and quadrilaterals Review questions Activity – Constructing triangles Answers

79 81 83 85 87 89 89 90

10 Coordinates and graphs Overview 10.1 Coordinates in four quadrants 10.2 Graphs from relationships 10.3 Predicting graphs from relationships 10.4 Graphs of fixed values of x and y, y = x and y = –x 10.5 Graphs of the form x + y = a 10.6 Graphs from the real world Review questions Challenge – Travelling abroad Answers

7 Algebra Overview 7.1 Expressions and substitution 7.2 Simplifying expressions 7.3 Using formulae 7.4 Writing formulae Review questions Problem solving – Winter sports Answers

91 93 95 97 99 101 101 102

8 Fractions Overview 8.1 Equivalent fractions 8.2 Comparing fractions 8.3 Adding and subtracting fractions 8.4 Mixed numbers and improper fractions 8.5 Calculations with mixed numbers Review questions Challenge – Fractional dissections Answers

Maths Frameworking 3rd edition Teacher Pack 1.3

133 135 137 139 141 143 145 147 147 148

11 Percentages 103 105 107

Overview 11.1 Fractions, decimals and percentages 11.2 Fractions of a quantity 11.3 Calculating simple percentages 11.4 Percentages with a calculator 11.5 Percentage increases and decreases Review questions Financial skills – Income tax Answers

109 111 113 115 115 116

9 Angles Overview 9.1 Measuring and drawing angles 9.2 Calculating angles 9.3 Corresponding and alternate angles 9.4 Angles in a triangle 9.5 Angles in a quadrilateral

129 131 131 132

149 151 153 155 157 159 161 161 162

12 Probability 117 119 121

Overview 12.1 Probability scales 12.2 Combined events 12.3 Experimental probability Review questions Financial skills – School Easter Fayre Answers

123 125 127

163 165 167 169 171 171 172

17 Ratio

13 Symmetry Overview 13.1 Line symmetry and rotational symmetry 13.2 Reflections 13.3 Rotations 13.4 Tessellations Review questions Activity – Rangoli patterns Answers

173 175 177 179 181 183 183 184

14 Equations Overview 14.1 Finding unknown numbers 14.2 Solving equations 14.3 Solving more complex equations 14.4 Setting up and solving equations Review questions Challenge – Number puzzles Answers

Overview 17.1 Introduction to ratios 17.2 Simplifying ratios 17.3 Ratios and sharing 17.4 Solving problems Review questions Problem solving – Smoothie bar Answers

217 219 221 223 225 227 227 228

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

229 246 252

185 187 189 191 193 195 195 196

15 Interpreting data Overview 15.1 Pie charts 15.2 Comparing range and averages of data 15.3 Statistical surveys Review questions Challenge – Ice skate dancing competition Answers

197 199 201 203 205 205 206

16 3D shapes Overview 16.1 Naming and drawing 3D shapes 16.2 Using nets to construct 3D shapes 16.3 3D investigations Review questions Problem solving – Packing boxes Answers

Maths Frameworking 3rd edition Teacher Pack 1.3

207 209 211 213 215 215 216

Using numbers

Learning objectives • • • • • • •

How to use number skills in real life How to use numbers in financial mathematics How to use a number line to compare negative numbers, including decimals How to use the symbols < (less than) and > (greater than) How to use a number line to calculate with negative numbers How to add and subtract negative numbers How to multiply negative numbers by both positive and negative numbers

Prior knowledge • How to write and read whole numbers. By Year 6, pupils read, write, order and compare numbers up to 10 000 000 and determine the value of each digit. • How to add and subtract positive numbers. In KS2, pupils perform increasingly complex calculations using a range of mental methods and more formal written algorithms. • Multiplication tables up to 12 × 12. By Year 5, pupils frequently apply the multiplication tables and related division facts, commit them to memory and use them confidently to make larger calculations. During Year 6, pupils continue to use these skills to maintain fluency. • How to use a calculator to do simple calculations. Calculators are introduced near the end of KS2 (if pupils’ written and mental arithmetic skills are secure), to support pupils’ conceptual understanding and exploration of more complex number problems.

