Pupil Book 2.3
Kevin Evans, Keith Gordon, Trevor Senior, Brian Speed, Chris Pearce
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Contents 1
Introduction
4
Working with numbers
6
1.1
Multiplying and dividing negative numbers 1.2 Factors and highest common factor (HCF) 1.3 Multiples and lowest common multiple (LCM) 1.4 Powers and roots 1.5 Prime factors Ready to progress? Review questions Challenge Blackpool Tower
2
Geometry
2.1 2.2
Parallel lines The geometric properties of quadrilaterals 2.3 Translations 2.4 Enlargements 2.5 Constructions Ready to progress? Review questions Challenge More constructions
3
Probability
7 10 13 16 18 22 22 24 26 27 32 35 39 45 50 50 52 54
3.1
Mutually exclusive outcomes and exhaustive outcomes 3.2 Using a sample space to calculate probabilities 3.3 Estimates of probability Ready to progress? Review questions Financial skills Fun in the fairground
4 4.1 4.2 4.3
2
55 60 63 68 68 70
Percentages
72
Calculating percentages Calculating percentage increases and decreases Calculating a percentage Change Ready to progress? Review questions
73 76 78 82 82
Challenge Changes in population
5
Congruent shapes
5.1 5.2 5.3
Congruent shapes Congruent triangles Using congruent triangles to solve problems Ready to progress? Review questions Investigation Using scale diagrams to work out distances
6
Surface area and volume of prisms
6.1 6.2 6.3
Metric units for area and volume Surface area of prisms Volume of prisms Ready to progress? Review questions Investigation A cube investigation
7
Graphs
7.1 7.2
Graphs from linear equations Gradient (steepness) of a straight line 7.3 Graphs from quadratic equations 7.4 Real-life graphs Ready to progress? Review questions Challenge The M25
8
Number
84 86 87 90 93 96 96
98 100 101 103 107 112 112 114 116 117 119 122 125 128 128 130 132
8.1 8.2 8.3 8.4
Powers of 10 133 Significant figures 135 Standard form with large numbers 139 Multiplying with numbers in standard form 141 Ready to progress? 144 Review questions 144 Challenge Space – to see where no one has seen before 146
Contents
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9
Interpreting data
9.1 9.2 9.3 9.4
Interpreting graphs and diagrams Relative sized pie charts Scatter graphs and correlation Creating scatter graphs Ready to progress? Review questions Challenge Football attendances
10 Algebra 10.1 10.2 10.3 10.4 10.5
Algebraic notation Like terms Expanding brackets Using algebraic expressions Using index notation Ready to progress? Review questions Mathematical reasoning Writing in algebra
11 Shape and ratio 11.1 11.2 11.3
Ratio of lengths, areas and volumes Fractional enlargement Map scales Ready to progress? Review questions Activity Map reading
12 Fractions and decimals 12.1 12.2 12.3 12.4
Adding and subtracting fractions Multiplying fractions and integers Dividing with integers and fractions Multiplication with large and small numbers 12.5 Division with large and small numbers Ready to progress? Review questions Challenge Guesstimates
13 Proportion 13.1 13.2
Direct proportion Graphs and direct proportion
148 149 151 156 160 164 164
13.3 13.4
Inverse proportion Comparing direct proportion and inverse proportion Ready to progress? Review questions Challenge Planning a trip
14 Circles 166 168 169 171 173 175 178 180 180 182 184 185 189 195 200 200 202 204 205 207 210 212 215 218 218 220
236 240 240 242 244
14.1 14.2
The circumference of a circle Formula for the circumference of a circle 14.3 Formula for the area of a circle Ready to progress? Review questions Financial skills Athletics stadium
15 Equations and formulae 15.1 15.2
Equations with brackets Equations with the variable on both sides 15.3 More complex equations 15.4 Rearranging formulae Ready to progress? Review questions Reasoning Using graphs to solve equations
16 Comparing data
245 247 250 254 254 256 258 259 261 263 265 268 268 270 272
16.1 16.2 16.3 16.4
Grouped frequency tables Drawing frequency diagrams Comparing sets of data Misleading charts Ready to progress? Review questions Problem solving Why do we use so many devices to watch TV?
273 276 279 282 286 286
Glossary
290
Index
295
288
222 223 227
Contents
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How to use this book Learning objectives See what you are going to cover and what you should already know at the start of each chapter. About this chapter Find out the history of the maths you are going to learn and how it is used in real-life contexts. Key words The main terms used are listed at the start of each topic and highlighted in the text the first time they come up, helping you to master the terminology you need to express yourself fluently about maths. Definitions are provided in the glossary at the back of the book. Worked examples Understand the topic before you start the exercises, by reading the examples in blue boxes. These take you through how to answer a question step by step. Skills focus Practise your problem-solving, mathematical reasoning and financial skills.
PS
FS
Take it further Stretch your thinking by working through the Investigation, Problem solving, Challenge and Activity sections. By tackling these you are working at a higher level.
