Maths Frameworking Teacher Pack 2.1 by Collins

Teacher Pack 2.1

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

7537846 TEACHER PACK 2.1 title.indd 1

11/04/2014 10:37

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

4 Percentages

Overview 4.1 Calculating percentages 4.2 Calculating the result of a percentage change 4.3 Calculating a percentage change Review questions Challenge – Changes in population

ix xvi

1 Working with numbers Overview 1.1 Adding and subtracting with negative numbers 1.2 Multiplying and dividing negative numbers 1.3 Factors and highest common factors (HCF) 1.4 Multiples and lowest common multiple (LCM) 1.5 Squares, cubes and roots 1.6 Prime factors Review questions Challenge – The Eiffel Tower

Maths Frameworking 3rd edition Teacher Pack 2.1

49 51 51

5 Sequences

Overview 5.1 The Fibonacci sequence 5.2 Algebra and function machines 5.3 The nth term of a sequence Review questions Investigation – Pond borders

5 7 9 11 13 15 15

53 55 57 59 61 61

6 Area Overview 6.1 Area of a rectangle 6.2 Areas of compound shapes 6.3 Area of a triangle 6.4 Area of a parallelogram Review questions Investigation – Pick’s formula

17 19 21 23 25 27 27

63 65 67 69 71 73 73

7 Graphs Overview 7.1 Rules with coordinates 7.2 Graphs from rules 7.3 Graphs from simple quadratic equations 7.4 Distance–time graphs Review questions Problem solving – The M60

3 Probability Overview 3.1 Probability scales 3.2 Collecting data for a frequency table 3.3 Mixed events 3.4 Using a sample space to calculate probabilities 3.5 Experimental probability Review questions Financial skills – Fun in the fairground

2 Geometry Overview 2.1 Parallel and perpendicular lines 2.2 Angles in triangles and quadrilaterals 2.3 Translations 2.4 Rotations Review questions Challenge – Constructing triangles

43 45

29 31 33 35

75 77 79 81 83 85 85

8 Simplifying numbers Overview 8.1 Powers of 10 8.2 Large numbers and rounding 8.3 Significant figures 8.4 Estimating answers 8.5 Problem solving with decimals

37 39 41 41

iii

87 89 91 93 95 97

13 Proportion

Review questions 99 Challenge – Space – to see where no one has seen before 99

Overview 13.1 Direct proportion 13.2 Graphs and direct proportion 13.3 Inverse proportion 13.4 The difference between direct proportion and inverse proportion Review questions Challenge – Coach trip

9 Interpreting data Overview 9.1 Information from charts 9.2 Reading pie charts 9.3 Creating pie charts 9.4 Scatter graphs Review questions Challenge – What should we eat?

101 103 105 107 109 111 111

Overview 14.1 The circle and its parts 14.2 Circumference of a circle 14.3 A formula to work out the approximate circumference of a circle Review questions Activity – Constructions

113 115 117 119 121 123 125

Overview 15.1 Equations 15.2 Equations with brackets 15.3 More complex equations 15.4 Substituting into formulae Review questions Reasoning – Old trees

125

127 129 131 133 135

Maths Frameworking 3rd edition Teacher Pack 2.1

173 175 175

177 179 181 183 185 187 187

16 Comparing data

135

12 Fractions and decimals Overview 12.1 Adding and subtracting fractions 12.2 Multiplying fractions and integers 12.3 Dividing with integers and fractions 12.4 Multiplication with powers of ten 12.5 Division with powers of ten Review questions Problem solving – Making estimates

167 169 171

15 Equations and formulae

11 Congruence and scaling Overview 11.1 Congruent shapes 11.2 Shape and ratio 11.3 Scale diagrams Review questions Financial skills – Carpeting a bungalow

163 165 165

14 Circles

10 Algebra Overview 10.1 Algebraic notation 10.2 Like terms 10.3 Expanding brackets 10.4 Using algebra 10.5 Using powers Review questions Mathematical reasoning – Strawberries

151 153 155 159

137 139 141 143

Overview 16.1 Frequency tables 16.2 The mean 16.3 Drawing frequency diagrams 16.4 Comparing data 16.5 Which average to use? Review questions Problem solving – Questionnaire

189 191 193 195 197 199 201 201

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

203 219 225

145 147 149 149 iv

Working with numbers

Learning objectives • • • •

How to multiply and divide negative numbers How to find the highest common factor and the lowest common multiple of sets of numbers How to use powers and find roots How to find the prime factors of a number

Prior knowledge • • • •

How to add and subtract negative integers How to order the operations, following the rules of BIDMAS What a factor is What a multiple is

Context •

Ask pupils what games they play. During the discussion, pupils should realise that they use numbers in many games. For example, darts players need to make quick calculations in order to work out the target numbers they need to score to finish the game. Darts players must also think about possible combinations of scores from three darts. Number skills are also important in field games such as rugby and cricket, and in many board games and computer games. In this chapter, pupils will develop their number skills for daily use.

