11 - 19 PROGRESSION
Edexcel GCSE (9-1)
Mathematics
Sample
unit
Higher
Student Book
Confidence • Fluency • Problem-solving • Reasoning
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We are seeking endorsement for use with the
Edexcel GCSE (9-1) Mathematics specification.
A LW AY S L E A R N I N G
Master
p.2
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
Problem solve
p.19
Check
p.20
Strengthen
p.22
Extend
p.25
Test
p.28
1 NUMBER
Estimate the amount of money taken by a
football club on match day.
1 Prior knowledge check
10 Probability
Numerical fluency
11 Multiplicative reasoning
1
12 Similarity and congruence
13 More trigonometry
14 Cumulative frequency and histograms
15 Equations and graphs
97 × 0.02
27 ÷ 0.09
0.4 × 0.6
0.09 × 0.09
0.9 ÷ 0.03
0.88 ÷ 0.04
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.
18 Vectors and geometric proof
4
19 Proportion and graphs
b
d
f
h
j
l
2
16 Circle theorems
17 More algebra
Work out
a 5 × 0.3
c 6 ÷ 0.2
e 4.2 ÷ 0.1
g 0.9 × 0.02
i 0.4 ÷ 0.2
k 0.45 ÷ 0.3
a
Factors of 20
b Write down the highest common
factor.
1
a
6
a
Write down the first 10 multiples of 6
and 9.
b Make a list of the common multiples.
c Write down the lowest common
multiple.
Copy and complete the Venn diagram to
show the first 10 multiples of 4 and 10.
Multiples of 4
Multiples of 10
b Write down the lowest common
multiple of 4 and 10.
7
Copy and complete the Venn diagram
to show the factors of 16 and 20.
Factors of 16
Draft, subject to endorsement
5
Work out
a 8–2×3
c 7 – (4 – 1) × 6
e 42 + 1
b (8 – 2) × 3
d 24 ÷ (8 – 2)
f (–6)2
8
Insert brackets to make this calculation
correct.
9 + 18 ÷ 3 = 9
9
Estimate
a 7.3 × 8.94
c 5.2 + 4.9
b 47 ÷ 2.1
d 7.9 – 2.4
Draft, subject to endorsement
1
Unit 1 Number
Unit 1 Number
10 Write the positive and negative square
roots of these numbers.
a 36
b 1
c 64
11 Work out
a 4 × 9 × 25
c 182 × 99
Challenge
14 How many different ways can these cards
be arranged?
R
b 102 × 48
d 27 × 6 + 27 × 4
B
Y
What about now?
12 Copy and complete.
a 6×6=6
b 3×3×3×3=3
R
13 Work out
a 42
c 113
4
B
Y
G
b 14
d 27
A restaurant offers a set menu for birthday parties.
a Write down all possible combinations of starters
and main courses.
b Reflect How did you order your list to make sure
you didn’t miss any starters or mains?
The restaurant decides to offer fish (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 different restaurant offers 2 starters, 4 mains
and 3 desserts.
e How many possible combinations are there now?
Q4a hint Use letters for
combinations, for example
VP for vegetable soup, pizza.
Starters
Vegetable So
up (V)
Salad (S)
Melon (M)
Mains
Pizza (P)
Spaghetti Bol
ognaise (B)
Cur ry (C)
Lasagne (L)
Key point 1
When there are m ways of doing one task and n ways of doing a different task, the total number
of ways the two tasks can be done is m × n.
1.1 Number problems and reasoning
Why learn this?
Objectives
• Work out the total number of ways of
performing a series of tasks.
5
5! in maths means ‘five factorial’ and is equal
to 5 × 4 × 3 × 2 × 1.
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)
Fluency
Work out
• 4×4×4
Warm up
1
• 5×4×3
• 10 × 9 × 8
• 4×3×2×1
a Copy and complete this list of all possible outcomes for rolling a dice and flipping a coin.
H, 1
H, 2
…
T, 1
…
…
6
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?
A
B
2
4
6
Questions in this unit are targeted at the steps indicated.
3
2
How many possible outcomes are there when
a rolling a dice
b flipping 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?
Homework, practice and support: Higher 1.1
1
9
3
7
Exam-style question
Q5 communication hint
Inclusive means that the
end numbers are also
included.
Three people, A, B and C, enter a race.
a Write down the different orders in which they can finish first, 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 different ways can people finish 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.
5
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?
Draft, subject to endorsement
3
Unit 1 Number
Unit 1 Number
9
1.2 Place value and estimating
Objectives
Why learn this?
• Estimate an answer.
• Use place value to answer questions.
Builders use estimates to give their clients an
idea of how much the work will cost.
Fluency
Warm up
2
Write each number to
i 1 significant figure
ii 2 significant figures.
a 873 209
b 2019
c 0.007 059
Work out
a 9 × (4 + 7)
d 30 – 5 × 8
b 5+3×8
e 72 – 9
c 7______
×5–4×2
√
f
29 − 4
√ 35 . 5
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?
7
8
__
Q12a iv hint The whole of the
numerator is being square rooted.
So work out the numerator before
square rooting.
__________
√ 3 . 93 × 5 . 42
___
Q13 hint Work out the value
of the card on the right first.
80 − √ 64
________
2×4
14 Problem-solving A large cubic dice has a side length of 9.2 cm.
Estimate the surface area of the dice.
Q5d hint Rewrite
the given calculation
as a division.
