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Edexcel GCSE (9-1)

MathematicsHigherStudent BookConfidence • Fluency • Problem-solving • Reasoning

11 - 19 PROGRESSION

A LWAY S L E A R N I NG

CVR_MATH_SB_KS4_0209_CVR.indd 1 11/11/2014 10:29

Sample unit

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1

Contents

1 Number

2 Algebra

3 Interpreting and representing data

4 Fractions, percentages, ratio and proportion

5 Angles and trigonometry

6 Graphs

7 Perimeter, area and volume

8 Transformations

9 Equations and inequalities

10 Probability

11 Multiplicative reasoning

12 Similarity and congruence

13 More trigonometry

14 Cumulative frequency and histograms

15 Equations and graphs

16 Circle theorems

17 More algebra

18 Vectors and geometric proof

19 Proportion and graphs

00-Higher Contents_001.indd 1 07/11/2014 11:05

1

1 NUMBER

Masterp.2

Checkp.20

Problem solvep.19

Strengthenp.22

Extendp.25

Testp.28

Estimate the amount of money taken by a football club on match day.

1 Prior knowledge check

Numerical fl uency

1 Work out a 5 × 0.3 b 97 × 0.02 c 6 ÷ 0.2 d 27 ÷ 0.09 e 4.2 ÷ 0.1 f 0.4 × 0.6 g 0.9 × 0.02 h 0.09 × 0.09 i 0.4 ÷ 0.2 j 0.9 ÷ 0.03 k 0.45 ÷ 0.3 l 0.88 ÷ 0.04

2 Choose the correct sign, < or >. a 2.7 2.5 b 3.04 3.3 c –2.9 –2.8 d –5.16 –5.5

3 a Write down all the factors of 12 and 18. b Make a list of the common factors. c Write down the highest common

factor.

4 a Copy and complete the Venn diagram to show the factors of 16 and 20.

Factors of 16 Factors of 20

b Write down the highest common factor.

5 a Write down the fi rst 10 multiples of 6 and 9.

b Make a list of the common multiples. c Write down the lowest common

multiple.

6 a Copy and complete the Venn diagram to show the fi rst 10 multiples of 4 and 10.

Multiples of 4 Multiples of 10

b Write down the lowest common multiple of 4 and 10.

7 Work out a 8 – 2 × 3 b (8 – 2) × 3 c 7 – (4 – 1) × 6 d 24 ÷ (8 – 2) e 4 2 + 1 f (–6) 2

8 Insert brackets to make this calculation correct.9 + 18 ÷ 3 = 9

9 Estimate a 7.3 × 8.94 b 47 ÷ 2.1 c 5.2 + 4.9 d 7.9 – 2.4

01-Higher CHAPTER 1_001-029.indd 1 20/11/2014 12:28

Draft, subject to endorsementDraft, subject to endorsement

2

Unit 1 Number

1 a Copy and complete this list of all possible outcomes for rolling a dice and fl ipping a coin.

H, 1 H, 2 …

T, 1 … …

b How many outcomes are there altogether?

2 a Copy and complete this list of all possible outcomes for spinner A and spinner B.

2, 1 4, 1 6, 1

2, 3 4, 3 …

2, 5 … …

b How many outcomes are there altogether?

Questions in this unit are targeted at the steps indicated.

3 How many possible outcomes are there when a rolling a dice b fl ipping a coin c spinning A in Q2 d spinning B in Q2?

Discussion What do you notice about your answers to i Q1b and Q3a and b above

ii Q2b and Q3c and d above?

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421 3

57

96

A B

Homework, practice and support: Higher 1.1

10 Write the positive and negative square roots of these numbers. a 36 b 1 c 64

11 Work out a 4 × 9 × 25 b 102 × 48 c 182 × 99 d 27 × 6 + 27 × 4

12 Copy and complete. a 6 × 6 = 6 b 3 × 3 × 3 × 3 = 3

13 Work out a 42 b 14 c 113 d 27

Challenge 14 How many diff erent ways can these cards

be arranged?

R B Y

What about now?

R B Y G

1.1 Number problems and reasoning

Fluency

Work out • 4 × 4 × 4 • 5 × 4 × 3 • 10 × 9 × 8 • 4 × 3 × 2 × 1

Objectives

• Work out the total number of ways of performing a series of tasks.

Why learn this?

5! in maths means ‘fi ve factorial’ and is equal to 5 × 4 × 3 × 2 × 1.


3

Unit 1 Number

4 A restaurant off ers a set menu for birthday parties. a Write down all possible combinations of starters

and main courses. b Refl ect How did you order your list to make sure

you didn’t miss any starters or mains? The restaurant decides to off er fi sh (F) as a main course. c How many possible combinations are there now? d Copy and complete.

3 starters and 4 mains: combinations 3 starters and 5 mains: combinations n starters and m mains: combinations

A diff erent restaurant off ers 2 starters, 4 mains and 3 desserts. e How many possible combinations are there now?

5

6 Three people, A, B and C, enter a race. a Write down the diff erent orders in which they can fi nish fi rst, second and third. Harry says that there are 3 possible winners, but then only 2 possibilities for second place and only one person left for third place. Discussion Is Harry correct? Explain your answer. b How many diff erent ways can people fi nish in

i a 4-person race ii a 6-person race iii a 10-person race?

Q6 communication hint A factorial is the result of multiplying a sequence of descending integers. For example 4! = 4 × 3 × 2 × 1. Find the factorial button on your calculator.

x!

!

7 Eddie needs to choose a 6-digit code for his computer password. a How many codes can Eddie create using

i 6 numbers ii 4 numbers followed by 2 letters

iii 1 number followed by 5 letters? Eddie decides that he does not want to repeat a digit or a letter. b How many ways are possible in parts i to iii now?

The restaurant decides to off er fi sh (F) as a main course. Starters Vegetable Soup (V)

Salad (S) Melon (M)

Mains Pizza (P) Spaghetti Bolognaise (B) Curry (C)

Lasagne (L)

Q4a hint Use letters for combinations, for example VP for vegetable soup, pizza.

Key point 1

When there are m ways of doing one task and n ways of doing a diff erent task, the total number of ways the two tasks can be done is m × n.

Q5 communication hint Inclusive means that the end numbers are also included.

Exam-style question

Jess has a 4-digit password for her mobile phone. Each digit can be between 0 and 9 inclusive.

a How many choices are possible for each digit of the code?

b What is the total number of 4-digit passwords that Jess can create?

Jess would like to choose an even number. The code can start with a zero.

c How many different ways are possible now? (5 marks)


Draft, subject to endorsement

4

Unit 1 Number

1.2 Place value and estimating

Fluency

Which two whole numbers does each square root lie between? √ 

__ 3 √ 

___ 17

Objectives

• Estimate an answer. • Use place value to answer questions.

Why learn this?

Builders use estimates to give their clients an idea of how much the work will cost.

1 Write each number to i 1 signifi cant fi gure

ii 2 signifi cant fi gures. a 873 209 b 2019 c 0.007 059

2 Work out a 9 × (4 + 7) b 5 + 3 × 8 c 7 × 5 – 4 × 2 d 30 – 5 × 8 e 72 – 9 f √ 

______ 29 − 4

3 Work out the mean of 3, 6, 7, 9, 15 and 20.

4 Work out a 32 × 6 b 16 × 12 c 8 × 24 d 4 × 48 Discussion What do you notice about your answers? Why has this happened?

5 3.7 × 9.86 = 36.482 Use this fact to work out the calculations below. Check your answers using an approximate calculation. a 37 × 9.86 b 3.7 × 0.0986 c 0.0037 × 98.6 d 36.482 ÷ 9.86 e 3648.2 ÷ 98.6 f 364.82 ÷ 370

6 Reasoning 54.8 × 7.29 = 399.492 a Write down three more calculations that have the same answer. b Write down a division that has an answer of 54.8. c Write down a division that has an answer of 0.729. d Charlie says that 54.8 × 72.9 = 3989.44. Explain why Charlie must be wrong.