4
How to use this book
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Progress indicators Track your progress with indicators that show the difficulty level of each question.
Ready to progress? Check whether you have achieved the expected level of progress in each chapter. The statements show you what you need to know and how you can improve. Review questions The review questions bring together what you’ve learnt in this and earlier chapters, helping you to develop your mathematical fluency.
Activity pages Put maths into context with these colourful pages showing real-world situations involving maths. You are practising your problem-solving, reasoning and financial skills.
Interactive book, digital resources and videos A digital version of this Pupil Book is available, with interactive classroom and homework activities, assessments, worked examples and tools that have been specially developed to help you improve your maths skills. Also included are engaging video clips that explain essential concepts, and exciting real-life videos and images that bring to life the awe and wonder of maths. Find out more at www.collins.co.uk/connect
How to use this book
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1
Working with numbers
This chapter is going to show you: • how to divide negative numbers • how to find the highest common factor and the lowest common multiple of sets of numbers • how to find the prime factors of a number.
You should already know: • how to add and subtract negative integers • how to multiply by a negative number • how to order operations, following the rules of BIDMAS • how to test numbers 2 and 3 for divisibility • what a factor is • what a multiple is.
About this chapter What games do you play? In many of them, you will certainly use numbers. For example, darts players need to make calculations very quickly. They must work out the target numbers they must score, to finish the game. Then they have to think about the possible combinations of scores from three darts. Number skills are important in field games, such as rugby or cricket. They are essential in many board games and, of course, computer games. This chapter will help you develop those skills further for use in everyday life.
6
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1.1 Multiplying and dividing negative numbers Learning objective
Key words
• To carry out divisions involving negative numbers
negative number positive number
This diagram shows the result of multiplying both positive and negative numbers by a positive number. The numbers are all multiplied by +2. The diagram also shows the result of dividing both positive and negative numbers by a positive number. The numbers are all divided by +2. It is the inverse operation of multiplying. –4
–3
–2
–4 2
–2 2
–2 2 –4
–3
–1
0 0 2
–1 2 –2
1
3
2 2 0 2
–1
2
0
4 2
1 2 1
4
2 2 2
3
4
This pattern will continue for all the numbers on the top line going to the right and to the left. This shows us that: • multiplying or dividing a positive number by another positive number results in a positive number • multiplying or dividing a negative number by a positive number results in a negative number. (+) × (+) = (+) (+) ÷ (+) = (+)
You can summarise this as:
and: (-) × (+) = (-) (-) ÷ (+) = (-)
Example 1 Work out the answers. a -12 × 5 b -14 ÷ 2 a -12 × 5 = -60
b -14 ÷ 2 = -7
What happens if you multiply a number by a negative number? This diagram shows positive and negative numbers multiplied by -2. It also shows that multiplying a positive number by a negative number gives a negative result, as in the first diagram. –6
–5
–4
3 –2
–6
–5
–4
–3
–2
1 –2
2 –2
–3
–1
–2
–1
0
0 –2
0
1
2
–1 –2
1
3
–2 –2
2
3
4
5
6
5
6
–3 –2
4
Here it is just shown the other way round. But this diagram also shows that multiplying a negative number by a negative number gives a positive number. 1.1 Multiplying and dividing negative numbers
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The inverse operation of multiplying is dividing. To see the result of dividing by a negative number, look again at the diagram and follow the arrows from the bottom line to the top line. You see now that a negative number multiplied or divided by a negative number gives a positive number. You can summarise this as:
(-) × (-) = (+)
(-) × (+) = (-)
and:
(-) ÷ (-) = (+)
(-) ÷ (+) = (-)
To help you remember . . .
Hint
• When multiplying or dividing numbers with a different sign, the answer is negative. • When multiplying or dividing numbers with the same sign, the answer is positive.
Don’t forget that any number that is written without a sign in front of it is always positive.
Example 2 Work out the answers. a 7 × (-3)
b -5 × -9
a 7 × (-3) = -21
c 15 ÷ -3
b -5 × -9 = 45
d -12 ÷ -4 c 15 ÷ -3 = -5
d -12 ÷ -4 = 3
Example 3 Work out the answers. a -4 × -3 - 6
b -4 × (-3 - 6)
c -4 - 15 ÷ -3
a Using BIDMAS, work out -4 × -3 first. -4 × -3 - 6 = 12 - 6 = 6 b This time you must do the calculation inside the brackets first. -4 × (-3 - 6) = -4 × -9 = 36 c -4 - 15 ÷ -3 It is important to recognise here that 15 is a positive number since, the minus (-) here is the operation and not the sign of the number 15. So -4 - 15 ÷ -3 = -4 - (15 ÷ -3) = -4 - -5 = -4 + 5 = 1
Exercise 1A 1
2
Work these out. a
-8 + 9
b -3 - 8
f
-4 - 8
g
7-3+4
h 8 - 13
d -7 - 2 + 8 i
e
-4 - 8 + -9 j
-4 + 5 - 8 -6 + -5 - -8
Work out the answers. a
3 × -4
b -4 × 5
c
-6 × 3
d -7 × -4
e
-4 × 9
f
-5 × 6
g
-4 × -5
h -7 × -2
i
27 ÷ -3
j
-24 ÷ 8
k
-16 ÷ -4
l
m -12 ÷ -1
8
-5 + -7
c
85 ÷ -5
n -8 ÷ -2 × -1 o -3 × 6 ÷ -2 p -5 × 6 ÷ -3
1 Working with numbers
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3
Copy these number walls. Multiply numbers in adjacent bricks to find the number in the brick above. Some bricks have been completed for you. a
b 54 –3 –6 –3
4
3 2
–1
4
1
The answer to the question on this board is -24.