__
Estimate the value to the nearest tenth.
___
___
___
a √ 47
b √ 22
c √ 84
___
___
____
d √ 127
e √ 10
f √ 40
42
Work out the missing number.
Write down three more calculations that have the same answer.
Write down a division that has an answer of 54.8.
Write down a division that has an answer of 0.729.
Charlie says that 54.8 × 72.9 = 3989.44. Explain why Charlie must be wrong.
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.
ii (1.98 × 3.14)2 ÷ 8.85
3.22
13 The sum of the values on these cards is 12.
Reasoning 54.8 × 7.29 = 399.492
a
b
c
d
32
b Use your calculator to work out each answer.
Give your answers correct to one decimal place.
c Reflect 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 first?
52 +
______
42
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
Q5a hint Compare with the given
b 3.7 × 0.0986
calculation.
c 0.0037 × 98.6
3.7 9.86 36.482
d 36.482 ÷ 9.86
e 3648.2 ÷ 98.6
37 9.86 f 364.82 ÷ 370
Q11a hint Use a number line to help.
√ 27 . 3 − 1 . 85
__________
iv ____________
88 . 72
− 21 . 9
_____
iii ___________
Work out the mean of 3, 6, 7, 9, 15 and 20.
6
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.
12 a Estimate answers to these.
_____
i (11.2 – √ 50 . 3 ) × 4.08
3
5
The total area is 3000 cm2.
a Estimate the side length of a tile.
b Use a calculator to check your answer.
11 Estimate to the nearest whole number.
a 3.22
b 4.72
c 1.72
2
2
d 7.1
e 6.3
f 9.82
Which
two___
whole numbers does each square root lie between?
__
√ 17
√ 3
1
Problem-solving A mosaic uses 150 square tiles.
Q7b hint Use a number line to help.
4
5
9
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.
Q8a hint Use a number line to help.
36
47
49
A
1
78
B
12
C
D
4
15
E
F
0
35
G
H
0
7
I
J
K
12
21 11.8
a Use estimates to show that Robert’s average is wrong.
b Work out Robert’s correct average to the nearest tenth.
4
Homework, practice and support: Higher 1.2
Draft, subject to endorsement
5
Unit 1 Number
Unit 1 Number
1.3 HCF and LCM
Objectives
7
Write each number as a product of its prime factors in index form.
a 18
b 42
Q7 communication hint In index form means to write a number
c 25
d 36
to a power or an index.
e 24
f 80
23 is written in index form. 3 is the power or index.
8
120 can be written as a product of its prime factors in the form 2m × n × p.
Work out m, n and p.
Why learn this?
• Write a number as the product of its prime factors.
• Find the HCF and LCM of two numbers.
Astronomers use the lowest common
multiple of patterns in the orbits of the
Sun and the Moon to predict solar eclipses.
Fluency
Example 1
Work out
• 2 × 3 × 52
3
2
• 2 ×3
• 5×5×5=5
• 7×7×7×7×7=7
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.
Warm up
1
Prime numbers between
1 and 20
5
3
2
Prime factors
of 24
2
2
2
3
5
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
12
9
a Copy and complete this factor tree for 40.
8
4
b Write 40 as a product of its prime factors.
40 =
×
×
×
=
×
Q3b hint Circle the prime
factors in your factor tree.
4
Write 75 as a product of its prime factors.
5
Steve and Ian are asked to find 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 final answer for Steve.
b Work out a final answer for Ian.
Discussion What do you notice about the two answers?
c Start the prime factor decomposition of 48 in two different ways: 6 × 8 and 12 × 4.
Discussion Does your first step in a prime factor decomposition affect your final answer?
Q4 hint Use the method from Q3.
Reflect a Write down your own short mathematical definition of these words.
i prime
ii factor
iii decomposition
b Use your definition to write down (in your own words) the meaning of prime factor
decomposition.
6
Prime factors
of 60
Draw a Venn diagram.
Factors of 24
40
6
Find the highest common factor and lowest common multiple of 24 and 60.
24 = 2 × 2 × 2 × 3
Write each number as a product of prime factors
60 = 2 × 2 × 3 × 5
Homework, practice and support: Higher 1.3
Multiply the common prime factors.
Multiply all the prime factors.
Find the highest common factor (HCF) and lowest common multiple (LCM) of
a 24 and 30
b 20 and 42
Q9 hint Draw a Venn diagram
c 8 and 18
d 15 and 45
for each question to help you.
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.
Q10 strategy hint Work out
At 2:30 pm both buses leave the bus station.
the LCM first.
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 finds square tiles with a side length of 10 cm, 12 cm or 18 cm.
Which of these tiles will fit the wall exactly?
Discussion How do you know whether to find 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.
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.
Draft, subject to endorsement
7
Unit 1 Number
15
Unit 1 Number
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.
1
2
(2 marks)
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
Q19c hint What factors do 700
ii 49
and 1960 have in common?
Any factor of this number will be
iii 20
2
2
a factor of 350 and 1960.
iv 2 × 5 × 7
d Which of these are multiples of 350 and 1960?
i 23 × 5 × 73
Q19d hint What multiples do 700 and 1960 have in common?
ii 26 × 52 × 72
Any multiple of this number will be a multiple of 350 and 1960.