Context • The questions are designed to activate prior knowledge and exercise mathematical ‘fluency’, that is, pupils’ ability to manipulate mathematical language and concepts and apply them in different contexts. • Tables and charts appear everywhere in real life. It is important that pupils are confident with extracting and using information from these, in increasingly unfamiliar and complex situations. • Pupils should also be confident with setting up their own charts and tables to help them solve problems both in mathematics and across a range of subjects. Pupils should be able to assess what types of charts and tables will best help them to clarify problems, and to identify when the representation used may be less helpful. • The first reference to negative numbers can be found in China in 200 BCE. Introduce the history of negative numbers with this unusual statement of the rules for negative numbers by an Indian mathematician named Brahmagupta (598–670) A debt minus zero is a debt. A fortune minus zero is a fortune. Zero minus zero is a zero. A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune.

Maths Frameworking 3rd edition Teacher Pack 1.3

The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt. Find information on the internet by doing a search for: history of negative numbers. • Money problems are everywhere in life and pupils should realise just how important their ability to interpret these problems and identify the mathematics involved is to their future financial wellbeing. This chapter provides many FS questions to give pupils practice.

Discussion points • Choose a number to enter into a calculator. Add 325 (or, for example, multiply by 16). What single operation will take you back to your starting number? • If someone has forgotten the 7 multiplication table, what tips would you give that person so that he or she can work out the answer? • What other links between tables are useful?

Associated Collins ICT resources • Chapter 1 interactive activities on Collins Connect online platform • Subtracting negative numbers video on Collins Connect online platform • Escape Wonder of Maths on Collins Connect online platform

Curriculum references Number • Understand and use place value for decimals, measures and integers of any size. • Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥. • Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative. Solve problems • Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems. • Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics. • Begin to model situations mathematically and express the results using a range of formal mathematical representations.

Fast track for classes following a 2-year scheme of work • Ensure that pupils have a good understanding of the rules they are applying throughout the chapter. As soon as they have demonstrated their understanding, move them on to the PS and MR questions.

Maths Frameworking 3rd edition Teacher Pack 1.3

Lesson 1.1 Charts and financial mathematics Learning objectives

Resources and homework

• To carry out calculations from information given in tables and charts • To understand and use financial language

• • • •

Links to other subjects

Key words

• Science – to use information given in tables and charts • History – to extract chronological information provided in tables and charts • Physical education – to compare results given in tables and charts

• • • •

Pupil Book 1.3, pages 7–12 Intervention Workbook 1, pages 60–63 Homework Book 1, section 1.1 Online homework 1.1, questions 1–10

balance credit chart debit

Problem solving and reasoning help • Many of the questions in Exercise 2A are either PS or FS questions. Pupils may not see immediately how the mathematical techniques they are learning apply to some of these problems. Make sure that pupils read each question carefully and discuss the mathematics involved. Encourage pupils to draw on their knowledge of bank statements, credit and debt, and profit or loss accounts. • Pupils may be unfamiliar with some contexts. A useful way to check this is to have class discussions at key points during the lesson.

Common misconceptions and remediation • When working with money, pupils often forget to convert, for example, between pence and pounds. Encourage them to realise that it is always good practice to check that their solutions make sense. • Also encourage pupils to develop the use of estimation across the different strands of mathematics.

Probing questions • Present pupils with a chart or table and ask them to design a question for it. Ask more able pupils to design an easy question and a hard question, and to identify the criteria for each. Encourage less able pupils to design an easy question. • ‘The number of days = seven times the number of weeks.’ Using time or money, ask pupils to write as many sentences like this as they can. More able pupils might recognise fractional relationships and the link to inverse relationships. This recognition is a good link to algebraic relationships.

Part 1 • Put an example of a local timetable on the board. • Ask pupils to work in pairs to prepare questions based on the timetable. • Encourage more able students to write questions that are easy, difficult and hard. They should also explain what makes the questions easy, difficult or hard.

Part 2 • Explain that data can be shown in various ways, for example, as lists, tables and charts. • Ask the class to think of different ways to do this. For example: Maths Frameworking 3rd edition Teacher Pack 1.3

• • • • • • •

•

o shopping lists o menus o television programme times o calendars o bus and train timetables o mileage charts o football or other sports results. Write all pupils’ suggestions on the board. It is important to be able to read and interpret data correctly from tables and charts. For example, many people find it difficult to read bus and train timetables. Show the class a selection of, for example, calendars, bus timetables, tables of sports results, television programme listings from newspapers and magazines. Encourage pupils to cut out some of these and to make a display of different ways to represent data. Discuss with the class what each table shows and the importance of being able to read the data correctly. Pupils can now do Exercise 1A from Pupil Book 1.3. If necessary, pupils can refer to the examples in the Pupil Book. Many of the questions are PS or FS problems, and while some contexts may be familiar to pupils, do not make assumptions about their knowledge. For example, having a mobile phone does not mean that pupils understand how a mobile phone contract works. Encourage pupils to discuss the details of the contexts in questions 2 and 3. Address any questions that pupils may have about the details and language used. More able pupils who are confident with simple time and money questions can start with question 4.