Using multiplication and/or division signs write down at least five different calculations that give this answer. 5
Work out the answers. a
15 ÷ -3
b -28 ÷ 4
c
-16 ÷ 2
d -63 ÷ -3
e
-40 ÷ 8
f
-45 ÷ 5
g
-36 ÷ -4
h -7 ÷ -1
i
9 ÷ -2
j
-9 ÷ 6
k
75 ÷ -10
l
m -15 ÷ -2 6
n -4 × -5 ÷ -2
p -5 × 6 ÷ -3
Find the missing number in each calculation. a d g j
21 × -3 =
a
b -3 ×
÷ -4 = -15
e
= 37
h
-74 ÷
× -16 = -48 = -10
m 5×8÷ 7
o -3 × 7 ÷ -2
-18 ÷ 4
k
= -18
-7 × -9 = × 9 = 36 -3 ×
n -2 ×
c
3×
= -39
f
-4 ×
i
-12 × 3 =
= -44
× 4 = 12
l
÷ -4 = 12
÷ -3 = -10
o
× -5 ÷ -2 = -8
Work out the answers. i -1 × -1
iii (-7)2
ii -5 × -5
iv (-9)2
b Explain why you cannot find the square root of a negative number. 8
9
Work out the answers. a
(-4)2 × 3
b
e
-5 × (-8 + 6)2
f (-4 + 10)2 ÷ -4
10
c
-12 ÷ ( -2)2
d
g -12 ÷ (-6 + 5)2
(-2 + 3)2 × -4
h -12 × (-6 + 5)2
Put brackets in each of these statements to make them true. a
PS
3 × (-5)2
3 × -6 + 2 = -12
b
-3 + -4 × 2 = -14
c
8-4-1=5
Find the missing numbers in these calculations. The boxes in each part should be filled with the same number. a
×
×
= -8
b
×
×
= -27
c
×
×
= -125
1.1 Multiplying and dividing negative numbers
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Challenge: Multiplication square Complete this multiplication square by finding the missing numbers in the outside cells, then the inside. × -16 -20
24 -56
21 -54
1.2 F actors and highest common factor (HCF) Learning objective • To understand and use highest common factors
Key words common factor divisible factor highest common factor (HCF) integer
Remember that the factors of an integer are all the numbers that divide exactly into it without leaving a remainder (or giving a decimal number as the answer). An integer is a whole number, whether it is positive or negative. Look at these examples. • The factors of 6 are 1, 2, 3 and 6. The numbers 4 and 5 are not factors of 6 because if you divided them into 6 you would get a remainder (or a decimal answer). • The factors of 25 are 1, 5 and 25. No other integers divide into 25 exactly. It is important to remember that every number apart from 1 has at least two factors, 1 and itself. For example: • 1 × 17 = 17 • 17 × 1 = 17. Another way of saying ‘can be divided by’ is ‘is divisible by’. The number 6 is divisible by the integers 1, 2, 3 and 6, and the number 25 is divisible by 1, 5 and 25. Every number is divisible by its factors. Sets of numbers always have common factors. These are numbers that will divide into all of them. For example, the factors of 15 are 1, 3, 5 and 15 and the factors of 20 are 1, 2, 4, 5, 10 and 20. So 5 is a common factor of 15 and 20. Some pairs of numbers, such as 2 and 3, or 4 and 5, only have 1 as a common factor. Common factors can help us in some mathematical problems such as simplifying fractions. If numbers have more than one common factor you should use the highest one, which is called the highest common factor (HCF). Using this helps you to simplify a fraction as far as you can.
10
1 Working with numbers
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Example 4 Simplify the fraction
36. 54
To simplify fractions, you need to divide the numerator (top) and denominator (bottom) by their HCF. To simplify
36 : 54
the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 the factors of 54 are 1, 2, 3, 6, 9, 18, 27, 54
The common factors are 2, 3, 6, 9 and 18. The HCF is 18, so divide numerator and denominator by 18 to give 32.