Discussion points • • •

Can all whole numbers be represented by a product of prime numbers? Why, when you multiply or divide two negative numbers, do you get a positive answer? Why, when you multiply or divide a negative number and a positive number, do you get a negative answer?

Associated Collins ICT resources •

Chapter 1 interactive activities on Collins Connect online platform

Curriculum references •

•

Solve problems Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems Number Use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

Maths Frameworking 3rd edition Teacher Pack 2.1

â&#x20AC;˘

Use integer powers and associated real roots (square, cube and higher), recognise powers of and distinguish between exact representations of roots and their decimal approximations

Fast-track for classes following a 2-year scheme of work Much of the material in this chapter will be new to Year 8 pupils. However, pupils could leave out Exercise 1A of the Pupil Book, which was covered in Year 7. If pupils are quick to grasp the concepts in this chapter they can move swiftly through the exercises, leaving out some of the questions.

Maths Frameworking 3rd edition Teacher Pack 2.1

ÂŠ HarperCollinsPublishers Ltd 2014

Lesson 1.1 Adding and subtracting with negative numbers Learning objective

Resources

• To carry out additions and subtractions involving negative numbers

• • • •

Links to other subjects

Key words

• Geography – to observe weather, climate and temperatures • Chemistry – to recognise properties of substances below zero

• brackets • positive number • negative number

Pupil Book 2.1, pages 7–9 Intervention Workbook 1, pages 23–25 Intervention Workbook 3, pages 20–21 Online homework 1.1, questions 1–10

Problem solving and reasoning help •

The activity at the end of Exercise 1A in the Pupil Book draws on pupils’ learning regarding addition and subtraction of negative numbers. It may be necessary to model an example for less able pupils. Encourage more able pupils to design their own number pyramids.

Common misconceptions and remediation •

Pupils often learn rules without really understanding the reasoning behind each rule. Pupils will benefit from visual images such as a number line and/or an understanding of the patterns that lead to the rules, in this case how we use the four operations with positive and negative numbers. Then, when pupils are in stressful situations such as examinations, they can fall back on these images to provide backup if they are uncertain.

Probing questions • • •

The answer to a calculation is –8. Make up some addition calculations to give this answer. Your calculator display shows –156. What keys could you have pressed for this answer? The answer to a calculation is –23. Can you make up some subtraction calculations?

Part 1 • •

• •

On the board, draw a number line with 21 segments. Start on the left and number each segment from –10 to 10, with 0 at the mid-point. Working in pairs, ask pupils to take turns to explain to each other how they would use the number line to calculate 7 – 3, and then 3 – 7. Pupils should provide constructive feedback, and challenge each other if the explanation is inaccurate or not strong enough. Encourage more able 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. Encourage pairs to share explanations. Establish that we start at 0 and move in the positive direction for 7; then in the negative direction for –3; the same for 3 – 7, but this time we move back through 0 to the negative numbers. This activity revisits adding and subtracting positive and negative numbers on a number line, thus preparing pupils for the next stage.

Maths Frameworking 3rd edition Teacher Pack 2.1

Part 2 •

• •

Write these two patterns on the board. Ask pupils to work in pairs to complete them. 5 + +1 = 6 –3 – +1 = –4 5+0=5 –3 – 0 = –3 5 + –1 = 4 –3 – –1 = –2 5 + –2 = … –3 – –2 = … 5+…=… –3 – … = … 5+…=… –3 – … = … Ask pupils to place the examples in context. Less able pupils should try to visualise the patterns on a number line. Pupils could repeat the patterns using different starting numbers. Ask more able pupils to complete these sentences: o ‘Subtracting a negative number gives the same result as …’ o ‘Adding a negative number gives the same result as …’ Use mini whiteboards to write some examples on the board, for example: –6 –7 – (–9) and 6 + 7 + (–9). Encourage pupils to see how brackets can help them: 5 + 6 + (–8). Pupils can now do Exercise 1A from Pupil Book 2.1.