7 a Write down the value of √ __

4 and √ __

9 . b Estimate the value of √ 

__ 5 , √ 

__ 6 , √ 

__ 7 and √ 

__ 8 .

Round each estimate to 1 decimal place. c Use a calculator to check your answer to part b.

8 Estimate the value to the nearest tenth.

a √ ___

47 b √ ___

22 c √ ___

84

d √ ____

127 e √ ___

10 f √ ___

40

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Q5a hint Compare with the given calculation.

� �

3.7

37

9.86

9.86

36.482�

�

�

�

Q5d hint Rewrite the given calculation as a division.

Q7b hint Use a number line to help.

4 5 9

Q8a hint Use a number line to help.

36 47 49



5

Unit 1 Number

9 Problem-solving A mosaic uses 150 square tiles. The total area is 3000 cm2. a Estimate the side length of a tile. b Use a calculator to check your answer.

10 a Write down the value of 82 and 92. b Estimate the value of 8.32 and 8.82. Round each estimate to the nearest whole number. c Use a calculator to check your answer to part b.

11 Estimate to the nearest whole number.

a 3.22 b 4.72 c 1.72

d 7.12 e 6.32 f 9.82

12 a Estimate answers to these. i (11.2 – √ 

_____ 50 . 3 ) × 4.08 ii (1.98 × 3.14)2 ÷ 8.85

iii 88 . 72 − 21 . 9 ___________ √ 

_____ 35 . 5 iv

√ __________

27 . 3 − 1 . 85 ____________

√ __________

3 . 93 × 5 . 42

b Use your calculator to work out each answer. Give your answers correct to one decimal place.

c Refl ect How did you decide what to round each number to? For iii and iv does it matter if you round the numerator or the denominator fi rst?

13 The sum of the values on these cards is 12.

5 2 +

______

4 2

80 − √ 

___ 64 ________

2 × 4

Work out the missing number.

14 Problem-solving A large cubic dice has a side length of 9.2 cm. Estimate the surface area of the dice.

15 Problem-solving The area of a square is 80 cm2. Estimate the perimeter of the square.

16 Problem-solving Pieces of turf are 1 m long by 0.5 m wide. Each piece costs £3.79. a Estimate the cost of turf required to cover these spaces.

i 9.6 m by 2.4 m ii 6.2 m by 1.9 m

iii 4.4 m by 2.1 m b Use a calculator to work out each answer.

How good were your estimates? Discussion Is it better to overestimate or underestimate a cost?

17 STEM Robert uses a spreadsheet to record his runs for 10 innings. His scores are in cells A1 to J1. His mean average is in cell K1.

1

A

78

B

12

C

4

D

15

E

0

F

35

G

0

H

7

I

12

J

21

K

11.8

a Use estimates to show that Robert’s average is wrong. b Work out Robert’s correct average to the nearest tenth.

Q11a hint Use a number line to help.

32 3.22 42

Q12a iv hint The whole of the numerator is being square rooted. So work out the numerator before square rooting.

Q13 hint Work out the value of the card on the right fi rst.



6

Unit 1 Number

1.3 HCF and LCM

Fluency

Work out • 2 × 3 × 52 • 23 × 32 • 5 × 5 × 5 = 5 • 7 × 7 × 7 × 7 × 7 = 7

Objectives

• Write a number as the product of its prime factors. • Find the HCF and LCM of two numbers.

Why learn this?

Astronomers use the lowest common multiple of patterns in the orbits of the Sun and the Moon to predict solar eclipses.

1 a Write down all the factors of 20. b Which of these factors are prime numbers?

2 a Write down all the prime numbers between 1 and 20. b Write down all the factors of 24. c Copy and complete this Venn diagram.

Prime numbers between1 and 20

Factors of 24

1225

3 a Copy and complete this factor tree for 40.

8

4

40

b Write 40 as a product of its prime factors.

40 = × × × = ×

4 Write 75 as a product of its prime factors.

5 Steve and Ian are asked to fi nd 60 as a product of its prime factors. Steve begins by writing 60 = 5 × 12 Ian begins by writing 60 = 6 × 10 a Work out a fi nal answer for Steve. b Work out a fi nal answer for Ian. Discussion What do you notice about the two answers? c Start the prime factor decomposition of 48 in two diff erent ways: 6 × 8 and 12 × 4. Discussion Does your fi rst step in a prime factor decomposition aff ect your fi nal answer?

6 Refl ect a Write down your own short mathematical defi nition of these words. i prime ii factor iii decomposition

b Use your defi nition to write down (in your own words) the meaning of prime factor decomposition.

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Q3b hint Circle the prime factors in your factor tree.

Q4 hint Use the method from Q3.



7

Unit 1 Number

7 Write each number as a product of its prime factors in index form . a 18 b 42 c 25 d 36 e 24 f 80

8 120 can be written as a product of its prime factors in the form 2m × n × p. Work out m, n and p.

9 Find the highest common factor (HCF) and lowest common multiple (LCM) of a 24 and 30 b 20 and 42 c 8 and 18 d 15 and 45 e 27 and 36 f 33 and 66

10 Real / Problem-solving One bus leaves the bus station every 15 minutes. Another bus leaves every 12 minutes. At 2:30 pm both buses leave the bus station. At what time will this next happen?

11 Real / Problem-solving Amber wants to tile her bathroom. It measures 1.2 m by 2.16 m. She fi nds square tiles with a side length of 10 cm, 12 cm or 18 cm. Which of these tiles will fi t the wall exactly?

Discussion How do you know whether to fi nd the HCF or LCM for Q10 and Q11?

12 Problem-solving The HCF of two numbers is 2. Write down three possible pairs of numbers.

13 Problem-solving The LCM of two numbers is 18. One of the numbers is 18. a Write down all the possibilities for the other number. b Describe the set of numbers you have created.

14 48 = 24 × 3 and 36 = 22 × 32 Write down, as a product of its prime factors, a the HCF of 48 and 36 b the LCM of 48 and 36.

Q7 communication hint In index form means to write a number to a power or an index. 23 is written in index form. 3 is the power or index.

Example 1

Find the highest common factor and lowest common multiple of 24 and 60. 24 = 2 × 2 × 2 × 3 60 = 2 × 2 × 3 × 5

Prime factorsof 60

Prime factorsof 24

5223

2

The highest common factor (HCF) of 24 and 60 = 2 × 2 × 3 = 12 The lowest common multiple (LCM) of 24 and 60 = 2 × 2 × 2 × 3 × 5 = 120

Write each number as a product of prime factors

Draw a Venn diagram.

Multiply the common prime factors.

Multiply all the prime factors.

Q9 hint Draw a Venn diagram for each question to help you.

Q10 strategy hint Work out the LCM fi rst.

Q12 hint First choose two numbers where 2 is a factor. Is 2 the highest common factor of these numbers?

Q14 hint You could draw a Venn diagram.



8

Unit 1 Number

15

16 Write 80 as a product of its prime factors. Discussion How can you use the prime factor decomposition of 80 to quickly work out the prime factor decomposition of 160? What about 40?

17 Problem-solving The prime factor decomposition of 2100 is 22 × 3 × 52 × 7. Write down the prime factor decompositions of a 75 b 24 c 12 d 30

18 a Harry says the prime factors of 75 appear in the prime factor decomposition of 2100, so 2100 is divisible by 75. Is 2100 divisible by 24, 12 or 30?

b Use prime factor decomposition to show that 792 is divisible by 12. c Is 792 divisible by 132? Explain your answer. d Is 792 divisible by 27? Explain your answer.