Example 5 Mr Evans is taking two classes on a school visit. One class has 28 pupils and the other has 21 pupils. He wants to split the classes into smaller groups to be looked after by helpers. He wants all the groups to be the same size but does not want to mix pupils from different classes in any group. What is the largest size group he can choose, to split the classes, so that each group is the same size? He can split the class with 28 pupils into groups of 1, 2, 4, 7, 14 and 28 (these are the factors of 28). He can split the class with 21 pupils into groups of 1, 3, 7 and 21 (these are the factors of 21). There are two common factors, 1 and 7. The highest (HCF) is 7 so the largest groups of the same size Mr Evans can create are groups of 7.
Example 6 Find the highest common factor (HCF) of each pair of numbers. a 18 and 27 a
b 24 and 32
Write out the factors of each number. 18: 1, 2, 3, 6, 9, 18 27: 1, 3, 9, 27 You can see that the HCF of 18 and 27 is 9.
b Write out the factors of each number. 24: 1, 2, 3, 4, 6, 8, 12, 24 32: 1, 2, 4, 8, 16, 32 You can see that the HCF of 24 and 32 is 8.
Exercise 1B 1
Write down all the factors of each number. a
2
16
b
22
c
36
d
45
e
75
Use your answer to Question 1 to help find the HCF of each pair. a
16 and 22
b 16 and 36
c
36 and 45
d 45 and 75
1.2 Factors and highest common factor (HCF)
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PS
3
Mrs Bishop takes 35 pupils on a visit to a museum. She doesn’t want to divide them into unequal size groups. What possible sized groups could she use?
4
Write down all the common factors for each pair of numbers. a 120 and 160 b 72 and 108 c 100 and 150
d 90 and 150
e 5
6
72 and 120
g
45 and 54
h 110 and 140
Find the HCF of each pair of numbers. a 30 and 36 b 36 and 96
c
48 and 88
d 40 and 60
e
g
36 and 54
h 63 and 99
8
PS
9
14 and 126
f
f
48 and 108
Find the largest number that each of these pairs of numbers are divisible by. a 40 and 160 b 54 and 90 c 144 and 280 d 280 and 112 e
7
300 and 500
150 and 120
f
72 and 360
g
200 and 260
h 150 and 225
By finding the HCF, simplify the fractions. a
36 96
b
30 75
c
48 80
d
400 650
e
24 132
f
90 105
g
45 60
h
54 90
Find the HCF of these sets of numbers. a 20, 50 and 170 b 36, 60 and 96
c
64, 80 and 112
d 28, 42 and 98
e
f
75, 120 and 180
g
h 34, 85 and 153
54, 144 and 216
150, 200 and 350
The teachers of two classes in a primary school need to put them into groups of equal size. They don’t want to mix the classes. Class A has 32 pupils and class B has 24 pupils. What is the largest size groups they can make and how many of these groups will there be?
PS
10
A room measures 4.5 metres by 3.5 metres. Work out the side length of the largest square tile that can be used to tile the floor without cutting any tiles to fit.
PS
11
The HCF of two numbers is 16. The smaller of the two numbers is 64 less than the larger. Both numbers are less than 100. Work out what the two numbers could be.
Challenge: Remainders Find the smallest number that when divided by: A 45 leaves the remainder of 4 B 454 leaves the remainder of 45 C 4545 leaves the remainder of 454 D 45 454 leaves the remainder of 4545.
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1 Working with numbers
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1.3 Multiples and lowest common multiple (LCM) Learning objective
Key words
• To understand and use lowest common multiples
multiple lowest common multiple (LCM)
A multiple of an integer is the result of multiplying that integer by another integer. For example, multiplying 3 by 1, 2, 3, 4 and 5 gives 3, 6, 9, 12 and 15. So 3, 6, 9, 12 and 15 are all multiples of 3. This also means that any integer that is divisible by 3, giving another integer without a remainder, must be a multiple of 3. You can find a common multiple for any pair of integers by multiplying one by the other. All pairs of integers will have many common multiples, but the lowest common multiple (LCM) is generally the most useful. For example 10 and 15 have a common multiple of 10 × 15 = 150, but the lowest common multiple is 30. Sometimes the LCM is one of the numbers. For example 15 is a multiple of 5 and of 15, but it is also the LCM of 5 and 15. • 15 = 15 × 1 • 15 = 5 × 3 You can use LCMs to help in calculations with fractions that have different denominators, as well as in some real life problems.
Example 7 Add the pairs of fractions by considering the lowest common multiple (LCM) of the denominators of each pair. 1 1 b 1 and a and 1 12 5 9 8 You need to find a common denominator to add the fractions. The LCM is the best to use. a The LCM is the product of 5 and 9 (45), because 5 and 9 have no common factors. Change each fraction to a denominator of 45 1 1× 5 5 9 1 1× 9 = = and = = 9 9 × 5 45 5 5 × 9 45 You can now add the fractions. 9 + 5 14 = 45 45 b 8 and 12 have a common factor, 4, so look at the multiples of each number. 8: 12:
8, 16, 24, 32, . . . 12, 24, 36, . . .