2
iii 2 × 5 × 7
3
Work out
• 62
8
• (–4)2
• 24
• 15
Homework, practice and support: Higher 1.4
3×3×3×3×3×3
b ________________
3×3×3
Copy and complete.
a 2 = 16
c 5 = 25
b
d
3
3
2
d 3___
× 5 ___
h √ 81 × √ 64
Q3a hint 2 × 2 × … = 16
= 64
= 27
Q3b hint
×
×
= 64
4
Work___
out
_____
_____
___
3
3
3
3
√
a  27
b √ −1
c √ 1000
d √ −125
Discussion Explain why it is possible for you to find the cube root of a negative number,
but not the square root.
5
Work______
out these. Use a calculator
to check your answers.
_______
3
2
2
2
a √ 4 + 3
b √ 10 + 5 2
____
_____
Q5a hint Use the priority of operations.
3
3
c 43 – √ −27
d 33 – √ −8 – (–4)2
____________
_____
3
e √ 5 2 + 3 × √ −27
g
___
3
− √ 64
−3__ _____
____
×
√ 9
f
___
√ −1
3
h
6
Work out
a [(33 – 52) × 2)]3
b 20 – [3 × 42 – (22 × 32__)]
c [72 ÷ (7 – 5)3 – 3] ÷ √9
7
Work out
___
4
a √ 16
b
2
3
____
5 × √ −27
_________
Q5e hint The square root applies to
the while calculation. Work out the
calculation first.
___
__
√ −8 − √ 9
_____
3
0 . 2 2 × √ −125
____________
__
3
√ 8
3
Q6 communication hint Square brackets [ ] make
the inner and outer brackets easier to see. Input
them as round brackets on your calculator.
___
√ 81
4
c
_________
√ 100 000
5
Q7a hint
4
= 16
___
√ 16 =
4
8
a Work out
ii 105
iii 106 × 102
iv 108
i 103 × 102
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
Q8c ii hint 10 = 101
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
A googol is a 1 followed by 100 zeros. It can be
written as 10100.
Fluency
Work out
4×4×4×4
a __________
4×4
2
c 4 × 4___
g 3 × √ 16
The inverse of a cube is the cube
__ root.
3
23 = 8, so the cube root of 8, √ 8 = 2
Why learn this?
• Use powers and roots in calculations.
• Multiply and divide using index laws.
• Work out a power raised to a power.
b (–1)3
f 0.23
Key point 2
1.4 Calculating with powers (indices)
Objectives
Work out
a 33
e 23 × 102
Warm up
Q15 strategy hint
Look at the common
prime factors.
c 9−3 × 9−4
Key point 3
To multiply, add the indices.
xm × xn = xm + n
Draft, subject to endorsement
9
Unit 1 Number
10 Find the value of a.
a 84 × 8a = 87
Unit 1 Number
5
a
3
b 6 ×6 =6
−3
a
−10
c 2 ×2 =2
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
Key point 4
You can only add the indices when multiplying powers of the same 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.
Key point 6
x0 = 1, where x is any non-zero number.
12 Reasoning
5×5×5×5×5
i Work out _____________ by cancelling. Write your answer as a power of 5.
5×5
ii Copy and complete. 55 ÷ 52 = 5
4×4×4×4×4×4
b Copy and complete. 46 ÷ 42 = ________________ = 4
4×4
c Work out 65 ÷ 64
Discussion How can you quickly find 79 ÷ 73 without writing all the 7s?
a
13 Work out
a 76 ÷ 72
b 4−5 ÷ 43
c 3–6 ÷ 3–7
19 Copy and complete.
×
×
×
=2
a (23)5 = 23 ×
b (64)3 =
×
×
=6
c (87)2 =
×
=8
Discussion What do you notice about the powers in the question and the powers in the
final answer?
Key point 7
To work out a power to another power, multiply the powers together.
(xm)n = xmn
Key point 5
To divide powers, subtract the indices.
xm ÷ xn = xm – n
14 Find the value of a.
a 96 ÷ 9a = 94
b 45 ÷ 4a = 4
20 Write as a single power.
a (23)4
b (62)5
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 different
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?
10
8 –2 × 8 –6
d _______
8 –7
Q17 communication hint
Backing up a hard drive
means copying its
contents on to another
storage device.
d (5−2)−6
21 Problem-solving Write each calculation as a single power.
16 × 64 × 16
58
a 8 × 32 × 8
b ____
c ___________
125
44
1.5 Zero, negative and fractional indices
Objectives
Why learn this?
• Use negative indices.
• Use fractional indices.
The smallest known time measurement is
approximately 10−43 seconds. Scientists call
this unit one Planck time, after Max Planck.
Fluency
• Work out 20
1
2
3
Work out
a 62
• Convert 2_12 to an improper fraction.
b 23
• Convert 0.3 to a fraction.
c 34
d 53
Write each calculation as a single power.
b 25 ÷ 23
a 34 × 36
c 16 × 8
d 7–3 × 73
Work out
1
___
a ____
√ 25
−3
c ____
___
3
√ 27
___
b
−8
√ ___
27
3
Homework, practice and support: Higher 1.5
Warm up
16 Work out these. Write each answer as a single power.
5 4 × 5 –3
a 53 × 57 ÷ 54
b 62 ÷ 6−3 × 67
c _______
55
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?
c (42)−3
___
4
d √ 81
Draft, subject to endorsement
11
Unit 1 Number
4
5
6
Unit 1 Number
Work out the value of n.
a 40 = 5 × 2n
b 3n × 3n = 38
2n
n
6
c 5 ÷5 =5
d _12 × 4n = 32
Q4a hint
40 = 5 ×
↗
How do you write
this number as 2 ?
