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Cambridge Online Mathematics, giving you access to auto-marked practice
questions and step-by-step walkthroughs.
• Understand what you need to know with the ‘Getting started’ feature
• Develop your ability to think and work mathematically with clearly identified
activities throughout each unit
• ‘Think like a mathematician’ provides investigation activities linked to the skills
you are developing
• ‘Summary checklist’ in each section and ‘Check your progress’ exercise at the
end of each unit help you reflect on what you have learnt
• Answers for all activities can be found in the accompanying teacher’s resource
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(0862) curriculum framework from 2020
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Mathematics
LEARNER’S BOOK 8
Lynn Byrd, Greg Byrd & Chris Pearce
LEARNER’S BOOK 8
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Cambridge Lower Secondary
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9781108771528 Byrd, Byrd and Pierce Lower Secondary Mathematics Learner’s Book 8 CVR C M Y K
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LEARNER’S BOOK 8
Greg Byrd, Lynn Byrd & Chris Pearce
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Introduction
Introduction
Welcome to Cambridge Lower Secondary Mathematics Stage 8
The Cambridge Lower Secondary Mathematics course covers the Cambridge
Lower Secondary Mathematics curriculum framework and is divided into three
stages: 7, 8 and 9.
During your course, you will learn a lot of facts, information and techniques.
You will start to think like a mathematician. This book covers all you need to
know for Stage 8.
The curriculum is presented in four content areas:
• Number
• Algebra
• Geometry and measures
• Statistics and probability.
This book has 16 units, each related to one of the four content areas.
However, there are no clear dividing lines between these areas of
mathematics; skills learned in one unit are often used in other units.
The book encourages you to understand the concepts that you need
to learn, and gives opportunity for you to practise the necessary skills.
Many of the questions and activities are marked with an icon that
indicates that they are designed to develop certain thinking and
working mathematically skills.
There are eight characteristics that you will develop and apply
throughout the course:
• Specialising – testing ideas against specific criteria;
• Generalising – recognising wider patterns;
• Conjecturing – forming questions or ideas about mathematics;
• Convincing – presenting evidence to justify or challenge a
mathematical idea;
• Characterising – identifying and describing properties of
mathematical objects;
• Classifying – organising mathematical objects into groups;
• Critiquing – comparing and evaluating ideas for solutions;
• Improving – Refining your mathematical ideas to reach
more effective approaches or solutions.
Your teacher can help you develop these skills, and you will also
develop your ability to apply these different strategies.
We hope you will find your learning interesting and enjoyable.
Greg Byrd, Lynn Byrd and Chris Pearce
3
Contents
Page
Unit
6
How to use this book
9–28
1 Integers
1.1 Factors, multiples and primes
1.2 Multiplying and dividing integers
1.3 Square roots and cube roots
1.4 Indices
Number
29–64
2 Expressions, formulae and equations
2.1 Constructing expressions
2.2 Using expressions and formulae
2.3 Expanding brackets
2.4 Factorising
2.5 Constructing and solving equations
2.6 Inequalities
Algebra
65
Project 1 Algebra chains
66–79
3 Place value and rounding
3.1 Multiplying and dividing by 0.1 and 0.01
3.2 Rounding
Number
80–103
4 Decimals
4.1 Ordering decimals
4.2 Multiplying decimals
4.3 Dividing by decimals
4.4 Making decimal calculations easier
Number
104
Project 2 Diamond decimals
105–125
5 Angles and constructions
5.1 Parallel lines
5.2 The exterior angle of a triangle
5.3 Constructions
Geometry and measure
126–136
6 Collecting data
6.1 Data collection
6.2 Sampling
Statistics
137–170
7 Fractions
7.1 Fractions and recurring decimals
7.2 Ordering fractions
7.3 Subtracting mixed numbers
7.4 Multiplying an integer by a mixed number
7.5 Dividing an integer by a fraction
7.6 Making fraction calculations easier
Number
171–196
8 Shapes and symmetry
8.1 Quadrilaterals and polygons
8.2 The circumference of a circle
8.3 3D shapes
Geometry and measure
197
Project 3 Quadrilateral tiling
198–223
9 Sequences and functions
9.1 Generating sequences
9.2 Finding rules for sequences
9.3 Using the nth term
9.4 Representing simple functions
4
Strand of mathematics
Algebra
Contents
Page
Unit
Strand of mathematics
224–234
10 Percentages
10.1 Percentage increases and decreases
10.2 Using a multiplier
Number
235–255
11 Graphs
11.1 Functions
11.2 Plotting graphs
11.3 Gradient and intercept
11.4 Interpreting graphs
Algebra; Statistics and probability
256
Project 4 Straight line mix-up
257–274
12 Ratio and proportion
12.1 Simplifying ratios
12.2 Sharing in a ratio
12.3 Ratio and direct proportion
Number
275–288
13 Probability
13.1 Calculating probabilities
13.2 Experimental and theoretical probabilities
Statistics and probability
289
Project 5 High fives
290–330
14 Position and transformation
14.1 Bearings
14.2 The midpoint of a line segment
14.3 Translating 2D shapes
14.4 Reflecting shapes
14.5 Rotating shapes
14.6 Enlarging shapes
Statistics and probability
331–351
15 Distance, area and volume
15.1 Converting between miles and kilometres
15.2 The area of a parallelogram and a trapezium
15.3 Calculating the volume of triangular prisms
15.4 Calculating the surface area of triangular prisms and pyramids
Geometry and measure
352
Project 6 Biggest cuboid
353–387
16 Interpreting and discussing results
16.1 Interpreting and drawing frequency diagrams
16.2 Time series graphs
16.3 Stem-and-leaf diagrams
16.4 Pie charts
16.5 Representing data
16.6 Using statistics
388–394
Glossary and Index
Statistics and probability
5
How to use this book
How to use this book
In this book you will find lots of different features to help your learning.
Questions to find out what you
know already.
What you will learn in the unit.
Important words to learn.
Step-by-step examples showing
how to solve a problem.
These questions will help you
develop your skills of thinking
and working mathematically.
6
How to use this book
These investigations, to be carried
out with a partner or in a group,
will help develop skills of thinking
and working mathematically.
Questions to help you think
about how you learn.
This is what you have
learned in the unit.
Questions that cover what you
have learned in the unit.
At the end of several units,
there is a project for you to
carry out, using what you
have learned. You might make
something or solve a problem.
7
Acknowledgements
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and are grateful for the permissions granted. While every effort has been made, it has
not always been possible to identify the sources of all the material used, or to trace all
copyright holders. If any omissions are brought to our notice, we will be happy to include
the appropriate acknowledgements on reprinting.
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8
1
Integers
Getting started
1
2
3
4
5
a
Find all the prime numbers less than 20.
b Show that there are two prime numbers between 20 and 30.
a
Find all the factors of 18.
b Find all the 2-digit multiples of 18.
c
Find the highest common factor of 18 and 12.
d Find the lowest common multiple of 18 and 12.
Work out
−6 + 3
a
b −6 − 3
c
−6 × 3
d −6 ÷ 3
e 8 + −10
f
−5 − − 9
Write whether each of these numbers is a square number, a cube number or both.
a
49
b 27
c
1000
d 64
e
121
f
225
Find
3
125
a
100
b
c
152 − 122
Prime numbers have exactly two factors, 1 and the number itself.
Some examples of prime numbers are 7, 31, 83, 239 and 953.
The number 39 is the product of two prime numbers (3 and 13).
It is quite easy to find these two numbers.
The number 2573 is also the product of two prime numbers (31 and 83).
It is much harder to find the two numbers in this case.
It is easy to multiply two prime numbers together using a calculator
or a computer.
9
1 Integers
It is much harder to carry out the inverse operation – that is, to find the
two prime numbers that multiply to a given product. This fact is the
basis of a system used to encode messages sent across the internet.
The RSA cryptosystem was invented by Ronald Rivest, Adi Shamir
and Leonard Adleman in 1977. It uses two
large prime numbers with about 150 digits
each. These numbers are kept secret, but
anybody can use their product, N, which
has about 300 digits.
If someone sends their credit card number
to a website, their computer does a
calculation using N to encode their credit
card number. The computer that receives
the coded number does another calculation
to decode it. Anyone who does not know
the two factors of N will not be able to do
this. Your credit card number is protected.
10
1.1 Factors, multiples and primes
1.1 Factors, multiples and primes
In this section you will …
Key words
•
write a positive integer as a product of prime factors
factor tree
•
use prime factors to find a highest common factor (HCF)
and a lowest common multiple (LCM).
highest common
factor (HCF)
index
Any integer bigger than 1:
is a prime number, or
•
can be written as a product of prime numbers.
•
Example:
46 = 2 × 23 47 is prime 48 = 2 × 2 × 2 × 2 × 3 49 = 7 × 7 50 = 2 × 5 × 5
You can use a factor tree to write an integer as a product of its
prime factors.
This is how to draw a factor tree for 120.
1
Write 120.
2
Draw branches to two numbers that have a product of 120. Do
not use 1 as one of the numbers. Here we have chosen 12 and 10.
120 = 12 × 10
3
Do the same with 12 and 10. Here 12 = 3 × 4 and 10 = 2 × 5
4
3, 2 and 5 are prime numbers, so circle them.
5
Draw two more branches from 4. 4 = 2 × 2. Circle the 2s.
6
Now all the end numbers are prime, so stop.
7
120 is the product of all the end numbers: 120 = 2 × 2 × 2 × 3 × 5
8
You can check that this is correct using a calculator.
You can also write the result like this: 120 = 23 × 3 × 5
23 means 2 × 2 × 2 and the small 3 is an index.
Now check that 75 = 3 × 52
You can use products of prime factors to find the HCF and LCM of two
numbers.
integer
lowest common
multiple (LCM)
prime factor
120
12
3
10
4
2
2
5
2
11
1 Integers
Worked example 1.1
a Find the LCM of 120 and 75.
b Find the HCF of 120 and 75.
Answer
a
Write 120 and 75 as products of their prime factors:
120 = 2 × 2 × 2 × 3 × 5
75 = 3 × 5 × 5
Look at the prime factors of both numbers.
For the LCM, use the larger frequency of each prime factor.
• 120 has three 2s and 75 has no 2s. The LCM must have three 2s.
• 120 has one 3 and 75 has one 3. The LCM must have one 3.
• 120 has one 5 and 75 has two 5s. The LCM must have two 5s.
The LCM is 2 × 2 × 2 × 3 × 5 × 5 = 23 × 3 × 52 = 8 × 3 × 25 = 600
b For the HCF use the smaller frequency of each factor: there are no 2s in 75, and there is one
3 and one 5 in both numbers.
Multiply these factors.
The HCF is 3 × 5 = 15
Exercise 1.1
Think like a mathematician
1
The factor tree for 120 in Section 1.1 started with 12 × 10.
a
b
c
d
2
12
a
b
c
d
120
Draw a factor tree for 120 that starts with 6 × 20.
Compare your answer to part a with a partner’s. Are your
trees the same or different?
Draw some different factor trees for 120. Can you say
6
how many different trees are possible?
Do all factor trees for 120 have the same end points?
Complete this factor tree for 108.
Draw a different factor tree for 108.
Write 108 as a product of its prime factors.
Compare your factor trees and your product of prime factors
with a partner’s. Have you drawn the same trees or different
ones? Are your trees correct?
20
108
2
54
1.1 Factors, multiples and primes
3
4
5
6
7
8
a
b
c
Draw a factor tree for 200 that starts with 10 × 20.
Write 200 as a product of prime numbers.
Compare your factor tree with a partner’s. Have you drawn
the same tree or different ones? Are your trees correct?
d How many different factor trees can you draw for 200 that start
with 10 × 20?
a
Draw a factor tree for 330.
b Write 330 as a product of prime numbers.
Match each number to a product of prime factors.
The first one has been done for you: a and i.
a
20
i
2² × 5
b 24
ii
2×3×7
c
42
iii 2² × 3² × 5
d 50
iv 2 × 5²
e
180
v
2³ × 3
Work out the product of each set of prime factors.
32 × 5 × 7
a
b 23 × 53
c
2 2 × 32 × 11
d 2 4 × 72
e 3 × 172
Write each of these numbers as a product of prime factors.
a
28
b 60
c
72
d 153
e
190
f
275
aCopy the table and write each number as a product of prime
numbers.
Number
35
70
140
280
b
9
a
b
c
10 a
b
c
Tip
You can use a
factor tree to help
you.
Product of prime numbers
5×7
Add more rows to the table to continue the pattern.
Write 1001 as a product of prime numbers.
Write 4004 as a product of prime numbers.
Write 6006 as a product of prime numbers.
Use a factor tree to write 132 as a product of prime numbers.
Write 150 as a product of prime numbers.
132 × 150 = 19 800. Use this fact to write 19 800 as a product
of prime numbers.
13
1 Integers
11 a
12
13
14
15
16
17
18
19
20
14
Write each of these numbers as a product of prime numbers.
i 15
ii 15²
iii 28
iv 28²
v 36
vi 36²
b What do you notice about your answers to i and ii, iii and iv,
v and vi?
c
If 96 = 25 × 3, show how to find the prime factors of 96 2 .
Will your method work for all numbers?
40 = 2 × 2 × 2 × 5 and 28 = 2 × 2 × 7
Use these facts to find
a
the HCF of 40 and 28
b the LCM of 40 and 28.
450 = 2 × 3 × 3 × 5 × 5 and 60 = 2 × 2 × 3 × 5
Use these facts to find
a
the HCF of 450 and 60
b the LCM of 450 and 60.
180 = 2² × 3² × 5 and 54 = 2 × 3³
Use these facts to find
a
the HCF of 180 and 54
b the LCM of 180 and 54.
a
Write 45 as a product of prime numbers.
b Write 75 as a product of prime numbers.
c
Find the LCM of 45 and 75.
d Find the HCF of 45 and 75.
a
Draw factor trees to find the LCM of 90 and 140.
b Compare your answer with a partner’s. Did you draw the same
factor trees? Have you both got the same answer?
a
Write 396 as a product of prime numbers.
b Write 168 as a product of prime numbers.
c
Find the HCF of 396 and 168.
d Find the LCM of 396 and 168.
a
Find the HCF of 34 and 58.
b Find the LCM of 34 and 58.
Show that the HCF of 63 and 110 is 1.
37 and 47 are prime numbers.
a
What is the HCF of 37 and 47?
b What is the LCM of 37 and 47?
c
Write a rule for finding the HCF and LCM of two prime
numbers.
d Compare your answer to part c with a partner’s answer.
Check your rules by finding the HCF and LCM of 39 and 83.
Tip
Use a calculator
to help you.
1.2 Multiplying and dividing integers
In this exercise you have:
•
used factor trees to write an integer as a product of prime factors
•
found the HCF of two integers by first writing each one
as a product of prime numbers
•
found the LCM of two integers by first writing each one
as a product of prime numbers.
a
Which questions have you found the easiest? Explain why.
b
Which questions have you found the hardest? Explain why.
Summary checklist
I can write an integer as a product of prime numbers.
I can find the HCF and LCM of two integers by first writing each one
as a product of prime numbers.
1.2 Multiplying and
dividing integers
In this section you will …
Key words
•
multiply and divide integers, in particular when both are
negative
brackets
understand that brackets, indices and operations follow a
particular order.
inverse
•
conjecture
investigate
You can add and subtract any two integers.
For example:
2 + −4 = −2
−2 + −4 = −6
−2 − 4 = −6
−2 − −4 = 2
You can also multiply and divide a negative integer by a positive one.
For example:
2 × −9 = −18
−6 × 3 = −18
−18 ÷ 3 = −6
20 ÷ −5 = −4
In this section you will investigate how to multiply or divide any two
integers. You will use number patterns to do this.
15
1 Integers
Worked example 1.2
Look at this sequence of subtractions.
3 − 6 = −3
3 − 4 = −1
3−2 =
3−0 =
3 − −2 =
3 − −4 =
a Copy the sequence and fill in the missing answers.
b Write the next three lines in the sequence.
c Describe any patterns in the sequence.
A sequence is a set of
numbers or expressions
made and written in
order, according to some
pattern.
Answer
a 3−2 =1
3−0 = 3
3 − −2 = 5
3 − −4 = 7
b 3 − −6 = 9
3 − −8 = 11
3 − −10 = 13
c The first number, 3, does not change.
The number being subtracted decreases by 2 each time.
The answer increases by 2 each time.
Exercise 1.2
Think like a mathematician
1
Here is the start of a sequence of multiplications.
−3 × 4 = −12
−3 × 3 =
−3 × 2 =
a
b
16
Copy the sequence and write six more terms. Use a pattern to fill in the answers.
Describe the patterns in the sequence.
1.2 Multiplying and dividing integers
Continued
c
d
e
f
2
3
4
Work out these multiplications.
a
5× −2
b −5 × 2
Work out these multiplications.
a
−6 × − 4
b −7 × − 7
6
7
c
−5 × − 2
d
−2 × − 5
c
−10 × −6
d
−8 × −11
Copy and complete this multiplication table.
×
4
−3
−6
5
Here is the start of another sequence of multiplications.
−5 × 4 =
−5 × 3 =
−5 × 2 =
Copy the sequence and write six more terms.
Describe any patterns in the sequence.
In the sequences in a and c, you have some products of two negative integers.
What can you say about the product of two negative integers?
Make up a sequence of your own like the ones in a and c.
Share your answers to parts d and e with a partner. Are your partner’s sequences
correct?
−5
3
−8
−9
30
Work out
b (−3 + −5) × −6
a
(3 + 5) × −4
c
−4 × (5 − 8)
d −6 × (−2 − −7)
Round these numbers to the nearest whole number to estimate
the answer.
b −11.2 × 2.95
3.9 × −6.8
a
2
c
(−6.1)
d (−4.88)2
a
Put these multiplications into groups based on the answers.
3 × −4 −6 × −2 12 × 1
−4 × −3 2 × −6 −12 × −1
b Find one more product to put in each group.
Tip
Do the calculation
in brackets first.
17
1 Integers
8
These are multiplication pyramids.
a
b
c
–8
2
–4
–3
–3
5
–1
–4
–5
–2
Each number is the product of the two numbers below it. For example,
in a, 2 × −4 = −8
Copy and complete the multiplication pyramids.
aDraw a multiplication pyramid like those in Question 8, with
the integers −2, 3 and −5 in the bottom row, in that order.
Complete your pyramid.
9
If you change the order
of the bottom numbers,
the number at the top of the
pyramid is the same.
b
Is Zara correct? Test her idea by changing the order of the
numbers in the bottom row of your pyramid.
10 Find the missing numbers in these multiplications.
a
−3 ×
= −12
b −5 ×
= 45
c
× −6 = 24
d
× −10 = 80
Think like a mathematician
Tip
11 A multiplication can be written as a division.
For example, 5 × 8 = 40 can be written as 40 ÷ 8 = 5 or 40 ÷ 5 = 8
A conjecture is
a possible value
based on what
you know.
a
b
c
d
18
Here is a multiplication: −4 × 6 = −24
Write it as a division in two different ways.
Write a multiplication of a positive integer and a
negative integer.
Then write it as a division in two different ways.
Here is a multiplication: −7 × −2 = 14
Write it as a division in two different ways.
Write a multiplication of two negative integers.
Then write it as a division in two different ways.
1.2 Multiplying and dividing integers
Continued
e
Can you make a conjecture about the answer when
you divide an integer by a negative integer?
Test your conjecture.
Compare your answer with a partner’s answers.
Have you made the same conjectures?
f
12 Work out these divisions.
a
18 ÷ −6
b −28 ÷ −4
d −30 ÷ −10
e
42 ÷ −6
g 60 ÷ −5
h
−25 ÷ −5
13 Here are three multiplication pyramids.
a
b
6
5
14
15
16
17
–1
12
c
f
30 ÷ −6
−24 ÷ −4
c
–200
–8
Tip
–20
–2
Copy and complete each pyramid.
Work out
a
(3 × −4) ÷ −2
b (2 − 20) ÷ −3
c
(−3 + 15) ÷ −4
d 24 ÷ (2 × −4)
Find the value of x.
a
x ÷ −4 = 8
b x ÷ −3 = −15
c
16 ÷ x = −2
d −15 ÷ x = 3
Round these numbers to the nearest whole number to estimate
the answer.
a
−8.75 ÷ 2.8
b 18.1 ÷ −5.9
c
−28.2 ÷ −3.8
d −35.2 ÷ −6.9
Round these numbers to the nearest 10 to estimate the answer.
a
−48 × −29
b −18.1 × 61.5
c
−71.4 ÷ −11.8
d −99.4 ÷ 19
–4
Remember,
division is the
inverse of
multiplication so
you will divide as
you work down
the pyramid.
Summary checklist
I can multiply two negative integers.
I can divide any integer by a negative integer.
19
1 Integers
1.3 Square roots and cube roots
In this section you will …
Key words
•
find the squares of positive and negative integers and their
corresponding square roots
cube root
find the cubes of positive and negative integers and their
corresponding cube roots
rational numbers
•
•
square root
learn to recognise natural numbers, integers and rational
numbers.
52 = 25
This means that the square root of 25 is 5. This can be written as 25 = 5.
This is the only answer in the set of natural numbers.
However (−5)2 = −5 × −5 = 25
This means that the integer −5 is also a square root of 25.
Every positive integer has two square roots, one positive and
one negative.
5 is the positive square root of 25 and −5 is the negative square root.
No negative number has a square root.
For example, the integer −25 has no square root because the equation
x2 = −25 has no solution.
53 = 125
This means that the cube root of 125 is 5. This can be written as 3 125 = 5.
You might think −5 is also a cube root of 125.
However (−5)3 = −5 × −5 × −5 = (−5 × −5) × −5 = 25 × −5 = −125
So 3 −125 = −5
Every number, positive or negative or zero, has only one cube root.
Worked example 1.3
Solve each equation.
a x2 = 64
b x3 = 64
c x3 + 64 = 0
20
natural numbers
Tip
The natural
numbers are the
counting numbers
and zero.
1.3 Square roots and cube roots
Continued
Answer
64 has two square roots. One is 64 = 8 and the other is − 64 = −8
So the equation has two solutions: x = 8 or x = −8
b 3 64 = 4. This means 43 = 4 × 4 × 4 = 64 and so x = 4
c If x3 + 64 = 0 then x3 = −64. So x = 3 −64 = −4
a
Exercise 1.3
1
2
3
4
5
6
7
8
9
Work out
a
72
b (−7)2
c
73
Find
3
3
3
−1
−27
a
125
b
c
Solve these equations.
a
x2 = 100
b x2 = 144
c
2
2
d x =0
e
x +9=0
Solve these equations.
a
x3 = 216
b x3 + 27 = 0
c
x3 + 1 = 0
d x3 + 125 = 0
272 = 93 = 729
Use this fact to find
3
729
−729
729
a
b
c
d
(−7)3
d
3
−8
3
−729
x2 = 1
d
A calculator shows that 82 − ( −8)2 = 0
Explain why this is correct.
b Find the value of 3 43 − 3 ( −4 )3 . Show your working.
The square of an integer is 100.
What can you say about the cube of the integer?
The integer 1521 = 32 × 132
Use this fact to
a
find 1521
b solve the equation x2 = 1521
a
How is −52 different from (−5)2?
b What is the difference between −53 and (−5)3?
a
21
1 Integers
10 a
b
Show that 32 + 42 = 52
Are these statements true or false?
Give a reason for your answer each time.
i (−3)2 + (−4)2 = (−5)2
ii (−13)2 = 122 + (−5)2
iii 82 = −102 − 62
Show your work to a partner.
Do they find your explanation clear?
c
Think like a mathematician
11 a
Here is an equation: x2 + x = 6
i Show that x = 2 is a solution of the equation.
ii Show that x = −3 is a solution of the equation.
b
Here is another equation: x2 + x = 12
i
ii
c
d
e
12 a
Find two solutions to this equation: x2 + x = 20
What patterns can you see in the answers to a, b and c?
Find some more equations like this and write down the solutions.
Compare your answers with a partner’s.
Copy and complete this table.
x
2
3
4
5
b
c
d
e
22
Show that x = 3 is a solution of the equation.
Find a second solution to the equation.
x−1
1
2
x3 − 1
7
x2 + x + 1
13
What pattern can you see in your answers?
Add another row to see if the pattern is still the same.
Add three rows where x is a negative integer.
Is the pattern still the same if x is a negative integer?
Compare your answers with a partner’s.
1.3 Square roots and cube roots
13 Any number that can be written as a fraction is a rational number.
Examples are 7 3 , −12 18 , 6, 1 , −2 9
4
25
10
15
Here is a list of six numbers:
5 − 1 −500 16 −4.8 99 1
5
2
Write
a
all the integers in the list
b all the natural numbers in the list
c
all the rational numbers in the list.
14 This Venn diagram shows the relationship
I
between natural numbers and integers.
N
N stands for natural numbers and I for
integers.
a
Copy the Venn diagram.
b Write each of these numbers in the
correct part of the diagram.
1 −3 7 −12 41 −100 2 1
2
c
Add another circle to your Venn diagram to show
rational numbers.
d Add these numbers to your Venn diagram.
−8 3 3 0 6.3 − 10
e
7
5
3
Give your diagram to a partner to check.
Tip
Integers and
fractions are
included in the
set of rational
numbers.
Tip
Remember, all
integers are
included in the
rational numbers.
Summary checklist
I can find and recognise square numbers and their two corresponding square roots.
I can find and recognise positive and negative cube numbers and their cube roots.
I can recognise natural numbers, integers and rational numbers.
23
1 Integers
1.4 Indices
In this section you will …
Key words
•
generalise
use positive and zero indices to represent numbers and in
multiplication and division.
In this section you will investigate numbers written as powers.
Look at these powers of 5
n
5n
0
1
2
25
3
125
4
625
5
3125
So 53 = 5 × 5 × 5 = 125 and 54 = 5 × 5 × 5 × 5 = 625 and so on.
As you move to the right the numbers in the bottom row multiply by 5.
As you move to the left the numbers in the bottom row divide by 5.
3125 ÷ 5 = 625, 625 ÷ 5 = 125, 125 ÷ 5 = 25
If you continue to divide by 5, 25 ÷ 5 = 5 so 51 = 5
There is another number missing in the table. What is 50?