Part 3 • • •

•

Problem 1: Before being paid, Sarah’s bank balance was –£275.34 (the minus sign showed an overdrawn account). After being paid, the 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. Ask pairs to present their solution to one problem (with an explanation) to another pair, with the second pair providing constructive feedback. The second pair should then present their solution to the second problem and repeat the process. Encourage more able pupils who finish this quickly to design a similar question but in a different context, and to present it in groups of four, as outlined in the previous point. Answers Exercise 1A 1 a −2 b3 c1 d4 e −8 f −4 g6 i6 j −10 k3 l0 m −1 n −10 o −4 2 a4 b2 c −6 d 14 e −3 f −1 g4 i −8 j7 k −12 l −2 m0 n0 o −14 3 a5 b −17 c −10 d −22 e 10 f0 g −5 i7 j −280 k −30 l −24 4 a 12 b 17 c 32 d −7 e0 5 a 12 b 11 c0 d 14 e −2 f7 g3 i 14 j −6 k0 l 10 m −6 n 10 o 20 6 a 21 b0 c 33 d −10 e 39 f 62 g −5 i 27 j −200 k 36 l8 7 a −4 b −5 c −7 d −6 e 16 f −12 g 20 8 a 10 b0 c 15 d −10 e 13 f −2 g −12 i −6 j −14 k 11 l 17 m1 n0 o −15 9 Mon 7, Tue −9, Wed 3, Thu 6, Fri −3, Sat 7 10 21 m Activity: Brick walls A a −6, 5 : 15, −11, 11 : 26, −22 b 5, −7 : −1, 12, −12 : −13, 24

Maths Frameworking 3rd edition Teacher Pack 2.1

h −8 p -13 h −7 p1 h −100

h −7 p4 h 150 h −14 h 15

Lesson 1.2 Multiplying and dividing negative numbers Learning objective

Resources and homework

• To carry out multiplications and divisions involving negative numbers

• Pupil Book 2.1, pages 9–12 • Homework Book 2, section 1.1 • Online homework 1.2, questions 1–10

Links to other subjects

Key words

• Science – to understand magnetism and electricity

• No new key words for this topic

Problem solving and reasoning help •

The challenge at the end of Exercise 1B in the Pupil Book introduces pupils to multiplication grids involving negative numbers. It may be necessary to explain to less able pupils how to work backwards. Encourage more able pupils to design their own multiplication grids.

Common misconceptions and remediation •

One of the main misconceptions pupils have when multiplying two negative numbers is giving a negative answer. Reinforce the fact that when multiplying two negative numbers, the answer will always be positive. And, when multiplying two numbers, pupils often think that the sign of the answer is determined by the sign of the largest number. Remind pupils not to rush through their work, as they need to have a clear understanding of the rules.

Probing questions • • •

Can you make up some multiplication/division questions to give an answer of –24? What keys could you have pressed on your calculator to get the answer of –144? Talk me through how you found the answers to the two questions.

Part 1 • • • • • • •

Use a number line that you have drawn on the board, or a ‘counting stick’ with 10 divisions marked on it. The right end should show the number 0. As pupils look at the number line, point out that the values to the left of zero are negative. Give a value to each segment, for example, –3. As a class, count down the number line in steps of –3 from zero. Point out the positions on the line as you say each number. Repeat with other values for the segments, for example: –4, –2, –1.5. Now give a value to each segment such as –6. Point to a position on the stick, for example, the fourth division, 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 to find the value at any position on the stick without counting down in steps. Move on to Part 2.

Part 2 •

•

Draw a number line on the board and mark it from –10 to +10. Recall the rules for dealing with directed number problems using the number line. Make sure you recall that two signs together can be rewritten as one sign, for example: + + is +; + – is –; – + is –; – – is +. Another way to emphasise this is to say that if the signs are the same, then the overall sign is plus and if the signs are different, then it is minus. Demonstrate this by using the number line to work out: 7 + –3 (= +4) and –4 – –5 (= +1).

Maths Frameworking 3rd edition Teacher Pack 2.1

• • •

•

• •

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). Ask pupils if they can see a quick way to work out products such as: –2 × + 3 or –5 × –4 or +7 × +3. Pupils should mention that the rule is the product of the numbers, combined with the rules they met earlier about combining signs. Explain that – × – = + can cause problems. Ask pupils to complete these: +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 to do these. As for multiplication, the numbers are divided; the sign of the final answer depends on the signs in the original division problem. Pupils can now do Exercise 1B from Pupil Book 2.1.