19 In prime factor form, 700 = 22 × 52 × 7 and 1960 = 23 × 5 × 72 a What is the HCF of 700 and 1960?

Give your answer in prime factor decomposition. b What is the LCM of 700 and 1960?

Give your answer in prime factor decomposition. c Which of these are factors of 350 and 1960?

i 2 × 5 × 7 ii 49

iii 20 iv 22 × 5 × 72

d Which of these are multiples of 350 and 1960? i 23 × 5 × 73

ii 26 × 52 × 72 iii 22 × 5 × 7

Q15 strategy hint Look at the common prime factors.

Exam-style question

Given that A = 23 × 34 × 52 and B = 22 × 36 × 5

Write down, as a product of its prime factors,

a the HCF of A and B

b the LCM of A and B. (2 marks)

Q19c hint What factors do 700 and 1960 have in common? Any factor of this number will be a factor of 350 and 1960.

Q19d hint What multiples do 700 and 1960 have in common? Any multiple of this number will be a multiple of 350 and 1960.

1.4 Calculating with powers (indices)

Fluency

Work out • 62 • (–4)2 • 24 • 15


Objectives

• Use powers and roots in calculations. • Multiply and divide using index laws. • Work out a power raised to a power.

Why learn this?

A googol is a 1 followed by 100 zeros. It can be written as 10100.


9

Unit 1 Number

1 Work out a 33 b (–1)3 c 4 × 42 d 32 × 5 e 23 × 102 f 0.23 g 3 × √ 

___ 16 h √ 

___ 81 × √ 

___ 64

2 Work out

a 4 × 4 × 4 × 4 __________ 4 × 4

b 3 × 3 × 3 × 3 × 3 × 3 ________________ 3 × 3 × 3

3 Copy and complete. a 2 = 16 b 3 = 64 c 5 = 25 d 3 = 27

4 Work out a

3 √ ___

27 b 3 √ ___

−1 c 3 √ _____

1000 d 3 √ _____

−125 Discussion Explain why it is possible for you to fi nd the cube root of a negative number, but not the square root.

5 Work out these. Use a calculator to check your answers.

a √ ______

4 2 + 3 2 b 3 √ _______

10 2 + 5 2

c 43 – 3 √ _____

−27 d 33 – 3 √ ____

−8 – (–4)2

e √ ____________

5 2 + 3 × 3 √ _____

−27 f 5 2 ×

3 √ ____

−27 _________

3 √ ___

−8 − √ __

9

g − 3 3 ____ √ 

__ 9 ×

− √ ___

64 _____

3 √ ___

−1 h

0 . 2 2 × 3 √ _____

−125 ____________

3 √ __

8

6 Work out a [(33 – 52) × 2)]3 b 20 – [3 × 42 – (22 × 32)] c [72 ÷ (7 – 5)3 – 3] ÷ √ 

__ 9

7 Work out

a 4 √ ___

16 b 4 √ ___

81 c 5 √ _________

100 000

8 a Work out i 103 × 102 ii 105 iii 106 × 102 iv 108

b How can you work out the answers to part a by using the indices of the powers you are multiplying?

c Check your rule works for i 103 × 104 ii 105 × 10 iii 10–2 × 10–3

Discussion Do the index laws work for negative indices?

9 Write each product as a single power. a 32 × 34 b 4−2 × 48 c 9−3 × 9−4

Warm

up

Q3a hint 2 × 2 × … = 16

Q3b hint × × = 64

Key point 2

The inverse of a cube is the cube root. 23 = 8, so the cube root of 8,

3 √ __

8 = 2

Q5a hint Use the priority of operations.

Q5e hint The square root applies to the while calculation. Work out the calculation fi rst.

Q6 communication hint Square brackets [ ] make the inner and outer brackets easier to see. Input them as round brackets on your calculator.

Q7a hint 4 = 16

4 √ ___

16 =

Q8c ii hint 10 = 101

Key point 3

To multiply, add the indices. xm × xn = xm + n



10

Unit 1 Number

10 Find the value of a. a 84 × 8a = 87 b 65 × 6a = 63 c 2−3 × 2a = 2−10

11 Write these calculations as a single power. Give your answers in index form. a 27 × 35 = 3 × 35 = 3 b 43 × 64 c 5 × 125 d 32 × 4 e 8 × 8 × 8 f 9 × 27 × 3

12 Reasoning

a i Work out 5 × 5 × 5 × 5 × 5 _____________ 5 × 5

by cancelling. Write your answer as a power of 5.

ii Copy and complete. 55 ÷ 52 = 5

b Copy and complete. 46 ÷ 42 = 4 × 4 × 4 × 4 × 4 × 4 ________________ 4 × 4

= 4

c Work out 65 ÷ 64 Discussion How can you quickly fi nd 79 ÷ 73 without writing all the 7s?

13 Work out a 76 ÷ 72 b 4−5 ÷ 43 c 3–6 ÷ 3–7

14 Find the value of a. a 96 ÷ 9a = 94 b 45 ÷ 4a = 4 c 76 ÷ 7a = 79

15 Problem-solving a Yu multiplies three powers of 9 together. 9 × 9 × 9 = 912 What could the three powers be when

i all three powers are diff erent ii all three powers are the same?

b Harvey divides two powers of 5. 5 ÷ 5 = 56 What could the two powers be when i both numbers are greater than 520

ii the power of one number is double the power of the other number?

16 Work out these. Write each answer as a single power.

a 53 × 57 ÷ 54 b 62 ÷ 6−3 × 67 c 5 4 × 5 –3 _______ 5 5

d 8 –2 × 8 –6 _______ 8 –7

17 Real / STEM The hard drive of Tom’s computer holds 238 bytes of data. He buys a USB memory stick that holds 234 bytes of data. a How many memory sticks does he need to back up his

computer? He buys an external hard drive that holds 239 bytes of data. b What fraction of the external hard drive does he use when

backing up his computer?

Key point 4

You can only add the indices when multiplying powers of the same number.

Key point 5

To divide powers, subtract the indices. xm ÷ xn = xm – n

Q17 communication hint Backing up a hard drive means copying its contents on to another storage device.


11

Unit 1 Number

18 Problem-solving a Copy and complete. 23 ÷ 23 = 2 b Write down 23 as a whole number. c 23 ÷ 23 = 8 ÷ = d Copy and complete using parts a and c. 23 ÷ 23 = 2 = e Repeat parts a and b for 75 ÷ 75. f Write down a rule for a0, where a is any number.

19 Copy and complete. a (23)5 = 23 × × × × = 2

b (64)3 = × × = 6

c (87)2 = × = 8 Discussion What do you notice about the powers in the question and the powers in the fi nal answer?

20 Write as a single power. a (23)4 b (62)5 c (42)−3 d (5−2)−6

21 Problem-solving Write each calculation as a single power.

a 8 × 32 × 8 b 58 ____

125 c 16 × 64 × 16 ___________

4 4

Key point 6

x0 = 1, where x is any non-zero number.

Key point 7

To work out a power to another power, multiply the powers together. (xm)n = xmn

1.5 Zero, negative and fractional indices

Fluency

• Work out 20 • Convert 2 1 _ 2 to an improper fraction. • Convert 0.3 to a fraction.

Objectives

• Use negative indices. • Use fractional indices.

Why learn this?

The smallest known time measurement is approximately 10−43 seconds. Scientists call this unit one Planck time, aft er Max Planck.

1 Work out a 62 b 23 c 34 d 53

2 Write each calculation as a single power. a 34 × 36 b 25 ÷ 23 c 16 × 8 d 7–3 × 73

3 Work out

a 1 ____ √ 

___ 25 b

3 √ ___

−8 ___ 27

c −3 ____ 3 √ ___

27 d

4 √ ___

81

Warm

up




12

Unit 1 Number

4 Work out the value of n.a 40 = 5 × 2n b 3n × 3n = 38

c 52n ÷ 5n = 56 d 1 _ 2 × 4n = 32

5 a Work out i 2−1 ii 4−1 iii 5−1 iv 10−1

b Use your answers to part a to write these negative indices as fractions.c Work out i 2−2 ii 4−2 iii 5−2 iv 10−2

d Use your answers to part c to write these negative indices as fractions.e Work out

i ( 1 __ 2

) −1

ii ( 3 __ 4

) −2

Discussion What is the rule for writing negative indices as fractions?