Part 3 • Ask the class to give examples of different tables and charts that can be used to represent data. • For more able pupils, you could provide examples of more effective and less effective tables; then ask pupils to explain why they think the tables are effective or not. Answers 1 £1040.82 2 £723 3 a 334.46, 280.84, 263.96, 241.17, 469.71, 469.71 b £93.29 4 210 270 360 576 432.22 464.63 493.95 672.80 Kenny 5 168 159 184 179 49 145 6 376 miles 7 a 895 kWh b £124.325 8 a 123 b £91.66

Maths Frameworking 3rd edition Teacher Pack 1.3

c £143.46

Lesson 1.2 Positive and negative numbers Learning objectives 



To use a number line to order positive and negative numbers, including decimals To understand and use the symbols < (less than) and > (greater than)

Links to other subjects 



Physical education – to place results on a number line compared to an optimal time limit Food technology – to compare calorific intake to a given dietary requirement

Resources and homework     

Pupil Book 1.3, pages 12–15 Intervention Workbook 1, pages 23–24 Intervention Workbook 3, pages 20–21 Homework Book 1, section 1.2 Online homework 1.2, questions 1–10

Key words    

greater than negative number less than positive number

Problem solving and reasoning help • Most of the questions in this lesson focus on temperature as the context, and the use of negative numbers to describe temperatures below zero. This section also uses the concept of difference to introduce negative numbers as an operator, as well as a point on the number line. This is demonstrated in the ‘Problem solving – Where in the world?’ activity at the end of the chapter. • Challenge pupils to explain how they can transfer their understanding to different contexts. Encourage them to explain that while the context may be different, the mathematics that they need to apply is the same. A suitable example would be to use negative numbers to describe credit and debt situations. Find suitable support material by searching on the internet, for example, for ‘finances’, ‘banking’ or look at this link: http://www.pfeg.org.

Common misconceptions and remediation • Pupils often confuse the operation of subtraction and negative numbers as numbers on a number line. The sign is the same for both, which makes it more confusing. Encourage pupils to visualise the number line when making calculations. This will help them to see the place of negative numbers and the effect of the operation of subtracting both positive and negative numbers, and to discuss the difference.

Probing questions • Explain whether the following are true or false: o –8 is less than –6 o –36 is greater than –34 • What image could you use to help someone who is not confident with negative numbers? • Can you write your own question using negative numbers?

Part 1 • It is best to teach this lesson using a number hoop. However, if you do not have a number hoop, draw a number line with 10 segments on the board and use that, or use a counting stick marked with 10 segments. • Point to one marker on the hoop and say: ‘This is 20.’ Point to the next marker and say: ‘This is 17.’ Then ask the class to count down in 3s until a few pupils get it wrong. You could ask pupils to do this individually. Maths Frameworking 3rd edition Teacher Pack 1.3

• Repeat with different starting numbers and different jumps, but always count into the negative integers. • You could start with a negative number and count up, or you could move backwards and forwards between counting up and down and vary the jump size.

Part 2 • Draw on the board (horizontally or vertically) a number line and mark 20 segments on it. Start at the mid-point and number the right half of the line from 0 to 10. • Ask the class how we could use the number line to calculate 7 – 3. • Establish that we start at 0 and move first in the positive direction for 7, and then in the negative direction for –3. Mark the number line as shown here.

• Repeat if necessary with other similar examples, making sure that in each case pupils obtain a positive answer. • Now ask pupils how they could use the same idea to find the answer to 3 – 7. • Using the same procedures, pupils should grasp the idea of extending the line in the negative direction.

Repeat with other examples and encourage the mental (or actual) use of the number line. Introduce the symbol >, or greater than, and <, or less than, with suitable number sentences such as –2 < 6 or –3 > –6, linking them to the number line as a visual reminder. • Pupils can now do Exercise 1B from Pupil Book 1.3. • More able pupils, who are confident with simple ordering of numbers, including negative numbers, can start with question 5.