Part 3 • • •

Ask some mental questions such as: How many negative fours make negative 16? What is: 6 – 9; –5 – +3; –4 – 3; –2 × +7; –32 ÷ –8; –3 squared? Encourage pupils to ‘say the problem to themselves’. For example, for +7 – –2, pupils should say ‘plus seven minus minus two’. Make sure that pupils overcome the confusion about ‘two negatives make a positive’. For example, pupils often say: –6 – 7 = +13. Answers Exercise 1AB 1 a −6 b −12 c −10 d −18 e −24 f −20 g −1 h −6 i −14 j 16 k −60 l 32 m −30 n −18 o −28 p −33 2 a −4 b −6 c −3 d −2 e −2 f −4 g −3 h −6 i −5 j2 k −6 l2 m −7 n −4 o −6 p −2 3 a −6 b4 c −3 d3 e −24 f8 g5 h −6 i −6 4 a 1 × 3 = 3, 0 × 3 = 0, −1 × 3 = −3, −2 × 3 = −6 b 1 × −2 = −2, 0 × −2 = 0, −1 × −2 = 2, −2 × −2 = 4 c −2 × 1 = −2, −1 × 1 = −1, 0 × 1 = 0, 1 × 1 = 1, 2 × 1 = 2 5 a8 b 15 c 25 d 24 e 21 f 20 g 12 h 18 i 14 j 18 k 70 l 40 m 24 n 30 o 33 p 48 Challenge: Algebraic magic squares

Maths Frameworking 3rd edition Teacher Pack 2.1

a3 e3 i5 m7 7 a −4 f 12 8 a −15 e −32 i −24 m −55 9 a5 e2 i4 m −12 10 a 32 e5 i 77 m −7

b4 f4 j2 n6 b3 g −2 b 24 f 30 j 30 n 42 b −8 f −5 j5 n −11 b −4 f −80 j9 n −9

c3 g3 k4 o2 c −3 h −36 c −36 g −20 k 88 o −48 c4 g3 k −3 o3 c −35 g7 k −36 o7

d4 h5 l2 p5 d 10 e5 i −6 d 35 h 28 l −54 p 60 d −6 h −7 l7 p −6 d −55 h −8 l 20 p −84

Lesson 1.3 Factors and highest common factors (HCF) Learning objective

Resources and homework

• To understand and use highest common factors

• Pupil Book 2.1, pages 13–15 • Homework Book 2, section 1.2 • Online homework 1.3, questions 1–10

Links to other subjects

Key words

• Design and technology – to split two different amounts into equal quantities

• common factor • factor • integer

• divisible • highest common factor (HCF)

Problem solving and reasoning help •

Encourage pupils to list all the factors of both numbers when finding the HCF. It will make answering the questions in Exercise 1C of the Pupil Book much easier.

Common misconceptions and remediation •

Pupils sometimes confuse factors and multiples. (Say that multiples come from multiplying.)

Probing questions • •

Two numbers have a highest common factor (HCF) of 7. What are the numbers? Convince me that the HCF of 30 and 45 is 15.

Part 1 • • •

Pupils should write their answers on mini whiteboards, and hold them up when asked. Ask pupils to write: an even number, a multiple of 6, a factor of 12, a prime number, a square number, a number that is a multiple of 3 and 4 at the same time, a triangle number. Go around the class each time, checking pupils’ answers. If necessary, discuss and define what was required. Emphasise factors, as these will be used in Part 2. Feedback suggests that one of the biggest barriers to learning is that pupils do not feel comfortable using mathematical terminology such as: congruent, perimeter, radius. Encourage simple activities to help pupils to remember these words.

Part 2 • •

• • •

Ask pupils to try and write down a number that is a factor of 12 and a factor of 18. Write all numbers on the board. Most possibilities should come up (1, 2, 3, 6, plus some that may be incorrect). Make sure that pupils understand the idea, and if any factors are missing ask what number is needed to complete the set. Ask: ‘What is special about 6?’ Emphasise that it is the highest common factor, or HCF. Repeat for 30 and 50. Then ask for the HCF of 16 and 20. Repeat for 15 and 30. (5 is a likely answer – make sure pupils realise that the HCF is 15.) Repeat for 7 and 9. Ask why the answer is 1. Prime numbers should have been defined in Part 1. If not, define prime numbers. Pupils can now do Exercise 1C from Pupil Book 2.1.