Divide by 5 again: 50 = 51 ÷ 5 = 5 ÷ 5 = 1
So 50 = 1
If n is any positive integer then n0 = 1.
Worked example 1.4
a Show that 73 = 343
b Work out
4
i 7
ii 70
Answer
a 73 = 7 × 7 × 7 = 49 × 7 = 343
b i 7 4 = 73 × 7 = 343 × 7 = 2401
ii 70 = 1
24
power
1.4 Indices
Exercise 1.4
1
Copy and complete this list of powers of 2.
Power
Number
2
20
1
21
2
22
23
8
24
25
26
64
34
35
27
28
29
512
210
Copy and complete this list of powers of 3.
Power
Number
30
31
3
32
33
27
36
37
2187
38
Think like a mathematician
3
Look at this multiplication: 4 × 16 = 64
You can write all the numbers as powers of 2: 22 × 24 = 26
a
b
c
d
4
5
6
7
Write each of these multiplications as powers of 2.
i 8 × 4 = 32
ii 16 × 8 = 128
iii 4 × 32 = 128
iv 2 × 128 = 256
v 16 × 32 = 512
Can you see a pattern in your answers? Make a
conjecture about multiplying powers of 2.
Test your conjecture on some more multiplications of
your own.
Make a conjecture about multiplying powers of 3.
Use some examples to test your conjecture.
Generalise your results so far.
Tip
‘Generalising’
means using a set
of results to come
up with a general
rule.
Write the answers to these calculations as powers of 6.
a
6 2 × 63
b 64 × 6
c
65 × 6 2
d 63 × 63
Write the answers to these calculations in index form.
a 103 × 102
b 205 × 20
c
153 × 153
d 55 × 53
a
38 = 6561
Use this fact to find 39 and show your method.
b 56 = 15 625 Use this fact to find 57 and show your method.
Find the missing power.
a
33 × 3
c
124 × 12
= 35
= 126
b
93 × 9
= 98
d
15
× 153 = 1510
25
1 Integers
8
Read what Sofia says.
42 is equal to 24 and 43
is equal to 34
Is Sofia correct? Give a reason for your answer.
9
A million is 106. A billion is 1000 million.
Write as a power of 10
a
one billion
b 1000 billion
10 Write in index form
a
22 × 23 × 2
b 33 × 34 × 32
c
5 × 53 × 53
d 103 × 102 × 104
11 a
(32)3 = 32 × 32 × 32 Write (32)3 as a single power of 3.
b Write in index form
i (23)2
ii (53)2
iii (42)3
iv (152)4
v (104)3
c
N is a positive integer. Write in index form
i ( N 2 )3
ii ( N 4 )2
iii ( N 5 )3
d Can you generalise the results of part c?
Think like a mathematician
12 Here is a division:
You can write this using indices:
a
b
c
d
32 ÷ 4 = 8
25 ÷ 22 = 23
Write each of these divisions using indices. All the numbers are powers of 2 or 3.
i 64 ÷ 4 = 16
ii 81÷ 3 = 27
iii 512 ÷ 16 = 32
iv 729 ÷ 9 = 81
v 9÷9 =1
Write some similar divisions using powers of 5.
Can you generalise your results from a and b?
Check with some powers of other positive integers.
Compare your results with a partner’s.
13 Write the answers to these calculations in index form.
a
2 7 ÷ 25
b 106 ÷ 103
c
10
9
15
11
d 8 ÷8
e
f
2 ÷2
14 Write the answers to these calculations in index form.
a
b 95 ÷ 92
c
(95)2
95 × 92
e 128 ÷ 123
f
(73)3
g (100 )4
26
158 ÷ 156
25 ÷ 25
d
55 × 54
1.4 Indices
15 Read what Zara says.
I think that ( 52 ) = ( 53)
3
2
a
Is Zara correct? Give a reason for your answer.
b Is a similar result true for other indices?
16 15 = 3 × 5
Use this fact to write as a product of prime factors
a
152
b 153
c
155
17 a
Write 56 ÷ 54 as a power of 5.
b Write 56 ÷ 56 as a power of 5.
c
Is it possible to write 54 ÷ 56 as a power of 5?
d
158
Summary checklist
I can use index notation for positive integers where the index is a positive integer
or zero.
I can multiply and divide numbers written as powers of a positive integer.
27
1 Integers
Check your progress
1
2
a
Draw a factor tree for 350.
b Write 350 as a product of prime factors.
c
Write 112 as a product of prime factors.
d Find the HCF of 350 and 112.
e
Find the LCM of 350 and 112.
Copy and complete this multiplication table.
×
−6
−10
3 −18
−7
3
4
5
6
7
8
9
28
−5
7
Are these calculations correct? If not, correct them.
a
b −9 × −11 = −99
( −5)2 = −25
3
c
d ( −10 ) = −1000
45 ÷ −9 = −6
Work out
a
b −36 ÷ −6
40 ÷ −5
c
d (12 − −18) ÷ −3
100 ÷ ( 2 – 7 )
Solve these equations.
a
x 2 = 36
b x 2 + 16 = 0
c
x3 = 8
d x3 + 27 = 0
Work out
a
( −5 )2 − ( −4 )2
Here is an expression: x3 + x 2
Find the value of the expression when
a
x=3
Write as a single power of 8
a
82 × 83
c
1
a
Write 46 as a power of 2.
b Write 94 as a power of 3.
64 + 3 −64
b
3
b
x = −3
b
d
86 ÷ 82
(83)3
2
Expressions,
formulae and equations
Getting started
1
2
3
4
5
6
Alex thinks of a number, n.
Write an expression for the number Alex gets when
a
she multiplies the number by 2
b she adds 5 to the number.
Work out the value of p − q when p = 15 and q = 3
Simplify these expressions by collecting like terms.
a
3c + 4c + 9d − 2d
b
Expand the brackets.
a
4(x + 3)
b
Solve these equations.
a
n + 12 = 15
b
c
3p = 27
d
Write the inequality shown by this number line.
4xy + 7yz − 2xy + zy
6(2 − 3y)
m−7=2
2r + 7 = 19
1
2
3
4
5
6
29
2 Expressions, formulae and equations
A formula is a set of instructions for working something out.
It is a rule written using letters or words.
The plural of formula is formulae.
People use formulae in everyday life to work out all sorts of things.
An employer may use a formula to work out how much to pay the
people who work for them. For example, they could use the formula
P = R × H, where P is the pay, R is the amount paid per hour and
H is the number of hours worked.
Doctors may use a formula to assess a person’s health. For example,
they could use a formula to find the person’s body mass index (BMI).
This formula is: BMI = mass 2 , where the person’s mass is measured
height
in kilograms and their height is measured in metres.
If a person’s BMI is too high or too low, the doctor may ask them
to lose or put on weight, to make them healthier.
2.1 Constructing expressions
In this section you will …
Key words
•
use letters to represent numbers
coefficient
•
use the correct order of operations in algebraic expressions
constant
•
use words or letters to represent a situation.
equivalent
You can write an algebraic expression by using a letter to represent an
unknown number.
In the expression 3n + 8 there are two terms. 3n is one term. The other
term is 8.
The letter n is called the variable, because it can have different values.
The coefficient of n is 3, because it is the number that multiplies the
variable.
The number 8 is called a constant.
Example:
Let n represent a mystery number.
You write the number that is 5 more than the mystery number as n + 5
or 5 + n.
You write the number that is three times the mystery number
as 3 × n or simply 3n.
30
expression
linear expression
term
unknown
variable
2.1 Constructing expressions
ou write the mystery number multiplied by itself as n × n or simply n2.
Y
n + 5 and 3n are called linear expressions because the variable is only
multiplied by a number.
n2 is not a linear expression because the variable is multiplied by itself.
Tip
n + 5 is the same
as 1 × n + 5.
Worked example 2.1
Tyler thinks of a number, x. Write an expression for the number Tyler gets when he
a doubles the number and subtracts 3
b divides the number by 3 and adds 2
c adds 2 to the number, then multiplies by 4.
Answer
a
2x − 3
Multiply x by 2, then subtract 3. Write 2 × x as 2x.
b
x
+2
3
Divide x by 3, then add 2. Write x ÷ 3 as x .
c
4(x + 2)
Add 2 to x, then multiply the answer by 4. Write x + 2 in brackets
to show this must be done before multiplying by 4.
3
Exercise 2.1
1
2
Copy and complete these sentences. Use the words from the cloud.
In the ................... 4x + 9, x is a ....................
4x and 9 are ................... of the expression.
4 is the ................... of x. 9 is a ....................
The expression is not equal to anything so cannot be ....................
a
Tanesha has a box that contains x DVDs.
Choose the correct expression from the cloud that shows
the total number of DVDs she has in the box when
i she takes 2 out
ii she puts in 2 more
iii she takes out half of the DVDs
iv she doubles the number of DVDs in the box.
b Tanesha starts with 12 DVDs in the box. Work out how
many she will have for Question 2a, parts i to iv.
solved
coefficient
terms
variable
constant
expression
x x + 2
2
x − 2 2x
31
2 Expressions, formulae and equations
3
a
Jake thinks of a number, n.
Write an expression for the number Jake gets when he:
i multiplies the number by 6, then adds 1
ii divides the number by 4, then adds 5
iii multiplies the number by 2, then subtracts 3
iv divides the number by 10, then subtracts 7.
bJake thinks of the number 20. Work out the
numbers he gets in Question 3a, parts i to iv.
4
Tip
Remember the order
of operations:
Brackets,
Indices,
Division,
Multiplication,
Addition,
Subtraction
Match each description with the correct expression.
The first one has been done for you: a and iv.
a
Multiply n by 5 and subtract 4
b
Add 4 and n, then multiply by 5
c
Multiply n by 5 and add 4
d
Add 5 and n, then multiply by 4
e
Subtract 4 from n, then multiply by 5
f
Subtract 5 from n, then multiply by 4
i 5(n + 4)
ii
4(n + 5)
iii 4(n − 5)
iv 5n − 4
v 5n + 4
vi 5(4 − n)
vii 5(n − 4)
Write a description for the expression that has not been matched.
Think like a mathematician
5
In pairs or in a small group, discuss.
Sofia and Zara discuss what to write for this problem.
‘I think of a number, n. I halve the number then add 4.’
I think the
expression is n + 4
2
I think the
expression is n + 4
2
What do you think?
Make a conjecture and convince the other members of your group.
6
32
Kia thinks of a number, x.
Write an expression for the number Kia gets when she:
a
divides the number by 3, then adds 1
b adds 1 to the number, then divides by 3
c
subtracts 1 from the number, then divides by 3
d divides the number by 3, then subtracts 1.
2.1 Constructing expressions
Think like a mathematician
7
In pairs or in a small group, discuss.
Sofia, Zara and Arun discuss what to write for this problem.
‘I think of a number, n. I divide by 3, then multiply by 2.’
I think the
expression
2n
is .
3
I think the
n
expression is × 2
If you divide by 3
then times by 2, you are
2
finding of the number,
3
2
so you can write n
3
3
What do you think?
Make a conjecture and convince the other members of your group.
8
9
a
Sort these cards into groups of equivalent expressions.
A
3×x
4
B
x+3
4
C
4x
3
D
4×x
3
E
3
×x
4
F
3x
4
G
3
+x
4
H
3x
4
I
4x
3
J
3+x
4
b Which card is in a group on its own?
This is part of Pedro’s classwork.
Are Pedro’s answers correct? If not, write the
correct answers for him.
Question
Write an expression for these.
a one-third of x add 4
b 5 subtract two-fifths of y
Answers
a x3 + 4
2y
b 5 −5
33
2 Expressions, formulae and equations
10 a
b
Write an expression for each description.
i one-half of x add 8
ii three-quarters of x subtract 12
iii 7 add four-fifths of x
iv 20 subtract five-ninths of x
Describe each expression in words.
5x
i x+2
ii
iii 8 − 2x
−4
6
3
7
iv
3 + 7x
8
11 This is part of Maya’s homework.
Question
Write an expression for
a the perimeter of this rectangle
4b cm
b the area of this rectangle.
Answers
a perimeter = 31 a + 31 a + 4b + 4b = 32 a + 8b cm
1
3 a cm
b area = 31 a × 4b = 34 ab cm 2
Use Maya’s method to write an expression for the perimeter and
area of each of these rectangles. Simplify each expression.
a
b
1
2 a cm
6b cm
3
5 d cm
7c cm
Activity 2.1
Work with a partner.
Take it in turns to say ‘Write an expression for …’ and give a description
like those in Question 10a.
For example, ‘Write an expression for two-thirds of x add 9.’
Your partner must write the expression correctly.
Check their expression. If it is correct, they score 1 point.
Write five expressions each, then check the scores!
34
2.1 Constructing expressions
12 The shortest side of a triangle is y cm.
Tip
The second side is 3 cm longer than the shortest side.
Start by writing
The third side is twice as long as the second side.
expressions for
Write an expression, in its simplest form, for the perimeter
the second and
of the triangle.
third sides.
13 The price of one bag of cement is $c.
The price of one bag of gravel is $g.
The price of one bag of sand is $s.
Write an expression for the total cost of
a
one bag of cement and three bags of sand
b three bags of cement, four bags of gravel and six bags of sand.
14 The price of one kilogram of apples is $a.
The price of one kilogram of bananas is $b.
The price of one kilogram of carrots is $c.
Write an expression for the total cost of
a
one kilogram of apples and half a kilogram of bananas
b two kilograms of bananas and three-quarters of a kilogram of carrots
c
three kilograms of apples, a quarter of a kilogram of bananas
and four-fifths of a kilogram of carrots.
15 Brad thinks of a number, y.
Choose the correct expression from the cloud for
when Brad
a
adds 5 to one-half of y, then multiplies by 6
b adds 6 to one-fifth of y, then multiplies by 2
c
adds 2 to five-sixths of y, then multiplies by 6
d adds 5 to two-fifths of y, then multiplies by 6.
y
5y
+ 2
6  + 5
 6
  2

6 
2y
y
+ 5
2  + 6
 5
  5

6 
35
2 Expressions, formulae and equations
Which statement best describes how you found the questions in
this exercise?
A
I found the questions very difficult to answer.
B
I found the questions difficult to answer.
C
I answered the questions but I had to think carefully.
D
I found the questions easy to answer.
What can you do to improve your knowledge and understanding
of this topic?
Summary checklist
I can use letters to represent unknown numbers.
I can use the correct order of operations in algebraic expressions.
I can use words or letters to represent situations.
36
2.2 Using expressions and formulae
2.2 Using expressions and formulae
In this section you will …
Key words
•
use the correct order of operations in algebraic expressions
•
represent a situation either in words or as a formula
changing the
subject
•
change the subject of a formula.
A formula is a mathematical rule that shows the relationship between
two or more quantities (variables). It is a rule that can be written in
letters or words. The plural of formula is formulae.
You can write, or derive, your own formulae to solve problems.
An example of a formula is F = ma.
In this formula, F is the subject of the formula. The variable F is written
on its own on the left hand side of the formula.
You may need to rearrange a formula to make a different
F = ma
variable the subject. This is called changing the subject of
F
=m
the formula.
a
For example, if you know the values of F and a, and
m= F
a
you want to find the value of m, you will rearrange the
equation like this:
When you substitute numbers into formulae and expressions, remember
the order of operations. Brackets and indices must be worked out before
divisions and multiplications. Additions and subtractions are always
worked out last.
derive
formula
formulae
inverse operation
solve
subject of a
formula
substitute
Tip
Examples of
indices are 22, 52,
43 and 73.
Worked example 2.2
a Work out the value of the expression 2x + 4y when x = 5 and y = −2.
b Work out the value of the expression 3x 2 + 4 when x = 10.
c Write a formula for the number of hours (h) in any number of days (d), using
words
ii
letters.
i
d Use the formula in part c to work out the number of hours in 7 days.
e Rearrange the formula in part cii to make d the subject.
37
2 Expressions, formulae and equations
Continued
Answer
2 × 5 + 4 × −2
= 10 + −8
= 10 − 8 = 2
Substitute x = 5 and y = −2 into the expression.
Work out 2 × 5 and 4 × −2.
Adding −8 is the same as subtracting 8.
b 3 × 102 + 4
= 3 × 100 + 4
= 300 + 4 = 304
Substitute x = 10 into the expression.
Work out 102 first.
Work out the multiplication before the addition.
c
There are 24 hours in every day.
Use h for hours and d for days.
a
i hours = 24 × days
ii h = 24d
d h = 24 × 7 = 168
Substitute d = 7 into the formula.
e
h = 24d
The formula is h = 24 × d
h
=d
24
h
d = 24
Use the inverse operation to make d the subject by dividing by 24.
Rewrite the formula with the subject, d, on the left hand side.
Exercise 2.2
1
2
Copy and complete the working to find the value of each
expression.
a
p + 5 when p = −3
p + 5 = −3 + 5 =
b
q − 6 when q = 4
q−6=4−6=
c
6h when h = −3
6h = 6 × −3 =
d
j
when j = −20
4
j
= −20 =
4
4
e
a + b when a = 6 and b = −3
f
c − d when c = 25 and d = 32
Work out the value of each expression.
a
8m − 5 when m = −2
c
2x + 3y when x = 4 and y = 5
e
38
u
− 5 when u = 4
2
a + b = 6 + −3 = 6 − 3 =
c − d = 25 − 32 =
b
d
3z + v when z = 8 and v = −20
20 − 3n when n = 9
f
p q
+ when p = 30 and q = −8
5 2
2.2 Using expressions and formulae
3
Work out the value of each expression.
a
x 2 + 5 when x = 4
b 10 − y2 when y = 5
c
g2 + h2 when g = 3 and h = 6 d m2 − n2 when m = 7 and n = 8
e
f
3r 3 when r = 1
4k 2 when k = 2
g 2y3 when y = 3
h x3 − 5 when x = 2
i
4
20 − w3 when w = 4
j
Tip
Remember that r3
means r × r × r.
y2
when y = 4
2
This is part of Dakarai’s homework. He has made a mistake in his working.
Question
Work out the value of x2 – 8 when x = –3.
Answer
x2 – 8 = (–3)2 – 8
= –3 × –3 – 8
= –9 – 8
= –17
5
a
Explain the mistake he has made.
b Work out the correct answer.
c
Work out the value of y2 + 4 when y = −5.
This is part of Oditi’s homework. She has made a mistake in her working.
Question
Work out the value of 5x3 when x = –2.
Answer
5x3 = 5 × (–2)3
= (–10)3
= –10 × –10 × –10
= –1000
6
a
Explain the mistake she has made.
b Work out the correct answer.
c
Work out the value of 2y3 when y = −3.
aWrite a formula for the number of months in any number of years, in
i words
ii letters.
b Use your formula in part aii to work out the number of
months in 8 years.
39
2 Expressions, formulae and equations
7
This is how a taxi company works out the cost of a journey for a customer:
There is a fixed charge of $6 plus $2 per kilometre.
a
Write a formula for the cost of a journey, in
i words
ii letters.
b Use your formula in part aii to work out the cost of a journey of 35 km.
8
Use the formula v = u + 10t to work out the value of v when
a
u = 5 and t = 12
b u = 8 and t = 15
c
u = 0 and t = 20.
9
Use the formula F = ma to work out F when
a
m = 6 and a = 2
b m = 18 and a = 3
c
m = 8 and a = −4.
10 The height of a horse is measured in hands (H) and inches (I).
This formula is used to work out the height of a horse in
centimetres (C ).
C = 2.5(4H + I) where: C is the number of centimetres
H is the number of hands
I is the number of inches.
Sasha has a horse with a height of 16 hands and 1 inch.
She uses the formula to work out the height of her horse,
in centimetres.
C = 2.5(4 × 16 + 1)
= 2.5(64 + 1)
= 2.5 × 65
= 162.5 cm
Work out the height, in centimetres, of a horse with height
a
14 hands and 2 inches
b 15 hands and 3 inches
c
13 hands and 1 inch
d 17 hands and 2 inches
e
16 hands
f
12 hands.
Tip
16 hands exactly
means 16 hands
and 0 inches.
Would it matter if the formula used the letters D, E and F instead of C, H and I?
Do the letters help you to understand a formula? Explain your answer.
11 Use the formula C = πd to
a
estimate the value of C when d = 19 m
b calculate the value of C when d = 19 m.
Give your answer to one decimal place.
40
Tip
Remember that π
is approximately
3.14.
2.2 Using expressions and formulae
12 Xavier uses this formula to work out the volume of a triangular
prism.
V = bhl where: V is the volume; b is the base; h is the height;
2
l is the length.
Xavier compares two prisms.
Prism A has a base of 8 cm, height of 5 cm and length of 18 cm.
Prism B has a base of 9 cm, height of 14 cm and length of 6 cm.
Xavier works out that Prism A has the larger volume by 12 cm3.
Is Xavier correct? Explain your answer.
Tip
Remember that
bhl means
b × h × l.
Think like a mathematician
13 Work with a partner to answer this question.
Discuss which answers are correct. Identify the mistakes
that have led to the incorrect answers.
Make x the subject of each formula.
Write which answer is correct, A, B or C.
a
y=x+9
A
x=y+9
B
x=y−9
C
x=9−y
C
y
x=
m
b
y = mx
A
x = my
B
m
x=
y
c
y=x−c
A
x=y+c
B
x=y−c
C
x=c−y
d
y=
x
k
A
x=
y
k
B
x = ky
C
x=
k
y
e
y = 7x − 3
A
x=
y+3
7
B
x=
C
x=
y−3
7
y
+3
7
14 aUse the formula T = mg to work out the value of
T when m = 4.5 and g = 10.
b Rearrange the formula T = mg to make m the subject.
c
Use your formula to work out the value of m when
T = 320 and g = 10.
15 aUse the formula h = k − d to work out the value of h
when k = 72 and d = 37.
b Rearrange the formula h = k − d to make k the subject.
c
Use your formula to work out the value of k when
h = 0.42 and d = 1.83
Tip
To answer this
question, you will
need to critique
the given answers
and improve
them.
Tip
Use inverse
operations on
T=m×g
to make the
formula
m=
41
2 Expressions, formulae and equations
16 aUse the formula f = w to work out the value of f when w = 60
p
and p = 12.
w
b Rearrange the formula f = p to make w the subject.
c
Use your formula to work out the value of w when f = 0.25
and p = 52.
17 Polly and Theo use different methods to work out the answer to a question.
This is what they write.
Question
Use the formula P = 3n + b to work out the value of n when P = 72 and b = 6.
Answers
Polly
Step 2: Substitute
Step 1: Make n
the subject of the in the numbers.
formula.
n = 723− 6
P = 3n + b
= 66
3
P − b = 3n
P−b
= 22
3 =n
n = P −3 b
a
b
c
42
Theo
Step 1: Substitute Step 2: Solve the
in the numbers.
equation.
72 − 6 = 3n
P = 3n + b
66 = 3n
72 = 3n + 6
66
3 =n
22 = n
n = 22
Look at Polly and Theo’s methods.
Do you understand both methods?
Do you think you would be able to use both methods?
Which method do you prefer and why?
Use your preferred method to answer these questions.
iUse the formula H = 6p − k to work out the value of p
when H = 40 and k = 14.
iiUse the formula y = mx + 7 to work out the value of m
when y = 25 and x = 3.
2.3 Expanding brackets
Summary checklist
I can substitute numbers into expressions.
I can write formulae.
I can understand and use formulae.
I can change the subject of formulae.
2.3 Expanding brackets
In this section you will …
Key words
•
expand brackets
expand brackets.
To expand brackets, multiply each term inside the brackets by the term
directly in front of the brackets.
Tip
Expanding brackets is sometimes called multiplying out brackets.
Worked example 2.3
a Expand these expressions. i 3(2b + 5) ii a(a − 3)
b Expand and simplify this expression. 4(2x + 3x2) − x(6 + x)
Answer
a
i 3(2b + 5) = 3 × 2b + 3 × 5
= 6b + 15
ii a(a − 3) = a × a − a × 3
= a2 − 3a
b 4(2x + 3x2) − x(6 + x)
= 8x + 12x2 − 6x − x2
= 2x + 11x2
Multiply 3 by 2b then multiply 3 by 5.
Simplify 3 × 2b to 6b and simplify 3 × 5 to 15.
Multiply a by a then multiply a by −3.
Simplify a × a to a 2 and simplify a × −3 to −3a.
Start by multiplying out both brackets.
So, 4 × 2x = 8x, 4 × 3x2 = 12x2, −x × 6 = −6x, −x × x = −x2
Collect like terms: 8x − 6x = 2x and 12x2 − x2 = 11x2.
43
2 Expressions, formulae and equations
Exercise 2.3
1
Copy and complete the working.
Expand the brackets.
a
3(x + 4) = 3 × x + 3 × 4
b 8(y − 2) = 8 × y − 8 × 2
= 3x + = 8y −
c
9(3q − 4) = 9 × 3q − 9 × 4
=
−
Expand each expression.
a
4(x + 6)
b 7(z − 2)
c
2(a + 8)
e
2(2p + 3q)
f
9(6t − 2s)
g 7(6xy − 2z)
Copy and complete the working.
Expand the brackets.
a
x(y + 3) = x × y + x × 3
b y(y − 2) = y × y − y × 2
= xy + = y2 −
c
p(3 + 4p) = p × 3 + p × 4p
d q(6q − 15) = q × 6q − q × 15
2
=
+ 4p =
−
Expand each expression.
a
y(y + 8)
b z(2w − 1)
c
m(m − 4)
e
n(9 − 8n)
f
a(1 − 3b)
g e(2e + 7f)
i
h(2h − 5k)
j
d(3c − 5e)
2
3
4
Think like a mathematician
5
In pairs or a small group, discuss what Zara and Arun say.
When I expand
2d(4c − 7a), I get
8cd − 14ad
I don’t, I get
8dc − 14da
so one of us must
be wrong!
Do you agree with Arun? Explain your answer.
44
d
h
6(3 − 4e)
5(2x + y + 4)
d
h
n(2n + 5)
g(3h + 7g)
2.3 Expanding brackets
6
Jing, Jun and Amira compare the methods they use to expand the bracket 5k(6m − 8k).
Jing uses the method shown
in Question 3.
Jun uses a multiplication Amira uses multiplication arcs.
box.