Part 3 • Place or write a list of formulae on the board. • Ask pupils to think about this question: ‘If you are substituting a negative value for the variable, which of these might be tricky?’ • Ask pupils to explain their answers. Answers 1 a 0, –4 b 8, –15 c –15, –20 2 a 25 b 12 c 21 3 1°C 4 a –8, –7, 2, 9, 13 b –12, –11, –10, –7, 8 c –11, –6, –4, 0, 4 d –14, –13, –9, 8, 9 e –18, –10, –9, –7, 10 f –17, –8, 5, 7, 19 5 a true b true c false d false e false f false g true h true 6 a< b> c> d< 7 a –7.5 b –1 c –15 8 15.8, 15.5, –3.5, –4.6, –4.9 9 a 10 b2 c 15 degrees 10 6° 11 a 187, 133.38, –95.16, –95.16 b taken out more than is in the account Challenge: Changing state A water, fresh; water, sea; glycerine; castor oil; linseed oil; mercury; turpentine; chloroform; carbon dioxide; ether; butane; nitrogen B butane, carbon dioxide, chloroform, ether, nitrogen C 131°

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Lesson 1.3 Simple arithmetic with negative numbers Learning objectives

Resources and homework

•

• • • •

•

To carry out additions and subtractions involving negative numbers To use a number line to calculate with negative numbers

Pupil Book 1.3, pages 16–18 Intervention Workbook 3, pages 20–21 Homework Book 1, section 1.3 Online homework 1.3, questions 1–10

Links to other subjects

Key words

•

• add • brackets



Geography – to refer to weather, climate and temperatures Chemistry – to use properties of substances below zero

Problem solving and reasoning help • The magic square is a type of problem that pupils will have come across before. • The 4  4 grid looks more difficult but explain that in fact, if the total is known, there is no difference.

Common misconceptions and remediation • Pupils often learn mathematical rules without understanding the reasoning behind them. Pupils usually benefit from having a visual image such as the number line and/or an understanding of the patterns that lead directly to the rules, in this case: how we use the four operations with both positive numbers and negative numbers. • Having a visual image will ensure that when pupils are in stressful situations such as examinations, they can use these images as backup if they are uncertain.

Probing questions • The answer to a calculation is –8. Can you make up some addition calculations with this answer? • The answer on your calculator is –156. What keys could you have pressed to get this answer?

Part 1 • ‘Addition makes numbers bigger.’ When is this statement true? When is it false? • ‘Subtraction makes numbers smaller.’ When is this statement true? When is it false?

Part 2 • Write the following pattern on the board. Ask pairs of pupils to complete the pattern. 5 + +1 = 6 5+0=5 5 + –1 = 4 5 + –2 = … 5+…=… 5+…=… • Encourage pupils to place the examples in context so that they grasp what is actually happening; ask less able pupils to visualise the examples on a number line.

Maths Frameworking 3rd edition Teacher Pack 1.3

• Ask pupils to repeat the pattern with a different starting number. • Ask pupils to complete the sentence: ‘Adding a negative number gives the same result …‘ • Using mini whiteboards, put some examples on the board, for example, 6 + 7 + (–9). Encourage pupils to see how brackets can help them: 5 + 6 + (–8). • Pupils can now do Exercise 1C from Pupil Book 1.3. • After a couple of simple examples, more able pupils could focus on the contextual MR question 7 and the PR activity at the end of the exercise. As extension work, ask pupils to design their own 3  3 magic square using negative numbers. You could give –2 as the starting point, and say to pupils that the sum of the rows will be three times this.

Part 3 • Using a context of their own choosing, ask pupils to work in pairs to design a test question based on the work they have done in this lesson and in Lesson 1.2. • Encourage pupils to review what makes questions about negative numbers easy or hard and to design multi-step problems of their own. They should also include a mark scheme. • Choose a pair of pupils at a time to present their questions to the class. The rest of the class could assess the questions according to an agreed set of criteria. The criteria may include the choice of context, accuracy and difficulty of the question. Answers 1 a –11 i –30 2 a9 3 a –2 i –3 4 a9 g –15 5 a4 6 a