Maths Frameworking 3rd edition Teacher Pack 2.1

Part 3 • • • •

Write numbers on the board (or use prepared cards), for example: 10, 12, 15, 20, 24, 25, 30, 35, 40, 48. Ask pupils to pick out one card and to list all the factors. Ask pupils to pick out two cards. Ask for the HCF. Alternatively, ask for a card that is the highest common factor of, for example, 15 and 20. Answers Exercise 1B 1 a 1, 2, 7, 14 b 1, 5, 25 c 1, 2, 4, 7, 14, 28 d 1, 3, 5, 9, 15, 45 e 1, 2, 4, 5, 8, 10, 20, 40 f 1, 2, 5, 10, 25, 50 g 1, 5, 7, 35 h 1, 2, 3, 4, 6, 9, 12, 18, 36 i 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 j 1, 2, 4, 5, 10, 20, 25, 50, 100 2 a and b any of 10, 14, 35, 70 3 1 and 4 or 5 and 20 4 a 1, 5 b 1, 2, 3, 6 c 1, 2, 4 d 1, 5 e 1, 2, 5, 10 g 1, 2, 3, 6 h 1, 3, 9 5 1, 2, 3, 5, 6, 10, 15, 30 6 a3 b8 c6 d2 e7 f4 g9 h1 7 a 10 b5 c6 d 14 e 15 f 12 g 25 h 20 8 a7 b8 c 27 d 28 2 2 1 9 a b c d 2 e

3 3 16

3 4

3 3 5

3 2 3

10 a 8s b7 Investigation: Tests for divisibility A ends in an even number B a and b pupils’ own work c digits add up to a multiple of 3 C last two digits divisible by 4 D ends in a 5 or 0 E digits all add up to a multiple of 3 and the number is even F digits add up to a multiple of 9 G a 90, 114, 120, 480, 1503, 111 111 b 120, 480, 716 c 90, 114, 120, 480 d 90, 1503

Maths Frameworking 3rd edition Teacher Pack 2.1

f 1, 2, 4

e 90, 120, 480

Lesson 1.4 Multiples and lowest common multiple (LCM) Learning objective

Resources and homework

• To understand and use lowest common multiples

• • •

Pupil Book 2.1, pages 16–18 Homework Book 2, section 1.3 Online homework 1.4, questions 1–10

Links to other subjects

Key words

•

• common multiple • multiple

Food technology – to calculate portions and quantities when preparing food

• lowest common multiple (LCM)

Problem solving and reasoning help •

The PS questions in Exercise 1D of the Pupil Book require pupils to work out the LCM from worded questions. Encourage pupils to pick out the key facts and apply the methods taught.

Common misconceptions and remediation • •

Pupils sometimes confuse factors and multiples. (Say that multiples come from multiplying.) Pupils should understand that the multiples of a number are the times table for that number.

Probing questions • •

How can we tell if two numbers have no common factors? Two numbers have an HCF of 5 and an LCM of 60. What numbers are they?

Part 1 • • • •

Pupils should write their answers on mini whiteboards, and hold them up when asked. Ask for examples of: an odd number, a multiple of 8, a factor of 12, a prime number, a square number, a number that is a multiple of 3 and 4 at the same time, a triangle number. Go around the class and check pupils’ answers, discussing and defining as necessary. Emphasise multiples, as these will be used in Part 2.

Part 2 •

• • • •

•

Still using mini whiteboards, ask pupils to write down a number that is a multiple of 3 and 4 (as asked in Part 1). Write all the answers on the board. Ask for more suggestions if many answers are the same. Hopefully, 12 will have been suggested. Ask pupils what is special about this. Emphasise that it is the lowest common multiple, or LCM. Repeat for a common multiple of 4 and 5. Ask for the LCM of 3 and 5. (15) Then ask for the LCM of 4 and 6. Many pupils will answer 24, as they will have seen that previous answers were the product of these two numbers. Make sure that all pupils understand that in fact 12 is the LCM of 4 and 6. To make it easier, encourage pupils to write down the multiples for each number and look for the first common value in each list. For example, the common value for 4 and 5 is 20: 4 8 12 16 20 24 28 … 5 10 15 20 25 30 35 … Pupils can now do Exercise 1D from Pupil Book 2.1.