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
−1
3 −2
1
i (__)
ii (__)
4
2
Discussion What is the rule for writing negative indices as fractions?
Match these negative indices to their fractions.
( 14 )
1
83
22
3
2
1
35
325
( )
1
24
2
3
328
21
16
523
1
53
224
j
8
f
(1 4 )
h (0.7)−1
i
(0.1)−5
k (5−1)0
l
(7−1)−1
b
c
d
e
Work out
_1
a 36 2
64 _1
e (___) 2
49
_1
_1 −1
Q7f strategy hint Convert
mixed numbers to improper
fractions.
Q7h strategy hint Convert
decimals to fractions.
_1
ii 16 2
iii 121 2
_1
b 81 2
f
_2
2
__
_1
−8 3
1 _1
c (__) 2
9
1 _1
___
g ( )3
27
_3
b 16 – 4
a 27 3
Use the rule (x m)n = x mn. Work out the cube
root of 27 first. Then square your answer.
1
__
a) 27 3 = (27 3 )2 = 32 = 9
3
__
1 = ___
1 = ______
1 = __
1
b) 16 − 4 = ____
3
1
__
__
3 8
3
16 4 (16 4 ) 2
1
Use x−n = ___n
x
_3
_3
b 10 000 4
e 27
c 16 2
− _2
f
3
−81
Q2a hint
___ 2
_2
_1 2
3
64 3 = (64 3 ) = (√ 64 ) =
− _3
4
2
=
m __
12 Work out
4 _1
iv (___) 2
25
_1
Copy and complete. a 2 is the same as the _______ _______ of a.
Work out
_1
_1
_1
1 _1
i 27 3
ii 1000 3 iii −1 3
iv (_____) 3
1000
_1
Copy and complete. a 3 is the same as the _______ _______ of a.
Copy and complete.
_1
_1
i 625 = 5 so 625 4 =
ii 32 = 5 so 32 5 =
_1
12
c 10
9 _1
c (___) − 2
25
_1
b 64 − 3
Work out the value of
n
__
a Work out
i 49 2
9
–5
1
Q10 hint x–n = ___n
x
_1
1
1
so 25 − 2 = ____ = ___
25
Example 2
_1
b 2
4 −3
e (__)
5
(0.4)−3
10 Work out
_1
a 25 − 2
8 _1
4 _3
b (___) − 2 × (___) 3
25
27
_3
a 27 − 3 × 9 2
–4
−2
__
x m = ( √x  )n
1
g (2_34 )
n
Key point 10
x–n = ___n for any number n, x ≠ 0
x
Work out
a 3–1
3 −1
d (__)
4
_1
x n = √ x
11 Work out
_2
a 64 3
4 _3
d (__) 2
9
a Write a matching tile for the two tiles that are left over.
a −1
2 −1
b Copy and complete. (__) = , so (__) = (__)
3
b
Key point 8
7
Key point 9
13 Find the value of n.
a 16 = 2n
__
d
√ __49 = (__94)
n
b
___
√ 27 = 27n
81 _3
9 _3
c (___) 4 × (___) − 2
25
16
1
c ____ = 10n
100
3
__
e (√ 3 ) 7 = 3n
Q12a hint
First work
_1
out 27 − 3 .
_3
Then work out 9 2 .
Then multiply these
numbers together.
__
(4√ 8 ) 7 = 8n
f
_1
_2
14 Problem-solving / Reasoning Will says that 25 – 2 × 64 3 = 80
a Show that Will is wrong.
b What mistake did he make?
15 Problem-solving / Reasoning
Match these indices to their fractions.
( )
8
27
16 _1
d (___) 2
25
−64 _13
____
h (
125 )
22
3
9
4
( )
81
16
21
2
1
16
( )
1
64
1
4
16
4
9
23
2
16
2
3
22
32 5
1
64
4
83
8
3
16 4
13
Unit 1 Number
Unit 1 Number
7
1.6 Powers of 10 and standard form
Objectives
Warm up
845.3 × 0.001
1
Copy and complete. If your answer is a fraction, write it as a decimal too.
a 100 =
b 10–1 =
c 10–2 =
–3
–4
d 10 =
e 10 =
f 10–5 =
2
Write down the value of x.
a 10x = 1000
d 10–1 = x
b 105 = x
e 10x = 0.0001
Copy and complete.
million
a 5 670 000 =
b 15 800 000 =
3
4
c 4 908 340 000 =
Copy and complete the table of prefixes.
Prefix
tera
giga
mega
kilo
deci
centi
milli
micro
nano
pico
Letter
T
G
M
k
d
c
m
μ
n
p
Power
1012
109
Number
1 000 000 000 000
billion
Q4 communication hint
Prefix is the beginning
part of a word.
1 000 000
3
10
0.1
10–2
Q4 communication hint
μ, the letter for the prefix
micro, is the Greek letter
mu.
0.001
10–6
0.000 000 001
10–12
6
mg:
1
3
g:
STEM Write these measurements in metres.
a The size of the influenza virus is about 1.2 μm.
b The radius of a hydrogen atom is 25 pm.
c A fingernail grows about 0.9 nm every second.
14
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
Homework, practice and support: Higher 1.6
c 6.78 × 103
f 4.04 × 10−3
Q10 communication hint
An ordinary number is a
whole number or a decimal
which is not multiplied by
a power of 10.