You can see that the LCM of 8 and 12 is 24. Change each fraction to a denominator of 24. 1 1× 3 3 1 1× 2 2 = = and = = 8 8 × 3 24 12 12 × 2 24 You can now add the fractions. 3+ 2 5 = 24 24 1.3 Multiples and lowest common multiple (LCM)
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Example 8 A confectioner makes small sweets, some of mass 12 g and some of mass 15 g. He wants to sell them in bags that all have the same mass. What is the smallest mass he could have in each of these bags? The 12 g sweet could be put into batches of mass 12 g, 24 g, 36 g, 48 g, 60 g, 72 g, ... (all multiples of 12) The 15 g sweet could be put into batches of mass 15 g, 30 g, 45 g, 60 g, 75 g, ... (all multiples of 15) The smallest mass will be the lowest common multiple (LCM) of these numbers which is 60 g.
Exercise 1C 1
Find the lowest common multiple (LCM) of each pair of numbers. a e
6 and 9 9 and 15
b 8 and 20 f 12 and 16
c g
d 9 and 16 h 14 and 21
2
Add the following pairs of fractions by considering the LCM of their denominators.
3
1 1 1 1 c b and and 6 5 6 8 Find the LCM of the groups of numbers. a
a e
PS
6 and 14 10 and 25
4
3, 4 and 6 4, 8 and 12
b 4, 5 and 6 f 6, 15 and 16
c g
1 and 1 9 12
d
6, 7 and 8 14, 9 and 21
d 7, 8 and 9 h 8, 9 and 12
1 and 1 15 10
In the first year of a large school, it is possible to divide the pupils into equal size classes of either 24, 28 or 32. Find the smallest number of pupils there could be in this first year group.
PS
5
On a model race track, two model cars leave the starting line at the same time and travel around tracks of equal lengths. One completes a circuit in 15 seconds, the other in 18 seconds. How long will it be before they are together again at the starting line?
PS
6
Three friends are walking along a straight promenade. Sean has a step size of 24 cm, Kiefer has a step size of 30 cm, and Mel a step size of 40 cm. They all set off, walking from the same point, next to each other. How far will they have gone before they are all in step with each other again?
14
1 Working with numbers
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PS
7
Three sisters regularly go to the same local supermarket at lunchtime. Jo goes every four days. Jessica goes every five days. Joy goes every six days. How many days in a year are they likely to all be in the supermarket on the same day?
PS
8
a
Write down two numbers that have an LCM of 54 and an HCF of 3.
b Write down two numbers that have an LCM of 24 and an HCF of 4. 9
c
Write down two numbers that have an LCM of 75 and an HCF of 5.
a
What is the HCF and the LCM of: i 5 and 8
ii 7 and 9
iii 3 and 13?
b What can you say about a pair of numbers that have a HCF of 1? 10
a
What is the HCF and LCM of: i 5 and 15
ii 9 and 27
iii 5 and 35?
b What can you say about the larger number when the HCF of two numbers is the smaller number? 11
a
Copy and complete the table. y
Product
HCF
LCM
4
14
56
2
28
9
21
12
21
18
24
x
b Describe any relationships that you can see in the table.
Investigation: Triangular numbers The triangular numbers are T1 = 1, T2 = 3, T3 = 6, T4 = 10, T5 = 15, T6 = 21, . . . The nth triangular number is Tn. Investigate whether these statements are always true. A The sum of two consecutive triangular numbers is always a square number, for example, T1 + T2 = 1 + 3 = 4 = 22. B If T is a triangular number then 9 × T + 1 is also a triangular number, for example, 9 × T1 + 1 = 9 × 1 + 1 = 10 = T4. C A triangular number can never end in 2, 4, 7 or 9. D If T is a triangular number then 8 × T + 1 is always a square number, for example, 8 × T1 + 1 = 8 × 1 + 1 = 9 = 32. E
If you keep on working out the sum of the digits of any triangular number until you obtain a single digit the answer is always 1, 3, 6 or 9.
1.3 Multiples and lowest common multiple (LCM)
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1.4 Powers and roots Key words
Learning objective • To understand and use powers and roots
cube
cube root
power
square
square root
Squares, cubes and powers Look at these 3D shapes. Is shape B twice as big, four times as big or eight times as big as shape A? How many times bigger is shape C than shape A? You calculate the area of each face of shape C by squaring the side length. 3 × 3 or 32 = 9 You say this as ‘three squared’.
A
B
C
You calculate the volume of the shape by cubing the side length. 3 × 3 × 3 or 33 = 27 You say this as ‘three cubed’. The small digits 2 and 3 are called powers. The power shows how many lots of the number are being multiplied together. A power can be of any size. For example: 34 is equal to 3 × 3 × 3 × 3 = 9 × 9 = 81 and you say it as ‘three to the power 4’.