Maths Frameworking 3rd edition Teacher Pack 2.1

Part 3 • • • •

Write numbers on the board (or have prepared cards) such as: 1, 2, 3, 4, 6, 8, 10, 12, 15, 20. Ask pupils to pick one card and then write down the first 10 multiples Ask pupils to pick two cards. Ask for the LCM of both cards. Alternatively, ask for a card that is the LCM of, for example, 5 and 6. Answers Exercise 1D 1 a 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 b 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 c 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 d 7, 14, 21, 28, 35, 42, 49, 56 ,63, 70 e 11, 22, 33, 44, 55, 66, 77, 88, 99, 110 2 a any 3 digit even number, for example 356 b any 3 digit even number, for example 704 3 any 2 multiples of 5 that add together to give a multiple of 10, for example 15 and 25 or 30 and 40 4 a 12, 24, 18, 72, 10, 54, 60, 100 b 12, 15, 24, 18, 45, 72, 54, 60 c 12, 24, 72, 60, 100 d 15, 45, 10, 55, 60, 100, 25 e 12, 24, 18, 72, 54, 60 f 18, 45, 72, 54 5 a 10, 20, 30, 40, 50 b 15, 30, 45, 60, 75 c 20, 40, 60, 80, 100 d 40, 80, 120, 160, 200 e 11, 22, 33, 44, 55 6 a i 6, 12, 18, 24, 30 ii 8, 16, 24, 32, 40 iii 9, 18, 27, 36, 45 iv 12, 24, 36, 48, 60 b i 18 ii 24 iii 36 iv 24 7 a 12 b 30 c 30 d 56 e 40 f 45 g 63 h 48 8 a 20 b 12 c 90 d 90 e 60 f 120 g 180 h 60 9 420 10 90 seconds 11 96 cm 12 15 Challenge: LCM and HCF A a 20, 4 b 18, 3 c 60, 5 B a i 5, 10 ii 3, 12 iii 4, 16 by C a i 1, 35 ii 1, 12 iii 2, 22 b xy

Maths Frameworking 3rd edition Teacher Pack 2.1

Lesson 1.5 Squares, cubes and roots Learning objectives

Resources and homework

• To understand and use squares and square roots • To understand and use cubes and cube roots

• • • • •

Links to other subjects

Key words

• Science – to use formulae in experiments

• cube • square

Pupil Book 2.1, pages 18–21 Intervention Workbook 2, pages 28–29 Intervention Workbook 3, pages 31–32 Homework Book 2, section 1.4 Online homework 1.5, questions 1–10 • cube root • square root

Problem solving and reasoning help •

The PS and MR questions in Exercise 1E of the Pupil Book require pupils to apply their learning to worded questions. Encourage pupils to pick out the key facts and apply the methods they have been taught.

Common misconceptions and remediation • •

Reinforce the fact that the square root of a number can be both positive and negative. Another problem is that pupils often think that n2 is n  2 or that n3 is n  3. Explain clearly that this is not the case.

Probing questions •

•

Are the following statements always, sometimes or never true? o Cubing a number makes it bigger. o The square of a number is always positive. o You can find the square root of any number. o You can find the cube root of any number. If sometimes true, precisely when is the statement true and when is it false?

Part 1 • • • •

Use a target board such as the one shown. Assign values to a and b. These need to be squares, for example: a = 1 and b = 4. Randomly select pupils and ask them to evaluate the expressions. Repeat with other values for a and b, for example: a = 4 and b = 9.

√a 2b

3√b 3a b

b 2

a 2

2 3√b a

b2 2

√b 2a

2√a

2√b 3b

Part 2 • • • • • • •

Following on from Part 1, one of the problems asked earlier was ‘If a2 = 9, what is a?’ Pupils may only have identified 3. If so, ask for another solution. Obtain the answer –3. Another problem was ‘What is √9?’ Is there another answer to this? Obtain the answer –3. Emphasise that a square root is generally accepted as the positive square root. Then explain the subtle point that the solution to the equation a2 = 9 can be positive or negative. Ask pupils what we mean by a3. For example: If a = 2, what is a3? If a = 3, what is a3? If a = 4, what is a3? If a = 5, what is a3? Pupils should have the mental skill to work out up to 53, but may find 63 difficult. You could write out the sequences, for example: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Maths Frameworking 3rd edition Teacher Pack 2.1

• • • • •

If a3 = 729, what is a? It is likely that answers of –9 and 9 will be given. Demonstrate that: –9 × –9 × –9 = –729. Hence, only one answer is possible for a3 = 729. Introduce the notation of cube root, 3√729 = 9. What is 3√64? What is 3√125? Ask pupils what 24 means. What is the value of 24? (16) What about 35? (Pupils may need to use a calculator to get: 243) Pupils can now do Exercise 1E from Pupil Book 2.1.