b –9 j –8 b 15 b6 j –14 b –55 h –118 b –20

c –3 k –26 c9 c –11 k0 c –28 i 15

d 32 l –12 d –53 d 10 l0 d –45 j –165

e8 f0 g 10 h –21 m –31 n –28 o –36 p –94 e –15 m –6 e 36 k –19

f –8 g6 h –15 n 10 o –11 p –15 f0 l –48

16 –10 –6

–22 0 22

6 10 –16

7 £–11 Problem solving: Magic squares A –27 –6 0

c –18 –21 –6

–3 –15 –27

–24 –9 –12

0 –20 –16

–28 –12 4

–8 –4 –24

B 15

–28

–26

–6

–21

–12

–22

–10

–12

–16

–9

–24

–3

–14

–18

–20

–8

–15

–18

–24

–4

–2

–30

Maths Frameworking 3rd edition Teacher Pack 1.3

Lesson 1.4 Subtracting negative numbers Learning objective 

To carry out subtractions involving negative numbers

Resources and homework    

Pupil Book 1.3, pages 19–20 Intervention Workbook 3, pages 20–21 Homework Book 1, section 1.4 Online homework 1.4, questions 1–10

Links to other subjects

Key word

• Geography – to compare temperature when doing country evaluations



subtract

Problem solving and reasoning help • Pupils solve problems involving the difference between negative numbers in more abstract examples. Pupils could design their own contextualised problem using their work on question 5 or by adapting the context in question 10. • You could challenge pupils to write exactly the same question in terms of mathematics, just in a different context. This will help them to see how important it is to be able to identify different types of questions in different contexts.

Common misconceptions and remediation • Pupils often learn mathematical rules without understanding the reasoning behind them. • Pupils may benefit from having a visual image such as the number line and/or an understanding of the patterns that lead directly to the rules, in this case, how we use the four operations with both positive numbers and negative numbers. • Pupils may also find questions involving relationships more challenging. For example, how much more than –2 is 17? Give pupils the opportunity to practise this, particularly within contextualised questions.

Probing questions • The answer to a calculation is –23. Can you make up some subtraction calculations? • The answer on your calculator is –148. What keys could you have pressed to get this answer?

Part 1 • On the board, draw a number line horizontally or vertically, and mark 20 segments on it. Starting at the mid-point, number the right half of the line from 0 to 10. • Working in pairs and taking turns, ask pupils to explain to each other how they would use a number line to calculate 7 – 3. Then they should do the same to find the answer to 3 – 7. Challenge pupils to provide constructive feedback to their partner. This could include challenging the partner if they think the explanation is inaccurate or not strong enough. Encourage pupils to adapt the number line to use a wider range of numbers and to include calculations with more than one step, for example, 10 − 12 + 6. • Take feedback, encouraging pairs to share their explanations. Establish that we start at 0 and move first in the positive direction for 7, and then in the negative direction for –3 and the same for 3 – 7; but this time we move back through 0 to negative numbers. This activity revisits ideas that pupils will have met when learning how to place numbers on the number line and to add and subtract positive and negative numbers. The activity also prepares pupils for the next stage. Maths Frameworking 3rd edition Teacher Pack 1.3

Part 2 • Building on the work in the last lesson, place the following pattern on the board. Ask pupils to work in pairs to complete the pattern. –3 – +1 = –4 –3 – 0 = –3 –3 – –1 = –2 –3 – –2 = … –3 – … = … –3 – … = … • Encourage pupils to place the examples in context. Pupils could repeat the pattern with a different starting number. • Ask pupils to complete this sentence: ‘Subtracting a negative number gives the same result as …’ • Use mini whiteboards to write a couple of examples on the board, for example: –6 –7 – (–9). Encourage pupils to see how brackets can help them: 5 + 6 + (–8). • Pupils can now do Exercise 1D from Pupil Book 1.3. • More able pupils could extend the work they did on magic squares.