6 Match these negative indices to their fractions.

14( )22

23( )211

83

135

124

1533

2

523325 328

224

16

a Write a matching tile for the two tiles that are left over.

b Copy and complete. ( 2 __ 3

) −1

= , so ( a __ b )

−1 = ( __ )

7 Work outa 3–1 b 2–4 c 10–5

d ( 3 __ 4

) −1

e ( 4 __ 5

) −3

f (1 1 _ 4 ) −1

g (2 3 _ 4 ) −2

h (0.7)−1 i (0.1)−5

j (0.4)−3 k (5−1)0 l (7−1)−1

8 a Work out

i 4 9 1 _ 2 ii 1 6

1 _ 2 iii 12 1 1 _ 2 iv ( 4 ___

25 )

1 _ 2

b Copy and complete. a 1 _ 2 is the same as the _______ _______ of a.

c Work out

i 2 7 1 _ 3 ii 100 0

1 _ 3 iii − 1 1 _ 3 iv ( 1 _____

1000 )

1 _ 3

d Copy and complete. a 1 _ 3 is the same as the _______ _______ of a.

e Copy and complete.

i 625 = 5 so 62 5 1 _ 4 = ii 32 = 5 so 3 2

1 _ 5 =

9 Work outa 3 6

1 _ 2 b 8 1 1 _ 2 c ( 1 __

9 )

1 _ 2 d ( 16 ___ 25

) 1 _ 2

e ( 64 ___ 49

) 1 _ 2 f − 8

1 _ 3 g ( 1 ___ 27

) 1 _ 3 h ( −64 ____

125 )

1 _ 3

Q4a hint 40 = 5 × ↗How do you write this number as 2 ?

Key point 8

x–n = 1 ___ x n

for any number n, x ≠ 0

Q7f strategy hint Convert mixed numbers to improper fractions.

Q7h strategy hint Convert decimals to fractions.


13

Unit 1 Number

10 Work outa 2 5 − 1 _ 2 b 6 4 − 1 _ 3 c ( 9 ___

25 ) − 1 _ 2

11 Work outa 6 4

2 _ 3 b 10 00 0 3 _ 4 c 1 6

3 _ 2

d ( 4 __ 9

) 3 _ 2 e 2 7 − 2 _ 3 f −8 1 − 3 _ 4

12 Work out

a 2 7 − 1 _ 3 × 9 3 _ 2 b ( 4 ___

25 ) − 3 _ 2 × ( 8 ___

27 )

1 _ 3 c ( 81 ___ 16

) 3 _ 4 × ( 9 ___

25 ) − 3 _ 2

13 Find the value of n.a 16 = 2n b

3 √ ___

27 = 27n c 1 ____ 100

= 10n

d √ __

4 __ 9

= ( 9 __ 4

) n

e ( √ __

3 ) 7 = 3n f ( 4 √ __

8 ) 7 = 8n

14 Problem-solving / Reasoning Will says that 2 5 – 1 _ 2 × 6 4 2 _ 3 = 80

a Show that Will is wrong.

b What mistake did he make?

15 Problem-solving / ReasoningMatch these indices to their fractions.

827( )2 8

43

1641

1694

14

49

8

16

23 81

16( )2 12 1

64( )23

1634

3225

2

1632

2

Key point 9

x 1 _ n = n √ 

__ x

Q10 hint x–n = 1 ___ x n

so 2 5 − 1 _ 2 = 1 ____ 2 5

= 1 ___

Example 2

Work out the value of a 2 7 2 _ 3 b 1 6 – 3 _ 4

a) 2 7 2 __ 3 = (2 7

1 __ 3 )2 = 32 = 9

b) 1 6 − 3 __ 4 = 1 ____ 1 6

3 __ 4 = 1 ______

(1 6 1 __ 4 )3

= 1 ___ 23

= 1 __ 8

Use the rule (x m)n = x mn. Work out the cube root of 27 first. Then square your answer.

Use x−n = 1 ___ xn

Q2a hint

6 4 2 _ 3 = (6 4

1 _ 3 ) 2 = ( 3 √

___ 64 )

2 =

2 =

Key point 10

x n __ m = ( m √ 

__ x )n

Q12a hint First work out 2 7 − 1 _ 3 .Then work out 9

3 _ 2 .Then multiply these numbers together.


14

Unit 1 Number

1.6 Powers of 10 and standard form

Fluency

• Work out 4.5 × 1000 0.0063 × 100 69.4 × 0.1 845.3 × 0.001

• Which of these are the same as ÷10? × 1 __ 10 × 0.01 × 10–1

Objectives

• Write a number in standard form. • Calculate with numbers in standard form.

Why learn this?

Scientists use standard form to write very small or very large numbers.

1 Copy and complete. If your answer is a fraction, write it as a decimal too. a 100 = b 10–1 = c 10–2 = d 10–3 = e 10–4 = f 10–5 =

2 Write down the value of x. a 10x = 1000 b 105 = x c 10x = 100 000 000 d 10–1 = x e 10x = 0.0001 f 10–6 = x

3 Copy and complete. a 5 670 000 = million b 15 800 000 = million c 4 908 340 000 = billion

4 Copy and complete the table of prefixes.

Prefi x Letter Power Number

tera T 1012 1 000 000 000 000

giga G 109

mega M 1 000 000

kilo k 103

deci d 0.1

centi c 10–2

milli m 0.001

micro μ 10–6

nano n 0.000 000 001

pico p 10–12

5 Convert a 15 mg into grams b 7 nm into metres c 1.7 g into kg d 7.3 ps into seconds.

6 STEM Write these measurements in metres. a The size of the infl uenza virus is about 1.2 μm. b The radius of a hydrogen atom is 25 pm. c A fi ngernail grows about 0.9 nm every second.

War

m u

p

Q4 communication hint Prefi x is the beginning part of a word.

Q4 communication hint μ, the letter for the prefi x micro, is the Greek letter mu.

Key point 11

Some powers of 10 have a name called a prefi x. Each prefi x is represented by a letter. For example, kilo means 103 and is represented by the letter k, as in kg for kilogram.

Q5a hint Use a number line.

331mg:

g: 0.001

15



15

Unit 1 Number

7 Copy and complete. a 45 000 = 4.5 × b 10 000 = 10 c 45 000 = 4.5 × 10

8 Which of these numbers are in standard form? A 4.5 × 107 B 13 × 104 C 0.9 × 10–2 D 9.99 × 10–3 E 4.5 billion F 2.5 × 10

9 Write these numbers in standard form. a 87 000 b 1 042 000 c 1 394 000 000 d 0.007 e 0.000 002 84 f 0.000 100 3

10 Write these as ordinary numbers . a 4 × 105 b 3.5 × 102 c 6.78 × 103 d 6.2 × 10−2 e 8.93 × 10−5 f 4.04 × 10−3

11 STEM / Reasoning a The distance from the Sun to Neptune is 4 500 000 000 000 m.

i Write this number in standard form. ii Enter the ordinary number in your calculator and press the = key.

Compare your calculator number with the standard form number. Explain how your calculator displays a number in standard form.

b The thickness of a sheet of a paper is 0.000 07 m. i Write this number in standard form.

ii Enter the ordinary number in your calculator and press the = key. Compare your calculator number with the standard form number.

c Reflect Why do you think that scientists use standard form for very large and very small numbers?