5k(6m – 8k)
= 5k × 6m – 5k × 8k
= 30km – 40k2
× 6m
–8k
5k 30km –40k2
5k(6m – 8k) = 30km – 40k2
So 5k(6m – 8k) = 30km – 40k2
So 5k(6m – 8k) =
30km – 40k2
a
b
What do you think about Jing, Jun and Amira’s methods?
Which method do you think is best for expanding brackets
correctly? Explain why.
c
Use your favourite method to expand
i 2x(x + 3y)
ii
3y(5y + 6)
iii 4b(6b − 2a)
iv 2f(2f + g − 3)
Here are some expression cards.
Sort the cards into groups of equivalent expressions.
7
A
2 x ( 8x 2 + 6 x )
D
G
C
2 x 2 (12 x + 9)
Tip
4 x ( 4 x + 3)
F
3x (6 + 8x )
For card A,
2x(8x2 + 6x)
= 16x3 + 12x2
6 x ( 3x + 4 x 2 )
I
x 2 (12 + 16 x )
B
10 x (3x 2 + 2 )
2 x (10 + 15x )
E
5 (6 x3 + 4 x )
H
2
2
2
Think like a mathematician
8
Work with a partner to discuss this question.
Look at this expansion.
x(2x + 5) + 3x(2x + 4) = 2x 2 + 5x + 6x 2 + 12x
a
How would the expansion change if the + changed to −?
Here is the expansion again.
b
c
x(2x + 5) + 3x(2x + 4) = 2x 2 + 5x + 6x 2 + 12x
How would the expansion change if both the + changed to −?
Copy these expansions and fill in the missing signs (+ or −).
45
2 Expressions, formulae and equations
9
Expand each expression and simplify by collecting like terms.
a
x(x + 2) + x(x + 5)
b z(2z + 1) + z(4z + 5)
c
u(2u + 5) − u(u + 3)
d w(6w + 2x) − 2w(2w − 9x)
10 This is part of Shen’s homework. He has made a mistake in every question.
Question
Expand and simplify
1 8(x + 5) – 3(2x + 7)
2 a(2b + c) + b(3c – 2a)
3 2y(y + 5x) + x(3x + 4y)
Answers
1 8(x + 5) – 3(2x + 7) = 8x + 40 – 6x + 21 = 2x + 61
2a(2b + c) + b(3c – 2a) = 2ab + ac + 3bc – 2ab = ac + 3bc = 3abc2
32y(y + 5x) + x(3x + 4y) = 2y2 + 10xy + 9x2 + 4xy = 9x2 + 2y2 + 14xy
a
b
Explain what Shen has done wrong.
Work out the correct answers.
Activity 2.3
Work with a partner to answer this question.
Here are six expressions.
x(5x + 2) + 3x(4x + 1)
B
y(y2 + 4) + 6y2(y + 8)
A
C
D
6k + 18 − 3k ( 4 − 5k 2 )
7p (2 p 2 + 7p − 1) + 9 p
E
5n( n2 − 4) − 2n2 ( n + 2)
F
8m ( m + 3) − 2m (4m − 3)
a
Choose one of the expressions and ask your partner to expand the brackets
and simplify the expression. While they are working, you work out the answer too.
Mark your partner’s work.
If your answers are different, discuss any mistakes that have been made.
b Now ask your partner to choose an expression for you.
Expand the brackets and simplify the expression.
Ask your partner to mark your work. Discuss any mistakes that have been made.
c
Do this twice each, so four of the expressions have been chosen altogether.
Summary checklist
I can multiply out a bracket and collect like terms.
46
2.4 Factorising
2.4 Factorising
In this section you will …
Key words
•
factorisations
use the HCF to factorise an expression.
factorise
To expand a term with brackets, you multiply
each term inside the brackets by the term
outside the brackets.
When you factorise an expression, you take the
highest common factor (HCF) and put it outside
the brackets.
highest common
factor (HCF)
4(x + 3) = 4x + 12
4x + 12 = 4(x + 3)
Worked example 2.4
Factorise these expressions.
a
2x + 10
b
8 − 12y
c
4a + 8ab
d
x 2 − 5x
Answer
a
2x + 10 = 2(x + 5)
The HCF of 2x and 10 is 2, so put 2 outside the brackets.
Divide both terms by 2 and put the results inside the brackets.
Check the answer by expanding: 2 × x = 2x and 2 × 5 = 10.
b 8 − 12y = 4(2 − 3y)
The HCF of 8 and 12y is 4, so put 4 outside the brackets.
Divide both terms by 4 and put the results inside the brackets.
Check the answer by expanding: 4 × 2 = 8 and 4 × −3y = −12y.
c
4a + 8ab = 4a(1 + 2b)
The HCF of 4a and 8ab is 4a, so put 4a outside the brackets.
Divide both terms by 4a and put the results inside the brackets.
Check the answer: 4a × 1 = 4a and 4a × 2b = 8ab.
d x2 −5x = x(x − 5)
The HCF of x2 and 5x is x, so put x outside the brackets.
Divide both terms by x and put the results inside the brackets.
Check the answer: x × x = x2 and x × −5 = −5x.
47
2 Expressions, formulae and equations
Exercise 2.4
Copy and complete these factorisations.
All the numbers you need are in the cloud.
1
2
2
3
4
5
a
3x + 15 = 3(x +
)
b
10y − 15 = 5(2y −
c
14 − 28x = 7(
− 4x)
d
12 − 9y = 3(
Copy and complete these factorisations.
All the numbers you need are in the cloud.
a
4x2 + 5x = x(4x +
c
7y − 7y2 = 7y(
1
2
)
− 3y)
5
7
)
b
6xy + 12y = 6y(x +
)
− y)
d
21x − 12xy = 3x(
− 4y)
Think like a mathematician
3
In pairs or a small group, discuss what Marcus and Arun say.
When I factorise
6x + 18 I get 3(2x + 6)
I don’t, I get
6(x + 3) so one of
us must be wrong!
Do you agree with Arun? Explain your answer.
4
5
6
48
Factorise each of these expressions.
Each one has a highest common factor of 2.
a
2x + 4
b 4b − 6
c
8 + 10y
d 18 − 20m
Factorise each of these expressions.
Each one has a highest common factor of 3.
a
18 + 21p
b 3y − 18
c
9 + 15m
d 12 − 27x
Factorise each of these expressions.
Make sure you use the highest common factor.
a
10z + 5
b 8a − 4
c
14 + 21x
d 18 − 24z
2.4 Factorising
Think like a mathematician
7
In pairs or a small group, discuss what Zara and Sofia say.
I think the highest
common factor of
6x and 9x2 is 3x.
I think the highest
common factor of
6x and 9x2 is 3.
a
b
8
What do you think? Explain your answer.
What is the highest common factor of
i
8y and 4y2
ii
12p2 and 15p
iii
4ab and 5a?
Each expression on a yellow card has been factorised to give an
expression on a blue card.
Match each yellow card with the correct blue card.
A
6x2 + 12x
B
6x2 + 15x
C
6x2 + 9x
D
6x2 + 18x
i
3x(2x + 5)
ii
6x(x + 3)
iii
6x(x + 2)
iv
3x(2x + 3)
9
Factorise each of these expressions.
a
3x2 + x
b 6y2 − 12y
e
18y − 9x
f
12y + 9x
10 Copy and complete these factorisations.
a
2x + 6y + 8 = 2(x + 3y +
)
c
9xy + 12y − 15 = 3(3xy +
e
9y − y2 − xy = y(
−
−
c
g
3b + 9b2
8xy − 4y
d
h
b
4y − 8 + 4x = 4(y −
− 5)
d
5x2 + 2x + xy = x(5x +
)
f
3y2 − 9y + 6xy = 3y(
12n − 15n2
15z + 10yz
+ x)
+
−
)
+
)
49
2 Expressions, formulae and equations
11 Read what Zara says.
When I expand
5(2x + 6) + 2(3x − 5), then
collect like terms and finally
factorise the result, I get
the expression
4(4x + 5)
Show that she is right.
12 Read what Marcus says.
When I expand
6(3y + 2) − 4(y − 2), then
collect like terms and finally
factorise the result, I get the
expression 2(7y + 2)
Show that he is wrong.
Explain the mistake Marcus has made.
13 The diagrams show two rectangles.
2a
A
length
3d
Tip
B
length
The area of rectangle A is 2a2 + 18a.
The perimeter of rectangle B is 14d − 10c.
Write an expression for the length of each rectangle,
in its simplest form.
You need to
factorise the
expressions to
find the lengths
of A and B.
You will need an
extra step first
for rectangle B.
Work with a partner.
Take it in turns to define the following terms.
What is a factor?
a
b What is the highest common factor?
c
What is factorising?
How did your answers to a and b help with your answer to c?
Summary checklist
I can use the highest common factor (HCF) to factorise an expression.
50
2.5 Constructing and solving equations
2.5 Constructing and
solving equations
In this section you will …
Key word
•
construct
write and solve equations.
You already know the difference between a formula, an expression and
an equation.
Remember
A formula is a rule that shows the relationship
between two or more quantities (variables). It must
have an equals sign.
An expression is a statement that involves one or
more variables, but does not have an equals sign.
Examples
F = ma
v = u + 10t
3x − 7
a2 + 2b
An equation contains an unknown number. It must 3x − 7 = 14
have an equals sign, and it can be solved to find the
4 = 6y + 22
value of the unknown number.
When you are given a problem to solve, you may need to construct, or
write, an equation.
Worked example 2.5
a
rite if each of the following is a formula, an expression or an equation.
W
i 4c + 3e ii P = 8h + b iii 9k − 2 = 16
b The diagram shows a rectangle.
3(x + 3) cm
Work out the values of x and y.
5y − 4 cm
3y + 8 cm
24 cm
Answer
a
i
expression
4c + 3e involves two variables but does not have an equals sign.
ii formula
P = 8h + b is a rule showing the relationship between three
quantities, P, h and b.
iii equation
9k − 2 = 16 contains an unknown number, k, it has an equals
sign, and it can be solved to find the value of k.
51
2 Expressions, formulae and equations
Continued
b 3(x + 3) = 24
3x + 9 = 24
The two lengths must be equal, so construct an equation by
writing one length equal to the other.
First, multiply out the brackets.
3x + 9 − 9 = 24 − 9
Then use inverse operations to solve the equation.
Start by subtracting 9 from both sides.
3x = 15
Simplify both sides of the equation.
x= 3 =5
15
Divide 15 by 3 to find the value of x.
5y − 4 = 3y + 8
The two widths must be equal, so write one width equal
to the other.
5y − 4 − 3y = 3y + 8 − 3y
Rewrite the equation by subtracting 3y from both sides.
2y − 4 = 8
Simplify.
2y − 4 + 4 = 8 + 4
Use inverse operations to solve the equation. Start by adding
4 to both sides.
2y = 12
Simplify both sides of the equation.
y = 12
=6
2
Divide 12 by 2 to work out the value of y.
Exercise 2.5
1
2
Write if each of the following is a formula, an expression or
an equation.
a
3y + 7 = 35
b 6(x + 5)
2
c
T = 3a − 8d
d 9u − vw
Copy and complete the workings to solve these equations.
a
3x + 5 = 26
(subtract 5 from both sides)
3x + 5 − 5 = 26 − 5
(simplify)
3x =
(divide both sides by 3)
b
4(x − 3) = 24
(simplify)
(multiply out the brackets)
(add 12 to both sides)
52
x=
3
x=
4x − 12 = 24
4x − 12 + 12 = 24 + 12
(simplify)
4x =
(divide both sides by 4)
x=
(simplify)
x=
4
2.5 Constructing and solving equations
c
y
− 10 = 1
4
y
4 − 10 + 10 = 1 + 10
y
=
4
(add 10 to both sides)
(simplify)
d
6y + 2 = 4y + 18
(multiply both sides by 4)
y=
(simplify)
y=
(subtract 4y from both sides)
6y − 4y + 2 = 4y − 4y + 18
(simplify)
y + 2 = 18
(subtract 2 from both sides)
y + 2 − 2 = 18
(simplify)
(divide both sides by
y=
y=
)
(simplify)
3
×4
For each learner
i
write an equation to represent what they say
ii
solve your equation to find the value of x.
The first one has been started for you.
a
My sister is 15 years old.
My Dad is x years old. Half
of my Dad’s age minus 3 is
the same as my sister’s age.
y=
x − 3 = 15
2
x − 3 + 3 = 15 + 3
2
x=
2
x=
×2
x=
b
My brother is 12 years old.
My Mum is x years old.
One-third of my Mum’s
age plus 1 is the same as
my brother’s age.
c
My Aunt is 30 years old.
My Gran is x years old.
One-quarter of my Gran’s
age plus 9 is the same as
my Aunt’s age.
53
2 Expressions, formulae and equations
Think like a mathematician
4
Marcus and Sofia are discussing what equation to
write to answer this question.
The diagram shows an isosceles triangle.
All measurements are in centimetres.
Work out the value of y.
2y + 7
5y − 17
I would write
the equation
5y − 17 = 2y + 7
because I think this
is easier to solve.
As the triangle is an
isosceles triangle, the
two sides shown are equal in length.
So I would write the equation
2y + 7 = 5y − 17
What do you think? Does it matter which way round you write the equation?
Work with a partner to discuss and explain your answers.
5
Work out the value of x in each isosceles triangle.
All measurements are in centimetres.
a
b
6x − 3
9
Tip
Think carefully
about which way
round you write
your equations.
x
2 + 20
27
c
x + 35
5x − 13
Think like a mathematician
6
7
Look back at your answers to Question 5.
Discuss with a partner how you can check your value for x in each part.
Work out the value of y in each shape. All measurements are
in centimetres.
Show how to check your answers are correct.
a
4(y − 3)
2y + 2
b
8y − 5
3(y + 5)
54
c
2(y + 6)
4(y − 3)
2.5 Constructing and solving equations
8
Work out the value of x and y in each diagram. All measurements
are in centimetres.
Show how to check your answers are correct.
3x + 1
5x − 3
a
b
20
2(y + 3)
4y + 5
37
c
2(x + 5)
d
5x − 3
3y + 16
8y − 4
3x + 11
2y + 15
25
16
8(y − 1)
x
+ 17
4
9
Work in a group of three or four. For each part of this question:
i
Write an equation to represent the problem.
ii
Compare the equation you have written with the equations
written by the other members of your group. Decide who
has written the correct equation in the easiest way.
iii Solve the equation you chose in part ii.
a
Emily thinks of a number. She multiplies it by 3 then adds 8.
The answer is 23. What number did she think of ?
b Anders thinks of a number. He divides it by 4 then subtracts 8.
The answer is 5. What number did he think of ?
c
Sasha thinks of a number. She multiplies it by 5 then subtracts 4.
The answer is the same as 2 times the number plus 20.
What number did Sasha think of ?
d Jake thinks of a number. He adds 5 then multiplies the result
by 2. The answer is the same as 5 times the number take away 14.
What number did Jake think of ?
10 The diagram shows the sizes of
the angles in a triangle.
6n °
a
Write an equation to
represent the problem.
b Solve your equation to find
n − 5°
2n + 5 °
the value of n.
c
Work out the size of each of
the angles in the triangle.
Tip
Start by writing an
expression for the
total of the angles
in the triangle.
Then write an
equation. Use the
fact that the angles
in a triangle add up
to …
55
2 Expressions, formulae and equations
11 The diagram shows the sizes of the two equal angles in an
isosceles triangle.
4x − 6 °
2x + 18 °
a
Write an equation to represent the problem.
b Solve your equation to find the value of x.
c
Work out the size of each of the angles in the triangle.
12 Solve these equations.
Use the Tip box to help.
a
5(2x + 3) + 2(x − 4) = 31
b 4(3x − 1) − 3(5 − 2x) = 35
c
2
y=8
3
d
3
y + 1 = 19
5
Tip
Start by expanding
the brackets and
simplifying the left
hand side.
2 y is the same as
3
2× y
so start by
3
multiplying both
sides by 3.
13 This is part of Mo’s homework.
Question
Solve the equation 4(2b – 3) = −8b
Answer
Divide both sides by 4 4(2b4− 3) = −8b
4
Add 3 to both sides
2b – 3 + 3 = −2b + 3
Add 2b to both sides
2b + 2b = −2b + 2b + 3
4b 3
Divide both sides by 4
4 =4
You can see that instead of multiplying out the bracket,
Mo’s first step is to divide both sides of the equation by 4.
Use Mo’s method to solve these equations.
a
6(3a + 4) = 12a
b 5(4c − 9) = 25
c
56
→
→
→
→
2b – 3 = −2b
2b = −2b + 3
4b = 3
3
b= 4
(
)
3 2 d + 4 = 18
5
2.6 Inequalities
14 Art has these cards.
2y + 14
8(y − 12)
y
− 18
4
=
4
−2
−20
He chooses one blue card, the red card and one yellow card to
make an equation.
Which blue and yellow card should he choose to give him the
equation with
a
the largest value for y
b the smallest value for y?
Explain your decisions and show that your answers are correct.
Summary checklist
I can understand equations and solve them.
I can write equations and solve them.
2.6 Inequalities
In this section you will …
Key words
•
closed interval
use letters and inequalities to represent open and
closed intervals.
You have already learned how to use a letter and an inequality sign
to represent lots of numbers.
Remember the inequality signs:
< means ‘is less than’
⩽ means ‘is less than or equal to’
> means ‘is greater than’
⩾ means ‘is greater than or equal to’
So if you see the inequality x > 5, this means that x can be any number
greater than 5.
If you see the inequality y ⩽ 2, this means that y can be any number
less than, or equal to, 2.
inequality
integer
Tip
Remember that
you use an
open circle ( )
for the < and >
inequalities and a
closed circle ( )
for the ⩽ and ⩾
inequalities.
57
2 Expressions, formulae and equations
If you see the inequality 3 ⩽ x < 9, this means that x is greater than
or equal to 3 and is also less than 9. This inequality represents a closed
interval. You can show this on a number line like this.
2
3
4
5
6
7
8
9
10
Worked example 2.6
a
i
ii
b i
ii
Show the inequality 2 ⩽ x < 6 on a number line.
List the possible integer values for.
Show the inequality −5 < y ⩽ −1 on a number line.
List the possible integer values for y.
Tip
This closed
interval includes
the endpoints 3
and 9.
Tip
Remember that
an integer is a
whole number.
Answer
a
i
1
2
3
4
5
6
7
8
Use a closed circle for the ⩽ sign and start the
line at 2.
Extend the line to 6, where you use an open
circle for the < sign.
x is greater than or equal to 2, so 2 is the
smallest integer.
ii 2, 3, 4, 5
x is less than 6, so 5 is the greatest integer.
b i
−6
−5
−4
−3
−2
−1
0
Use an open circle for the < sign and start the
line at −5.
Extend the line to −1, where you use a closed
circle for the ⩽ sign.
ii −4, −3, −2, −1
y is greater than −5, so −4 is the smallest
integer.
y is less than or equal to −1, so −1 is the greatest
integer.
Exercise 2.6
1
58
Write in words what each of these inequalities means.
Part a has been done for you.
a
6 < x < 11
x is greater than 6 and less than 11
b 12 ⩽ x ⩽ 18
c
0 < x ⩽ 20
d −9 ⩽ x < −1
2.6 Inequalities
Write each statement as an inequality. Part a has been done for you.
a
y is greater than or equal to 3 and less than 17
3 < y <17
b y is greater than 15 and less than 25
c
y is greater than −2 and less than or equal to 5
d y is greater than or equal to −9 and less than
or equal to −3
Copy each number line and show each inequality on the
number line.
4 < x < 7 a
b 6 ⩽ x ⩽ 9
2
3
3
5
6
7
8
5
6
7
8
9
10
−1
0
1
2
1
2
3
4
5
2
4
6
8
10
−4 < x ⩽ 1 d −2 ⩽ x < 2
c
−5
4
4
−4
−3
−2
−1
0
1
2
−3
−2
3
Write the inequality that each of these number lines shows.
Use the letter x.
a
b 11
12
13
14
15
16
c
17
0
d −3
−2
−1
0
1
2
0
12
Think like a mathematician
5
Sofia and Zara are looking at the inequality x > 5.
In pairs or small groups, discuss Sofia’s and Zara’s comments.
The inequality
x > 5 is equivalent to
2x > 10.
a
b
c
The inequality
x > 5 is equivalent
to x − 2 > 3.
How can you show that Sofia and Zara are correct?
Write two different inequalities that are equivalent
to x > 5.
Is it possible to say how many different inequalities
there are that are equivalent to x > 5? Explain your
answer.
Tip
Remember that
‘equivalent to’
means ‘the same
as’ or ‘equal to’.
59
2 Expressions, formulae and equations
6
Copy and complete these equivalent inequalities.
a
x > 8 is equivalent to 3x >
b x < 3 is equivalent to
y > 7 is equivalent to y + 3⩾
c
This is part of Ryan’s homework.
7
d
x < 15
y ⩽ 2 is equivalent to y − 4 ⩽
Question
Use the inequality 12 ⩽ x < 18 to write
i the smallest integer that x could be
ii the largest integer that x could be
iii a list of the integer values that x could be.
Answer
i smallest integer is 13
ii largest integer is 18
iii x could be 13, 14, 15, 16, 17, 18
a
b
8
Explain the mistakes Ryan has made and write the correct
solutions.
Discuss your answers to part a with a partner.
Make sure you have corrected all of Ryans’s mistakes.
For each of these inequalities, write
i
the smallest integer that y could be
ii
the largest integer that y could be
iii a list of the integer values that y could be.
3< y<8
a
b 4<y⩽7
c
0⩽y<6
Think like a mathematician
9
Arun and Sofia are looking at the inequality 2 < y < 9.
In pairs or small groups discuss Arun’s and Sofia’s comments.
I think the
inequality 2 < y < 9
can be written as
9>y>2
What do you think? Explain your answers.
60
I think the
inequality 2 < y < 9
can be written as
2>y>9
d
−10 ⩽ y ⩽ −6
2.6 Inequalities
10 Write true (T) or false (F) for each statement.
Part a has been done for you.
a
7 ⩾ y > 3 means the same as 3 < y ⩽ 7 T
b 15 > y ⩾ 5 means the same as 5 < y ⩽ 15
c
10 ⩾ y ⩾ −6 means the same as −6 ⩽ y ⩽ 10
d 8 > y ⩾ −8 means the same as −8 < y ⩽ 8
11 Samir combines two inequalities into one.
The two inequalities m < 10 and m > 2 can be combined like this:
Step 1:
m > 2 is the same as 2 < m
Step 2:
The two inequalities are now
2 < m and m < 10
I can write these as one inequality:
2 < m < 10
I can use a number line to help me:
m < 10
m > 2 or 2 < m
0
2
4
6
8
10
12
10
12
0
2
4
6
8
10
12
2 < m < 10
0
a
b
2
4
6
8
Use Samir’s method to combine each pair of inequalities
into one.
Use a number line to help if you want to.
i m < 15 and m > 8
ii m ⩽ 10 and m > 7
iii m > 0 and m < 6
Is it possible to write m > 14 and m < 8 as one inequality?
Give a reason for your answer.
Discuss your answer with a partner.
61
2 Expressions, formulae and equations
12 This is part of Sandeep’s classwork.
Question
1
1
a Show the inequality 2 2 ⩽ m < 7 4 on a number line.
b Write
i the smallest integer that m could be
ii the greatest integer that m could be
iii a list of the integer values that m could be.
Solution
a
1
2
3
4
5
6
7
8
b i smallest integer is 2
ii largest integer is 7
iii m could be 2, 3, 4, 5, 6, 7
a
b
Sandeep has made two mistakes. What are they?
For each of these inequalities, write
i the smallest integer that m could be
ii the greatest integer that m could be
ii a list of the integer values that m could be.
A 53 ⩽ m < 91
B −6 1 ⩽ m < −2 1
4
2
3
8
13 Zara is looking at this question.
Draw a line linking each inequality on the left with: the correct
smallest integer; the correct largest integer; and the correct list of
integers. The first one has been done for you: a and ii and D and Z
Smallest
integer
Largest
integer
a: 1.5 ⩽ x ⩽ 4
i: 3
A: 5
U: 4, 5, 6
b: 0.8 < x < 5.9
ii: 2
B: 6
V: −5, −4, −3, −2, −1, 0
c: 3 < x ⩽ 6.1
iii: −5
C: 1
W: 1, 2, 3, 4, 5
d: 2.2 ⩽ x < 3.9
iv: 4
D: 4
X: −4, −3, −2, −1, 0, 1
e: −4.5 < x < 1.1
v: 1
E: 0
Y: 3
f: −5.01 < x ⩽ 0
vi: −4
F: 3
Z: 2, 3, 4
Inequality
62
List of integers
2.6 Inequalities
Read Zara’s comments.
The method I am going
to use is to identify the smallest
integer for each inequality first. Then I’ll
identify the largest integer for each
inequality. Then I’ll work out the list of
integers for each inequality.
a
b
What do you think of Zara’s method? Can you improve
her method, or suggest a better one?
Use what you think is the best method to answer
the question.
Summary checklist
I can understand inequalities.
I can draw inequalities.
63
2 Expressions, formulae and equations
Check your progress
1
2
3
4
5
Jin thinks of a number, x.
Write an expression for the number Jin gets when he divides the number by 2
then adds 5.
a
Use the formula K = mg to work out K when m = 12 and g = 4 .
b Rearrange the formula K = mg to make m the subject.
c
Use your formula in part b to work out m when K = 75 and g = 10.
Expand
a
x(x + 3)
b 5y(7y − 4w)
Factorise
a
6x + 9
b 2y2 − 12y
Work out the value of x and y in this diagram.
All measurements are in centimetres.
6(x + 1)
y
3 + 16
20
3x + 21
6
Write the inequality shown by this number line. Use the letter x.
0
64
5
10
15
20
25
Project 1
Algebra chains
An algebra chain is a sequence of expressions where an input number is
substituted as the value of x in the first expression, and then the output of each
expression is substituted as the value of x in the following expression.
So, in the algebra chain below:
3 is substituted for x in the expression 4x − 10, giving the output 2
Then 2 is substituted for x in the expression 8x + 4, giving the output 20
3
4x − 10
8x + 4
20
Can you find a way to arrange the eight cards below into four algebra chains
that take the inputs 1, 2, 3 and 4 and give the outputs 40, 30, 20 and 10?