Part 3 Problem 1: Immediately before Sarah was paid, her bank balance was –£275.34. The minus sign showed that her account was overdrawn. After being paid, Sarah’s bank balance was £1752.48. How much was Sarah paid? Problem 2: The lowest winter temperature in a city in Norway was –20 °C. The highest summer temperature was 49° higher. What was the summer temperature? • Ask pupils to work in pairs on these two contextual problems. Then they should present their solution to one of the problems, together with an explanation, to another pair. The second pair should provide constructive feedback. • The second pair should then present their solution to the second problem and then they should repeat the process. • Pupils who finish this quickly can design a similar question but in a different context, and present it in groups of four, as outlined above. Answers 1 a 28 b 29 c 40 d –9 e 11 2 a 15 b 15 c –5 d 15 e –4 f –2 g6 i 13 j –7 k –7 l –6 m –8 n 11 o 11 3 a 48 b 0 c 132 d 0 e 47 f 72 g 22 h –60 i 184 j –45 k 81 l 31 4 a 81 b –55 c 372 d –21 5 largest 38, smallest –38 6 a –11 b 18 c 33 d –42 7 a –1, –15, –6, –20, –11 b –3, –6, –1, –8, 1 8 a1 b –11 c 43 d –74 e 24 f –14 g 0 i –18 j –29 k –2 l 72 m –33 n 23 o –28 9 5, –9, –4, 11, 10, 11 10 29 Challenge: Marking a test A Eve 68, Sophia 10, Oliver 25 B 40 correct, 6 incorrect and 4 not attempted

Maths Frameworking 3rd edition Teacher Pack 1.3

h8 p –6

c 8, –8, 8, –4, –12, 8 h 35

Lesson 1.5 Multiplying negative numbers Learning objectives

Resources and homework

• To carry out multiplications involving negative numbers

  

Links to other subjects 

Food technology – to compare calorific intake to a given dietary requirement

Pupil Book 1.3, pages 21–23 Homework Book 1, section 1.5 Online homework 1.5, questions 1-10

Key words    

balance credit chart debit

Problem solving and reasoning help •

The PS questions in this chapter take examples that many pupils will have met before when working with positive numbers, and asks them to extend their reasoning to include negative numbers.

Common misconceptions and remediation •

Pupils often learn mathematical rules without understanding the reasoning behind the rules. Pupils need a visual image such as the number line and/or an understanding of the patterns that lead directly to the rules, in this case, how we use the four operations with both positive numbers and negative numbers.

Probing questions • The answer to a calculation is –24. Can you make up some multiplication calculations? • The answer on your calculator is –148. What keys could you have pressed to get this answer?

Part 1 • Use a number line drawn on the board, or a ‘counting stick’ with 10 divisions marked on it. Say that the right end is the number zero. • Point out that as the students look at the line, the values to the left of zero are negative. • Give a value to each segment, say –3. As a group or with one pupil, count down the line in steps of –3 from zero. You could point out the positions on the line until you reach the end; then continue without prompts. • Repeat with other values for the segments, for example, –4, –2, –1.5, and so on. • Now give a value to each segment, for example, –6, and point to a position on the stick, for example, the fourth division, while asking what value it represents. • Repeat with other values for each segment and different positions on the stick. • Explain that there is an easy way of finding the value at any position on the stick without counting down in steps. This leads to the main lesson activity.

Part 2 • Draw a number line on the board and mark it from –10 to +10. • Place the rules for dealing with directed number problems using the number line. It is important to recall that two signs together can be rewritten as one sign, that is: + + is +, + – is –, – + is – and – – is +. • Demonstrate this by using the number line to work out 7 + –3 (= +4) and –4 – –5 (= +1). Maths Frameworking 3rd edition Teacher Pack 1.3

• Now ask for the answer to: –2 + –2 + –2 + –2 + –2 (= –10). Ask if there is another way to write this, that is 5 × –2 (recall that multiplication is repeated addition). • Repeat with other examples such as: – –4 – –4 – –4 = –3 × –4 (= +12). • Make the link to multiplication as repeated addition by asking pupils if they can see a quick way to work out products such as: –2 × + 3 or –5 × –4 or +7 × +3. • Pupils should come up with the rule that they are the products of the numbers, combined with the rules they met earlier about combining signs. • The – × – = + can cause problems. Ask pupils to complete this pattern: +2 × –3 = –6 +1 × –3 = –3 0×…=… –1 × … = … , and so on. • Link this to division. For example, if –3 × +6 = –18, then –18 ÷ –3 = +6, if +5 × –3 = –15, then –15 ÷ +5 = –3. • Ask pupils to explain a quick way of doing these. As for multiplication, the numbers are divided as normal and the sign of the final answer depends on the combination of signs in the original division problem. • Pupils can now do Exercise 1E from Pupil Book 1.3.