12 Write these numbers from a calculator display i in standard form ii as ordinary numbers.

a 4E–3 b 2.5E10 c 6.9E–7 d 3.009E4

Key point 12

A number is in standard form when it is in the form A × 10n, where 1 < A , 10 and n is an integer (A is a number between 1 and 10 multiplied by 10 to a power). For example, 6.3 × 104 is written is standard form because 6.3 is between 1 and 10. 63 × 104 is not is standard form because 63 does not lie between 1 and 10. Standard form is sometimes also called scientifi c notation .

Q9 hint Write the number between 1 and 10 fi rst. Then multiply by a power of 10.

Q10 communication hint An ordinary number is a whole number or a decimal which is not multiplied by a power of 10.

Example 3

Work out (5 × 103) × (7 × 106)

5 × 7 × 103 × 106

35 × 109

35 = 3.5 × 101

35 × 109 = 3.5 × 101 × 109 = 3.5 × 1010

Rewrite the multiplication grouping the numbers and the powers.

Simplify using multiplication and the index law xm × xn = xm+n. This is not in standard form because 35 is not between 1 and 10.

Write 35 in standard form.

Work out the fi nal answer.



16

Unit 1 Number

13 Work out these. Use a calculator to check your answers. a (3 × 102) × (2 × 105) b (5 × 103) × (4 × 107) c (8 × 10–2) × (6 × 107) d (8 × 106) ÷ (4 × 103) e (9 × 10–2) ÷ (3 × 106) f (2 × 103) ÷ (8 × 107) g (5 × 103)2 h (4 × 10–2)3

14 STEM / Problem-solving The Sun is a distance of 1.5 × 108 km from the Earth. Light travels at a speed of 3 × 105 km per second. How many seconds will it take for light from the Sun to reach the Earth?

15 STEM / Problem-solving A water molecule has a mass of 3 × 10−29 kg. A bottle contains 1.7 × 1028 molecules of water. Calculate the mass of water in the bottle.

16 a Write these numbers as ordinary numbers. i 8 × 104

ii 3 × 102 b Work out (8 × 104) + (3 × 102), giving your answer in

standard form.

17 Work out these. Give your answers in standard form. a 3.4 × 105 + 6.7 × 104 b 9.8 × 104 – 2.2 × 102 c 7.2 × 102 + 6.2 × 10–1 d 8.3 × 105 – 7 × 10–1

18

Q13g hint (5 × 103)2 = (5 × 103) × (5 × 103)

Q16b hint Use your answers from part a to write the answer as an ordinary number. Then convert this to standard form.

Exam hint Don’t just write down the possible values – give your working to show how you worked out the values.

Exam-style question

(7 × 10x) + (7 × 10y) + (7 × 10z) = 700 070.07

Write down a possible set of values for x, y and z. (3 marks)

1.7 Surds

Fluency

• What does the dot above the 1 mean in 0 . 1 ̇ ? • What are the missing numbers?

147 = 3 × 125 = × 25 180 = 5 × 96 = × 16

Objectives

• Understand the diff erence between rational and irrational numbers.

• Simplify a surd. • Rationalise a denominator.

Why learn this?

Surds are used to express irrational numbers in exact form.

1 Write each number as a product of its prime factors. a 8 b 18 c 24 d 48

2 Write each number as a fraction in its simplest form. a 0.6 b 0.85 c 1.625 d 4.25 e 0 . 3 ̇ f 1 . 5 ̇

War

m u

p



17

Unit 1 Number

3 a Work out i √ 

__ 2 × √ 

__ 3 ii √ 

__ 6

b Work out i √ 

__ 3 √ 

__ 5 ii √ 

___ 15

c What do you notice about your answers to parts a and b? d Find the missing numbers.

i √ __

2 × √ __

6 = √ __

ii √ __

2 × √ __

= √ ___

10 iii √ __

√ __

7 = √ ___

35

4 Find the value of the integer k. a √ 

____ 150 = √ 

__ √ 

__ 6 = k √ 

__ 6 b √ 

___ 40 = √ 

__ √ 

___ 10 = k √ 

___ 10

c √ ____

128 = k √ __

2 d √ ____

108 = k √ __

3

5 Simplify these surds. a √ 

___ 20 b √ 

____ 300 c √ 

___ 44

d √ ____

250 e 4 √ ___

50 f 6 √ ___

56

6 Reasoning / Problem-solving Trish types √ ___

72 into her calculator. Her display shows 6 √ 

__ 2 .

a Show how the calculator worked this out. Trish presses the S⇔D button on her calculator. The display shows 8.485281374. b What does the S⇔D button do? c Louie types √ 

___ 90 into his calculator.

i What number will be shown on the calculator display? ii Estimate the number that will be shown when Louie presses the S⇔D button.

iii Use a calculator to check your answers to parts i and ii.

7 a A surd simplifi es to 4 √ __

5 . What is the original surd? b Refl ect How did you fi nd the surd?

8 Copy and complete the table using the numbers below.

Rational Irrational

3 √ __

6 3 _ 8 √ _____

6 . 25 – 4 – √ __

8 √ ___

17 1 . 4 ̇ √ ___

4 ___ 49

0.3

Key point 13

√ ____

mn = √ __

m √ __

n

Key point 14

A surd is a number that cannot be simplifi ed by removing the square root sign. For example, √ 

__ 5 is a surd, but √ 

__ 4 is not a surd, because √ 

__ 4 = 2.

All surds are irrational numbers.

Q5 hint Find a factor that is also a square number.

Q6a hint ‘Show’ means show your working.

Key point 15

Rational numbers can be written as a fraction in the form a __ b , where a and b are integers

and b ? 0. 2 is rational as it can be written as 2 __

1 .

0 . 2 ̇ is rational as it can be written as 2 __ 9

.

√ __

2 is irrational.



18

Unit 1 Number

9 Simplify

a √ __

7 __ 4

= √ 

__ 7 ___

√ __

4 = b √ 

__

5 __ 9

c √ ___

12 ___ 49

= √ 

___ 12 ____

√ ___

49 = d √ 

___

18 ___ 25

10 Solve the equation x2 – 90 = 0, giving your answer as a surd in its simplest form. Discussion Can you solve the equation x2 + 90 = 0 in the same way? Explain your answer.

11 Solve these equations, giving your answer as a surd in its simplest form. a 4x2 = 200 b 1_

2x2 = 80 c 3x2 = 36 d 2x2 – 14 = 42

12 The area of a square is 60 cm2. Find the length of one side of the square. Give your answer as a surd in its simplest form.

13 a Work out

i 5 √ __

2 × 4 √ ___

27 ii 4 √ __

5 × 6 √ ___

12

iii 9√___10 × 4√

__5 iv 8√

___12 × 3√

__3

b Use a calculator to check your answers to parts i to iv.

14 Rationalise the denominators. Simplify your answers if possible.

a 1 ___ √ 

__ 7 b 1 ___

√ __

3 c 1 ___

√ __

5 d 1 ____

√ ___

20

e 2 ___ √ 

__ 8 f 3 ____

√ ___

15 g 32 ____

√ ___

40 h 11 ____

√ ___

11

Key point 16

√ __

m __ n = √ 

__ m ____

√ __

n

Q10 hint x2 = x = √ 

__

x = √ __

Q13a hint Multiply the integers together and the surds together. Simplify.

Key point 17

To rationalise the denominator , the denominator should be a rational number and all the square roots should have no factors that are square numbers.

Example 4

Rationalise the denominator.

a 1 ___ √ 

__ 2

b 5 ____ √ 

___ 75

a) 1 ____ √ 

__ 2 = 1 ____

√ __

2 ×

√ __

2 ___ √ 

__ 2

= √ 

__ 2 ___

√ __

4 =

√ __

2 ___ 2

b) √ ___

75 = √ ___

25 √ __

3 = 5 √ __

3

5 ____ √ 

___ 75 = 5 ____

5 √ __

3 = 1 ___

√ __

3 ×

√ __

3 ___ √ 

__ 3 =

√ __

3 ___ √ 

__ 9 =

√ __

3 ___ 3

√ __

2 is an irrational number. We need to make the number of the denominator rational.