Part 3 •

Here is a quick factual recall test of, for example, squares, cubes, and so on. 1 What is the cube root of 64? 6 What is 5 cubed? 2 What is 1000 as a power of 10? 7 What is 1 million as a power of 10? 3 What is 3 cubed? 8 What is –2 squared? 4 What is the square root of 196? 9 If x squared equals x cubed, what is x? 10 What are the values of x if x2 = 25? 5 What is the cube root of 1000? Answers Exercise 1E 1 a9 b 25 c 4 d 36 e1 f 16 g 49 h 64 i 81 j 100 k 144 l 121 2 a 169 b 441 c 225 d 289 e 529 f 961 g 196 h 324 i 625 j 2025 k 361 l 1089 3 a 27 b −1 c 125 d 64 e 343 f 512 g8 h 729 i 216 j 1000 k −8 l 1331 4 a 64 cm² b 169 cm² c 49 cm² d 441 cm² 5 a 125 cm³ b 1728 cm³ c 29 791 cm³ d 729 cm³ 6 a 10 b 15 c 7 d 20 e 36 f 16 g 80 h 30 i 28 j 54 7 52 + 122 = 132; 62 + 82 = 102; 102 + 242 = 262; 7² + 24² = 25² 8 a6 b4 c3 d 120 e 15 f 21 g 90 h 15 9 a2 b2 c2 d 50 e 45 f 21 g 40 h 60 3 10 27 cm 11 36 cm2 12 You cannot have the same sign multiplied by itself without getting a positive number. 13 for example −1 × −1 × −1 = −1 Challenge: Square roots with a calculator A a 3.5 b 182 c 17.1 d 9.9 B a 22.4 b 31.4 c 8.7 d 6.5

Maths Frameworking 3rd edition Teacher Pack 2.1

Lesson 1.6 Prime factors Learning objectives

Resources and homework

• To understand what prime numbers are • To find the prime numbers of an integer

• • •

Pupil Book 2.1, pages 21–23 Homework Book 2, section 1.5 Online homework 1.6, questions 1–10

Links to other subjects

Key words

• IT – to use internet security for key encryption

• factor tree • prime number

• prime factor

Problem solving and reasoning help •

The PS and MR questions in Exercise 1F of the Pupil Book require pupils to apply their learning to worded questions. Encourage pupils to pick out the key facts and apply the methods they have been taught.

Common misconceptions and remediation •

Remind pupils to include the multiplication signs when writing a number as a product of its prime factors. (These are often replaced incorrectly by addition signs or commas.)

Probing questions • • • •

Give reasons that none of the following are prime numbers: 4094, 1235, 5121. How do you go about finding the prime factors of a given number? What are the prime factors of 125? 81? 343? What do you notice? Can you think of a number that has one repeated prime factor? Or all different prime factors?

Part 1 • • •

Using a target board like the one shown, point to a number and ask a randomly picked pupil to give the factors of the number. Recall the rule for factors, that is, they come in pairs, except for square numbers. The number itself and 1 are always factors. Prime numbers only have two factors.

25 36 70 64 75 81 18 50 20 45 30 63 80 92 16 32 15 10 28 60

Part 2 • • • • • •

•

Ask for the answer to: 2 × 3 × 3. (18) What about: 2 × 2 × 5; 3 × 5 × 5; 3 × 3 × 7? (20, 75 and 63 respectively) What can you say about the numbers in the multiplication? (They are all prime numbers.) This is the prime factor form of a number (the number broken down into a product of primes). How can we find this if we start with the number 30, for example? Explain the tree method: split 30 into a product such as 2 × 15; then continue splitting any number in the product that is not a prime. We can see this easily in the form of a ‘tree’. For example, find the prime factors of 120.

Maths Frameworking 3rd edition Teacher Pack 2.1

• •

An alternative is the division method where the number is repeatedly divided by any prime that will go into it exactly. Demonstrate this with 50. Continue to divide by primes until the answer is 1. 2 50

5 25 55 • • •

1 Repeat with 96 (2 × 2 × 2 × 2 × 2 × 3), 60 (2 × 2 × 3 × 5). Now it may be useful to introduce the index notation, that is: 96 = 25 × 3, 60 = 22 × 3 × 5. Pupils can now do Exercise 1F from Pupil Book 2.1.