2x + 3
2(x + 1)
5x − 7
12 − 2x
3x − 4
7x + 5
3(x − 4)
8x − 2
1
40
2
30
3
20
4
10
Now choose any two cards and an input number.
Work out what the output of your algebra chain would be.
Tell a partner the input you chose and the output you got.
Can your partner work out which cards you used?
Can they still work it out when you make an algebra
chain with three cards?
65
3
Place value
and rounding
Getting started
1
2
3
4
Work out
a
4.5 × 10
b 18 × 10
c
82 × 100
d
Work out
a
70 ÷ 10
b 342 ÷ 10
c
140 ÷ 100
d
Write the correct answer, A or B, for each part.
Round each number to one decimal place.
a
7.23
A 7.2
B
7.3
b 12.45
A 12.4
B
12.5
c
0.793
A 0.7
B
0.8
Round each number to the given degree of accuracy.
a
4.587 (2 d.p.)
b 0.672 315 (4 d.p.)
c
54.788 99 (3 d.p.)
d 12.050 299 7 (5 d.p.)
Today, there are hundreds of different languages in use in the world.
However, all over the world people write numbers in the same way.
Everyone uses the decimal system to write numbers.
The decimal system was first developed in India.
•
It was adopted by Persian and Arab mathematicians in the
•
9th century.
It was introduced to Europe about 1000 years ago.
•
At first it was banned in some European cities because people
•
did not understand it and thought they were being cheated.
66
4.6 × 100
3120 ÷ 100
3 Place value and rounding
One system that was used in the past is Roman numerals.
2000 years ago the Romans used letters to represent numbers.
You can still see them on clock faces and carvings.
Their use continued in Europe for over 1000 years.
Here are some examples.
Roman
Decimal
III
3
VII
7
IX
9
XX C MCMXXX S
20 100
1930
0.5
Here are some calculations, multiplying or dividing by 10 (X)
or 100 (C), written using Roman numerals.
III × X = XXX V × C = D M ÷ C = X LV ÷ X = VS
Can you work out what D, M and L represent?
You can see that arithmetic with Roman numerals is very difficult.
You keep needing new letters.
The decimal system uses place value. That is why it only needs ten
symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It makes the arithmetic we do
today much easier than the arithmetic the Romans did!
67
3 Place value and rounding
3.1 Multiplying and dividing by 0.1
and 0.01
In this section you will …
Key words
•
multiply numbers by 0.1 and 0.01
decimal number
•
divide numbers by 0.1 and 0.01.
equivalent
calculations
1
The decimal number 0.1 is the same as 10
.
So when you multiply a number by 0.1, it has the same effect as dividing
the number by 10.
Example:
1
1
8 × 0.1 = 8 × 10
and 8 × 10
= 8 ÷ 10
inverse operation
The decimal number 0.01 is the same as 1 .
100
So when you multiply a number by 0.01, it has the same effect as
dividing the number by 100.
Example:
1
1
8 × 0.01 = 8 × 100
and 8 × 100
= 8 ÷ 100
When you divide a number by 0.1, it has the same effect as multiplying
the number by 10.
Example:
1
1
8 ÷ 0.1 = 8 ÷ 10 and 8 ÷ 10 = 8 × 10
When you divide a number by 0.01, it has the same effect as multiplying
the number by 100.
Example:
1
1
8 ÷ 0.01 = 8 ÷ 100 and 8 ÷ 100 = 8 × 100
Worked example 3.1
Work out
a
32 × 0.1
b
4.2 × 0.01
c
6 ÷ 0.1
d
4.156 ÷ 0.01
Answer
a
68
32 × 0.1 = 3.2
Multiplying by 0.1 is the same as dividing by 10,
and 32 ÷ 10 = 3.2
3.1 Multiplying and dividing by 0.1 and 0.01
Continued
b 4.2 × 0.01 = 0.042
Multiplying by 0.01 is the same as dividing by 100, and
4.2 ÷ 100 = 0.042
c
Dividing by 0.1 is the same as multiplying by 10, and 6 × 10 = 60
6 ÷ 0.1 = 60
d 4.156 ÷ 0.01 = 415.6
Dividing by 0.01 is the same as multiplying by 100, and
4.156 × 100 = 415.6
Exercise 3.1
1
Copy and complete these calculations.
All the answers are in the cloud.
a
20 × 0.1 = 20 ÷ 10 =
b
c
2000 × 0.1 = 2000 ÷ 10 =
d
Copy and complete these calculations.
All the answers are in the cloud.
a
400 × 0.01 = 400 ÷ 100 =
b 40 000 × 0.01 = 40 000 ÷ 100 =
c
40 × 0.01 = 40 ÷ 100 =
d 4000 × 0.01 = 4000 ÷ 100 =
2
200 × 0.1 = 200 ÷ 10 =
2 × 0.1 = 2 ÷ 10 =
20 200 0.2 2
0.4 40 400 4
Think like a mathematician
3
Work with a partner or in a small group to discuss this question.
Sofia and Arun are discussing the best way to work out 56 × 0.1
When I multiply
56 by 0.1, I move the
digits 5 and 6 one place to
the right in the place value
table. This gives me an
answer of 5.6
a
b
When I multiply
56 by 0.1, I move the
decimal point one place to
the left. This gives me an
answer of 5.6
Can you explain how both of these methods work?
Whose method do you prefer?
Describe how you would work out 56 × 0.01 using Sofia’s method
and using Arun’s method.
69
3 Place value and rounding
4
Work out
a
62 × 0.1
b 55 × 0.1
e
37 × 0.01
f
655 × 0.01
Copy and complete these calculations.
All the answers are in the cloud.
a
2 ÷ 0.1 = 2 × 10 =
5
b
20 ÷ 0.1 = 20 × 10 =
c
200 ÷ 0.1 = 200 × 10 =
c
g
125 × 0.1
750 × 0.01
d
h
3.2 × 0.1
4 × 0.01
200 2000 2 20
d 0.2 ÷ 0.1 = 0.2 × 10 =
Copy and complete these calculations.
All the answers are in the cloud.
a
4 ÷ 0.01 = 4 × 100 =
6
40 40 000 400 4000
b
40 ÷ 0.01 = 40 × 100 =
c
400 ÷ 0.01 = 400 × 100 =
d
0.4 ÷ 0.01 = 0.4 × 100 =
Think like a mathematician
7
Work with a partner or in a small group to discuss this question.
Win uses equivalent calculations to work out 3.2 ÷ 0.1 and 12.8 ÷ 0.01
This is what she writes.
a
b
8
70
i
3.2 3.2 × 10 32
So 3.2 ÷ 0.1 = 32 ÷ 1 = 32
=
=
0.1 0.1 × 10
1
ii
12.8 12.8 × 10 128 128 × 10 1280
=
=
=
=
0.01 0.01 × 10
0 .1
0.1 × 10
1
So 12.8 ÷ 0.01 = 1280 ÷ 1 = 1280
Can you explain how Win’s method works? Do you like her method?
Explain your answer.
Work out 0.45 ÷ 0.1 and 78 ÷ 0.01 using Win’s method.
Work out
a
7 ÷ 0.1
e
2 ÷ 0.01
b
f
4.5 ÷ 0.1
8.5 ÷ 0.01
c
g
522 ÷ 0.1
0.32 ÷ 0.01
d
h
0.67 ÷ 0.1
7.225 ÷ 0.01
3.1 Multiplying and dividing by 0.1 and 0.01
9
Jake works out 23 × 0.1 and 8.3 ÷ 0.01
He checks his answers by using an inverse operation.
i 23 × 0.1 = 23 ÷ 10 = 2.3
ii 8.3 ÷ 0.01 = 8.3 × 100 = 8300
Check: 2.3 × 10 = 23 ✓ Check: 8300 ÷ 100 = 83 ✘
Correct answer: 830
Work out the answers to these questions.
Check your answers by using inverse operations.
a
18 × 0.1
b 23.6 × 0.01
c
10 Which symbol, × or ÷, goes in each box?
a
6.7
0.1 = 67
b
4.5
d 550
0.01 = 5.5
e
0.23
11 Which of 0.1 or 0.01 goes in each box?
a
26 ×
= 0.26
b
3.4 ÷
d
0.6 ÷ 0.1
0.01 = 0.045
c
0.9
0.1 = 0.09
0.1 = 2.3
f
12
0.01 = 1200
= 34
c
0.06 ×
d 7÷
= 70
e
8.99 ×
= 0.899
f
52 ÷
12 A jeweller uses this formula to work out the mass of copper in green gold.
C = 0.1G
where: C is the mass of copper
G is the mass of green gold
a
Work out the mass of copper in 125 g of green gold.
The jeweller uses this formula to work out the mass of zinc
in yellow gold.
Z = 0.01Y
4.5 ÷ 0.01
where: Z is the mass of zinc
Y is the mass of yellow gold
= 0.0006
= 520
Tip
Remember, 0.1G
means 0.1 × G.
Tip
b Work out the mass of zinc in 80 g of yellow gold.
Remember,
The jeweller says, ‘I think that 10% of green gold is copper.’
‘percent’ means
c
Is the jeweller correct? Explain your answer.
‘out of 100’, so
d What percentage of yellow gold is zinc?
10% = 10 .
100
Explain how you worked out your answer.
13 a
Sort these expressions into groups of the same value.
There should be one expression left over.
B 240 × 0.1
C 2.4 ÷ 0.01
D 24 ÷ 0.01
E 2.4 ÷ 0.1
A 24 × 0.1
F
240 × 0.01 G 24 ÷ 0.1
H 0.24 ÷ 0.01 I
2400 × 0.1 J
0.24 ÷ 0.1
b Write two new expressions that have the same value as the expression that is left over.
71
3 Place value and rounding
14 Razi thinks of a number. He multiplies his number by 0.1, and then
divides the answer by 0.01.
Razi then divides this answer by 0.1 and gets a final answer of 12 500.
What number does Razi think of first?
Explain how you worked out your answer.
15 This is part of Harsha’s homework.
Question
Write one example to show that this statement is not true.
‘When you multiply a number with one decimal place
by 0.01 you will always get an answer that is smaller
than zero.’
Answer
345.8 × 0.01 = 3.458 and 3.458 is not smaller than zero
so the statement is not true.
Write down one example to show that each of these statements
is not true.
a
When you multiply a number other than zero by 0.1 you will
always get an answer that is greater than zero.
b When you divide a number with one decimal place by 0.01
you will always get an answer that is greater than 100.
Look at these two questions.
a
56 × 0.1
b 3.2 ÷ 0.01
Explain to a partner the methods you would use to work out the
answers to these questions.
Explain why you would use these methods.
Does your partner use the same methods?
If they use different methods, do you understand their methods?
Summary checklist
I can multiply numbers by 0.1 and 0.01.
I can divide numbers by 0.1 and 0.01.
72
3.2 Rounding
3.2 Rounding
In this section you will …
Key words
•
decimal places
(d.p.)
round numbers to a given number of significant figures.
You already know how to round decimal numbers to a given number
of decimal places (d.p.).
You also need to know how to round numbers to a given number
of significant figures (s.f.).
The first significant figure in a number is the first non-zero digit in
the number.
For example:
In the number 450, 4 is the first significant figure, 5 is the second
•
significant figure and 0 is the third significant figure.
In the number 0.008 06, 8 is the first significant figure, 0 is the
•
second significant figure and 6 is the third significant figure.
To round a number to a given number of significant figures, follow
these steps:
Look at the digit in the position of the degree of accuracy. The
•
‘degree of accuracy’ is the number of significant figures you are
working to. So, if you have been asked to round to 3 significant
figures, look at the third significant figure in the number.
If the number to the right of this digit is 5 or more, increase the
•
digit by 1. If the number is less than 5, leave the digit as it is.
degree of
accuracy
round
significant figures
(s.f.)
Worked example 3.2
a Round 4286 to one significant figure.
b Round 0.080 69 to three significant figures.
c Round 0.7963 to two significant figures.
Answer
a
4286 = 4000 (1 s.f.)
The first significant figure is 4. The digit to the right of it is 2.
2 is less than 5, so 4 stays the same.
Replace the 2, the 8 and the 6 with zeros to keep the place value consistent.
In this case, rounding to one significant figure is the same as rounding
to the nearest 1000.
The letters ‘s.f.’ stand for ‘significant figure’.
73
3 Place value and rounding
Continued
b 0.080 69 = 0.080 7 (3 s.f.)
The first significant figure is 8, the second is 0 and the
third is 6.
The digit to the right of the 6 is 9.
9 is more than 5 so round the 6 up to 7.
You must keep the zeros at the start of the number to keep
the place value consistent.
In this case, rounding to 3 s.f. is the same as rounding to 4 d.p.
c
0.7963 = 0.80 (2 s.f.)
The first significant figure is 7 and the second is 9.
The digit to the right of the 9 is 6.
6 is more than 5 so round the 9 up to 10. This has the effect of
rounding ‘79’ up to ‘80’.
You must keep the zero after the 8 to show that you have
rounded to 2 s.f.
In this case, rounding to 2 s.f. is the same as rounding to 2 d.p.
because the first significant figure is also the first decimal place.
Exercise 3.2
1
2
74
Round each of these numbers to one significant figure (1 s.f.).
Choose the correct answer: A, B or C.
a
352
A 4
B 40
C 400
b 7.291
A 7
B 7.3
C 7.29
c
11 540
A 12 000
B 10 000
C 11 000
d 0.0087
A 9
B 0.09
C 0.009
Round each of these numbers to two significant figures (2 s.f.).
All the answers are in the cloud.
a
243
b 0.235
0.0024 0.24 2.4 24
c
24.15
d 0.002 380 1
240 2400
e
2396
f
2.3699
3.2 Rounding
Think like a mathematician
3
Work with a partner to discuss the answers to this question.
This is part of Harry’s homework.
Question
Round these numbers to 2 s.f.
a 45 150
b 0.032 84
Answer
a 45
b 0.03
Harry has rounded one large number and one small number to
two significant figures.
Both of his answers are wrong.
a
b
c
4
5
6
Explain the mistakes he has made.
Write the correct answers.
What must you remember to do when you round a large number
to a given number of significant figures?
What must you remember to do when you round a small number
to a given number of significant figures?
Round each number to the stated number of significant figures (s.f.).
a
135 (1 s.f.)
b 45 678 (2 s.f.)
c
18.654 (3 s.f.)
d 0.0931 (1 s.f.)
e
0.7872 (2 s.f.)
f
1.409 48 (4 s.f.)
g 985 (1 s.f.)
h
0.697 (2 s.f.)
i
8.595 (3 s.f.)
Which answer is correct: A, B, C or D?
a
2569 rounded to 1 s.f.
A 2
B 3
C 2000
D 3000
b 47.6821 rounded to 3 s.f.
A 47.6
B 47.682 C 47.7
D 48.0
c
0.0882 rounded to 2 s.f.
A 0.08
B 0.088
C 0.09
D 0.1
d 3.089 62 rounded to 4 s.f.
A 3.089 B 3.0896 C 3.09
D 3.090
e
19.963 rounded to 3 s.f.
A 2
B 20
C 20.0
D 19.96
Round the number 209.095 046 to the stated number of significant figures (s.f.).
a
1 s.f.
b 2 s.f.
c
3 s.f.
d 4 s.f.
e
5 s.f.
f
6 s.f.
75
3 Place value and rounding
Activity 3.2
You are going to write a question for a partner to answer.
On a piece of paper, write a question of your own similar to Question 4.
Make sure:
you have parts a to d
•
you use four different numbers
•
•
you ask for the numbers to be rounded to different degrees of accuracy
•
you write the answers on a different piece of paper.
Exchange questions with your partner. Answer their question, then exchange back
and mark each other’s work. Discuss any mistakes that have been made.
Use a calculator to work out the answer to 262 + 58 .
Write all the numbers on your calculator display.
b Round your answer to part a to the stated number of
significant figures (s.f.).
i 1 s.f.
ii 2 s.f.
iii 3 s.f.
iv 4 s.f.
v 5 s.f.
vi 6 s.f.
8
At a football match there were 63 475 Barcelona supporters
and 32 486 Arsenal supporters.
How many supporters were there altogether?
Give your answer correct to two significant figures.
9
Ahmad has a bag of peanuts that weighs 150 g.
There are 335 peanuts in the bag.
Work out the average (mean) mass of one peanut.
Give your answer correct to one significant figure.
10 The speed of light is approximately 670 616 629 miles per hour.
This formula changes a speed in miles per hour into a speed in
metres per second.
7
a
metres per second =
miles per hour
2.25
Work out the speed of light in metres per second.
Give your answer correct to three significant figures.
76
3.2 Rounding
11 Zara and Sofia are looking at this question.
Tip
Choose a
sensible degree
of accuracy for
the context.
Think about how
accurate you
need your answer
to be.
Work out the area of this rectangle.
0.87 m
9.6 m
Give your answer to an appropriate degree of accuracy.
Read Zara’s and Sofia’s comments.
area = 9.6 × 0.87
= 8.352 m2
I think we should
give 8.352 m2 as
our answer.
9.6 and 0.87
are both written to 2 s.f.,
so I think we should round
our answer to 2 s.f. I think
our answer should
be 8.4 m2.
What do you think? Explain your answers.
12 A rugby club sells, on average, 12 600 tickets to a match each week.
The average cost of a ticket is $26.80
How much money does the club get from ticket sales, on average,
each week?
Round your answer to an appropriate degree of accuracy.
13 This formula is often used in science.
F = ma
Work out the value of a when F = 32 and m = 15.
Round your answer to an appropriate degree of accuracy.
Tip
Change the
subject of the
formula first.
77
3 Place value and rounding
14 This is part of Jake’s homework.
He works out an estimate by rounding each number to one
significant figure.
Question
× 576
a Work out an estimate of 0.238
39.76
b Work out the accurate value.
c Compare your estimate with the accurate value.
Answer
a 0.238 ≈ 0.2, 576 ≈ 600, 39.76 ≈ 40
0.2 × 600 = 120 and 120 ÷ 40 = 3
Estimate = 3
b 0.238 × 576 = 137.088
137.088 ÷ 39.76 = 3.45 (3 s.f.)
Accurate value = 3.45 (3 s.f.)
cMy estimate is close to the accurate value, so my
accurate answer is probably correct.
Tip
The symbol
≈ means ‘is
approximately
equal to’.
Follow these steps for each of the calculations below.
i
Use Jake’s method to work out an estimate of
the answer.
ii
Use a calculator to work out the accurate answer.
Give this answer correct to three significant figures.
iii Compare your estimate with the accurate answer.
Decide if your accurate answer is correct.
0.3941 × 196 b
4.796
a
4732 + 9176 c
19.5166
2.764 × 84.695
d
9.687 − 4.19
58 432 × 0.08
0.2 × 348
Summary checklist
I can round numbers to a given number of significant figures.
78
3 Place value and rounding
Check your progress
1
2
3
4
Work out
a
90 × 0.1
b 552 × 0.1
c
135 × 0.01
d 8 × 0.01
e
6 ÷ 0.1
f
23.5 ÷ 0.1
g 5.2 ÷ 0.01
h
0.68 ÷ 0.01
Which calculation, A, B, C or D, gives a different answer from the others?
Show your working.
B 5.2 ÷ 0.01
C 0.052 ÷ 0.1
D 52 × 0.01
A 5.2 × 0.1
Round each of these numbers to the given degree of accuracy.
a
78.023 (2 s.f.)
b 0.067 91 (3 s.f.)
c
1.549 62 (4 s.f.)
d 12 452 673 (5 s.f.)
Use a calculator to work out the answer to 892
48
Give your answer correct to two significant figures.
79
4
Decimals
Getting started
1
2
Write the correct symbol, < or >, between each pair of
decimal numbers.
a
4.5
4.1
b 6.57
6.68
c
10.52
10.59
d
Here are four decimal number cards.
0.763
3
4
21.75
80
0.761
2.781
0.759
Write the numbers in order of size, starting with the smallest.
Write true (T) or false (F) for each of these.
a
6 × 0.1 = 0.6
b 12 × 0.7 = 0.84
c
0.03 × 2500 = 750
d 0.04 × 25 = 1
Match each blue question card with the correct yellow
answer card.
12 × 1.8
5
0.756
2.784
19 × 1.2
22.4
Work out.
a
12.3 ÷ 3
c
152.88 ÷ 6
9 × 2.5
21.6
25 × 0.87
22.5
b
d
22.8
44.1 ÷ 7
28.86 ÷ 12
320 × 0.07
Tip
Remember:
< means ‘is less
than’ and
> means ‘is
greater than’.
4 Decimals
There are many situations in everyday life where we have to
calculate with decimals.
If you go to an airport, you will always see a currency
exchange desk. This is where you can change money from
one currency to another. The exchange rates are shown on
a board. They tell you how much of one currency you can
exchange for another.
An architect is a person who designs buildings, and in many
cases, supervises their construction. They need to measure
accurately and also calculate using decimals.
An incorrect decimal calculation could result
in a disaster for the building!
81
4 Decimals
4.1 Ordering decimals
In this section you will …
Key words
•
compare
compare and order decimals.
To order decimal numbers, compare the whole-number part first.
When the numbers you are ordering have the same whole-number part,
look at the decimal part and compare the tenths, then the hundredths,
and so on.
Look at the three decimal numbers on
the right.
1 Highlight the whole numbers.
You can see that 7.4 is the smallest
number, so 7.4 goes first.
2The other two numbers both have
8 units, so highlight the tenths.
3They both have the same number
of tenths, so highlight the
hundredths.
4You can see that 8.518 is smaller
than 8.56, so in order of size the
numbers are:
Mass
1000 g = 1 kg
1000 kg = 1 t
order
term-to-term rule
Tip
8.56
7.4
8.518
8.56
7.4
8.518
7.4
8.56
8.518
7.4
8.56
8.518
7.4
8.518
8.56
When you order decimal measurements, you must make sure they are
all in the same units.
You need to remember these conversion factors.
Length
10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km
decimal number
Capacity
1000 ml = 1 l
The number of
digits after the
decimal point is
the number of
decimal places
(d.p.) in the
number.
Tip
When you
compare decimal
numbers you
can use these
symbols.
= means ‘is equal
to’
≠ means ‘is not
equal to’
> means ‘is
bigger than’
< means ‘is
smaller than’
82
4.1 Ordering decimals
Worked example 4.1
a
Write these decimal numbers in order of size.
5.6, 4.95, 5.68, 5.609
b Write the correct symbol, = or ≠, between these measures.
7.5 m
75 cm
c
4.5 kg
450 g
Write the correct symbol, > or <, between these measures.
Answer
a
4.95, 5.6, 5.609, 5.68
The smallest number is 4.95 as it has the smallest whole-number
part.
The other three numbers have the same whole-number part and
the same number of tenths, so compare the hundredths.
5.68 has 8 hundredths compared with 5.6 and 5.609 which
have 0 hundredths, so 5.68 is the biggest number.
Now compare the thousandths in 5.6 and 5.609
5.6 has 0 thousandths and 5.609 has 9 thousandths, so 5.6 is
smaller.
b 7.5 m ≠ 75 cm
There are 100 cm in 1 m. 7.5 m × 100 = 750 cm, so the measures
are not equal. Use the ‘≠’ symbol.
c
There are 1000 g in 1 kg. 4.5 kg × 1000 = 4500 g, so 4.5 kg is greater.
Use the ‘>’ symbol.
4.5 kg > 450 g
Exercise 4.1
1
2
Write these decimal numbers in order of size, starting with
the smallest.
They have all been started for you.
a
5.49, 2.06, 7.99, 5.91
2.06,
,
,
b
3.09, 2.87, 3.11, 2.55
2.55,
c
12.1, 11.88, 12.01, 11.82
11.82,
,
,
,
,
d 9.09, 8.9, 9.53, 9.4
8.9,
,
,
Write the correct sign, < or >, between each pair of numbers.
a
4.23
4.54
b 6.71
6.03
c
0.27
0.03
d
27.9
27.85
e
8.55
8.508
f
5.055
5.505
83
4 Decimals
Think like a mathematician
3
Maya uses this method to order decimals.
Question
Write these numbers in order of size, starting with the smallest.
26.5 26.41 26.09 26.001 26.92
Answer
The greatest number of decimal places in the numbers is 3.
Step 1: Write all the numbers with 3 decimal places.
26.500 26.410 26.090 26.001 26.920
Step 2: Compare only the numbers after the decimal point.
500 410 090 001 920
Step 3: Write these numbers in order of size.
001 090 410 500 920
Step 4: Now write the decimal numbers in order.
26.001 26.09 26.41 26.5 26.92
Discuss the answers to these questions with a partner.
a
b
c
4
5
Use your preferred method to write these decimal numbers in order
of size, starting with the smallest.
a
23.66, 23.592, 23.6, 23.605
b 0.107, 0.08, 0.1, 0.009
c
6.725, 6.78, 6.007, 6.71
d 11.02, 11.032, 11.002, 11.1
Write the correct sign, = or ≠, between each pair of measurements.
a
6.7 l
670 ml
b 4.05 t
4500 kg
c
6
84
Do you understand how Maya’s method works?
Do you like Maya’s method?
Do you prefer Maya’s method to the method shown in Worked example 4.1?
Explain your answer.
0.85 km
850 m
d
0.985 m
985 cm
e
14.5 cm
145 mm
f
2300 g
0.23 kg
Write the correct sign, < or >, between each pair of measurements.
a
4.5 l
2700 ml
b 0.45 t
547 kg
c
3.5 cm
345 mm
d
0.06 kg
550 g
e
7800 m
0.8 km
f
0.065 m
6.7 cm
Tip
Start by
converting
one of the
measurements
so that both
measurements
are in the
same units.
4.1 Ordering decimals
7
Write these measurements in order of size, starting
with the smallest.
a
2.3 kg, 780 g, 2.18 kg, 1950 g
b 5.4 cm, 12 mm, 0.8 cm, 9 mm
c
12 m, 650 cm, 0.5 m, 53 cm
d 0.55 l, 95 ml, 0.9 l, 450 ml
e
6.55 km, 780 m, 6.4 km, 1450 m
f
0.08 t, 920 kg, 0.15 t, 50 kg
Tip
Make sure all the
measurements
are in the same
units before
you start to
order them.