Multiplying by √ 

__ 2 ___

√ __

2 is the same as multiplying

by 1, so this does not change the value.

First simplify √ ___

75

Simplify the fraction before rationalising.


19

Unit 1 Number

15 Reasoning / Problem-solving

Ben types 1 ___ √ 

__ 7 into his calculator. His display shows

√ __

7 ___

7 .

a Show that 1 ___ √ 

__ 7 =

√ __

7 ___

7 .

b Use your calculator to check your answers from Q14.

16 The area of a rectangle is 20 cm2. The length of one side is √ 

__ 5 cm.

Work out the length of the other side. Give your answer as a surd in its simplest form.

17 Work out the area of these shapes.

3 2

1 6 4

7

4 7

a b c

10 2

3 8

Give your answer as a surd in its simplest form.

Q16 hint Multiply the numerators and denominators separately. Then rationalise the denominator.

Objective

• Use pictures or lists to help you solve problems.

1 Problem solving

Example 5

A furniture maker orders 22 metal legs. He uses the legs to make three-legged stools and four-legged chairs. Describe the diff erent ways he can use all the legs.

1 stool

2 stools 4 chairs

6 stools 1 chair

He can use 22 legs to make 2 stools and 4 chairs or 6 stools and 1 chair .

An alternative method is to use a list.

Number of stools 1 2 3 4 5 6

Number of chairs 4 4 3 2 1 1

Legs le� over 3 1 2 3

Draw a picture to represent 22 metal legs.

Circle 3 legs for one stool. Can you use the remaining 19 legs to make complete chairs? No.

Circle another 3 legs for two stools. Can you use the remaining 16 legs to make complete chairs? Yes.

Draw the diagram again. Are there any other ways you can circle the legs to make complete chairs and stools?

Write your answer.



20

Unit 1 Number

1 There are 20 chairs in a conference room. The conference organiser can put 4, 5 or 6 chairs at a table. a Describe the diff erent ways the room can be arranged so

that all the chairs are used. b What is the maximum number of tables required in the room?

2 A play park is 18 m wide and 31.5 m long. The council plans to enclose it with a fence, using a supporting post every 2.25 m. How many posts does the council need?

3 When two plant stakes are placed end to end, their total length is 1.45 m. When the two stakes are placed side by side, one is 0.15 m longer than the other. What lengths are the stakes? Give your answer in cm.

4 Finance In a canteen, a starter costs £0.80, a main costs £2.40 and a dessert costs £1.20. Three friends bought lunch and paid £10 in total. They each had at least two courses. How many starters, mains and desserts did they buy?

5 Finance A bicycle shop hires road bikes for £25 per day and tandems for £40 per day. One day a family pays £155. a Which type of bicycles did they hire? b How many people are in the family?

6 A tour company off ers three diff erent walking tours. The landmark tour leaves every 15 minutes. The parks tour leaves every 20 minutes. The museum tour leaves every 45 minutes. All walking tours start at 9 am. When do the landmark, parks and museum tours next leave at the same time?

7 Refl ect How can you solve Q6 without making a list? Discussion Does it matter how you solve a maths problem?

Q1 hint Draw a picture or use a list.

Q2 hint Draw a picture to help you see what you do to solve the problem.

Q3 hint Draw one picture to represent the fi rst sentence. Draw another picture to represent the second sentence.

Q4 hint Find numbers that add to £10.

Q5 Literacy hint A tandem is for two cyclists.

Q7 hint List all the diff erent times each tour leaves.

1 Check up Log how you did on your Student Progression Chart.

Calculations, factors and multiples

1 16.7 × 9.2 = 153.64 Use this fact to work out the calculations below. Check your answers using an approximate calculation. a 167 × 9.2 b 1.5364 ÷ 1.67

2 Estimate the value of √ ___

54 to the nearest tenth.

3 a Estimate i ( √ _____

65 . 1 – 6.17) × 1.98 ii √ 

__________ 8 . 19 × 6 . 43 ____________

6 . 84 × √ _____

3 . 97

b Use your calculator to work out each answer. Give your answers correct to 1 decimal place.

Homework, practice and support: Higher 1 Check up


21

Unit 1 Number

4 Write 90 as a product of its prime factors in index form.

5 Find the highest common factor (HCF) and lowest common multiple (LCM) of 14 and 18.

6 In prime factor form, 2450 = 2 × 52 × 72 and 68 600 = 23 × 52 × 73. a What is the HCF of 2450 and 68 600? Give your answer in prime factor form. b What is the LCM of 2450 and 68 600? Give your answer in prime factor form.

Indices and surds

7 Copy and complete.

a 10 = 1000 b 43 = c 2 = 16

8 Work out

a 3 √ _______

9 2 − 1 2 ______ 10

b [(62 – 25) × 3]2 c 3 √ __________

√ ___

81 + (−2) 3

9 Write each product as a single power. a 9–3 × 97 b 27 × 9 × 27 c 57 ÷ 52

d 2 8 × 16 ______ 2 5

e (24)3 f (42)–1

10 Work out a 2–4 b 2 5

3 _ 2 c ( 16 ___ 81

) 3 _ 4 d 1 6 − 1 _ 2

11 Simplify a √ 

___ 54 b 5 √ 

_____ 1000

12 Rationalise the denominators. Simplify your answers if possible.

a 1 ____ √ 

___ 10 b 4 ___

√ __

8

Standard form

13 Write these numbers in standard form. a 32 040 000 b 0.0007

14 Write these as ordinary numbers. a 5.6 × 104 b 1.09 × 10–3

15 Work out a (5 × 104) × (9 × 107) b (3 × 108) ÷ (6 × 105) c (7 × 104)2

16 Refl ect How sure are you of your answers? Were you mostly Just guessing Feeling doubtful Confi dent What next? Use your results to decide whether to strengthen or extend your learning.

*Challenge

17 The diagram shows a warehouse (W) and fi ve destinations (A, B, C, D, E), and the times it takes to drive between each of them. A delivery driver has to deliver packages to A, B, C, D and E. He starts and ends at W. a How many diff erent ways could he do this? b Which is the quickest route? c Write your own delivery driver question.

Refl ect

A B40 min

25 min

20 min5 min

10 min

15 min

20 min20 min

30 min

50 min

C

E D

W



22

Unit 1 Number

Calculations, factors and multiples

1 Copy and complete these number patterns. a 0.38 × 29.4 = 11.172 b 6011.545 ÷ 94.67 = 63.5

3.8 × 29.4 = 60 115.45 ÷ 94.67 = 38 × 29.4 = 601 154.5 ÷ 94.67 = 380 × 29.4 = 6 011 545 ÷ 94.67 = 3800 × 29.4 = 60 115 450 ÷ 94.67 =

2 8.9 × 7.21 = 64.169 Use this fact to work out the calculations below. Check your answers using an approximate calculation. a 8.9 × 72.1 b 8.9 × 7210 c 0.89 × 0.721 d 0.089 × 0.721 e 64.169 ÷ 72.1 f 6416.9 ÷ 7.21

3 Copy and complete this square root number line.

1 4 16

3 5 6 7 9

64

4 Estimate the value to the nearest tenth. a √ 

___ 52 b √ 

___ 60 c √ 

___ 75

d √ ___

38 e √ ___

14 f √ ____

103

5 a Estimate i √ 

__________ 4 . 09 × 8 . 96 ii 25.76 – √ 

__________ 4 . 09 × 8 . 96

iii 3 √ ______

26 . 64 + √ _____

80 . 7 iv ( 3 √ _____

26 . 64 + √ _____

80 . 7 ) × 7.54

v √ __________

6 . 91 × 9 . 23 vi √ 

__________ 6 . 91 × 9 . 23 ____________

3 . 95 2 ÷ 2 . 03 3

vii ( √ ____

50 . 2 – 3 √ ____

63 . 9 ) × 5.87 viii √ 

__________ 7 . 91 × 9 . 89 ____________

2 . 98 3 ÷ 3 . 21 2

b Use a calculator to work out each answer. Give your answer correct to 1 decimal place.