Part 3 • • • • • • •

Choose a number, for example, 70. Find the factors. (1, 2, 5, 7, 10, 14, 35, 70) Find the prime factors. (2 × 5 × 7) Choose another number and repeat, for example, 90. The factors are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; the prime factors are 2 × 32 × 5. Ask pupils if they can spot a connection. This is simply that only the prime numbers in the list of factors appear in the prime factors. Discuss how to find the HCF and LCM of 70 and 90. (HCF 10, LCM 630) Repeat with 48 and 64. (HCF 16, LCM 192) Answers Exercise 1F c 28 d 45 e 50 1 a 20 b 30 f 75 g 105 h 175 i 63 j 125 2 a 2×3×7 b3×5×5 c2×2×5×7 d2×5×5×5 e2×2×2×2×2×3×5 f2×2×2×3×3 g2×2×2×2×2×3 h2×2×2×2×2×2×2×2 3 a2×3 b3×3 c2×2×3 d2×3×3 e 2 × 11 f2×3×5 g2×2×2×5 h2×5×5 i 5 × 11 j2×2×2×3×5 4 a2×2×2×3 b2×2×7 c2×2×3×5 d2×2×5×5 e2×3×5×5 5 2, 3, 4 = 2 × 2, 5, 6 = 2 × 3, 7, 8 = 2 × 2 × 2, 9 = 3 × 3, 10 = 2 × 5, 11, 12 = 2 × 2 × 3 6 a 2, 3, 5, 7, 11 b prime numbers 7 a2×2×5 b2×5×5 c2×2×5×5 d2×2×2×5×5×5 8 2×2×2×2×2×2×5×5×5×5×5×5 9 because 1 is not a prime number Challenge: LCM and HCF diagrams pupils’ own diagrams

Maths Frameworking 3rd edition Teacher Pack 2.1

Review questions • •

(Pupil Book pages 24–25)

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 – The Eiffel Tower •

•

• •

(Pupil Book pages 26–27)

This activity encourages pupils to think about a tourist attraction with different facilities and what is involved in running them. The topic could lead to class discussion about environmental issues such as electricity and water usage. Look at the information in the Pupil Book. Ask pupils some questions relating to the activity. Topics could include: o periods of time (for example: When did it open? Who built the Eiffel Tower and why? How long has it been since the foundation stone was laid?) o conversion factors (for example: imperial to metric, square metres to square centimetres, £ to €) Data from the internet will be useful in order to answer the questions fully (do a search for ‘Eiffel Tower’). Encourage pupils, in small groups or individually, to suggest possible questions. Then pupils could present the questions to the class for answers. This will be particularly useful if pupils have access to the internet. Pupils can now work through the questions in the Pupil Book. Ask pupils to develop questions for another tower they have researched, for example, the Empire State Building, the Shard or Blackpool Tower.

Maths Frameworking 3rd edition Teacher Pack 2.1

Answers to Review questions a i and ii any 3 digit multiple of 5, for example: 115, 135 b i and ii any one of 10, 12, 15, 20, 30, 60 2 18 and 2 3 a 45 b 30 c 75 d 99 4 a i 30, 15, 6, 12, 24 ii 30, 15, 5 iii 30 b i 5, 1, 2 ii 30, 15, 5, 1, 2, 6 iii 1, 2, 6, 12 5 a 35 b −5 c −32 d −48 e8 f −63 g 3 h −9 i 72 j5 k −44 l 20 m −6 n −64 o9 p −96 6 a for example: 21 is odd but a multiple of 7 b An even number is exactly divisible by 2. c itself and 1 d 23 and 29 are the primes between 20 and 30, both odd 7 3√27 , √25 , 23 , 72 8 a 36 cm² b 1024 cm² 9 a 64 cm³ b 2744 cm³ 10 5, 6, 10, 12, 15, 20, 30 11 a 35 b 18 c 72 1

Answers to Challenge – The Eiffel Tower 1 2 3 4 5 6 7 8 9

1789 1989 2039 4 times 19 178 37 500 kg 1350 tonnes 325 m €14 = £11.67, so the Eiffel Tower was cheaper by £18.33.

Maths Frameworking 3rd edition Teacher Pack 2.1