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.
4E–3
5 × 7 × 103 × 106
15
3
0.001
Q9 hint Write the number between 1 and 10 first.
Then multiply by a power of 10.
b
2.5E10
c
6.9E–7
d
3.009E4
Work out (5 × 103) × (7 × 106)
Q5a hint Use a number line.
b 7 nm into metres
d 7.3 ps into seconds.
9
Example 3
Some powers of 10 have a name called a prefix. Each prefix is represented by a letter.
For example, kilo means 103 and is represented by the letter k, as in kg for kilogram.
Convert
a 15 mg into grams
c 1.7 g into kg
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
a
Key point 11
5
8
10 Write these as ordinary numbers.
a 4 × 105
b 3.5 × 102
−2
d 6.2 × 10
e 8.93 × 10−5
c 10x = 100 000 000
f 10–6 = x
million
c 45 000 = 4.5 × 10
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 scientific notation.
Scientists use standard form to write very
small or very large numbers.
Fluency
• Work out
4.5 × 1000
0.0063 × 100
69.4 × 0.1
• Which of these are the same as ÷10?
1
× __
× 0.01
× 10–1
10
b 10 000 = 10
Key point 12
Why learn this?
• Write a number in standard form.
• Calculate with numbers in standard form.
Copy and complete.
a 45 000 = 4.5 ×
Rewrite the multiplication grouping the numbers and the powers.
35 × 109
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.
35 = 3.5 × 101
Write 35 in standard form.
35 × 109 = 3.5 × 101 × 109 = 3.5 × 1010
Work out the final answer.
Draft, subject to endorsement
15
Unit 1 Number
Unit 1 Number
13 Work out these. Use a calculator to check your answers.
a (3 × 102) × (2 × 105)
b (5 × 103) × (4 × 107)
–2
7
c (8 × 10 ) × (6 × 10 )
d (8 × 106) ÷ (4 × 103)
e (9 × 10–2) ÷ (3 × 106)
f (2 × 103) ÷ (8 × 107)
3 2
g (5 × 10 )
h (4 × 10–2)3
3
Q13g hint (5 × 103)2
= (5 × 103) × (5 × 103)
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.
18
Exam-style question
(7 × 10x) + (7 × 10y) + (7 × 10z) = 700 070.07
Write down a possible set of values for x, y and z.
____
√ mn
4
(3 marks)
Surds are used to express irrational numbers in
exact form.
Warm up
1
2
Find ____
the value
of the integer
k.
__
__ __
√
√
√
√
a  150 =    6 = k  6
____
__
c √ 128 = k√ 2
Simplify
these surds.
___
a √____
20
d √ 250
___
__
___
___
= √  √ 10 = k√ 10
____
__
√ 108 = k√ 3
√ 40
____
___
b √ 300
___
e 4√ 50
c √ 44
___
f 6√ 56
Q5 hint Find a factor that
is also a square number.
___
Reasoning / Problem-solving__ Trish types √ 72 into
Q6a hint ‘Show’ means show
her calculator. Her display shows 6√ 2 .
your working.
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 simplifies to 4√ 5 . What is the original surd?
b Reflect How did you find the surd?
__
96 =
a
Rational numbers can be written as a fraction in the form __, where a and b are integers
b
and b ? 0.
2
2 is rational as it can be written as __.
1
2
0 . 2̇ is rational as it can be written as __.
9
__
√ 2 is irrational.
× 16
Write each number as a product of its prime factors.
a 8
b 18
c 24
Write each number as a fraction in its simplest form.
a 0.6
b 0.85
c 1.625
d 4.25
e 0 . 3̇
f 1 . 5̇
d 48
8
Copy and complete the table using the numbers below.
Rational
3
__
√6
16
b
d
Key point 15
Fluency
• What does the dot above the 1 mean in 0 . 1̇ ?
• What are the missing numbers?
147 = 3 ×
125 =
× 25
180 = 5 ×
__
6
Why learn this?
• Understand the difference between rational
and irrational numbers.
• Simplify a surd.
• Rationalise a denominator.
__
= √ m √ n
A surd is a number
that cannot__be simplified 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.
1.7 Surds
Objectives
___
= √ 35
Key point 14
5
Exam hint
Don’t just write down the
possible values – give
your working to show how
you worked out the values.
__ __
√  √ 7
Key point 13
Q16b hint Use your answers
from part a to write the
answer as an ordinary
number. Then convert this to
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
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
Homework, practice and support: Higher 1.7
_3
8
Irrational
_____
√ 6 . 25
–4
__
– √ 8
___
√ 17
1 . 4̇
___
4
√ ___
49
0.3
Draft, subject to endorsement
17
Unit 1 Number
Unit 1 Number
15 Reasoning / Problem-solving
__
√ 7
1__
___
___
into his calculator. His display shows .
Ben types
7
√ 7
__
√
7
1
__ = ___.
a Show that ___
7
√ 7
Key point 16
__
__
√ m
m ____
__
√ __
n = √ n
9
Simplify
__
__
√ 7
7 ___
__
a
= __ =
4 √ 4
__
√
b
___
√ 9
5
__
c
√ 49
___
√ 12
12 ____
___
= ___ =
√ 49
b Use your calculator to check your answers from Q14.
___
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.
2
Q10 hint
x =
__
x = √  __
x = √
11 Solve these equations, giving your answer as a surd in its simplest form.
a 4x2 = 200
b _12 x2 = 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.