Think like a mathematician
8
Look at Arun’s solution to this question.
Write these decimal numbers in order of size, starting with the smallest.
−4.52 −4.31 −4.05 −4.38
All the numbers start with
−4 so I will just compare the
decimal parts: 52, 31, 05 and 38.
In order, they are 05, 31, 38, 52.
So the order is −4.05, −4.31,
−4.38, −4.52
Discuss the answers to these questions with a partner.
a
b
9
Is Arun correct? Explain your answer.
What do you think is the best method to use to order negative decimal
numbers?
Write the correct sign, < or >, between each pair of numbers.
a
−4.27
−4.38
b −6.75
−6.25
c
−0.2
−0.03
d −8.05
−8.9
10 Write these decimal numbers in order of size, starting with
the smallest.
a
−4.67, −4.05, −4.76, −4.5
b −11.525, −11.91, −11.08, −11.6
Tip
Draw a number
line to help if you
want to.
85
4 Decimals
11 Shen and Mia swim every day. They record the distances they
swim each day for 10 days.
These are the distances that Shen swims each day.
250 m 1.25 km 0.5 km 2500 m 2 km 1.75 km
750 m 1500 m 25 km 0.75 km
a
Shen has written down one distance incorrectly.
Which one do you think it is? Explain your answer.
These are the distances that Mia swims each day.
1.2 km
240 m 0.4 km 1.64 km 820 m 640 m
0.2 km 1.42 km 960 m 0.88 km
b
Mia says that the longest distance she swam is more than
eight times the shortest distance she swam.
Is Mia correct? Explain your answer.
Shen and Mia swim in different swimming pools. One of the
swimming pools is 25 m long. The other swimming pool is 20 m
long. Shen and Mia always swim a whole number of lengths.
c
Who do you think swims in the 25 m swimming pool?
Explain how you made your decision.
12 Each of the cards describes a sequence of decimal numbers.
86
A
First term: 0.5
Term-to-term rule: ‘add 0.5’
B
First term: 0.15
Term-to-term rule: ‘multiply by 2’
C
First term: −1.7
Term-to-term rule: ‘add 1’
D
First term: 33.6
Term-to-term rule: ‘divide by 2’
E
First term: 1.25
Term-to-term rule: ‘add 0.25’
F
First term: 10.45
Term-to-term rule: ‘subtract 2’
a
b
Work out the fifth term of each sequence.
Write the numbers from part a in order of size, starting with
the smallest.
4.2 Multiplying decimals
13 Zara is looking at this inequality:
3.27 ⩽ x <3.34
If x is a number with
two decimal places,
there are 8 possible
numbers that x
could be.
Is Zara correct? Explain your answer.
14 y is a number with three decimal places, and –0.274 < y ⩽ –0.27
Write all the possible numbers that y could be.
In this exercise you have written positive decimal numbers, negative
decimal numbers and decimal measurements in order of size.
a
Which have you found the easiest? Explain why.
b
Which have you found the hardest? Explain why.
Summary checklist
I can compare and order positive decimal numbers.
I can compare and order negative decimal numbers.
I can compare and order decimal measurements.
4.2 Multiplying decimals
In this section you will …
Key words
•
estimation
multiply decimals by whole numbers and decimals.
mentally
Follow these steps when you multiply a decimal by a whole number
or a decimal.
First, work out the multiplication without the decimal points.
•
Finally, put the decimal point in the answer. There must be the
•
same number of digits after the decimal point in the answer as
there were in the question.
place value
written method
87
4 Decimals
Worked example 4.2
Work out mentally
i 0.02 × −12
ii 3.2 × 0.04
b Use a written method to work out 5.96 × 0.35
Check your answer using estimation.
a
Answer
a
i 2 × −12 = −24
0.02 × −12
= −0.24
Work out the multiplication without the decimal points.
ii 32 × 4 = 128
3.2 × 0.04 = 0.128
Work out the multiplication without the decimal points.
5 9 6
b
×
3 5
Put the decimal point back in the answer. There are 2 digits after
the decimal point in the question, so there must be 2 digits after
the decimal point in the answer.
Put the decimal point back in the answer. There are 3 digits in total
after the decimal points in the question, so there must be 3 digits
after the decimal point in the answer.
Work out the multiplication without the decimal points.
(Use your preferred method for multiplication.)
2 9 8 0
Work out 596 × 5
+ 1 7 8 8 0
Work out 596 × 30
2 0 8 6 0
5.96 × 0.35 = 2.0860
5.96 × 0.35 = 2.086
Add together the two lines above.
Put the decimal point back in the answer. There are 4 digits
in total after the decimal points in the question, so there must be
4 digits after the decimal point in the answer.
The zero at the end of the number can now be ignored.
Check
Round both of the numbers in the question to one significant figure.
6 × 0.4 = 2.4 ✓
2.4 is close to 2.086, so the answer is probably correct.
Exercise 4.2
1
88
Use a mental method to work out
a
0.1 × −8
b 0.2 × 3
d 0.7 × 8
e
0.9 × −4
c
0.3 × −7
4.2 Multiplying decimals
2
Use a mental method to work out
a
−6 × 0.03
c
−18 × 0.001
All the answers are in the cloud.
Here are five calculation cards.
3
b
d
−9 × 0.2
−20 × 0.9
A
0.6 × −12
B
0.039 × −180
D
0.44 × −16
E
0.04 × −182
a
b
−18 −0.18
C
−1.8
−0.018
0.85 × −9
Work out the answers to the calculations on the cards.
Write the answers in order of size, starting with the smallest.
Think like a mathematician
4
Work with a partner or in a small group to discuss this question.
Look at what Arun says.
I don’t understand
why 0.2 × 0.3 is 0.06
and not 0.6
Zara shows Arun this pattern.
2×3=6
0.2 × 3 = 0.6
0.2 × 0.3 = 0.06
0.2 × 0.03 = 0.006
0.2 × 0.003 = 0.0006
How can you use the place value
of the digits in 0.2 and 0.3 to
explain to Arun why the
answer is 0.06 and not 0.6?
89
4 Decimals
5
a
Copy and complete these patterns.
i 2×4=8
ii
0.2 × 4 = 3 × 5 = 15
0.3 × 5 =
0.2 × 0.4 =
0.3 × 0.5 =
0.2 × 0.04 =
0.3 × 0.05 =
0.2 × 0.004 = 0.3 × 0.005 =
b Work out.
i 0.1 × 0.09
ii 0.6 × 0.8
iv 0.03 × 0.05
v 0.12 × 0.3
Fill in the missing spaces in this spider diagram.
All the calculations give the answer in the middle.
All the answers are in the yellow rectangle on the right.
6
0.6 3 0.6
3
0.4
0.9 3
3 0.12
iii 0.07 × 0.4
vi 0.06 × 0.11
0.3
0.36
63
3 0.01
36
0.04
39
3 1.2
43
0.06
0.09
Think like a mathematician
7
Work with a partner or in a small group to discuss this question.
Jan works out that 42 × 87 = 3654
a
b
c
90
Use this information to write the answers to these multiplications.
i 42 × 8.7
ii 42 × 0.87
iii 4.2 × 87
Tip
iv 4.2 × 8.7
An example
Explain why your answers to ai and aiii are the same,
calculation in
and why your answers to aii and aiv are the same.
part c could be
Use generalising to describe a method that someone
0.42 × 0.87 or
could follow to work out the answer to any change in
0.042 × 8.7
42 × 87 = 3654 that involves decimals.
4.2 Multiplying decimals
8
a
b
Work out 158 × 46
Use your answer to part a to write the answers to these multiplications
i 15.8 × 46
ii 158 × 4.6
iii 15.8 × 4.6
iv 1.58 × 4.6
v 15.8 × 0.46
vi 1.58 × 0.046
Activity 4.2
a
On a piece of paper, write a question of your own similar to Question 8.
On a different piece of paper, write the answers to your question.
Exchange questions with a partner and work out the answers to their question.
Exchange back and mark each other’s work. Discuss any mistakes.
b
c
9
Sam uses this method to work out and check her answer.
Question
Work out 0.67 × 4.28
Answer
First work out 67 × 428
×
400
20
60
24 000 1200
140
7
2 800
Totals 26 800 1340
8
480
56
536
26 800
1 340
+
536
28 676
So 0.67 × 4.28 = 2.8676
Check
Round 0.67 to 0.7
Round 4.28 to 4
0.7 × 4 = 2.8, which is close to 2.8676
a
b
c
Write the advantages and disadvantages of Sam’s method.
Can you improve her method?
Which method do you prefer to use to multiply decimals?
Write why you prefer this method.
10 Work out these multiplications.
Show how to check your answers.
a
6.7 × 9.4
b 0.56 × 8.3
c
0.23 × 8.15
d 0.69 × 0.254
Tip
Use estimation
to check your
answers by
rounding all the
numbers in the
question to one
significant figure.
91
4 Decimals
11 This is part of Syra’s homework.
Question
Work out
a 0.45 × 2.8
Answer
a 12.6
b 7.8 × 0.0093
c 0.065 × 0.043
b 0.072 54
c 0.027 95
Use estimation to check if Syra’s answers could be correct.
If you don’t think they are correct, explain why.
12 A vet needs to work out how much medicine
to give to a cat.
The instructions for the medicine say:
Give 7.3 mg (medicine) per kg (mass of cat)
The cat has a mass of 5.8 kilograms.
a
Work out an estimate of the number of
milligrams (mg) of medicine that the cat needs.
b Calculate the accurate number of
milligrams (mg) of medicine that the cat needs.
13 A coin is made from silver and copper.
The mass of the coin is 4.2 g.
You can use this formula to work out the mass of silver in the coin:
mass of silver = 0.775 × mass of coin
a
b
Work out an estimate of the mass of the silver in this coin.
Calculate the accurate mass of the silver in this coin.
Summary checklist
I can multiply decimals by whole numbers.
I can multiply decimals by decimals.
92
4.3 Dividing by decimals
4.3 Dividing by decimals
In this section you will …
Key words
•
divide decimals by numbers with one decimal place.
equivalent
calculation
When you divide a number by a decimal, you can use the place value
of the decimal to work out an easier equivalent calculation. An easier
equivalent calculation is to divide by a whole number instead of a
decimal.
For example, you can write 5.67 ÷ 0.7 as 5.67
0.7
Multiplying the numerator and denominator of the fraction by 10 gives
reverse
calculation
short division
5.67 × 10
= 56.7
0.7 × 10
7
This makes an equivalent calculation that is much easier to do because
dividing by 7 is much easier than dividing by 0.7.
Worked example 4.3
Work out
a −32 ÷ 0.2
b 3.468 ÷ 0.8
Answer
a
−32 ÷ 0.2 = −32
0.2
−32 × 10
= −320
0.2 × 10
2
−320 ÷ 2 = −160
b 3.468 ÷ 0.8 = 3.468
0.8
3.468 × 10
34.68
=
0.8 × 10
8
8
3
4
4
.
.
3 3 5
2
6 28 40
First of all write −32 ÷ 0.2 as a fraction.
Multiply numerator and denominator by 10.
Finally work out the division.
First, write 3.468 ÷ 0.8 as a fraction.
Multiply the numerator and denominator by 10.
Now use short division to work out 34.68 ÷ 8
So 3.468 ÷ 0.8 = 4.335
93
4 Decimals
Exercise 4.3
1
Copy and complete these divisions.
2
a
2.4 ÷ 0.4 = 2.4
2.4 × 10
=
0.4 × 10
=
b
7.2 ÷ 0.9 = 7.2
7.2 × 10
=
0.9 × 10
=
c
−42 ÷ 0.6 = −42
−42 × 10
=
0.6 × 10
=
d
−45 ÷ 0.5 = −45
−45 × 10
=
0.5 × 10
=
0.4
0.6
0.9
0.5
Which of these calculation cards is the odd one out?
Explain why.
A
6.3 ÷ 0.9
B
1.4 ÷ 0.2
D
4.8 ÷ 0.6
E
5.6 ÷ 0.8
Think like a mathematician
3
Discuss this question with a partner or in a small group.
Arun is working out 3 ÷ 0.6. This is what he says.
I understand that I must
do 3 = 3 × 10 = 30 = 5 because that
0.6 0.6 × 10
6
makes the calculation a lot easier.
But I don’t understand why we don’t divide the answer
at the end by 10 as we multiplied the numbers at the
start by 10.
I think the answer should be
5 ÷ 10 = 0.5.
What explanation
could you give to Arun to show that he is wrong?
94
C
4.9 ÷ 0.7
4.3 Dividing by decimals
4
Work out
a
0.92 ÷ 0.4
b
5.74 ÷ 0.7
c
−774 ÷ 0.9
d −288 ÷ 0.3
Artur pays $1.08 for a piece of string 0.8 m long.
Artur uses this formula to work out the cost of the
string per metre.
5
cost per metre =
price of piece
length of piece
Tip
Follow these steps
1Write the division as a
fraction.
2Multiply the numerator
and denominator by 10.
3Use short division to work
out the answer.
What is the cost of the string per metre?
Think like a mathematician
6
7
Discuss in pairs or small groups the answer to this question.
What calculations can you do to check that the answer to a division
is probably correct?
For example, how can you check that 20.504 ÷ 0.8 = 25.63 is probably correct?
This is part of Jamal’s homework.
Question
i Estimate an answer to 498 ÷ 0.6
ii Work out the answer to 498 ÷ 0.6
Answer
498 × 10 = 4980
0.6 × 10
6
i Round 4980 to 1 s.f. to give 5000
5000 ÷ 6 ≈ 800 because 6 × 800 = 4800
Estimate = 800
8 3 0
ii
6 4 19 18 0
Answer = 830
Tip
Remember, the
symbol ≈ means
‘is approximately
equal to’.
Use Jamal’s method to work out each of these divisions.
i
First, estimate the answer.
ii
Then calculate the accurate answer.
a
27.6 ÷ 0.3
b −232 ÷ 0.4
c
306 ÷ 0.9
−33972 ÷ 0.6
d −483 ÷ 0.7
e
43.76 ÷ 0.8
f
95
4 Decimals
8
Isla works out 50.46 ÷ 1.2. This is what she writes.
50.46 × 10 504.6
1.2 × 10 = 12
4 2 .
12 5 0 24 .
9
5
6 60
So, 50.46 ÷ 1.2 = 42.5
a
Explain the mistake that Isla has made.
b Write the correct answer.
Raffa works out 461.7 ÷ 1.8. This is what he writes.
461.7 × 10 4617
1.8 × 10 = 18
This is my 18 times table:
1 2 3
18 36 54
4 5
6
7 8 9
72 90 108 126 144 162
I can use the table to work out the division like this.
2 5 6 r9
So, 461.7 ÷ 1.8 = 256 remainder 9
18 4 6 101 117
a
b
10 a
What should Raffa have done, instead of stopping the division
and writing ‘remainder 9’?
Work out the correct answer.
Copy and complete the table below showing the 19 times table.
1 2 3
19 38 57
b
c
11 a
5
6
7
8
9
Use the table to help you work out 59.375 ÷ 1.9
Show how to check your answer to part b is correct.
Use estimation with a reverse calculation.
Complete the table below showing the 25 times table.
1 2 3
25 50 75
96
4
4
5
6
7
8
9
Tip
In part c, your
rounded answer
to part b × 2
should equal
about 60.
4.3 Dividing by decimals
b
Helen buys a piece of wood for $58.90
The piece of wood is 2.5 m long.
Work out the cost per metre of the wood.
c
Show how to check your answer to part b is correct.
Use estimation with a reverse calculation.
12 The diagram shows a rectangle with an area of 50.15 m2.
The width of the rectangle is 3.4 m.
3.4 m
Tip
You can use the
same formula as
in Question 5.
Area = 50.15 m2
length
Work out the length of the rectangle.
13 Work with a partner to answer this question.
a
Balsem works out that 425 × 27 = 11 475
Use this information to work out
i 11 475 ÷ 27
ii 11 475 ÷ 425
iii 11 475 ÷ 2.7
iv 11 475 ÷ 42.5
b Explain the method you used to work out the answers to part a.
c
Work out
i 1147.5 ÷ 2.7
ii 114.75 ÷ 2.7
iii 11.475 ÷ 2.7
iv 1.1475 ÷ 2.7
d Explain the method you used to work out the answers to part c.
e
Check your answers with some other learners in your class to see if you agree.
If you disagree about any of the answers, discuss where mistakes have been made.
14 This is part B of Marcus’s homework.
Question
Work out 1.798 ÷ 0.7
Give your answer to 1 d.p.
Answer
I must give
my answer to 1 d.p. so
I only need to work out the
division to 2 d.p. and then
I can round.
1.798 × 10 17.98
0.7 × 10 = 7
2 . 5 6 …
7 1 17 . 39 48 60
1.798 ÷ 0.7 = 2.56… = 2.6 (1 d.p.)
97
4 Decimals
Use Marcus’s method to work out these calculations.
Round each of your answers to the given degree
of accuracy.
a
3.79 ÷ 0.6 (1 d.p.)
b 82.35 ÷ 1.1 (2 d.p.)
c
−5689 ÷ 2.3 (1 d.p.)
Tip
You only need
to work out the
division to one
decimal place
more than
the degree of
accuracy you
need.
Summary checklist
I can divide decimals by numbers with one decimal place.
4.4 Making decimal
calculations easier
In this section you will …
Key word
•
factor
simplify calculations containing decimals.
When you are calculating using decimals, there are several methods
you can use to make a calculation easier, such as
using the place value of the decimal
•
breaking a decimal into parts using factors
•
using the correct order of operations.
•
98
4.4 Making decimal calculations easier
Worked example 4.4
Work out mentally
a
(0.6 − 0.2) × 0.8 b
22 × 0.9 c
4 × 12.6 × 2.5 d
12 × 0.15
Answer
a
0.6 − 0.2 = 0.4
Work out the brackets first.
0.4 × 0.8 = 0.32
4 × 8 = 32 so 0.4 × 0.8 = 0.32
b 22 × 0.9 = 22 × (1 − 0.1)
c
Replace 0.9 with (1 − 0.1)
= 22 × 1 − 22 × 0.1
Work out 22 × 1 and 22 × 0.1
= 22 − 2.2
Now subtract 2.2 from 22 by
= 22 − 2 − 0.2
subtracting the 2 first
= 20 − 0.2
and then subtracting the 0.2
= 19.8
This gives an answer of 19.8
4 × 12.6 × 2.5 = 4 × 2.5 × 12.6
You can do the multiplications in any order.
= 10 × 12.6
Notice that 4 × 2.5 = 10, so rearrange the question.
= 126
Replace 4 × 2.5 with 10 and work out 10 × 12.6
d 12 × 0.15 = 12 × 0.5 × 0.3
Use factors of 15 to make the calculation easier:
0.15 = 0.5 × 0.3
= 6 × 0.3
12 × 0.5 is the easier calculation so do this first:
= 6 × 3 ÷ 10
(12 × 0.5 = 6) Then work out 6 × 0.3
= 18÷ 10
Replace 0.3 with 3 ÷ 10, then do 3 × 6 = 18
= 1.8
Finally 18 ÷ 10 = 1.8
Exercise 4.4
1
2
Copy and complete the workings for these questions.
a
(0.3 + 0.4) × 0.2 =
× 0.2
b (1.1 − 0.8) × 0.4 =
× 0.4
= =
Use the same method as in Question 1 to work out
a
(0.7 + 0.1) × 0.6
b (0.3 + 0.9) × 0.5
c
(0.6 − 0.4) × 1.2
d (1.8 − 1.5) × 1.1
99
4 Decimals
3
Complete the workings to make these calculations easier.
a
52 × 0.9 = 52 × (1 − 0.1)
b 8.3 × 0.9 = 8.3 × (1 − 0.1)
= 52 × 1 − 52 × 0.1 = 8.3 × 1 − 8.3 × 0.1
=
− =
−
= =
Use the same method as in Question 3 to work out
a
28 × 0.9
b 17 × 0.9
c
4
4.9 × 0.9
Think like a mathematician
5
Work in pairs or small groups to discuss this question.
Look again at the working and method for Question 3a.
a
b
c
6
7
8
The diagram shows a rectangle.
6m
The width is 6 m and the length is 9.9 m.
Work out the area of the rectangle.
9.9 m
A new medicine was given to 3200 patients.
Tip
It made 99% of them feel better. For the other 1%
the medicine made no difference.
99% = 0.99, so
3200 × 0.99 =
Work out how many of the patients the new medicine made better.
Work out the answers to these calculations. Look for different ways
to make the calculations easier.
There are some tips in the cloud.
4 × 2.5 =
a
2.5 × 32.7 × 4
0.2 × 2.3
0.2 = 0.2 × 10
b
=
c
d
e
100
Describe a similar method you could use to work out 52 × 9.9
Describe a similar method you could use to work out 52 × 0.99
Use your methods to work out
i
26 × 9.9
ii
26 × 0.99
Check your methods and answers with other learners in the class.
Discuss any mistakes that have been made.
0.1
(420 + 360) × 0.7
8 × 1.32 × 2.5
(720 − 120) × 0.07
0 .1
0.1 × 10
780 × 0.7 = 78 × 7 =
8 × 2.5 =
4.4 Making decimal calculations easier
9
Complete the workings. Use factors to make these calculations
easier.
a
24 × 0.35 = 24 × 0.5 × 0.7
b 32 × 0.45 = 32 × 0.5 × 0.9
=
× 0.7 =
× 0.9
=
× 7 ÷ 10 =
× 9 ÷ 10
=
÷ 10 =
÷ 10
= =
10 Use the same method as in Question 9 to work out
a
80 × 0.15
b 116 × 0.25
11 This is part of Pedro’s classwork.
Question
Work out 280 × 0.12
Answer
Use 0.12 = 0.2 × 0.6
280 × 0.12 = 280 × 0.2 × 0.6
= 280 × 2 ÷ 10 × 6 ÷ 10
= 560 ÷ 10 × 6 ÷ 10
= 56 × 6 ÷ 10
= 336 ÷ 10
= 33.6
Use Pedro’s method to work out
a
180 × 0.14 (use 0.14 = 0.2 × 0.7)
b 120 × 0.16
c
250 × 0.18
d 450 × 0.24
Tip
For part b, you
could use 0.2 ×
0.8 or 0.4 × 0.4
Think like a mathematician
12 In questions 9 and 10, all the decimals had a factor of 0.5
Did you always multiply by 0.5 first? If you did, explain why you did this.
Look at Question 11. Pedro has used a factor of 0.2 and multiplied by this first.
Why do you think he has not used 0.5 in these questions?
101
4 Decimals
13 A grandmother gives $12 600 to be shared between her
grandchildren.
Abdul gets 0.35 of the money.
Zhi gets 0.25 of the money.
Paula gets 0.22 of the money.
Yola gets 0.18 of the money.
a
Work out how much they each get.
b Show how you can check that your answers are correct.
14 Zara and Sofia are trying to solve this puzzle.
What is the missing number in this calculation?
I think the missing
number is 0.4
(
32 × 0.8 −
0.02
) = 800
I think the missing
number is 0.3
Who is correct? Explain how you worked out your answer.
Summary checklist
I can use different methods to make decimal calculations easier.
102
4 Decimals
Check your progress
1
2
3
4
5
6
7
Write the correct sign, < or >, between these.
a
6.75
6.7
b −4.87
−4.81
c
Write these decimals in order of size, starting with the smallest.
−3.482 −3.449 −3.6 −3.44 −3.06
Use a mental method to work out
a
0.2 × 0.4
b 0.7 × 0.3
Work out 0.57 × 4.62
Sam works out that 365 × 24 = 8760
Use this information to write the answers to these calculations.
a
365 × 2.4
b 36.5 × 2.4
c
d 8760 ÷ 24
e 8760 ÷ 2.4
f
Work out 5.4 ÷ 0.9
a
Complete the table below showing the 15 times table.
1 2 3
15 30 45
4
5
6
7
8
0.65 kg
67 g
3.65 × 0.24
876 ÷ 2.4
9
Use the table to help you work out 38.205 ÷ 1.5
Show how to check your answer to part b is correct.
Use a reverse calculation.
Work out the answers to these calculations.
Use the methods you have learned
to make the calculations easier.
a
(0.7 − 0.4) × 0.12
b 0.9 × 27
c
14 × 0.35
b
c
8
103
Project 2
Diamond decimals
The Decimal Diamond on the right was generated from the diamond on the left.
Work out how the values in the squares were calculated.
4.3
4.3
7.2
10.1
6.7
10.1
5.5
8.4
6.7
3.3
w
x
y
z
6.7
5
3.3
Look at this Decimal Diamond. Choose a value for the top circle
w, and choose a decimal g, between 0 and 1, which will fill the
gaps between the values, so that:
w+g=x
x+g=y
y+g=z
Work out the values that go in the five squares.
Try this a few times with different values for w and g.
What do you notice?
If someone told you the values they chose for w and g, can you
find a quick way to work out the values in their squares?
Now look at this Decimal Diamond. Choose a number to go in the
square marked a, and choose a decimal g, between 0 and 1, which
will fill the gaps between the values, so that:
a+g=b
b+g=c
c+g=d
d+g=e
Can you work backwards to find out what numbers need to go in
the circles to complete the Decimal Diamond?
Is it possible to complete a Decimal Diamond if you are given only
two of the entries?
104
a
b
c
d
e
5
Angles and
constructions
Getting started
1
2
Two angles of a triangle are 55° and 70°.
a
Work out the third angle.
b What is the name of this type of triangle? Choose the correct word.
equilateral isosceles right-angled perpendicular
a
Work out the size of angle D.
b Work out the value of x.
C
D
127°
50°
42°
A
B
x°
154° x°
3
There are two parallel lines in this diagram.
One of the angles is 76°.
Copy the diagram and write in the values of the other angles.
You do not have to draw the angles accurately.
4
aUse the measurements
P
shown to make an accurate
100°
drawing of this shape.
b Measure the length
5 cm
of RS.
4 cm
Q
140°
76°
2.5 cm
R
S
105
5 Angles and constructions
The sum of the angles in a triangle is 180°. Can you remember the first
time you were shown this?
You may have measured the angles and added them.
You may have cut a triangle out of paper and folded it.