•
Example 9 Calculate the value of each number. a
53
b
7.52
c
(-2)4
a 53 = 5 × 5 × 5 = 125 b 7.52 = 7.5 × 7.5 = 56.25 You could also do this on a calculator. Most calculators have a button for squaring, usually marked x 2 . They also have a button for cubing, usually marked x 3 . c (-2)4 = −2 × −2 × −2 × −2 = 4 × 4 = 16
Square roots and cube roots To find the area of a square you ‘square’ the side. The inverse process, to work out the side length of a square from its area, is finding the square root. If you know that the area of a square is 25 cm2 then the side length will be the square root of 25, written as 25. This is the number that will give 25, when you multiply it by itself.
25 = 5, because 5 × 5 = 25 If you know that the volume of a cube is 64 cm3, and you want to know its side length, you need to find the cube root. This will be the number that, when multiplied by itself and then multiplied by itself again (three ‘lots’ of the number are multiplied together), gives 64. It is written as 3 64 . 3
16
64 = 4, because 4 × 4 × 4 = 64 1 Working with numbers
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Note that you can write the square root simply as , with no small number 2 in front of it, but the cube root must always have a small 3 in front of it, like this: 3 . Square roots can be positive or negative and a square number is always positive. A positive cube number can only have a positive cube root and a negative cube number can only have a negative cube root. 3
64 = 4, because 4 × 4 × 4 = 64 but 3 −64 = −4, because −4 × −4 × −4 = −64
Your calculator should have both a square root button, shown as shown as 3 .
, and a cube root button,
Practise using your calculator and these buttons by working through the next example, before doing the exercise.
Example 10 Use a calculator to work these out. a
33124
b
12.25
3
c
2197
Depending on your calculator, you might need to select the square root or cube root key before the number. Make sure you know how to use the functions on your calculator. a
12.25 = 3.5
33124 = 182
b
c
3
2197 = 13
Exercise 1D 1
2
Copy and complete this table. x
1
2
3
2
x
1
4
9
x3
1
8
27
f
4
5
5
6
7
8
9
10
11
12
13
Use your table in Question 1 to work out one value for each number. a
3
4
3
9
b
8
g
3
25
c
64
h
3
64
d
216
i
3
121
e
729
j
169 3
1331
Find two values of x that make each equation true. a
x2 = 49
b x2 = 169
c
x2 = 81
d x2 = 1.21
e
x2 = 225
f
x2 = 1.44
g
x2 = 1.69
h x2 = 400
Use a calculator to find the value of each number. a
172
b 173
c
192
d 193
e
232
f
233
g
1.92
h 1.93
i
2.73
j
3.52
k
153
l
1.43
e k
45 212
f l
53 311
Use a calculator to find the value of each number. a g
25 64
b 36 h 73
c i
35 37
d 27 j 210
1.4 Powers and roots
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6
Without using a calculator write down the value of each number. a
7
502
b 403
603
c
d 305
e
802
f
3003
102 = 100, 103 = 1000 Copy and complete this table. Number
100
Power of 10
8
a
10
2
1000
10 000
1 000 000
1 000 000 000
3
10
Work out the value of each number. i 12
ii 13
iii 14
iv 15
v 16
b Write down the value of 1567. 9
a
Work out the value of each number. i (−1)2
ii (−1)3
iii (−1)4
b Write down the value of: i (−1)523
PS
10
a
iv (−1)5
v (−1)6
ii (−1)524.
Find a square number that is also a cube number.
b Find another three square numbers that are also cube numbers.
Investigation: Square numbers A Try to find a square number ending in 11. B Try to find a square number ending in 22. C Try to find a square number ending in 33. D Try to find a square number ending in 44. E
Try to find square numbers ending in 55, 66, 77, 88 and 99.
What have you found out from this investigation?
1.5 Prime factors Learning objective • To find the prime factors of an integer
Key words factor tree
index form
prime factor
prime number
Venn diagram You may remember that a prime number is an integer that has only two factors, itself and one. These are the first ten prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A prime factor of an integer is a factor that is also a prime number.
18
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Therefore the prime factors of an integer are the prime numbers that will multiply together to give that integer. For example, 12 can be written as a product of its prime factors, as 2 × 2 × 3. You can see that 12 has repeated prime factors. One useful way to find the prime factors of any integer is to draw up a factor tree.
Example 11 Find the prime factors of 18. Using a prime factor tree, start by splitting 18 into 3 × 6. Then split the 6 into 3 × 2. You could also split 18 into 2 × 9 and then split the 9 into 3 × 3.
18 3
18 6
3
2 2
9 3
3
Keep splitting the factors up until you only have prime numbers at the ends of the ‘branches’. Whichever pair of factors you start with, you will always finish with the same set of prime factors. So, 18 = 2 × 3 × 3 = 2 × 32 When the number is written as 2 × 32, it is in index form. There is another way to calculate the prime factors of a number. • Start with the smallest prime number that is a factor of the number. • Divide that prime number into the integer as many times as possible. • Then try the next smallest prime number that is a factor of the number. • Carry on until you reach 1.
Example 12 Find the prime factors of 24. Use the division method. 2 24 2 12 2 6 3 3 1 So, 24 = 2 × 2 × 2 × 3 = 23 × 3.