6 Copy and complete these calculations in index form a 2 × 2 × 2 × 3 × 3 = 23 × 3 b 2 × 2 × 3 × 5 = 2 × × c 3 × 3 × 3 × 3 × 7 × 7 =

7 a Copy and complete this factor tree for 60 until you end up with just prime factors.

6

60

b Write 60 as a product of its prime factors. c Write your answer to part b in index form.

Q1a hint 3.8 = 0.38 × 10 3.8 × 29.4 = 11.172 × 10

Q2 hint Write out a number pattern to help you.

Q4a hint Use your number line from Q3. Which two square roots does √ 

___ 52 lie between?

Which is it closer to?

Q5a i hint Round each number to the nearest whole number. Which square root is it closest to on your square root number line?

Q5a iii hint Draw a cube root number line.

Q7b hint Write all the prime factors from your tree multiplied together.

Homework, practice and support: Higher 1 Strengthen

1 Strengthen


23

Unit 1 Number

8 Write each number as a product of its prime factors in index form. a 24 b 80 c 45 d 30 e 16 f 72

9 a Write 18 as a product of its prime factors. 18 = × ×

b Write 45 as a product of its prime factors. 45 = × ×

c Copy and complete this diagram.

Prime factorsof 45

Prime factorsof 18

i Put any prime factors of both numbers where circles overlap.

ii Put the remaining prime factors of 18 in the left -hand part of the left circle.

iii Put the remaining prime factors of 45 in the right-hand part of the right circle.

d Work out the HCF. e Work out the LCM.

10 Find the highest common factor (HCF) and lowest common multiple (LCM) of a 20 and 30 b 21 and 28 c 15 and 25 d 44 and 36

Indices and surds


a 2 × 2 × 2 = 2 = b 5 × 5 = 5 = c –3 × –3 × –3 = (–3) =

2 Work out a 25 = b 53 = c 4 = 64 d 10 = 1000

3 Work out a 3 × √ 

___ 81 b 23 × √ 

___ 16

c 3 √ _____

−27 × √ ___

64 d 102 ÷ – √ ___

25

e (–3)2 × √ ___

49 f √ __

9 × 3 √ ______

−125 × 3 √ _______

−1000

4 Work out a 5 × (42 – 32) – 23

b [(92 ÷ 32) + 22]2

c 60 – [43 – (82 ÷ 24)]

d √ ________________

√ ___

25 + 3 √ ______

−125 + √ ___

16

e √ ________

2 3 − 3 √ ____

−1

f √ _________

2 √ ___

49 + 3 √ __

8

Q9d hint HCF

3

Q9e hint LCM

33 3

Q10 hint Use a Venn diagram as in Q9.

Q2c hint 41 = 4 42 = 4 × 4 = 43 = 4 × 4 × 4 =

Q3 hint First work out any powers or roots. Then multiply or divide.

Q4b hint First work out the round brackets. Then work out the square brackets.

Q4d hint First work out the calculation underneath the square root sign. Then fi nd the square root.



24

Unit 1 Number

5 Copy and complete. a (3 × 3 × 3 × 3 × 3) × (3 × 3 × 3) = 3 × 3 = 3

b (4 × 4 × 4 × 4) × (4 × 4 × 4 × 4 × 4 × 4) = 4 × 4 = 4

c 6 × 6 × 6 × 6 __________ 6 × 6

= 62 ___

6 = 6

d 7 × 7 × 7 × 7 × 7 × 7 ________________ 7 × 7 × 7

= 7 ___ 7

= 7

e To multiply powers, _____ the indices. To divide powers, _____ the indices.

6 Work out a 56 × 53 = b 72 × 79 = c 58 ÷ 53 = d 98 ÷ 92 = e 84 × 8–6 = f 73 ÷ 7–4 =

7 Write these as a single power of a prime number.

a 16 × 8 = 2 × 2 = b 25 × 125 × 25 c 16 × 64 × 8 d 27 × 27 × 9

8 Copy and complete. a (42)3 = 42 × 42 × 42 = 4

b (63)4 = 63 × 63 × 63 × 63 = 6

c (75)2 =

d (83)7 = e To work out a power raised to a power, _________ the indices.

9 a Work out using a calculator i √ 

____ 169 ii 169

1 _ 2 b Use your answer to part a to work these out without a calculator.

i 6 4 1 _ 2 ii 2 5

1 _ 2 iii 8 1 1 _ 2 iv 14 4

1 _ 2 c Work out using a calculator

i 3 √ ____

512 ii 51 2 1 _ 3

d Use your answer to part c to work these out without a calculator. i 12 5

1 _ 3 ii 2 7 1 _ 3 iii 100 0

1 _ 3 iv 8 1 _ 3

e Copy and complete.

1 6 1 _ 4 = √ 

___ 16 =

10 a Work out i 6 4

1 _ 3 ii 6 4 2 _ 3

b Use your answer to Q9d to work out

i 12 5 2 _ 3 ii 2 7

2 _ 3 iii 100 0 2 _ 3 iv 8

2 _ 3 c Use your answer to Q9e to work out 16

3 _ 4 .

11 a Copy and complete.

i 4 −3 = 1 ___ 4

ii 1 ___ 10 5

= 10 iii 1 _ 2 = 2

iv 3 − 1 _ 3 = 1 ___ 3

v ( 7 __ 6

) = 36 ___ 49

vi a −n = 1 ___ a

b Work out i 4–1 ii 10–2 iii 3 6 − 1 _ 2 iv 12 5 − 1 __ 3

Q6 hint Use the rules from Q5.

Q9e hint Use what you have learned in parts a to d.

Q10a hint 6 4 2 _ 3 = (6 4

1 _ 3 ) 2

Work out 6 4 1 _ 3 and square

your answer.

Q10c hint 16 3 _ 4 = ( 16

1 _ 4 ) Use the same strategy as in part a.


25

Unit 1 Number


a 50 = × 2, so √ ___

50 = √ ___

× √ __

2 = √ __

2

b 84 = × 21, so √ ___

84 = √ ___

× √ ___

= √ ___

Simplify

c √ ___

96 d √ ____

175 e √ ____

128

13 Work out

a √ __

4 × √ __

4 b √ ___

25 × √ ___

25 c √ ___

17 × √ ___

17 d √ ___

21 × √ ___

21

14 Rationalise the denominator. Simplify your answer if possible.

a 1 ____ √ 

___ 17 b 3 ____

√ ___

21

c 1 ___ √ 

__ 8 d 6 ____

√ ___

20

Standard form

1 A number written in standard form looks like this.

numberbetween1 and 10

timessign

powerof 10

A 3 10n

Are these numbers in standard form? If not, give reasons why. a 9.004 × 10–3 b 32 × 105 c 7.3 million d 0.8 × 107

2 Write each number using standard form. a 68 000 = 6.8 × 10 b 94 000 000 c 801 000 d 0.000 004 e 0.0039 f 0.000 000 053

3 Work out a (4 × 102) × (2 × 107) = × 10 b (3 × 109) × (2 × 105) c (6 × 104) × (1 × 10–2) d (6 × 108) × (8 × 104) e (7 × 103) × (8 × 106) f (8 × 10–4) × (6 × 10–2)

4 Real A gymnastics fl oor is a square with side length 1.2 × 103 cm. Work out the area of the fl oor.

Q12c hint Write a list of square numbers up to 100. Find a square number that is a factor of 96.