Q13a hint Multiply the
integers together and the
surds together. Simplify.
10
2
1 Problem solving
• Use pictures or lists to help you solve problems.
___ __
is an irrational number. We need to make
the number of the
__ denominator rational.
√
2
__ is the same as multiplying
Multiplying by ___
√ 2
by 1, so this does not change the value.
___
__
First simplify √ 75
= √ 25 √ 3 = 5√ 3
__
__
Circle 3 legs for one stool. Can you use the
remaining 19 legs to make complete chairs? No.
√ 3
√ 9
2 stools
4 chairs
6 stools
Circle another 3 legs for two stools. Can you use the
remaining 16 legs to make complete chairs? Yes.
1 chair
__
√ 3 ___
√ 3 √ 3
5__ = ___
5
1__ × ___
____
___ = ____
__ = __ = ___
√ 3
1 stool
__
√ 2
√ 2
1
1
__ = ____
__ × ___
__
a) ____
√ 2
√ 2
√ 2
__
__
√ 2 ___
√ 2
__ =
= ___
2
√ 4
3
Simplify the fraction before rationalising.
14 Rationalise the denominators. Simplify your answers if possible.
1__
1__
1__
a ___
c ___
b ___
√ 7
√ 5
√ 3
3
32
2__
___
___
f ____
g ____
e ___
√ 15
√ 8
√ 40
18
4
7
3
8
Draw a picture to represent 22 metal legs.
__
5√ 3
c
A furniture maker orders 22 metal legs. He uses the legs to make three-legged stools and
four-legged chairs. Describe the different ways he can use all the legs.
Rationalise the denominator.
1__
a ___
√ 2
5
___
b ____
√ 75
√ 75
1
6
4
7
Example 5
Example 4
___
2
Objective
To rationalise the denominator, the denominator should be a rational number and all the
square roots should have no factors that are square numbers.
√ 75
17 Work out the area of these shapes.
a
b
3
Q16 hint Multiply the numerators
and denominators separately. Then
rationalise the denominator.
Give your answer as a surd in its simplest form.
Key point 17
b)
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.
Draw the diagram again. Are there any other ways
you can circle the legs to make complete chairs and
stools?
He can use 22 legs to make 2 stools and 4 chairs or 6 stools and 1 chair.
1
___
d ____
√ 20
11
___
h ____
√ 11
Write your answer.
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
Draft, subject to endorsement
19
Unit 1 Number
1
2
3
4
5
6
7
Unit 1 Number
There are 20 chairs in a conference room. The conference
Q1 hint Draw a picture or
organiser can put 4, 5 or 6 chairs at a table.
use a list.
a Describe the different 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?
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?
Q2 hint Draw a picture to
help you see what you do
to solve the problem.
When two plant stakes are placed end to end,
Q3 hint Draw one picture to represent
their total length is 1.45 m.
the first sentence. Draw another picture
When the two stakes are placed side by side,
to represent the second sentence.
one is 0.15 m longer than the other.
What lengths are the stakes? Give your answer in cm.
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?
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?
Q4 hint Find numbers
that add to £10.
Q5 Literacy hint A tandem
is for two cyclists.
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
8
Copy and complete.
a 10 = 1000
c 2 = 16
b [(62 – 25) × 3]2
c
Work out
_______
a
9
b 43 =
9 −1
√ ______
10
3
2
2
Write each product as a single power.
b 27 × 9 × 27
a 9–3 × 97
8
2 × 16
e (24)3
d ______
25
10 Work out
a 2–4
b 25 2
11 Simplify
___
a √ 54
b 5√ 1000
3
__________
___
√ √ 81 + (−2) 3
c 57 ÷ 52
f
(42)–1
16 _3
c (___) 4
81
_3
_1
d 16 − 2
_____
Reflect How can you solve Q6 without making a list?
Discussion Does it matter how you solve a maths problem?
14 Write these as ordinary numbers.
a 5.6 × 104
b 1.09 × 10–3
Log how you did on your
Student Progression Chart.
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
___
Estimate the value of √ 54 to the nearest tenth.
a Estimate
_____
i (√ 65 . 1 – 6.17) × 1.98
__________
√ 8 . 19 × 6 . 43
_____
ii ____________
6 . 84 × √ 3 . 97
b Use your calculator to work out each answer.
Give your answers correct to 1 decimal place.
20
Find the highest common factor (HCF) and lowest common multiple (LCM) of 14 and 18.
Homework, practice and support: Higher 1 Check up
Standard form
13 Write these numbers in standard form.
a 32 040 000
b 0.0007
15 Work out
a (5 × 104) × (9 × 107)
b (3 × 108) ÷ (6 × 105)
c (7 × 104)2
16 Reflect How sure are you of your answers? Were you mostly
Just guessing 
Feeling doubtful 
Confident 
What next? Use your results to decide whether to strengthen or extend your learning.
Reflect
3
5
12 Rationalise the denominators. Simplify your answers if possible.
4__
1
___
b ___
a ____
√ 10
√ 8
Calculations, factors and multiples
2
Write 90 as a product of its prime factors in index form.
A tour company offers three different walking tours.
Q7 hint List all the different
The landmark tour leaves every 15 minutes.
times each tour leaves.
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?
1 Check up
1
4
*Challenge
17 The diagram shows a warehouse (W) and five
A
B
40 min
destinations (A, B, C, D, E), and the times it takes
10 min
25 min
to drive between each of them.