1.5 Prime factors
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Example 13 Use prime factors to find the highest common factor (HCF) and lowest common multiple (LCM) of 24 and 54. 24 = 2 × 2 × 2 × 3, 54 = 2 × 3 × 3 × 3 You can see that 2 × 3 is common to both lists of prime factors. You can show these factors on a Venn diagram. Put the common factors in the centre, overlapping, part of the diagram. Put the other prime factors in the outer part of each set.
24
The product of the centre numbers, 2 × 3 = 6, is the HCF. The product of all the numbers, 2 × 2 × 2 × 3 × 3 × 3 = 216, is the LCM.
2
2
2
3
3
54
3
Exercise 1E 1
2
The numbers have been written as products of their prime factors. What are the numbers? a
2×3×3
b 2×2×3×5
e
23 × 32 × 52
f
d 2 2 × 33 × 5
g
2 × 52 × 7
h 32 × 5 × 7
b 60 g 45
84 44
c 36 h 72
d 48 i 120
e j
52 150
Use the division method to work out the prime factors of each number. a
4
2 × 32 × 5
Use a prime factor tree to work out the prime factors of each number. a f
3
32 × 5
c
b 90
144
c
d 350
243
e
450
100 can be written as a product of its prime factors as 100 = 2 × 2 × 5 × 5 = 22 × 52. a
Write down the prime factors of 200, in index form.
b Write down the prime factors of 50, in index form. c
Write down the prime factors of 1000, in index form.
d Write down the prime factors of one million, in index form. 5
Using the Venn diagrams below, work out the HCF and LCM of the number pairs. a
b
30 5
2 3 30 and 72
20
72
2 2 3
c
50 5
2 5 50 and 90
3 3
90
48
2 2
2 3 2
84 7
48 and 84
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6
The prime factors of 120 (including repeats) are 2, 2, 2, 3 and 5. The prime factors of 150 are 2, 3, 5 and 5. Put these numbers into a Venn diagram like those in Question 5. Use your diagram to work out the HCF and LCM of 120 and 150.
7
The prime factors of 210 are 2, 3, 5 and 7. The prime factors of 90 are 2, 3, 3 and 5. Put these numbers into a Venn diagram like those in Question 5. Use your diagram to work out the HCF and LCM of 210 and 90.
8
The prime factors of 240 are 2, 2, 2, 2, 3 and 5. The prime factors of 900 are 2, 2, 3, 3, 5 and 5. Put these numbers into a Venn diagram like those in Question 5. Use your diagram to work out the HCF and LCM of 240 and 900.
9
Use prime factors to work out the HCF and LCM of each pair of numbers. a
10
200 and 175
b 56 and 360
c
42 and 105
c
18 and 32
Find the LCM of each pair of numbers. a
56 and 70
b 28 and 38
Challenge: Factors A Show that 60 has 12 factors. Find two more numbers less than 100 that also have 12 factors. B Show that 36 has nine factors. There are seven other numbers greater than 1 and less than 100 with an odd number of factors. Find them all. What sort of numbers are they? C There are 69 three-digit multiples of 13. The first is 104 and the last is 988. Five of these have a digit sum equal to 13. For example, 715 is a multiple of 13 and 7 + 1 + 5 = 13. Find the other four. You may find a computer spreadsheet useful for this activity.
1.5 Prime factors
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Ready to progress? I can find square and cube numbers and square and cube roots. I can use a calculator to work out powers of numbers. I can find common factors for pairs of numbers. I can multiply and divide negative numbers. I can find the lowest common multiple (LCM) for pairs of numbers. I can find the highest common factor (HCF) for pairs of numbers. I can write a number as the product of its prime factors. I can work out the LCM and the HCF of two numbers using prime factors.
Review questions 1 a Two numbers multiplied together give −15.
They add together to give 2.
What are the two numbers?
b Two numbers multiplied together give −15, but added together they give −2.
What are the two numbers?
c The square of 5 is 25. The square of another number is also 25.
What is that other number?
2 a Put these values in order of size, with the smallest first.
52, 32, 33, 24
b What is the value of 57? 3 Hamza chooses a prime number. He multiplies it by 10 and then rounds it to the nearest hundred. His answer is 400. Write down all the possible prime numbers Hamza could have chosen. 4 What is the cube root of 15, correct to one decimal place?
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5 In these brick walls, two numbers next to each other are multiplied together to give the number above. Copy and complete each one. –300 15 –4
3
–2
6 a Find the shortest length of material that can be divided exactly into equal parts of length 9 m, 12 m, 36 m or 48 m. b Find the longest length that can be divided exactly into each of these lengths.
PS
7 Aysha paid £64 for a 27 cm tall figurine mounted on an 8 cm cube. When she looked at the price and the dimensions, she noticed something. a What is special about the price and the dimensions? b What would be the price and dimensions of the next smaller and next larger figurine that are special in the same way as this one?