Q14a hint Multiply both the numerator and denominator by √ 

___ 17 .

Q14c hint First rewrite the denominator. √ 

__ 8 = √ 

__ 2

Q2d hint 4 lies between 1 and 10. Divide by how many 10s to get 0.000 004?

Q2a hint 6.8 lies between 1 and 10. Multiply by how many 10s to get 68 000?

Q3a hint

4 3 2 3 102 3 107

3 10 Q3d hint 48 = 4.8 × 10

1 Extend

1 Problem-solving Square A has a side length of 9.2 cm. Square B has a perimeter of 34.4 cm. Square C has an area of 80 cm2. a Which square has the greatest perimeter? b Which square has the smallest area?



26

Unit 1 Number

2 Show that 272 = 93 = 36.

3

4 a Write 48, 90 and 150 as products of their prime factors. b Use a Venn diagram to work out the HCF and LCM of

48, 90 and 150. Discussion Explain how the diagram can be used to fi nd the HCF and LCM of any two of the numbers.

5 Real A new school is deciding whether their lessons should be 30, 50 or 60 minutes. Each length of lesson fi ts exactly into the total teaching time of the school day. How long is the teaching time of the school day? Discussion Ryan says there is more than one answer to this question. Is Ryan correct? Explain your answer.

6 Reasoning a Use prime factors to explain why numbers ending in a zero must be divisible by 2 and 5. b How many zeros are there at the end of 24 × 37 × 56 × 72? c Use prime factors to work out 32 × 9 × 3125. Write your answer as an ordinary number

and in standard form.

7 Write each of these as a simplifi ed product of powers. a 105 × 23 × 54 = (2 × 5)5 × 23 × 54 = 2 × 5 × 23 × 54 = 2 × 5 b 63 × 24 × 33 c 153 × 104 × 62 d 304 × 242 × 153

8 STEM Write each answer i as an ordinary number ii in standard form.

a Saturn has a diameter of 120 536 000 m. Convert this to kilometres.

b The distance from the Sun to Mars is 227 900 000 km. Convert this distance to metres.

c The diameter of a grain of sand is 4 μm. Convert this to metres.

d The wavelength of an X-ray is 0.1 nm. Convert this to metres.

9 Every six months, new licence plates are issued in the UK. A licence plate consists of two letters, then two numbers, then three letters. The numbers are fi xed, but the letters vary. a If all letters can be used, how many possible combinations are there? b If only 21 letters can be used, how may possible combinations are there?

10 STEM / Problem-solving A container ship carries 1.8 × 108 kg. An aeroplane can carry 3.8 × 105 kg. What is the diff erence in their mass? Write your answer in standard form.

Q2 communication hint ‘Show that’ means show your working.

Exam-style question

Here are some properties of a number. • It is a common factor of 216 and 540. • It is a common multiple of 9 and 12. Write two numbers with these properties. (6 marks)

Q4b hint

48 90

150

Put prime factors of all three in the very centre fi rst.


27

Unit 1 Number

11

12 A square has an area of 1 _ 2 cm2. a Work out the length of one side. b Work out the perimeter. Give your answers as surds in their simplest forms.

13 Write 3 − 1 _ 2 as a surd and rationalise the denominator.

14

15 What is the value of x?

a 3x = 1 ___ 27

b 2 × 2x = 1 __ 8

c 32x = 1 __ 9

Exam-style question

Work out

a 5 ___

√ __

2 +

8 ____

√ ___

32 b

7 ____

√ ___

72 –

3 ___

√ __

8

Write each answer in the form a √ __

2 . (3 marks)

Exam-style question

A restaurant offers 5 starters, 7 mains and 3 desserts. A customer can choose • just one course • any combination of two courses • all three courses. Show that a customer has 191 options altogether. (3 marks)

1 Knowledge check

◉ When there are m ways of doing one task and n ways of doing a diff erent task, the total number of ways the two tasks can be done is m × n. Mastery lesson 1.1

◉ You can round numbers to 1 or 2 signifi cant fi gures to estimate the answer to a calculation. Mastery lesson 1.2

◉ You can use a prime factor tree to write a number as the product of its prime factors. Mastery lesson 1.3

◉ You can use a Venn diagram of prime factors to work out the highest common factor and lowest common multiple of two numbers. Mastery lesson 1.3

◉ The prime factor decomposition of a number is the number written as the product of its prime factors. It is usually written in index form. Mastery lesson 1.3

◉ When multiplying powers, add the indices: xm × xn = xm + n

When dividing powers, subtract the indices: xm ÷ xn = xm – n

To raise a power to another power, multiply the indices.

x –n = 1 ___ xn x

1 __ n = n √ __

x x m __ n = ( n √ 

__ x )m Mastery lesson 1.4, 1.5

◉ A number in standard form is written in the format A × 10n, where A is a number between 1 and 10 and n is an integer. Mastery lesson 1.6



28

Unit 1 Number

◉ To write a number in standard form: • work out the value of A • work out how many times A must be multiplied or divided by 10.

This is the value of n. Mastery lesson 1.6

◉ To simplify a surd, identify any factors that are square numbers. Mastery lesson 1.7

◉ To rationalise a denominator, multiply the numerator and the denominator by the surd in the denominator and simplify. Mastery lesson 1.7

1 Unit test Log how you did on your Student Progression Chart.

1 6.23 × 5.4 = 33.642 a Write down two more multiplications with an answer of 33.642. b Write down a division with an answer of 0.623. (3 marks)

2

3 a Estimate (17.9 – √ ______

36 . 13) × 3.89 b Use a calculator to work out the answer. Give your answer correct to 1 decimal place. (2 marks)

4 a Write 42 as a product of its prime factors. b Use your answer to write 843 as a product of its prime factors in index form. (4 marks)

5 Work out the HCF and LCM of 75 and 30. (3 marks)

6 Real Ben and Sadie are doing a sponsored walk around a circuit. Ben takes 25 minutes to do one circuit and Sadie takes 45 minutes. They start together at 9:30 am. When will they next cross the start line together? (2 marks)

7 Find the value of a. a 53 × 5a = 59 b 6a ÷ 6–5 = 68 c 8a × 8a = 84 (3 marks)

8 Write (38)4 as a single power. (1 mark)

9 Write each number in standard form. a 0.000 000 65 b 0.9 million c 320 × 107 (3 marks)

10 918 = 27x Work out the value of x. (2 marks)

11 Use prime factors to determine whether 2520 is divisible by 18. (2 marks)

12 Write ( 4 __ 3

) −2

as a fraction in its simplest form. (2 marks)

13 Work out the area of this shape. Write your answer as a simplifi ed surd.

(3 marks)

Exam-style question

List these numbers in order, starting with the smallest. Show your working.

3.22 3 √ ___

27 √ ___

69 13.74 (3 marks)

30

45

Homework, practice and support: Higher 1 Unit test


29

Unit 1 Number

14

15 Rationalise the denominator. 8 ___ √ 

__ 6 (2 marks)

16 Let x = 6 × 105 and y = 8 × 104. Work out

a x + y b x – y c x y d y __ x

Write your answers in standard form. (2 marks)

Sample student answer

The maths is correct, but the student will only get 2 marks. Why?

9 × 10−3 × 500

0.009 × 500

9 × 500 = 4500, ÷1000 = 4.5

Exam-style question

How many different 4-digit odd numbers are there, where the fi rst digit is not zero? (3 marks)

Exam-style question

One sheet of paper is 9 × 10−3 cm thick.

Mark wants to put 500 sheets of paper into the paper tray of his printer.

The paper tray is 4 cm deep.

Is the paper tray deep enough for 500 sheets of paper?

You must explain your answer. (3 marks)

June 2013, Q15, 1MA0/1H



T972a

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