20 min
A delivery driver has to deliver packages to
20 min
5 min
C
W
20
min
A, B, C, D and E. He starts and ends at W.
30 min
15 min
a How many different ways could he do this?
b Which is the quickest route?
50 min
E
D
c Write your own delivery driver question.
Draft, subject to endorsement
21
Unit 1 Number
Unit 1 Number
1 Strengthen
Calculations, factors and multiples
1
2
3
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 =
601 154.5 ÷ 94.67 =
38 × 29.4 =
6 011 545 ÷ 94.67 =
380 × 29.4 =
60 115 450 ÷ 94.67 =
3800 × 29.4 =
4
16
a
5
Estimate
__________
i √ 4 . 09 × 8 . 96
v
6
7
9
______
_____
3
√ 26 . 64 + √ 80 . 7
__________
√ 6 . 91 × 9 . 23
c
f
Prime factors
of 18
___
____
√ 103
__________
ii 25.76 – √ 4 . 09 × 8 . 96
3
_____
_____
iv (√ 26 . 64 + √ 80 . 7 ) × 7.54
__________
√ 6 . 91 × 9 . 23
vi ____________
2
3
Q2 hint Write out a
number pattern to
help you.
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.
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=
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
2
4
6
Q7b hint Write all the prime
factors from your tree multiplied
together.
Copy and complete.
a 2×2×2=2 =
b 5×5=5 =
Work out
a 25 =
c 4 = 64
b 53 =
d 10 = 1000
Work out___
a 3 × √ 81
_____
___
3
c √ −27 × √ 64
___
e (–3)2 × √ 49
b 23 × √ 16
___
d 102 ÷ –√ 25
__ 3 ______ 3 _______
f √ 9 × √ −125 × √ −1000
___
Work out
a 5 × (42 – 32) – 23
b [(92 ÷ 32) + 22]2
c 60
– [43 – (82 ÷ 24)]
________________
___
______
___
3
√ 25 + √ −125 + √ 16
d √________
____
3
2 3 − √ −1
e √_________
f
Homework, practice and support: Higher 1 Strengthen
Q9e hint LCM
3
3
3
Q10 hint Use a Venn
diagram as in Q9.
Indices and surds
1
3
60
22
Prime factors
of 45
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.
a Copy and complete this factor tree for 60 until you end up with just prime factors.
b Write 60 as a product of its prime factors.
c Write your answer to part b in index form.
Q9d hint HCF
3
Q4a hint Use your
number line from Q3.
Which___
two square roots
does √52
lie between?
Which is it closer to?
√ 75
3 . 95 ÷ 2 . 03
__________
____ 3 ____
√ 7 . 91 × 9 . 89
____________
√
√
vii ( 50
. 2 – 63 . 9 ) × 5.87
viii
2 . 98 3 ÷ 3 . 21 2
b Use a calculator to work out each answer.
Give your answer correct to 1 decimal place.
7
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.
64
Estimate
the value to the nearest
tenth.
___
___
52
b √___
60
a √___
d √ 38
e √ 14
iii
6
9
Copy and complete this square root number line.
3
5
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
Q1a hint 3.8 = 0.38 × 10
3.8 × 29.4 = 11.172 × 10
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
1
4
8
___
Q4b hint First work
out the round brackets.
Then work out the
square brackets.
c –3 × –3 × –3 = (–3) =
Q2c hint
41 = 4
42 = 4 × 4 =
43 = 4 × 4 × 4 =
Q3 hint First work out
any powers or roots.
Then multiply or divide.
Q4d hint First work out the
calculation underneath the
square root sign.
Then find the square root.
__
√ 2√ 49 + 3√ 8
Draft, subject to endorsement
23
Unit 1 Number
5
Unit 1 Number
12 Copy and complete.
___
___
__
a 50 =
× 2, so √ 50 = √  ___× √ 2___
=
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
b 84 =
Simplify
___
c √ 96
6 × 6 × 6 × 6 62
c __________ = ___ = 6
6×6
6
6
7
8
9
= √
× √
√ 175
b
√ 25
13 Work out
__
__
a √ 4 × √ 4
Work out
a 56 × 53 =
c 58 ÷ 53 =
e 84 × 8–6 =
14 Rationalise the denominator.
Simplify your answer if possible.
3
1
___
___
a ____
b ____
√ 17
√ 21
6
1__
___
d ____
c ___
√ 8
√ 20
b 72 × 79 =
d 98 ÷ 92 =
f 73 ÷ 7–4 =
Q6 hint Use the rules
from Q5.
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
1
a Work out using a calculator
____
_1
i √ 169
ii 169 2
b Use your answer to part a to work these out without a calculator.
_1
_1
_1
_1
i 64 2
ii 25 2
iii 81 2
iv 144 2
c Work____
out using a calculator
_1
3
i √ 512
ii 512 3
d Use your answer to part c to work these out without a calculator.
_1
_1
_1
_1
i 125 3
ii 27 3
iii 1000 3
iv 8 3
e Copy and complete.
___
√ 16 =
10 a Work out
_1
_2
i 64 3
ii 64 3
b Use your answer to Q9d to work out
_2
_2
_2
i 125 3
ii 27 3
iii 1000 3
_3
c Use your answer to Q9e to work out 16 4 .
11 a Copy and complete.