This does not prove that the sum of the angles of any triangle is 180°.
It only shows that it is true for the triangles you have drawn and that
it is a reasonable conclusion.
A proof is a logical argument in which a reason is given for each step.
Over 2000 years ago the Greek mathematician Euclid wrote a book
called The Elements. He used logical arguments to prove many facts in
geometry and arithmetic. His book was the most successful textbook
ever written. It is still in print today.
Euclid started by defining basic things such as a point and a straight
line. He also made a set of statements which he thought everyone could
agree with. These were called axioms.
An example of one of his axioms is:
Things that are equal to the same thing are equal to one another.
From this simple starting point, he proved many complicated results.
In this unit you will look at several proofs.
5.1 Parallel lines
In this section you will …
•
use geometric vocabulary for equal angles formed when
lines intersect.
This diagram shows two straight lines.
Angles a and c are equal. They are called
a b
vertically opposite angles.
d c
Angles b and d are equal. They are also
vertically opposite angles.
Vertically opposite angles are equal.
Angles a and b are not equal (unless they
are both 90°).
They add up to 180° because they are angles on a straight line.
106
Key words
alternate angles
corresponding
angles
geometric
transversal
vertically
opposite angles
5.1 Parallel lines
The arrows on this diagram show that these
two lines are parallel. The perpendicular
distance between parallel lines is the same
wherever you measure it.
Here, there is a third straight line crossing
two parallel lines. It is called a transversal.
Where the transversal crosses the parallel
lines, four angles are formed.
Angles a and e are called corresponding
angles. Angles d and h are also corresponding
angles. So are b and f. So are c and g.
Corresponding angles are equal.
Angles d and f are called alternate angles.
Angles c and e are also alternate angles.
Alternate angles are equal.
These are important properties of parallel
lines.
To help you remember:
Tip
We usually
label pairs of
parallel lines with
matching arrows.
a
b
d
c
•
for vertically opposite angles, think of the letter X
•
for corresponding angles, think of the letter F
•
for alternate angles, think of the letter Z.
e
f
h
g
Tip
Alternate angles
are always
between the
parallel lines.
Exercise 5.1
1
2
Look at the diagram.
a
Write four pairs of corresponding angles.
b Write two pairs of alternate angles.
aOne angle of 62° is marked in the
diagram.
Copy and complete these sentences.
iBecause corresponding angles
= 62°
are equal, angle
iiBecause alternate angles are equal,
= 62°
angle
b Write the letters of a pair of vertically
opposite angles.
t
p
s
w
q
u
v
r
a
62°
d
b
c
107
5 Angles and constructions
3
4
5
The sizes of two angles are marked in
the diagram.
a
Which other angles are 105°?
b Which other angles are 75°?
Angle APY is marked on the diagram.
Complete these sentences.
a
APY and CQY are .................. angles.
b APY and XQD are .................. angles.
c
APX and ......... are corresponding
A
angles.
d CQX and ......... are alternate angles. C
e
CQP and ......... are vertically
opposite angles.
PQ and RS are parallel lines.
Q
c 136°
S
P
a d
b
R
6
Find the sizes of angles a, b, c and d.
Give a reason in each case.
Look at this diagram.
C
A
X
50°
S
130°
40°
T
140°
Y
D
B
Explain why AB and CD cannot be parallel lines.
108
105° 75°
p q
r s
t u
X
P
B
D
Q
Y
5.1 Parallel lines
7
This diagram has three parallel lines and a transversal.
a
Write a set of three corresponding angles that includes
angle f.
b Write a pair of alternate angles that includes angle c.
c
Write another pair of alternate angles that includes angle c.
Look at this diagram.
Write whether these are corresponding angles, alternate
angles or neither.
a
a and d
f h
b b and f
e g
c
c and g
b d
a c
d d and e
e
a and h
8
a b
d c
e f
h g
i j
l k
Think like a mathematician
9
Arun gives this explanation of why angles h and d are equal.
a
b
h g
h = b because they
are corresponding angles.
c
d
f e
b = d because they are
alternate angles.
Therefore h = d.
a
b
Arun’s explanation is not correct. Write a correct version.
Write a different explanation of why h = d that does not
use corresponding angles.
10 AB and CD are parallel.
a
Give a reason why a and d are equal.
b Give a reason why b and e are equal.
c
Use your answers to a and b to show that the sum of
the angles of triangle ABC must be 180°.
C
d c e
a
A
D
b
B
109
5 Angles and constructions
11 Show that the sum of the angles of triangle XYZ must be 180°.
X
Tip
Use your answer
to Question 10 as
a guide.
Y
Z
Think like a mathematician
12 ABCD is a trapezium.
Two sides are extended to make the triangle AXB.
a
b
c
X
Show that the angles of triangles ABX and DCX are the same size.
Show that angles A and D of the trapezium
add up to 180°.
D
C
D
What can you say about angles B and C
of the trapezium?
A
B
A
Give a reason for your answer.
13 ABCD is a parallelogram.
a
Show that opposite angles of the
parallelogram are equal.
b Compare your answer to part a
D
with a partner’s answer. Can you
improve his or her answer? Can you
improve your own answer?
Tip
A
B
Extend the
sides of the
parallelogram.
C
Imagine you have to explain corresponding angles to someone who does not
know about them. How can you convince him or her that corresponding angles
are equal?
Summary checklist
I can recognise vertically opposite angles and I know that they are equal.
I can identify corresponding angles and alternate angles between parallel lines.
110
C
B
5.2 The exterior angle of a triangle
5.2 The exterior angle
of a triangle
In this section you will …
Key word
•
learn to identify the exterior angle of a triangle
•
use the fact that the exterior angle of a triangle is equal to
the sum of the two interior opposite angles.
exterior angle of
a triangle
A
Here is a triangle ABC.
The side BC has been extended to X.
Angle ACX is called the exterior angle of the
triangle at C.
B
C
The angles marked at A and B are the angles
opposite C.
We know that a + b + c = 180°, the sum of
A
the angles in a triangle.
a
So a + b = 180° − c
Also d + c = 180°, the sum of the angles on
b
c d
a straight line.
B
C
So d = 180° − c
Compare these two results and you can see that d = a + b
This shows that:
The exterior angle of a triangle = the sum of the two interior
opposite angles.
This is true for any triangle.
X
X
111
5 Angles and constructions
Worked example 5.2
Work out x and y.
B
40° 45°
y°
C
105°
D
x°
A
Answer
x is an exterior angle of triangle ABD so x = 40 + 105 = 145°
y is an exterior angle of triangle BCD
Angle BDC = 180° − 105° = 75° because angles on a straight line = 180°
So y = 75 + 45 = 120°
Exercise 5.2
The diagrams in this exercise are not drawn to scale.
1
Calculate the sizes of angles a, b and c.
80°
20°
45°
2
134°
b
20°
a
c
86°
aWork out each of the exterior angles
shown in this triangle.
b
Work out the size of the exterior
angle x in this quadrilateral.
x
c
70°
65°
67° b
a
112
80°
5.2 The exterior angle of a triangle
3
An exterior angle of a triangle is 108°.
One of the interior angles of the triangle is 40°.
a
Work out the other two interior angles of the triangle.
b Work out the other two exterior angles of the triangle.
4
PBC is a straight line. AQ is parallel to PC.
a
Explain why y = c
b Explain why x = a + y
c
Use your answers to a and b to prove
that the exterior angle at B of triangle
ABC is the sum of the two interior
opposite angles.
5
6
108°
A
a
Q
y
x b
B
P
c
C
DX is parallel to BC.
ZD is parallel to AB.
B
BDY is a straight line.
a
Explain why angles BAD and ADZ are equal.
b Explain why angles ABD and ZDY are equal.
A
c
Use the diagram to prove that the angle sum
of quadrilateral ABCD is 360°.
Do not use the fact that the angle sum of a triangle is 180°.
C
30°
C
150°
160°
X
D
Y
Z
AB and CD are straight lines.
A
40°
D
20°
B
Explain why the angles cannot all be correct.
7
8
Look at the diagram.
a
Explain why d = a + c
b Write similar expressions for e and f.
c
Show that the sum of the exterior angles of a triangle is 360°.
ABC is an isosceles triangle.
AB = AC.
AB is parallel to DE.
Angle ABC = 68°
Work out the size of angle EDC.
Give a reason for your answer.
f
a
d
b
c e
B
68°
E
A
D
C
113
5 Angles and constructions
9
This pentagon is divided into a triangle and a quadrilateral.
a
Show that the angle sum of the pentagon is 540°.
b Compare your explanation with a partner’s.
Do you both have a similar explanation?
10 PQRS is a parallelogram.
P
a
Explain why x must be 22°.
b Work out angle y.
11 ABCD is a parallelogram.
Show that p + q = r
39°
Q
y
22°
S
A
p
x
R
B
r
D
q
C
12 a
Show that w + y = a + b + c + d
w
b
Show that w + x + y + z = 360°
w
c
x
d
a b
z
y
y
13 Work out angles a, b and c.
Tip
40°
80°
c
a
95°
b
114
Use the exterior
angle property of
a triangle for each
angle.
5.3 Constructions
Think like a mathematician
14 a
b
c
Explain why x = b + d
Explain why y = c + e
Show that the sum of the angles in the
points of the star, a + b + c + d + e = 180°
a
e
x
y
b
d
c
Summary checklist
I can identify the exterior angles of a triangle.
I can use the fact that the exterior angle of a triangle is equal to the sum
of the two interior opposite angles.
5.3 Constructions
In this section you will …
Key words
•
construct triangles
arc
•
learn how to draw the perpendicular bisector of a line
bisector
•
learn how to draw the bisector of an angle.
construct
(geometry)
You need to be able to draw a triangle when you know some of
the sides and angles.
You can do this using computer software. You can also do it using a
ruler and compasses.
Here are four different examples of how to construct triangles.
hypotenuse
115
5 Angles and constructions
1
When you know two angles and the side
between them, this is known as ASA.
C
50°
60°
A
8 cm
B
Step 1: Draw the side.
Draw an angle at one end.
A
60°
Step 2: Draw the angle at the
other end.
Where the two lines cross is
the third vertex of the triangle.
C
A
2
B
8 cm
50°
60°
8 cm
When you know two sides and the angle
between them, this is known as SAS.
B
C
10 cm
42°
A
12 cm
Step 1: Draw the angle first.
42°
A
116
B
5.3 Constructions
Step 2: Open your compasses to 12
cm. Put the point of the
compasses on A and draw an
arc to mark B. Mark C in a
similar way. Draw the side BC.
C
Tip
An arc is part
of a circle.
10 cm
42°
A
3
B
12 cm
When you know the three sides but no
angles, this is known as SSS.
C
4.5 cm
4 cm
A
5 cm
B
A
5 cm
B
Step 1: Draw one side.
Open your compasses to the
length of a second side.
Put the point of the
compasses on one end of the
side and draw an arc.
Step 2: Open your compasses to the
length of a second side.
Put the point of the
compasses on the other end of
the side and draw another arc.
Where the arcs cross is the
third vertex. Draw the other
two sides.
C
4.5 cm
A
4 cm
5 cm
B
117
5 Angles and constructions
4
When one angle is a right angle, and you
know the length of the hypotenuse and one
other side, this is known as RHS.
C
Tip
The hypotenuse is
the side opposite
the right angle.
9 cm
7 cm
A
B
Step 1: Draw the side.
Draw a right angle at one end.
7 cm
A
Step 2: At the other end, draw an arc
equal to the hypotenuse.
Draw the third side.
B
C
9 cm
7 cm
A
B
There are two other constructions that you need to be able to do using a
ruler and compasses:
1
Construct the bisector of a line segment.
This is a line through the mid-point of the line segment and
perpendicular to it.
Step 1: D
raw the line segment. Open the
­compasses to about the same length
as the line. (You do not need to
measure this exactly.)
Draw arcs from one end of the line
on both sides of the line.
118
A
B
5.3 Constructions
Step 2: Do the same thing at the other
end of the line segment.
Do not change the angle
between the arms of the
compasses.
A
2
B
Construct the bisector of an angle.
This is a line that divides an angle
into two equal parts.
Step 1: Open the compasses to a
few centimetres.
You do not need to
measure this.
Put the point of the compasses
on the angle and draw arcs that
cross each of the lines.
Step 2: Put the compass point on each
of the crosses and draw an arc
between the two lines.
Do not change the angle
between the arms of the
compasses. Draw a line through
the angle and the last cross.
This is the perpendicular
bisector of the angle. The two
angles marked are equal.
Tip
When you do any
construction, do
not rub out your
construction lines.
Draw them faintly
and leave them
on your drawing.
119
5 Angles and constructions
Exercise 5.3
1
a
Draw an accurate copy of this triangle.
C
42°
65°
6 cm
A
2
b
a
B
Measure the length of AC and BC.
Draw an accurate copy of this triangle.
Y
36°
5 cm
Z
100°
X
3
b
a
Measure the length of XY and XZ.
Draw an accurate copy of this triangle.
Q
7 cm
P
50°
10 cm
R
4
b
a
Measure angle Q
Draw an accurate copy of this triangle.
D
10 cm
E
117°
6 cm
F
b
120
Measure angle F.
5.3 Constructions
Activity 5.3
All the angles and sides of this triangle are shown.
Choose either 2 sides and the angle between them
a
(SAS) or 2 angles and the side between them (ASA).
Use your chosen measurements to draw an
accurate copy of the triangle.
Measure the three values you did not choose.
b
Was your drawing accurate? If not, where did you
go wrong?
5
73°
5.9 cm
4.6 cm
63°
44°
6.3 cm
The hypotenuse of a right-angled triangle is 12.5 cm.
One of the other sides is 10 cm.
a
Make an accurate drawing of the triangle.
b Measure the third side.
c
Measure the other two angles.
The sides of a triangle are 7 cm, 8.5 cm and 9.7 cm.
a
Make an accurate drawing of the triangle.
b Measure the largest angle of the triangle.
The sides of a triangle are 5.8 cm, 7.8 cm and 7.1 cm.
a
Make an accurate drawing of the triangle.
b Give your triangle to a partner to check the accuracy of
your drawing.
If necessary, correct your drawing.
6
7
Think like a mathematician
8
Two sides of a right-angled triangle are 10.5 cm and 8.3 cm.
Zara and Arun give different answers.
The third side is
6.4 cm
The third side is
13.4 cm
Use accurate drawings to show that both of them could be correct.
121
5 Angles and constructions
9
a
b
Draw this diagram accurately.
Construct the perpendicular
bisector of AB.
The perpendicular bisector
of AB intersects AC at D.
Label D on your diagram and
measure AD.
c
C
7.5 cm
A
39°
10 cm
B
Think like a mathematician
10 RST is a triangle. RS = 5 cm, RT = 6 cm.
a
b
If ST = 9 cm, use a diagram to show
that angle R is obtuse.
Write the size of angle R.
If angle R is obtuse what can
you say about the length of ST?
Give reasons for your answer.
11 a
b
c
Draw this triangle accurately.
Construct the bisector of angle A.
The bisector of angle A intersects
BC at X.
Mark X on your triangle and
measure BX.
12 Read what Marcus and Sofia say.
If you draw the
perpendicular bisectors
of each side of a
triangle, they intersect
at a single point.
a
b
c
d
122
R
5 cm
6 cm
S
T
A
7 cm
B
10 cm
8 cm
C
The bisectors of
each angle of a
triangle meet at a
single point.
Draw a triangle. Construct the perpendicular bisector of each
side to test Marcus’s theory.
Look at the triangles that other learners have drawn.
Do you think Marcus is correct?
Draw another triangle. Construct the bisector of each angle
to test Sofia’s theory.
Look at the triangles that other learners have drawn.
Do you think Sofia is correct?
5.3 Constructions
You can draw a triangle if you know three side (SSS) or 2 sides and the
angle between them (SAS) or two angles and the side between them (ASA).
Can you draw a triangle if you only know the 3 angles?
Why is this different from the other examples?
Summary checklist
I can draw a triangle when I know two angles and the side between them (ASA).
I can draw a triangle when I know two sides and the angle between them (SAS).
I can draw a triangle when I know the three sides (SSS).
I can draw a right-angled triangle when I know the hypotenuse and
one other side (RHS).
I can draw the perpendicular bisector of a line segment.
I can draw the bisector of an angle.
123
5 Angles and constructions
Check your progress
1
2
aWrite the correct words to complete
these sentences.
i c and h are ................... angles.
ii f and l are ................... angles.
iii g and k are ................... angles.
b Explain why e = j + k
c
Explain why c = i + j
ABCD is a trapezium.
C
D
105°
64°
B
3
A
Work out the angles of the trapezium.
Give a reason for each answer.
Work out x and y.
50°
118°
60°
x
y
124
b
a
e
c
d
g
h
f
i
j k
l
n m
5 Angles and constructions
4
Show that triangle ABC is isosceles.
D
A
40°
B
5
6
70°
C
E
The sides of a triangle are 5.1 cm, 6.8 cm and 8.5 cm.
a
Draw the triangle.
b Construct the bisector of the smallest angle.
c
The angle bisector divides one of the other sides into two parts.
How long is each part?
a
Draw triangle ABC accurately.
A
6 cm
75°
B
8 cm
C
b Construct the perpendicular bisector of AC.
The perpendicular bisector meets BC at P.
c
Measure PC.
125
6
Collecting data
Getting started
1
2
3
4
Give an example of
a
discrete data
b continuous data
c
categorical data.
You want to choose a sample of 3 boys and 3 girls from your class.
Describe three different ways to do this.
a
Write one advantage of a large sample size.
b Write one disadvantage of a large sample size.
You want to find the number of brothers and sisters of a group of children.
Describe two different ways you could collect this data.
In the United States of America, elections are held every two years to
choose members for the House of Representatives. This is one of the
groups of people who run the country.
Each American state chooses representatives. The number they can
choose depends on how many people live in that state. The more people
live in a state, the more representatives they can choose. It is therefore
very important to keep accurate records of the number of people living
in each state.
To maintain accurate records, a census is held every 10 years. A census
is a way of collecting data. It is a questionnaire that must be filled in by
every household in the USA.
126
6 Collecting data
There was a census in 2010. The questionnaire had only 10 questions,
which people were able to answer in about 10 minutes. The results of
the 2010 census showed that there were 308 745 538 people living
in the USA.
There are 435 representatives in the House of Representatives.
This means there is one representative for roughly every 710 000 people
in the USA.
The map shows the number of representatives in each state of the USA.
You can see that California has the largest number of representatives,
even though it is not the largest state by area. This is because more
people live in California than in any other state in the USA.
WA
(12)
NH (4)
MT (3)
ND (3)
OR (7)
ID (4)
SD (3)
WY (3)
NV (6)
CA
(55)
AZ
(11)
CO (9)
NM (5)
HI (3)
MN
(10)
WI
(10)
NY
(29)
MI
(16)
IA (6)
PA (20)
IL IN OH
(20) (11) (18) WV
VA
MO
(5)
(13)
KS (6)
(10)
KY (8)
NC (15)
TN (11)
AR
OK (7)
SC
(6)
(9)
MS AL GA
(9) (16)
TX
LA (6)
NE (5)
UT
(6)
VT (3)
(38)
ME
(4)
MA (11)
RI (4)
CT (7)
NJ (14)
DE (3)
MD (10)
DC (3)
(8)
AK (4)
FL
(29)
In this unit, you will learn more about collecting data.
127
6 Collecting data
6.1 Data collection
In this section you will …
•
select a method to collect data to answer a number of
linked questions
•
consider the different types of data
•
consider different sampling methods.
To answer questions in statistics, you must collect data.
First, you decide which data you need to collect. You need to know if it
is discrete, continuous or categorical.
Next, you must decide how to collect the data. If you need to question
people, you could use a questionnaire that they fill in by themselves.
Alternatively, you could interview them and write down the answers.
Sometimes, you need to make observations. For example, you might
record times or count vehicles. In this case, you need a sheet to record
your observations.
You may not be able to interview or give a questionnaire to everyone.
In this case, you need to take a sample. You must think carefully about
the best way to choose your sample.
Whenever you collect data, you need to choose a method and explain
why you think that method is the best one to use.
Worked example 6.1
The head teacher wants answers to the following questions:
• Are learners happy with the length of lessons? Should they be longer or shorter?
• Are learners happy with the length of their lunch break?
Your task is to investigate these questions using a sample of the learners in the school.
a What data will you collect?
b How will you choose your sample?
c How will you collect the data?
Give reasons for your answers.
128
6.1 Data collection
Continued
Answer
a
ou will need to collect data for: opinion about length of lessons; opinion about whether
Y
lessons should be longer, shorter or the same; opinions about length of lunch break. There
could be differences of opinion between boys and girls and between younger and older
learners so you also need to collect data about the age and gender of the learners sampled.
b If there are equal numbers of boys and girls in the school, you could choose 2 girls and
2 boys from each tutor group. Use the register and choose two names at random (for
example, in one tutor group, you might speak to the third and sixth girl and the third
and sixth boy in the register). A random sample is more likely to be representative.
c Individual interviews would be best because this will allow you to get an answer from
everyone. You could also talk to a group of students together so they can discuss their
ideas. However, this might take too long or be inconvenient, in which case you could
use a questionnaire.
Exercise 6.1
1
2
How would you collect data to answer these questions? State the type of data each time.
a
When a drawing pin is dropped, is it more likely to land point up or point down?
b How many people visit a particular shop before 09:00?
c
How many brothers and sisters do the members of your class have?
d How long do learners spend doing homework each night?
e
What is the average number of words in a sentence in a book?
f
How long do learners take to get to school each day?
Ahmad uses a gym. He asks these questions.
Do people visit this gym every week?
•
Do people come at a particular time of day?
•
Is there a difference between the habits of men and women?
•
He interviews a sample of people at the gym.
a
What data does he need to collect?
b What type of data is it?
c
Here is his data collection sheet.
Name
How often do you Do you prefer to visit in the
visit the gym?
morning or the evening?
What is wrong with this data collection sheet?
129
6 Collecting data
d
e
Design a better data collection sheet.
He asks the first people who come into the gym each morning
for a week. Why is this not a good way to choose his sample?
Describe a better way.
A cinema manager asks these questions.
How often do people visit the cinema?
•
Do younger people visit more often than older people?
•
What type of film do people like?
•
a
What data is required?
b Describe two ways to collect the data.
c
The manager decides to give a questionnaire to a sample of
customers. She gives it to all the customers on one night.
Why is this not a good way to choose the sample?
d Describe how the manager can get a representative sample.
e
Compare your answer to part d with another group’s answer.
Can you improve your answer? Can you improve theirs?
Xavier has a simple puzzle for children.
He asks these questions:
How long does it take to solve the puzzle?
•
Can girls solve it more quickly than boys?
•
Can older children solve it more quickly than younger ones?
•
a
What data must Xavier collect? What type of data is it?
b Xavier gives the puzzle to a sample of children.
Design a data collection sheet for Xavier.
3
4
Activity 6.1
You are going to plan and trial the collection of data from learners in your school.
Think of three questions you can ask about aspects of school life.
a
b What data do you need to answer these questions?
c
Decide how you will collect the data you need.
You could use a questionnaire or a data collection sheet.
d Test your data collection method on a few learners. Does it work well?
Can you improve it?
You need to choose a sample of learners. How can you do this?
e
What is the best way and why?
f
Compare your answers with those of another group.
Can you see a way to improve your work or their work?
130
6.1 Data collection
5
6
Sofia surveys cars using a busy road.
She wants to answer these questions:
•
What percentage of drivers are male?
•
What percentage of cars carry only one person?
•
What is the average number of people in a car?
a
What data does Sofia need to collect?
b What type of data is this?
c
Design a data collection sheet for Sofia.
Anders is comparing two books, X and Y.
He thinks book X is harder to read than book Y.
a
What things make a book hard or easy to read?
b What statistical questions can you ask to compare how easy it is
to read each book?
c
What data can you collect to answer your questions?
d How would you collect data to answer your questions?
e
Choose a book and use it to test your data collection method.
Does it give you the data you need? Can you improve your
method?
f
Compare your answers with those of another group. If you have
chosen different approaches, which do you prefer?
131
6 Collecting data
In this exercise, you have thought about collecting data to answer questions.
a
What is the most important thing to remember?
b
Explain in your own words what a representative sample is.
Summary checklist
I can choose a method to collect data to answer a number of questions and
justify my choice.
I can decide on the best way to select a sample and justify my choice.
6.2 Sampling
In this section you will …
•
understand the advantages of different sampling methods.
The word ‘population’ usually refers to the people living in a town or
country. In a statistical investigation, however, it means the people you
are interested in.
If you are investigating your class, the population means the people in
your class. If you are investigating your school, the population is all the
learners in your school.
Often you cannot question the whole population. In this case, you need
to choose a sample. There are different ways to choose a sample. In any
investigation you need to decide on the best way to choose your sample.
Sometimes, your investigation is not about people. For example, you
might be investigating the traffic going past your school. If you collect
data about some of the vehicles you are still taking a sample. In this
case, the population is all the vehicles passing your school.
132
Key word
population
6.2 Sampling
Worked example 6.2
A palace has visitors every day. You are doing a survey to find out what visitors think
of the visit. You want to talk to a sample of 100 people. The survey must be done on
one day.
a What factors might affect a person’s opinion of the visit?
b Describe how you can choose a sample. Take account of the factors you identified
in part a. Give any advantages and disadvantages of your method.
Answer
a Age and gender are two factors.
b Choose about five different age bands. Choose people arriving and ask them what age
band they are in as one of the questions. Make sure you speak to an equal number of
men and women. Select people at several different times during the day.
An advantage is that this includes a range of people. If you spoke to mostly older
people or mostly men, for example, the answers would not represent all the visitors.
A disadvantage is that this method will take longer than simply asking the first 100
people you see. You might need to speak to more than 100 people to make sure you
cover all the different age bands.
Exercise 6.2
1
A manager wants to find customers’ opinions about his shop.
The manager wants to choose a sample of 50 customers.
a
The sample could be the first 50 customers in the shop after
it opens.
i Write one advantage of this method.
ii Write one disadvantage of this method.
b The sample could be 10 customers chosen at random every
2 hours until 50 have been chosen.
i Write one advantage of this method.
ii Write one disadvantage of this method.
c
The manager thinks that the opinions of men and women
could be different.