Part 3 • Ask some mental questions such as: How many negative fours make negative sixteen? What are these squared: 6 – 9; –5 – +3; –4 – 3; –2 × +7; –32 ÷ –8; –3? • Encourage pupils to: ‘Say the problem to yourself’. For example, for +7 – –2, pupils should say: ‘plus seven minus minus two’. • Make sure that pupils overcome their confusion about ‘two negatives make a positive’. For example, pupils will often say: –6 – 7 = +13. Answers 1 a2 b 12 c –42 d –2 e 12 f –16 g–10 h 12 i –3 j –40 k 9 l 72 2 a 14 b 27 c –56 d –12 e 15 f –48 g –44 h 16 i –5 j –39 k 96 l 63 3 a –2, 8 b –12, 20, –240 c 8, –12, –96 4 a (–3, 8), (–4, 6), (3, –8), (4, –6) b (2, 6), (–2, –6),(3, 4), (–3, –4) 5 –5 –10 –15 –20 9 27 45 63 a b –6 –12 –18 –24 11 33 55 77 –7 –14 –21 –28 13 39 65 91 –8 –16 –24 –32 15 45 75 105 6 a –126 b –9 c –154 d 20 e –117 f –66 g 13 h 88 i 132 j –8 k –26 l –126 m 124 n 16 o –300 p 1 q 11 r 144 7 a 16, –32, 64 b –81, –243, –729 c –625, 3125, –15 625 d 256, –1024, 4096 8 a i (–8, 2, –3), (–16, –6), (96) ii (–2, –4, –12), (8, 48), (384) b Yes, because negative multiplied by negative always gives a positive and with three sets of multiplication, these always result in the last being either two negatives to multiply together or two positives. c If one of the end numbers is the opposite sign to the other three numbers you will end up with a negative number at the top. Challenge: Number puzzle They always end up with the same number started with, but without the negative sign.

Maths Frameworking 3rd edition Teacher Pack 1.3

Review questions

(Pupil Book pages 24–25)

• The review questions will help to determine pupils’ abilities with regard to the material within Chapter 1.

Problem solving – Where in the world?

(Pupil Book pages 26–27)

• This activity is designed to use both the mathematical reasoning and problem-solving outcomes covered in this chapter in a very common real-life problem set in a slightly less familiar context. • All the information required to answer the questions is given but pupils will need to read the think carefully about how they access the information. Remind them to highlight the key information they will need. • The second part, ‘B How shall we get there?’ involves the use of timetables. This is a common everyday problem but not a context that pupils met explicitly while working on this chapter. As pupils often make mistakes when using timetables, try to provide more familiar examples such as train times or cinema times. Working with different modulo such as units of seconds, minutes, hours and days can be confusing, so a class discussion about these units might be useful. This activity could also be an opportunity for some rich activities involving, in particular, modular arithmetic. Search on the internet for ‘modular arithmetic’ or look at: http://en.wikipedia.org/wiki/Modular_arithmetic • As a warm-up before pupils work on the questions, ask other questions such as: o Does anyone know what time it is in …? o Show images of clocks in different parts of the world. • Pupils can now work on the questions individually or in groups. • Pupils can develop this topic further by using the internet to research their own journeys across different time zones.

Maths Frameworking 3rd edition Teacher Pack 1.3

Answers 1 34 m 2 22 3 a £240 b £130 4 31 5 a 1080.25, 950.25, 700.25, 952.96, 952.96 b debit is money taken out; credit is money put in 6 a 19 759 m b 2063 m 7 a Item Buying Selling Profit (£) price (£) price (£) Silver dish 59 41 –18 Umbrella 33 47 +14 Toy car 45 55 +10 Total 137 143 +6 b Item Buying Selling Profit (£) price (£) price (£) Silver dish 38 55 +17 Umbrella 49 85 +36 Toy car 110 58 –52 Total 197 198 +1 c The Red team won by £5.00. d Red team Red team with without the the bonus buy bonus buy Blue team without the Red won Red won bonus buy Blue team with the bonus buy 8

a –5 g –60

b –27 h 96

Blue won c 56 i 23

d –132 j –1400

Red won e 63 k 300

f –24 l 720

Answers A Where shall we go? 1 San Francisco 2 1 hour 3 3 hours 4 5pm 5 4:30am B How shall we get there? 1 £262 2 10:00, 22nd; 18:00 23rd; 07:15, 22nd 3 09:30, Sept 1st; 10:50, Sept 1st; 14:00, Sept 1st 4 a 19:15 from Heathrow and 1400 from Sydney on Sept 1st b 16:00 from Heathrow and 0930 from Sydney on Sept 1st

Maths Frameworking 3rd edition Teacher Pack 1.3