PS
8 a Write down 240 as a product of its prime factors, in index form. b Write down 432 as a product of its prime factors, in index form. c Find the HCF of 240 and 432. d Find the LCM of 240 and 432. 9 Here is the rule to find the geometric mean of two numbers. Multiply the two numbers together, then find the square root of the result. Example: geometric mean of 4 and 9 = a
4×9
= 36 =6 Find the geometric mean of 8 and 12, correct to one decimal place.
b For the two numbers 10 and x, the geometric mean is 30. What is the value of x? c
eather says: ‘For the two numbers −2 and 8, it is impossible to find the geometric H mean.’
Is Heather correct? Explain your answer.
Review questions
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Challenge Blackpool Tower Blackpool Tower is a tourist attraction in Blackpool, Lancashire, England. It opened to the public on 14 May 1894. Inspired by the Eiffel Tower in Paris, it rises to 518 ft 9 inches. The foundation stone was laid on 29 September 1891. The total cost for the design and construction of the Tower and buildings was about £290 000. Five million bricks, 2500 tonnes of steel and 93 tonnes of cast steel were used to construct the Tower. The Tower buildings occupy a total of 6040 sq yards. When the Tower opened, 3000 customers took the first rides to the top. Tourists paid 6 old pence for admission, a further 6 old pence for a ride in the lifts to the top, and a further 6 old pence for the circus. Inside the Tower there is a circus, an aquarium, a ballroom, restaurants, a children’s play area and amusements. In 1998 a ‘Walk of Faith’ glass floor panel was opened at the top of the Tower. Made up of two sheets of laminated glass, it weighs half a tonne and is two inches thick. Visitors can stand on the glass panel and look straight down 380 ft to the promenade. Use the information to help you answer these questions.
1 In what year did the Tower celebrate its centenary (100th birthday)? 2 How many years and months did it take to build the Tower? 3 The Tower is painted continuously. It takes seven years to paint the Tower completely. How many times has it been painted since it opened? 4 The aquarium in the Tower opened 20 years earlier than the Tower. What year did the aquarium celebrate its 100th birthday? 5 The largest tank in the aquarium holds 32 000 litres of water. There are approximately 4.5 litres to a gallon. How many gallons of water does the tank hold? 6 The water in the tropical fish tanks is kept at 75 °F. This rule is used to convert from degrees Fahrenheit to degrees Centigrade. °F
Subtract 32
Divide by 9
Multiply by 5
°C
Use this rule to convert 75°F to °C.
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7
The circus in the base of the Tower first opened to the public on 14 May 1894. The admission fee was 6 old pence. Before Britain introduced decimal currency in 1971 there were 240 old pence in a pound. a What fraction, in its simplest form, is 6 old pence out of 240 old pence? b What is the equivalent value of 6 old pence in new pence?
8
Over 650 000 people visit the Tower every year. The Tower is open every day except Christmas day. Approximately how many people visit the Tower each day on average?
9
a In January 2008, it cost €12 to visit the Eiffel Tower and £9.50 to visit Blackpool Tower. The exchange rate in January 2008 was £1 = €1.35. Which Tower was cheapest to visit and by how much (answer in pounds and pence)? b The Eiffel Tower is 325 m high. Blackpool Tower is 519 ft high. 1 m ≈ 3.3 ft. How many times taller is the Eiffel Tower than the Blackpool Tower? c The Eiffel Tower gets 6.7 million visitors a year. How many times more popular is it with tourists than the Blackpool Tower? d The Eiffel Tower celebrated its centenary in 1989. How many years before the Blackpool Tower did it open?
10 Animals have not appeared in the Tower Circus performances since 1990. For how many years did animals appear in the circus? 11 The top of the Tower is 518 ft and 9 inches above the base. There are 12 inches in a foot and 2.54 cm in an inch. Calculate the height of the Tower in metres. 12 The ‘Walk of Faith’ can withstand the weight of five baby elephants. One baby elephant weighs on average 240 kg. One adult human weighs on average 86 kg. How many adults should be allowed on the ‘Walk of Faith’ at any one time (if they could fit)? 13 The Tower and buildings cost approximately £290 000 to construct. Today it is estimated that the cost would be £230 million. By how many times has the cost of building gone up since the Tower was built? 14 The Tower lift makes about 75 trips up and down each day. Each ascent and descent is approximately 350 ft. There are 5280 ft in a mile. In a year (assume 360 days) approximately how many miles does the lift travel? 15 The Ballroom floor measures 36.58 m by 36.58 m. It comprises 30 602 separate blocks of mahogany, oak and walnut. Assuming that every block is equal in area, what is the area, in square centimetres, of each block? Give your answer to the nearest square centimetre. 16 When it is lit up, the tower has 10 000 light bulbs using an average of 15 watts per hour each. The cost of electricity is 12p per kilowatt hour (1000 watts per hour). Calculate the approximate yearly electricity bill for the lights assuming they are lit for 12 hours per day.
Challenge
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