Explain how he should choose the sample to take account
of this.
d Can you think of another factor that might affect customers’
opinions?
133
6 Collecting data
Think like a mathematician
2
3
4
5
134
Work with a partner to answer this question.
You are going to find the lengths of the words in a novel. Choose a book to use.
You want a sample of 50 words.
a
Describe three different ways of choosing a
sample of 50 words.
b Use one of your methods from part a to
sample 50 words from your chosen novel.
Use a tally chart to record the number of
letters in each word.
c
Did your sampling method give you a sample
that was representative of the whole book?
Could you improve your method?
d Try one of your other sampling methods.
e
Compare your first method with your second
method. Can you improve the second method?
Was one better than the other?
Zalika is investigating the number of people in each car on a busy
road. She predicts that most cars will contain only one person,
the driver.
Zalika says, ‘I will start at 08:00 and observe 200 cars.’
a
Write one advantage and one disadvantage of
Zalika’s method.
b Describe a better way to take a sample of 200 cars.
Explain why your method is better than Zalika’s.
You have been asked to carry out an investigation. You want to find
out if learners would like to change the school homework policy.
You will choose a sample of about 50 learners.
Explain how you will choose your sample. Explain why you have
chosen this method.
Arun and Sofia carry out a survey of parents about school
homework. One conjecture is that parents want more homework.
To test this, each asks this question of a sample of 50 parents:
Is the amount of homework your child gets too little / about right /
too much ? (choose one)
6.2 Sampling
6
7
8
Frequency
Frequency
This chart shows the results:
25
a
Do Arun’s results support this conjecture?
20
Give a reason for your answer.
Arun
15
b Do Sofia’s results support this conjecture?
10
Sofia
Give a reason for your answer.
5
c
Give a possible reason why the results of
0
Too
About
Too
the two surveys are different.
little
right
much
A large factory has a restaurant where employees
go for lunch. Arun is investigating ways to improve
the restaurant.
a
Give one disadvantage of Arun’s method.
I will give a
b Describe a better way of doing the survey.
questionnaire to
the first 50 workers
Here is a conjecture about the cars using a
visiting
the restaurant
particular road:
this lunchtime.
The modal number of people in a car is 1.
Marcus, Zara and Sofia each do a survey of cars on the road.
Tip
They count the number of people in each car, including the driver.
Each person does the survey for 15 minutes.
The mode is the
They do their surveys at different times of day.
number with the
highest frequency.
The results are in the graph on the right.
a
Do the results of each survey support this conjecture?
b Describe any similarities or differences between
60
the surveys.
50
c
Why do the samples give different results?
Marcus
40
An examiner has marked 250 examination papers.
Zara
30
Sofia
To check the accuracy of her marking, a sample of
20
10 papers will be re-marked.
10
a
Describe three different ways of choosing the sample.
0 1 2 3 4 5 6
b Which of the three ways do you think is the best?
Number of people
Explain why you think so.
Summary checklist
I can describe different sampling methods.
I understand the advantages and disadvantages of different sampling methods.
135
6 Collecting data
Check your progress
You want to answer these questions:
•
Are girls or boys better at throwing a ball through a basketball hoop?
•
Does height make a difference?
1
What data could you collect to answer these questions?
What type of data is it?
2
How would you collect the data you need?
3
Explain how you could select a sample from your school.
Justify your choice.
136
7
Fractions
Getting started
1
2
Write the correct symbol, = or ≠, between each pair of fractions.
1
10
5
a
b 2
c
11
5
3
21
4
6
b
21
2
9
4
c
2
5
22 + 34
9
b
9
3
7
12 + 5 3
3
4
Work these out. Give each answer in its simplest form.
a
5
2
3
35
Work these out. Give each answer as a mixed number in its simplest form.
a
4
3
4
Write the correct symbol, < or >, between each pair of fractions.
a
3
3
2
2 7
×
3 8
b
1
× 600
8
b
2 7
÷
3 8
Work these out using a method to make the calculation easier. Show all your working.
a
2
× 320
5
c
19
× 4000
20
Fractions are used in everyday life more often than you might think.
One important use of fractions is in music.
Here is an example of a few bars of music.
one bar
A bar lasts a particular length of time, measured in a number of beats.
Different types of musical note last for different numbers of beats.
This means that the number of notes that can fit into each bar depends on
the type of notes.
Imagine a bar is like a cake. The number of slices (notes) into which it can be
cut depends on how large the slices are (how many beats each note lasts for).
137
7 Fractions
This table shows the names of some of
the different types of note.
It also shows the length of time (number
of beats) for which each note lasts.
You can see that two minims last for the
same length of time as one semibreve.
You can also see that four semiquavers
last for the same length of time
as two quavers or one crotchet.
In the piece of music below, each bar
must contain three beats.
Try to think of a combination of notes
that would fill the third bar.
Note
Name
Fraction
Number
of beats
semibreve
whole note (1)
4
minim
half note 2
crotchet
quarter note 1
1
quaver
eighth note 1
8
1
2
semiquaver sixteenth note 1
1
4
1
2
4
16
Tip
1+1+1
=3
1 + 1 +1 + 1
2 2
=3
can be written as
can be written as
can be written as
7.1 Fractions and recurring decimals
In this section you will …
Key words
•
equivalent
decimal
recognise fractions that are equivalent to recurring decimals.
You already know how to use equivalent fractions to convert a fraction
with a denominator that is a factor of 10 or 100 to a decimal.
For example: 3 = 6 = 0.6 and 3 = 15 = 0.15
improper fraction
mixed number
recurring decimal
5 10
20 100
You can also use division to convert a fraction to an equivalent decimal.
terminating
The fraction 5 is ‘five eighths’, ‘five out of eight’ or ‘five divided by eight’.
decimal
8
To work out the fraction as a decimal, divide 5 by 8: 5 ÷ 8 = 0.625
unit fraction
The decimal 0.625 is a terminating decimal because it comes to an end.
When you convert the fraction 71 to a decimal you get: 71 ÷ 99 = 0.71717171…
99
138
7.1 Fractions and recurring decimals
The number 0.71717171… is a recurring decimal as the digits 7 and 1
carry on repeating forever.
You can write 0.71717171… with the three dots at the end to show that
the number goes on forever.
..
You can also write the number as 0.71, with dots above the 7 and the 1,
to show that the 7 and 1 carry on repeating forever.
When you convert the fraction 1 to a decimal, you get 1 ÷ 14 =
14
0.0714285714285714285…
You. can see
. that 714285 in the decimal is repeating, so you write this as
0.0714285
You put a dot above the 7 and the 5 to show that all the digits from
7 to 5 are repeated.
Tip
You can use a
written method or
a calculator to do
this.
Tip
A recurring
decimal can
always be written
as a fraction.
Worked example 7.1
Use division to convert each fraction to an equivalent decimal.
a
3
b
8
5
c
11
11
12
Answer
3 ÷ 8 = 0.375
..
b 5 ÷ 11 = 0.45
This answer is a terminating decimal, so write down all the digits.
..
This answer is a recurring decimal, so write it as 0.45
c
This answer is a recurring decimal,
but only the 6 is recurring,
.
so write 0.916666… as 0.916
a
.
11 ÷ 12 = 0.916
Exercise 7.1
1
Tip
Use a written method to convert these unit fractions into decimals.
Write if the fraction is a terminating or recurring decimal.
The first two have been done for you.
a
1
2
b
0 . 5
2 1 . 10
1
2 = 0.5
Terminating decimal
c
h
1
4
1
9
d
i
1
5
1
10
e
j
1
6
1
11
1
3
0 . 3 3 3 ...
3 1 . 10 10 10
.
1
=
0.3
3
Recurring decimal
f
k
1
7
1
12
g
1
8
A unit fraction
has a numerator
of 1, e.g. 1 , 1 , 1 , ...
2 3 4
Tip
In part f, you will
need to keep
going with the
division for quite
a long time!
139
7 Fractions
Think like a mathematician
2
Work with a partner or in a small group to answer these questions.
a
Copy and complete this table. Use your answers to Question 1.
Unit fraction
Decimal
Terminating (T)
or recurring (R)
b
1
2
1
3
.
0.5 0.3
T
R
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
1
12
Read what Zara says.
The denominators of the
fractions 21 , 41 and 81 are all powers of 2.
The powers of 2 are 21 = 2, 22 = 4, 23 = 8, etc. There is
a pattern in the equivalent decimals for 1 , 1 and 1 .
2 4
8
They are all terminating decimals. The decimals are 0.5,
0.25 and 0.125. The pattern is 0.5, 0.25, 0.125. I think
that all unit fractions with a denominator which is a
power of 2 will be a terminating decimal that
ends in 25, apart from
ends in 5.
1
which just
21
iDo you think Zara is correct? Test her idea
1 and 1
on 16
32
Explain your decisions.
iiWhat other patterns can you see in the table in
Question 1?
Test your ideas to see if they work.
iiiDiscuss your ideas with other groups of
learners in your class.
Tip
24 = 16, 25 = 32
140
7.1 Fractions and recurring decimals
3
Here are five fraction cards.
5
8
A
B
3
4
7
10
C
11
20
D
E
3
5
a
Without doing any calculations, do you think these fractions are
terminating or recurring decimals? Explain why.
b Use a written method to convert the fractions to decimals.
c
Write the fractions in order of size, starting with the smallest.
Here are five fraction cards.
4
5
6
A
a
B
2
3
7
12
C
5
9
D
E
3
11
Without doing any calculations, do you think these fractions are
terminating or recurring decimals? Explain why.
Use a written method to convert the fractions to decimals.
Write the fractions in order of size, starting with the smallest.
b
c
Think like a mathematician
5
Work with a partner or in a small group to discuss these questions.
Maddie converts four fractions to recurring decimals on her calculator.
These are the answers she gets.
0.111111111
a
b
0.733333333
0.888888889
0.388888889
Why has the calculator put a 9 at the end of two of the decimals?
7 , 1 , 11, 8
Match each of the decimals to its equivalent fraction:
18 9 15 9
5
When Maddie converts the fraction
on her calculator, this is what she types.
18
5
÷
1
Maddie then presses the button: S
⇔
8
=
5
The answer she gets is: 18
c
d
D
What does this button do to the fraction?
What happens when you press the same button again?
Use a calculator to work out the decimal equivalent of
7
i 15
ii 8
11
141
7 Fractions
6
Use a calculator to convert these fractions to decimals.
a
c
7
7
9
2
15
b
d
13
20
9
40
Marcus and Sofia are discussing the fraction 5 .
13
My calculator
tells me that 5 ÷ 13 =
0.38461538, so I think that
5 is a recurring decimal
13
which I .can write
.
as 0.384615
8
What do you think? Explain your answer.
Use a calculator to convert these fractions to recurring decimals.
a
9
I don’t think the
calculator shows
you enough decimal
places to decide
it is a recurring
decimal.
2
7
b
9
13
c
11
14
This is part of Kim’s homework.
Question
Write these fractions as decimals.
5
6
1
i 12
ii 10
iii
iv
7
37
11
Answer
.
.
5
10
i 12 = 0.416
ii 11 = 0.90
. .
..
6
1
iii 7 = 0.857142
iv 37 = 0.027
a
b
Use a calculator to check Kim’s homework.
Explain any mistakes she has made and write the
correct answers.
10 Without using a calculator, write these fractions as decimals.
142
a
4
3
b
13
6
c
19
9
d
45
11
Tip
Remember, when
several digits
repeat in the
decimal, you only
put a dot over
the first and the
last digit of the
sequence that
repeats, e.g.
.
.
1
=
0.142857
7
Tip
Change the
improper
fractions into
mixed numbers
first. Then use
your answers to
Question 1 to
help.
7.1 Fractions and recurring decimals
11 This is part of Ada’s homework.
Question
Write 2 hours and 10 minutes as a recurring decimal.
Answer
10 = 1 of an hour and 1 = 0.16.
10 minutes is the same as 60
6.
6
So, 2 hours and 10 minutes = 2.16 hours.
Use Ada’s method to write these lengths of time as recurring
decimals.
a
4 hours 20 minutes
b 1 hour 40 minutes
c
6 hours 10 minutes
d 3 hours 50 minutes
12 Rajim has 8 weeks holiday a year.
There are 52 weeks in a year.
What fraction of the year does he have on holiday?
Write your answer as a decimal.
..
.
13 Sasha is told that 1 = 0.06 and that 1 = 0.045
15
22
Without using a calculator, she must match each yellow fraction
card with the correct blue decimal card.
4
15
.
0.26
7
22
..
0.318
.
..
Sasha thinks that 4 = 0.26 and that 7 = 0.318
15
22
Do you think she is correct? Explain your answer.
In this lesson, you have used a lot of mathematical words and phrases.
Write a short explanation, in your own words, for each of the
following terms.
a
terminating decimal
b
recurring decimal
c
unit fraction
d
equivalent fractions and decimals
Summary checklist
I can recognise fractions and recurring decimals.
143
7 Fractions
7.2 Ordering fractions
In this section you will …
Key words
•
advantages
compare and order positive and negative fractions.
When you write fractions in order of size, you must first compare them.
You can compare fractions in two ways.
1
Write them as fractions which have the same denominator.
2
Write them as decimals.
disadvantages
improve
Worked example 7.2
7 8
a
Write these fractions in order of size, starting with the smallest: 3 , 9 and 12
b
Use decimals to decide which is smaller: −3 4 or − 43
7
12
5
Answer
a
b
7
= 2 1 and 12 = 2 2
3
3
5
5
8, … , …
9
2 1 = 2 5 and 2 2 = 2 6
3
15
5
15
First, write the improper fractions as mixed numbers.
8 1 2
,2 ,2
9 3 5
−3 4 and − 43 = −3 7
7
12
12
4
=4÷7
7
2 5 < 2 6 so 2 1 is smaller than 2 2
0 .
7 4 .
5
0
4
7 1
0 10
5
15
3
5
First of all, write any improper fractions as mixed numbers.
Use division to work out 4 as a decimal.
7
4 ...
0
3
−3 4 as a decimal is −3.5714…
7
Use division to work out 7 as a decimal.
5
7
0
8 3 ...
10
0 40
12
7 as a decimal is −3.583…
−3
12
−3.583… < −3.5714…
As the numbers are negative, −3.58 is smaller than −3.57
− 43 < −3 4
Finally, write the answer using the original fractions given
in the question.
12
144
Now compare the other two fractions by writing them with
a common denominator of 15.
15
7
= 7 ÷ 12
12
0 .
12 7 .
8
is smaller than 2 1 and 2 2 so write that down first.
9
3
5
7
7.2 Ordering fractions
Exercise 7.2
For questions 1 to 4, use the common denominator method.
1
This is part of Seren’s homework.
She uses the symbol = to show that one fraction is equal to
another.
She uses the symbol ≠ to show that one fraction is not equal
to another.
Question
Write the correct sign, = or ≠, between each pair of
fractions.
2 147
b – 113
–3 94
a 52
Answer
a 52 = 221 and 2 147 = 221
so 52 = 2 147
2
b – 113 = –3 3 = –3 69
so – 113 ≠ –3 94
Write the correct sign, = or ≠, between each pair of fractions.
a
c
2
11
4
− 15
8
2 16
20
−2 1
8
b
45
6
71
d
−8 4
2
− 132
5
15
Write the correct symbol, < or >, between each pair of fractions.
Parts a and e have been done for you.
a
13
2
6 5 Tip
Working: 13 = 6 1 = 6 4 and 6 4 < 6 5
8
2
2
8
8
8
Answer: 13 < 6 5
2
b
17
3
e
− 17
−4 5
f
−7
−2 5
8
67
c
12
4
53
9
d
19
4
44
5
Working: − 17 = −4 1 = −4 3 and −4 3 > −4 5
12
17
Answer: − > −4 5
4
12
3
82
15
5
4
g
− 21
5
4
12
−4 2
h
15
12
−8
5
12
Change the
improper
fractions to mixed
numbers first.
Then compare
the fractions by
using a common
denominator.
−1 5
7
145
7 Fractions
3
Zara and Sofia compare the methods they use to work out which is
larger, − 8 or −2 4
3
7
They both use this method to start with.
Step 1:Write both fractions as mixed numbers
first: – 83 = –2 32
Step 2:Write –2 32 and –2 47 using a common
denominator of 21:
12
4
–2 32 = –2 14
and
–2
=
–2
21
21
7
Read what Zara and Sofia do next.
My step 3 is:
Without the negative signs 2 14 > 2 12
21
21
So with the negative signs −2 14 < −2 12 ,
21
21
so − 8 < −2 4
3
7
My step 3 is to sketch a number line.
−3 −2 −1
−2 14 −2 12
21
21
I can see that −2 14 < −2 12 ,
21
8
so − < −2 4
7
3
21
Write the advantages and disadvantages of each method.
Can you improve either method?
What is your preferred method for comparing negative
fractions? Explain why.
Work out which is larger.
a
b
c
4
a
146
− 7 or −113
4
16
b
− 21 or − 83
5
20
c
−6 2 or − 37
9
6
7.2 Ordering fractions
Think like a mathematician
5
Work with a partner or in a small group to answer this question.
a
With each pair of fractions, decide which is larger.
i
b
1
or 53
5
ii
7 or 5
9
9
iii
13
or 19
11
11
Discuss your answers to part a.
Copy and complete this sentence. Use either ‘larger’ or ‘smaller’.
When the denominators are the same, the larger the numerator the
................... the fraction.
c
With each pair of fractions, decide which is larger.
i
d
1
or 71
5
ii
2
or 23
9
iii
13
or 13
4
7
Discuss your answers to part c.
Copy and complete this sentence. Use either ‘larger’ or ‘smaller’.
When the numerators are the same, the larger the denominator the
................... the fraction.
6
Write the correct symbol, < or >, between each pair of fractions.
a
3
11
5
11
b
7
18
5
18
c
12
7
10
7
d
8
17
8
19
e
9
13
9
10
f
15
4
15
7
Think like a mathematician
7
Work with a partner. Discuss different methods you could use to answer this question.
Put these fraction cards in order of size, starting with the smallest.
− 17
−3 1
8
4
− 13
6
−7
13
What do you think is the best method to use? Explain why.
8
Put these fraction cards in order of size, starting with the smallest.
−3 1
6
−9
11
− 19
5
−4 2
5
147
7 Fractions
9
Three sisters sat a maths test on the same day.
Tip
13
Adele scored 16 , Belle scored 20 and Catrina scored 63%.
25
Who had the highest percentage score?
10 Two driving instructors compare the pass rates for their
students in January.
Steffan had 34 out of 40 students pass.
Irena had 87% of students pass.
Who had the higher pass rate for their students in January?
Show how you worked out your answer.
For questions 11 to 13, use the division method.
11 a
Complete the workings to write each fraction as a decimal.
Work out the first four decimal places.
i
− 11
− 14
ii
7
11
− = −1 4
7
7
0 .
5
7
1
4
7 4 .
4
5
1
3
0
0
0
0
0
iii
9
14
− = −1 5
9
9
.
5
5
5
5
9 5 .
5
5
5
5
4
=
7
−1 4 = −1.571…
7
0
0
0
0
12 a
Match each fraction with the correct decimal.
− 37
− 25
9
−4.18
− 209
6
− 47
50
−4.27...
11
−4.16...
−4.11...
Write the fractions − 37 , − 25 , − 209 and − 47 in order of size,
b
9
6
50
11
starting with the smallest.
13 Write these fractions in order of size, starting with the smallest.
− 107
20 148
− 37
−5 3
7 − 82
8 15
− 19
12
− 19 = −1 7
12
12
0
.
12 7 .
7
=
12
−1 7 =
12
5
=
9
−1 5 =
9
Write the fractions − 11, − 14 and − 19 in order of size, starting
7
9
12
with the smallest.
b
Change the fractions
into percentages by
writing equivalent
fractions with a
denominator of 100.
5
8
3
3
7
10
4
4
0
0
0
0
7.2 Ordering fractions
14 One day, a farmer sells 92% of her eggs.
The following day, she sells 56 out of 62 eggs.
Use a calculator to work out on which day she sold the greater
percentage of eggs.
15 Arun takes two English tests.
In the first test he scores 65 . In the second test he scores 35
72
38
Read what Arun and Sofia say.
a
Use Arun’s method to compare the scores.
b Use Sofia’s method to compare the scores.
c
Which method do you prefer and why?
d In which test did Arun get the better score?
16 In a science experiment, two groups of seeds are planted.
In group A, 175 seeds are planted and 156 start to grow.
In group B, 220 seeds are planted and 189 start to grow.
Use a calculator to work out which group is better at growing.
17 Li has 5 improper fraction cards. He puts them in order, starting
with the smallest.
There are marks on two of his cards.
7
− 25
9
56
Change 62 into
a decimal, then
multiply the
answer by 100 to
get a percentage.
I think you
should use a calculator
and change your scores into
decimals or percentages.
Then it will be easy to
compare your scores.
If I compare
my scores using a
common denominator,
I will have to use a
common denominator
of 1368!
− 20
Tip
Tip
Change 156
and
175
189 into decimals
220
or percentages to
compare.
− 13
5
What fractions could be under the marks?
Give two examples for each card.
Summary checklist
I can compare and order fractions.
149
7 Fractions
7.3 Subtracting mixed numbers
In this section you will …
•
subtract mixed numbers.
You already know that you can only subtract fractions when the
denominators are the same.
If the denominators are different, you must write the fractions as
equivalent fractions with a common denominator, then subtract the
numerators.
Here is a method for subtracting mixed numbers.
1
Change each mixed number into an improper fraction.
2
Subtract the improper fractions and cancel this answer to its
simplest form.
3
If the answer is an improper fraction, change it back to a mixed
number.
Worked example 7.3
Work out
a 3 1 − 1 4 b
5
61 − 24
5
3
9
Answer
a 3 1 = 16 and 1 4 = 9
5
5
16 9
− =7
5
5
5
5
5
7
= 12
5
5
b 6 1 = 19 and 2 4 = 22
3
3
9
9
19 22
−
= 57 − 22 = 35
3
9
9
9
9
35
= 38
9
9
150
Change both the mixed numbers into improper fractions.
Subtract the fractions. They already have a common
denominator of 5.
The answer is an improper fraction so change it back to a
mixed number.
Change both the mixed numbers into improper fractions.
Subtract the fractions, using the lowest common
denominator of 9.
The answer is an improper fraction so change it back to a
mixed number.
7.3 Subtracting mixed numbers
Exercise 7.3
1
Copy and complete these subtractions.
51 − 2 2
a
3
b
3
12
Step 1:
Step 2: 16 − 8 =
Step 2:
Step 3:
3
3
=2
3
Step 3:
3
53 − 35
4
6
d
Step 1: 23 −
4
Step 2: 23 −
4
Step 3:
2
6
Step 1: 16 − 8
3
3
3
c
91 − 3 5
12
6
=1
=
12
−
12
=
12
Step 2:
Step 3:
12
6
12
12
− 41 =
12
=
4
12
=5
4
4
20
−
5
8
12
12
4
−
=2
5
5
=
20
10
3
12
−
20
=
20
20
Work out these subtractions. Show all the steps in your working.
4 2 − 1 11
a
2 3 − 15
b 33 −1 7
c
d
8
− 41 =
4 1 − 13
4
5
Step 1:
6
6
− 41
52 − 31
3
4
Think like a mathematician
3
Work with a partner or in a small group to discuss this question.
Look at the different methods that Anders and Xavier use to work out 9 4 − 3 6
7
Anders
Change 9 47 into
8 + 1 + 47 = 8 + 77 + 47 = 8 117
So 9 47 – 3 67 is the same as 8 117 – 3 67
So 8 – 3 = 5 and 117 – 67 = 57 , so answer is 5 57
a
b
7
Xavier
Subtract whole numbers: 9 – 3 = 6
Subtract fractions: 47 – 67 = – 27
So 9 47 – 3 67 = 6 – 27 = 5 57
What are the advantages and disadvantages of:
i
Ander’s method
ii
Xavier’s method?
Which method do you prefer: Anders’ method, Xavier’s method or the method
in the worked example? Explain why.
151
7 Fractions
4
Work out these subtractions. Show all the steps in your working.
Use your preferred method.
3 3 − 14
a
b 71 − 2 7
c
82 − 41
d
14
7
3
12
3
4
6 7 − 4 17
12
18
Think like a mathematician
5
Work with a partner or in a small group to answer this question.
Marcus is looking at the question 9 2 − 3 8
a
b
c
d
6
7
9
Is Marcus correct? Explain your answer.
Without
subtracting
the
Choose two mixed numbers of your
fractions,
I
know
own but don’t subtract them yet.
the answer is going to
Write between which two whole
be between 5
numbers your total will be.
and 7.
Check that your answer is correct.
Think of subtracting any two mixed numbers.
Write a rule for working out between which two whole numbers the total will be.
How would you change this rule if you were subtracting 3, 4 or 5 mixed
numbers?
Shen has two pieces of fabric.
One of the pieces is 1 3 m long. The other is 2 3 m long.
4
8
a estimate, then b calculate, the difference in length between the
two pieces of material.
7
8
2 38 m
Zalika has a length of wood that is 5 1 m long.
4
First, Zalika cuts a piece of wood 1 3 m long from the length
5
of wood.
Then she cuts a piece of wood 2 9 m long from the piece of
10
wood she has left.
How long is the piece of wood that Zalika has left over?
The diagram shows the lengths of the three sides of a triangle.
a estimate, then b calculate, the difference in length between the
longest and shortest sides of the triangle.
6
57m
2
33m
3
74m
Write your answer to part b as a mixed number in its simplest form.
152
1 34 m
1
54 m
3
15 m
9
2 10 m
?m
7.3 Subtracting mixed numbers
Think like a mathematician
9
Work with a partner or in a small group to discuss this question.
What is the quickest method to use to work out the answer to 6 5 − 3 1?
8
2
10 Sami drives 16 5 km from his home to work.
8
2
Sami drives 11 km from his home to the supermarket.
5
What is the difference between the distance he drives from his home
to work and from his home to the supermarket?
11 Fina has two bags of lemons.
One bag has a mass of 4 7 kg.
10
The other bag has a mass of 2 4 kg.
7
4
4 10 kg
15
What is the difference in mass between the two bags
of lemons?