1
1
ii ___5 = 10
i 4 −3 = ___
4
10
_1
36
1
7
iv 3 − 3 = ___
v (__) = ___
3
6
49
b Work out
ii 10–2
i 4–1
____
e
√ 128
c
√ 17
___
___
× √ 17
d
___
√ 21
___
× √ 21
Q14a hint Multiply both the
___
numerator and denominator by √ 17 .
Q14c hint First rewrite the denominator.
__
__
√ 2
√ 8 =
_2
iv 8 3
2
3
Q9e hint Use what
you have learned in
parts a to d.
_3
_1 2
4
_1
power
of 10
c 7.3 million
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
Work out
a (4 × 102) × (2 × 107) =
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)
× 10
d 0.8 × 107
Q2a hint 6.8 lies between 1 and 10.
Multiply by how many 10s to get 68 000?
Q2d hint 4 lies between 1 and 10.
Divide by how many 10s to get 0.000 004?
Q3a hint
4 3 2 3 102 3 107
3
10
Q3d hint 48 = 4.8 × 10
Real A gymnastics floor is a square with side length 1.2 × 103 cm.
Work out the area of the floor.
1 Extend
1
1
__
times
sign
Are these numbers in standard form?
If not, give reasons why.
a 9.004 × 10–3
b 32 × 105
Q10c hint 16 4 = (16 4 )
Use the same strategy
as in part a.
iv 125 − 3
A number written in standard form looks like this.
n
number
between
1 and 10
_2
1
vi a −n = ___
a
_1
___
× √ 25
Q12c hint Write a list of
square numbers up to 100.
Find a square number that
is a factor of 96.
A 3 10
Q10a hint 641 3 = (64 3 )
_
Work out 64 3 and square
your answer.
iii _12 = 2
iii 36 − 2
___
___
√
Standard form
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.
16 =
=
____
d
7×7×7×7×7×7 7
d ________________ = ___ = 7
7×7×7
7
e To multiply powers, _____ the indices. To divide powers, _____ the indices.
_1
4
24
___
× 21, so √ 84
__
√ 2
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?
Draft, subject to endorsement
25
Unit 1 Number
2
3
Show that 272 = 93 = 36.
Unit 1 Number
Q2 communication hint ‘Show that’ means show your working.
5
6
(6 marks)
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
find the HCF and LCM of any two of the numbers.
Q4b hint
48
90
Real A new school is deciding whether their lessons should
be 30, 50 or 60 minutes.
150
Each length of lesson fits exactly into the total teaching time
Put prime factors of all three
of the school day.
in the very centre first.
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.
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
8
9
Write each of these as a simplified 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
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.
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 fixed, 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 difference in their mass? Write your answer in standard form.
26
Exam-style question
Work out
8
7
3__
5__ ____
___ – ___
+ ___
b ____
a ___
√ 2
√ 32
√ 72 √ 8
__
Write each answer in the form a√ 2 .
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.
4
11
(3 marks)
12 A square has an area of _12 cm2.
a Work out the length of one side.
b Work out the perimeter.
Give your answers as surds in their simplest forms.
_1
13 Write 3 − 2 as a surd and rationalise the denominator.
14
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.
15 What is the value of x?
1
a 3x = ___
27
1
b 2 × 2x = __
8
(3 marks)
1
c 32x = __
9
1 Knowledge check
◉ When there are m ways of doing one task and n ways of doing a different
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 significant figures 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.
m
1
__
__
__
1
n __
x n = n√ x
x n = ( √ x )m
Mastery lesson 1.4, 1.5
x–n = ___n
x
◉ 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
Draft, subject to endorsement
27
Unit 1 Number
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
14
◉ 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
1
2
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.
4
(3 marks)
8__
15 Rationalise the denominator. ___
√ 6
(2 marks)
16 Let x = 6 × 105 and y = 8 × 104. Work out
a x+y
b x–y
c xy
Write your answers in standard form.
(2 marks)
The maths is correct, but the student will only get 2 marks. Why?
(3 marks)
Exam-style question
One sheet of paper is 9 × 10−3 cm thick.
List these numbers in order, starting with the smallest.
Show your working.
Mark wants to put 500 sheets of paper into the paper tray of his printer.
___
√ 27
3
___
√ 69
13.74
The paper tray is 4 cm deep.
(3 marks)
Is the paper tray deep enough for 500 sheets of paper?
______
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 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)
Find the value of a.
a 53 × 5a = 59
(3 marks)
b 6a ÷ 6–5 = 68
8
Write (38)4 as a single power.
9
Write each number in standard form.
a 0.000 000 65
b 0.9 million
y
d __
x
Sample student answer
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.
7
How many different 4-digit odd numbers are there, where
the first digit is not zero?
Exam-style question
3.22
3
Log how you did on your
Student Progression Chart.
Exam-style question
c 8a × 8a = 84
You must explain your answer.
(3 marks)
June 2013, Q15, 1MA0/1H
9 × 10−3 × 500
0.009 × 500
9 × 500 = 4500, ÷1000 = 4.5
(1 mark)
c 320 × 107
10 918 = 27x
Work out the value of x.
(3 marks)
(2 marks)
11 Use prime factors to determine whether 2520 is divisible by 18.
4 −2
12 Write (__) as a fraction in its simplest form.
3
13 Work out the area of this shape.
45
Write your answer as a simplified surd.
(2 marks)
(2 marks)
30
(3 marks)
28
Homework, practice and support: Higher 1 Unit test
Draft, subject to endorsement
29
T972a
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