12 This is part of Rio’s homework. He has made a mistake in his
solution.
2 15 kg
Tip
If you cannot see
Rio’s mistake,
work through
the question
yourself and then
compare your
answer with his.
Question
9
Work out 435 – 10
Answer
6
435 = 4 10
6 – 9 =4 3
4 10
10
10
a
Explain the mistake Rio has made.
b Work out the correct answer.
13 In this pyramid, you find the mixed number in each block by
adding the mixed numbers in the two blocks below it.
Complete the pyramid.
3
12 4
3
2
12 4 − 8 3
2
83
4
25
5
19
153
7 Fractions
14 The perimeter of this quadrilateral is 35 13 m.
36
1
59 m
2
83 m
5
96 m
Work out the length of the missing side.
Tip
The perimeter
of a shape is the
distance around
the edge of the
shape.
Summary checklist
I can subtract mixed numbers.
7.4 Multiplying an integer
by a mixed number
In this section you will …
Key words
•
mean
multiply an integer by a mixed number.
You already know how to multiply a fraction by an integer.
For example: 2 × 12 Solution: 12 ÷ 3 = 4 and 2 × 4 = 8
3
You also know how to multiply two integers together using partitioning.
For example: 8 × 23 = 8 × 20 + 8 × 3 Solution: 160 + 24 = 184
You can now combine these methods to multiply an integer by a
mixed number.
Worked example 7.4
Work out i an estimate and ii the accurate answer to
a
154
2 1 × 16 b
2
4 2 × 20
3
partitioning
simplified
7.4 Multiplying an integer by a mixed number
Continued
Answer
a
i
Estimate: 3 × 16 = 48
ii 2 1 × 16 = 2 × 16 + 1 × 16
2
b i
2
= 32 + 8
= 40
Use partitioning to split the multiplication into two parts.
2 × 16 = 32 and 1 × 16 = 8
2
Add the two numbers together to get the total.
Estimate: 5 × 20 = 100
ii 4 2 × 20 = 4 × 20 + 2 × 20
3
3
Round the fraction to the nearest whole number.
= 80 + 40
Round the fraction to the nearest whole number.
Use partitioning to split the multiplication into two parts.
4 × 20 = 80 and 2 × 20 = 40
3
3
1
= 80 + 13 3
1
= 93 3
3
Change 40 into a mixed number.
3
Add the two numbers together to get the total.
Exercise 7.4
1
Copy and complete these multiplications.
a
31 ×8=3×8+ 1 ×8
2
2
=
b
2 1 × 12 = 2 × 12 + 1 × 12
4
+
=
=
c
3
3
d
8 3 × 10 = 8 × 10 + 3 × 10
5
+
=
2
+
=
42×9=4×9+ 2×9
=
4
This rectangle has length 15 m and width 2 1 m.
3
Work out
a
an estimate for the area of the
rectangle
b the accurate area of the rectangle.
5
=
+
=
1
23m
15 m
155
7 Fractions
3
Lin has 20 containers.
The mean amount of water the containers hold is 2 2 litres.
5
Lin uses this formula to work out the total amount of water that
the containers hold.
total amount
mean amount
number of
=
×
of water
of water
containers
4
Lin thinks the total amount of water the containers can hold
is 46 litres.
Is Lin correct? Explain your answer.
Copy and complete these multiplications. Use estimation to check
your answers.
a
41 ×9=4×9+ 1 ×9
2
2
=
+9
=
+41
b
3 3 × 11 = 3 × 11 + 3 × 11
4
4
2
2
=
c
156
+ 33
=
+
4
For the estimate
in part a, round
4 1 to 5, then
2
work out 5 × 9.
=
52 ×7=5×7+ 2 ×7
3
3
d
22 ×6=2×6+ 2 ×6
5
5
=
+
=
+
=
+
=
+
=
5
=
Tip
=
The diagram shows a square joined to a rectangle.
5 cm
Work out
a
an estimate for the area of the shape
b the accurate area of the shape.
5 cm
4
12 9 cm
7.4 Multiplying an integer by a mixed number
6
Martha is going to lay paving slabs on part of her garden.
This part of her garden is a rectangle with length 3 3 m
5
and width 2 m.
a
Martha estimates that the area of the rectangle is
6 m2. Is Martha correct? Explain your answer.
b Work out the area of the rectangle.
Martha buys paving slabs that cost $42 per
square metre.
She can only buy a whole number of square metres.
Martha works out that the paving slabs will cost
her $294.
c
Is Martha correct? Explain your answer.
Think like a mathematician
7
Work with a partner or in a small group to discuss this question.
Look at the different methods that Anders and Xavier use to work out 3 2 × 8
3
Anders
2
2
33 × 8 = 3 × 8 + 3 × 8
16
Xavier
Change 3 32 into 113
11 × 8 = 88
3
3
= 24 + 3
1
= 24 + 5 3
1
= 29 3
1
= 29 3
a
b
What are the advantages and disadvantages of
i Ander’s method
ii Xavier’s method?
Use both methods to work out
i
c
23 ×6
7
ii
6 5 × 12
9
Which method do you prefer, Anders’ or Xavier’s? Explain why.
157
7 Fractions
This is how Zara works out 2 1 × 15.
8
6
2 61 × 15 = 2 × 15 + 61 × 15
= 30 + 15
6
= 30 + 2 63
= 30 +2 21
= 32 21
15
Sofia says, ‘You changed 6 to a mixed number and then
simplified 3 to 1 . I would have simplified 15 to 5 before changing
6
2
6
2
it to a mixed number.’
a
Do you prefer Zara’s method or Sofia’s method? Explain why.
b Use your preferred method to work these out. Write your
answer in its simplest form.
i
9
3 3 × 10
8
4 3 × 14
ii
iii 2 7 × 12
4
10
Jamal works in a garden centre.
Tip
Seedlings are
seeds that are just
starting to grow
into plants.
It takes him 5 1 minutes to plant one tray of seedlings.
4
How long will it take him to plant 50 trays of seedlings?
Give your answer in hours and minutes.
Think like a mathematician
10 Work with a partner or in a small group to answer this question.
a
Work out
i 31×2
5
b
c
ii 3 1 × 3
5
What is the smallest integer that you must multiply by 3 1 to get
5
a whole number answer?
What is the smallest integer that you must multiply by 3 2 to get
a whole number answer? What about 3 3 and 3 4 ?
d
158
5
5
What do you notice about your answers to b and c?
5
7.5 Dividing an integer by a fraction
Continued
e
What is the smallest integer that you must multiply by 3 1 to get a whole
f
number answer? What about 3 2 , 3 3 , 3 4 , 3 5 and 3 6 ? What do you notice about
7
7 7
7
7
your answers?
Try starting with fractions with different denominators such as 6, 8, 9 and 11, for
example 2 1 or 4 1, etc. Do the patterns you noticed in parts d and e work
6
8
for these fractions as well? Explain your answers.
7
11 Work out
a
an estimate for the area of the blue section of this
rectangle
b the accurate area of the blue section of this rectangle.
3
12 5 m
2m
2
43 m
Look back at this section on multiplying an integer by a mixed number.
a
What did you find easy?
b
What did you find hard?
c
Are there any parts that you think you need to practise more?
Summary checklist
I can multiply an integer by a mixed number.
7.5 Dividing an integer by a fraction
In this section you will …
Key words
•
reciprocal
divide an integer by a proper fraction.
upside down
Look at this diagram.
It shows three rectangles, each divided in
half.
When you work out 3 ÷ 1 , the question
2
is asking you ‘How many halves are in three?’
3
1
2
159
7 Fractions
You can see that there are six, so 3 ÷ 1 = 6
2
Another method you can use is to turn the fraction upside down,
then multiply by the integer.
This is called multiplying by the reciprocal of the fraction.
So, 3 ÷ 1 = 3 × 2 = 6 = 6
2
1
1
Worked example 7.5
Tip
The reciprocal of
a fraction is the
fraction turned
upside down. So
the reciprocal of
1 is 2
2
1
Work out
a
4 ÷ 1 b
10 ÷ 3
3
4
Answer
a
You can use this diagram to work out how many thirds are in four.
4÷ 1 = 4×3
3
= 12
b 10 ÷ 3 = 10 × 4
4
3
= 40
3
Use the reciprocal method to answer this question.
Turn the fraction upside down to write the reciprocal and multiply.
The answer is an improper fraction which cannot be cancelled down.
= 13 1
3
Write the answer as a mixed number.
Exercise 7.5
1
Work out the answers to these calculations. Use the diagrams to
help you.
a
2÷1
b 4÷ 1
3
c
2
160
2
3÷ 1
d
4
5÷ 1
5
Read what Sofia says about dividing an integer by a unit fraction.
a
Can you explain why Sofia’s method works?
b Check your answers to Question 1 using Sofia’s method.
c
Use Sofia’s method to work out
i
12 ÷ 1
iii
8÷ 1
9
3
ii
25 ÷ 1
4
The quick way to
divide an integer by a unit
fraction is to multiply the integer
by the denominator of
the fraction.
7.5 Dividing an integer by a fraction
3
The area of a rectangle is 16 m2.
The width of the rectangle is 1 m.
5
What is the length of the rectangle?
Kai uses this formula to work out the average speed of a car in
kilometres per hour (km/h), when he knows the distance it has
travelled and the time it has taken.
4
Tip
area of rectangle
= length × width,
so length = area
÷ width.
speed = distance ÷ time
Work out the average speed of these cars. The first one has been done for you.
1
a
distance = 30 km, time = 4 hour So, speed = 30 ÷ 1 = 30 × 4 = 120 km/h
b
1
distance = 45 km, time = 2 hour
1
distance = 16 km, time = 6 hour
4
c
Think like a mathematician
5
Work with a partner or in a small group to answer this
question.
a
How can you use this diagram to work out 2 ÷ 2 ?
3
Tip
Think of the question as
‘How many 2 are in 2?’
3
b
c
6
How can you use this diagram to work out 3 ÷ 3 ?
4
Discuss your methods with other learners in
the class.
Write the method that you like better.
Tip
Think of the question as
‘How many 43 are in 3?’
Work out the answers to these calculations. Use the diagrams to help you.
a
4÷ 2
b
6÷ 3
c
4÷ 2
d
3
4
5
8÷ 4
7
161
7 Fractions
Think like a mathematician
7
Work with a partner or in a small group to answer this question.
Read what Zara says.
a
Use the diagram to show that Zara is correct.
b
Use the diagram to work out 4 ÷ 3
4
c
Complete the reciprocal method to check your answer to part c is correct.
The answer to
3 ÷ 2 is 4 1
3
2
4 ÷ 3 = 4 × 4 = 16 =
4
d
e
8
9
3
If your answers to parts b and c are different, explain the mistake you have made.
Discuss your answers with other learners in the class.
Discuss when you think it is easier to use the diagram method and the
reciprocal method.
Work out the answers to these calculations. Use the reciprocal method.
The first two have been started for you.
a
11 ÷ 3 = 11 × 4 =
c
7÷ 4
4
3
3
=
b
d
5
12 ÷ 7
10
9÷ 5 =9× 6 =
6
5
e
This is part of Anil’s homework.
You can see that he simplified the improper fraction to its lowest
terms before he changed it into a mixed number.
Question
Work out 10 ÷ 45
Answer
10 ÷ 45 = 10 × 54
= 50
4
= 25
2
1
= 12 2
162
3
=
10 ÷ 4
11
7.5 Dividing an integer by a fraction
Read what Marcus says.
a
Whose method do you prefer, Anil’s
I use a different method.
or Marcus’s? Explain why.
I change the improper
fraction
to a mixed number,
b Work out these calculations.
and then simplify the fraction
Give each answer as a mixed
to its lowest terms like this.
number in its lowest terms.
i
6÷ 4
7
ii
iii 12 ÷ 9
iv
10
4÷ 6
11
12
9÷
13
50 = 12 2 = 12 1
4
4
2
Activity 7.5
a
b
c
d
On a piece of paper, write four division questions: two like those in Question 1
and two like those in Question 8.
You must use an integer and a proper fraction.
On a separate piece of paper, work out the answers.
Exchange your questions with a partner and answer their questions.
Exchange back and mark each other’s work.
Discuss any mistakes that have been made.
10 Sofia is looking for patterns in the
1. When you divide an
division questions.
integer
by a proper fraction,
She has come up with two ideas.
the answer is always bigger than the
Are Sofia’s ideas correct?
integer you started with.
Explain your answers.
2. When you divide an integer by two
Look back at the questions
different proper fractions, the larger
you have done in this exercise
fraction will give you the
to help you explain.
larger answer.
11 a
Here is a sequence of calculations.
1 ÷ 1, 2 ÷ 1 , 3 ÷ 1 , 4 ÷ 1 , …
6
6
6
6
i Work out the sequence of answers.
ii Write the next two terms of the sequence.
iii Describe the sequence of answers in words.
163
7 Fractions
b
Here is a different sequence of calculations.
1 ÷ 2 , 2 ÷ 2, 3 ÷ 2, 4 ÷ 2, …
6
c
d
6
6
6
i Work out the sequence of answers.
ii Write the next two terms of the sequence.
iii Describe the sequence of answers in words.
Compare your sequences of answers in parts a and b.
What do you notice?
Explain why this happens.
Look at your answers to part bi and, without actually
completing the calculations, write down the sequence of
answers for this sequence of calculations.
1 ÷ 3, 2 ÷ 3 , 3 ÷ 3 , 4 ÷ 3 , …
6
e
6
6
6
Explain how you worked out your answer.
Here is another sequence of answers for a sequence of
calculations.
Calculations: 1 ÷ 1 , 2 ÷ 1 , 3 ÷ 1 , 4 ÷ 1 , …
15
15
15
15
Answers: 15, 30, 45, 60, …
Use this information to write down the sequence of answers
for this sequence of calculations.
1÷ 5 , 2 ÷ 5 , 3 ÷ 5 , 4 ÷ 5 , …
15
15
15
15
Explain how you worked out your answer.
Summary checklist
I can divide an integer by a proper fraction.
164
7.6 Making fraction calculations easier
7.6 Making fraction
calculations easier
In this section you will …
Key word
•
strategies
simplify calculations containing fractions.
When you are calculating using fractions, you can often make a
calculation easier by using different strategies. These strategies
3+3=?
will help you to work with fractions mentally. This means
4
8
you should be able to do simple additions, subtractions,
multiplications and divisions ‘in your head’. You should
also be able to solve word problems mentally. This section
will help you to practise the skills you need.
For harder questions, it may help you to write down some of the steps in
the working. These workings will help you to remember what you have
done so far, and what you still need to do.
With all calculations, you must remember the correct order of
operations.
Worked example 7.6
Work out mentally
a
3 3
+ b
4 8
4 3
− c
5 4
6 ÷ 3 d
4
(
2
× 2+1
5
3 2
)
Answer
a
6 3
+ =9
8 8
8
= 11
8
Change 3 to 6 so you can add it to 3
b
4 3
− = 4×4−3×5
5 4
5×4
Numerator: Multiply the diagonal pairs of numbers, shown by
the red and blue arrows – so, work out 4 × 4 and 3 × 5.
4
8
8
9
Then change to a mixed number.
8
Denominator: multiply the denominators, so work out 5 × 4.
= 16 − 15
Finally do the subtraction.
= 1
This gives a numerator of 1, with a denominator of 20.
20
20
165
7 Fractions
Continued
6÷ 3
6 × 4 = 24
Multiply the 6 by the 4.
24 ÷ 3 = 8
Then divide the answer by 3. This is equivalent to turning
the fraction upside down and multiplying by 6.
d 2 + 1 = 2 × 2 + 3 ×1
Using the correct order of operations, brackets come first.
c
4
3
3×2
2
= 4+3
Work out the addition.
=7
Leave the answer as an improper fraction.
6
6
Now work out the multiplication.
2 7 14
× =
5 6
30
Multiply the numerators and multiply the denominators.
Write the answer in its simplest form.
= 7
15
Exercise 7.6
In this exercise, work out as many of the answers as you can mentally.
Write each answer in its simplest form and as a mixed number when
appropriate.
1
Work out these additions and subtractions.
Some working has been shown to help you.
2
166
a
1 1 2 1
+ = + =
3 6 6 6
6
c
4
− 1 = 8 − 1 =
5 10 10 10
=
b
1 1 1
+ = +
8 4 8
8
=
d
5 1 5
− = −
6 3 6
6
=
=
Work out these additions and subtractions.
Use the same method as in part a of the worked example.
a
1 1
+
2 6
b
3 1
+
4 8
c
3
+ 1
5 10
d
1 3
+
2 8
e
3
+ 5
4 12
f
7
+4
15 5
g
1 1
−
3 9
h
1 1
−
4 8
i
1
− 1
5 15
j
2 1
−
3 6
k
4
− 1
5 10
l
11 2
−
20 5
7.6 Making fraction calculations easier
3
Work out these additions and subtractions.
Use the same method as in part b of the worked example.
a
1 1
+
3 5
b
1 1
+
4 7
c
2 1
+
9 5
d
3 2
+
4 3
e
5 1
+
8 5
f
1 5
+
4 6
g
1 1
−
2 3
h
4 1
−
5 4
i
5 1
−
7 2
j
3 2
−
4 7
k
7
−3
12 8
l
8 3
−
9 4
Think like a mathematician
4
aWork with a partner. Discuss the best method to use to work out
the answer to this question.
1
b
In a box of chocolates, 51 of the chocolates are white chocolate, 2 are milk
chocolate, and the rest are dark chocolate.
What fraction of the chocolates are dark chocolate?
Compare your methods with those of other learners in the class.
Do you think your method was the best method?
5
In a hockey squad, 1 of the players are short, 1 of the players are
3
4
medium height and the rest are tall.
What fraction of the squad are tall?
6
In a box of fruit, 2 are apples, 1 are guavas and the rest are coconuts.
5
6
What fraction of the fruit in the box are coconuts?
Work out these calculations. Use the same method as in part c of
the worked example.
Some working has been shown to help you with the first two.
7
a
4÷ 2= 4×3÷2=
c
9÷ 1
3
2
d
6÷ 2
5
b
8÷ 4=8×5÷ 4=
e
9÷ 3
5
4
f
10 ÷ 5
6
Think like a mathematician
8
With a partner, work out how to use the fractions button on
a calculator.
The fractions button looks like this.
167
7 Fractions
Continued
Work out the answer to 18 ÷ 5 . Write your answer as an
7
improper fraction.
Use the calculator to turn the improper fraction into
a mixed number.
You will need to use this button.
9
a bc ⇔ dc
S⇔D
a
Work out mentally
i 9÷ 4
ii
7÷3
iii 11 ÷ 2
iv 8 ÷ 5
5
3
7
5
b Use a calculator to check your answers to part a.
c
Did you get your answers to part a correct?
If not, what mistakes did you make?
10 The diagram shows a path.
The area of the path is 10 m2.
The width of the path is 3 m.
4
length
What is the length of the path?
11 This is how Marcus mentally works out 1 × 5 − 1
(
3
6
2
3
m
4
)
First, I work out 5 − 1
6 2
4
which equals which cancels
4
down to 1. Then I work out
1 × 1, which equals 1
3
3
a
Explain the mistake Marcus has made.
b Work out the correct answer.
12 Work out these calculations. If you cannot do them mentally,
write down some workings to help you.
)
6× 5 −1
c
11
− 3−1
12
4 2
e
168
(
a
(
6
(
1 1
+
4 2
6
( )
d
−( + )
f
( + )÷( + )
b
)
)×( − )
5
9
2
9
4÷ 1+1
3
3
11
12
1
2
1
3
9
4
1
3
2
3
2
5
3
10
Tip
Remember
brackets come
first, then indices,
then division and
multiplication,
then addition and
subtraction.
7.6 Making fraction calculations easier
13 Zara works out the answers to these calculation cards.
A
(
3× 3 + 3
4
4
)
B
(
2− 7 −1
10
5
)
C
(
4 × 2 2 − 11
3
6
Read what Zara says.
Is Zara correct?
Write the first term and the
term-to-term rule of the sequences
you can find.
)
D
(
6 ÷ 34 − 1 7
9
9
)
The answers
to these cards can be
rearranged to form two
different sequences of
numbers.
In this exercise, you have used mental methods to work out fraction
calculations.
Look back at the worked example and the types of question shown
in parts a, b, c and d.
a
Which type of questions have you found
i the easiest ii the hardest to work out mentally?
b
Which type of questions are you confident working out mentally?
c
Which type of questions do you need more practice with
working out mentally?
Summary checklist
I can use different methods to make fraction calculations easier.
169
7 Fractions
Check your progress
Progress
Nimrah
1
Usethinks
a written
of method
a number,
to convert
n.
these fractions into decimals.
Write ifanthe
expression
the number
Nimrah gets
each time.
Write
fraction is for
a terminating
or recurring
decimal.
a
She
multiplies the number by 4.
b b She4subtracts 6 from the number.
3
a
8
c
She multiplies the
by 3
She9divides the number by 6
12 number
27
17
38 d
2
Writethen
the fractions
−
,
−
,
−
and
−
in
order
of subtracts
size, starting
adds 5.
then
1. with the smallest.
5
10
6
15
3
Work
out number line and show the inequality on the number line.
Copy the
6 3 −the
2 5 inequality that this number linebshows.
45 × 9
Writea down
4
6
6
4
Work
out
Work out the value of each expression.
3
a
9with
÷ 3 3 friends. They share the electricity
b bill
8 ÷equally
Loli lives
between the four of
8
5
5
Work out mentally
them.
2
3
6 2
a
+ formula
b each
−
Write
to work out the amount they
5 a 10
7 3pay, in:
i 2
ii letters
c
d 9× 3 −1
4 ÷ words
7
4 3
Use your formula in part a ii to work out the
amount they each pay when the electricity bill is $96.
6
Simplify these expressions.
a
n+n+n
b 3c + 5c
c 9x − x
7
Simplify these expressions by collecting like terms.
a
5c + 6c + 2d b 6c + 5k + 5c + k c 3xy + 5yz − 2xy + 3yz
8
Work these out.
a
3 + (x × 2)
b 6 (3 − w)
c 4 (3x + 2)
d 3 (7 − 4v)
9
Solve each of these equations and check your answers.
a
n+3=8
b m − 4 = 12
c 3p = 24
d x=3
5
10 Shen has set a puzzle. Write an equation for
the puzzle. Solve the equation to find the
value of the unknown number.
(
170
)
8
Shapes and
symmetry
Getting started
1
a
b
Write the number of lines of symmetry for each of these shapes.
Write the order of rotational symmetry for each of these shapes.
i
ii
iii
2
3
iv
a
The diagram shows a cuboid.
Write the number of
i faces
ii edges
iii vertices of the cuboid.
b Draw the top view, front view and side view of this cuboid.
Make h the subject of each formula.
a
x=t+h−p
b x=h
c
4
4
y = 3xh
Label the parts of the circle shown. All the words you need are in the cloud.
centre chord
diameter tangent
radius circumference
171
8 Shapes and symmetry
Continued
5
Match each 3D shape with its name.
a
b
c
d
e
f
g
h
cylinder tetrahedron sphere equilateral triangular prism
cone cube square-based pyramid cuboid
6
A scale drawing of a building uses a scale of 1 : 20
a
The height of the building on the drawing is 25 cm.
What is the height, in metres, of the building in real life?
b The length of the building in real life is 12 m.
What is the length, in centimetres, of the building on the drawing?
Wherever you look, you will see objects of different shapes and sizes.
Many are natural, but many have been designed by someone.
An architect is a person who plans and designs buildings. Architects
make scale drawings, and often scale models too, of the buildings they
plan. They make sure their designs follow local rules and regulations,
and they make sure the people who build their buildings follow the plans
correctly.
Towns and cities all over the world have buildings designed to meet the
needs of the people who live and work there.
172
8 Shapes and symmetry
The Burj Khalifa in Dubai is the tallest building in the
world (as of 2018). It is over 828 metres tall and contains
163 floors. It holds the record for having an elevator with
the longest travel distance in the world.
Construction began in September 2004 and the building
was officially opened in January 2010. The Burj Khalifa
cost $1.5 billion to construct.
173
8 Shapes and symmetry
8.1 Quadrilaterals and polygons
In this section you will …
Key words
•
identify the symmetry of regular polygons
hierarchy
•
identify and describe the hierarchy of quadrilaterals.
lines of symmetry
quadrilateral
You already know how to describe the side length and symmetry
properties of a regular polygon.
For example, a regular pentagon has:
5 sides the same length
•
5 lines of symmetry
•
rotational symmetry of order 5.
•
A quadrilateral is a 2D shape with four straight sides.
These are the seven quadrilaterals you need to know.
square
rectangle
parallelogram
kite
rhombus
trapezium
isosceles trapezium
You can describe quadrilaterals using the properties of their sides
and angles.
For example, a square has:
all sides the same length
•
two pairs of parallel sides
•
all angles 90°.
•
174
regular polygon
rotational
symmetry
8.1 Quadrilaterals and polygons
Worked example 8.1
a S
ketch a regular octagon. Describe the side length and symmetry properties of the octagon.
b Sketch a parallelogram. Describe the side and angle properties of a parallelogram.
Answer
a
•
•
•
The octagon is regular, so all sides are the same
length.
The diagram shows the lines of symmetry.
In one complete turn, the octagon will fit onto itself
exactly 8 times.
A regular octagon has:
8 sides the same length
8 lines of symmetry
rotational symmetry of order 8.
b
•
•
•
A parallelogram has:
two pairs of sides the same length
two pairs of parallel sides
opposite angles that are equal.
Exercise 8.1
1
This diagram shows some regular polygons.
a
b
c
d
f
e
175
8 Shapes and symmetry
a
Copy and complete this table.
Name of regular
polygon
pentagon
hexagon
heptagon
octagon
nonagon
decagon
b
c
Number of sides
Number of lines
of symmetry
Order of rotational
symmetry
5
7
9
10
What do you notice about the number of sides, the number
of lines of symmetry, and the order of rotational symmetry
for each of the polygons?
Copy and complete these sentences.
The number of sides of a regular polygon is ........................... the number of
lines of symmetry.
The number of sides of a regular polygon is ........................... the order of
rotational symmetry.
Use your answers to part c to answer these questions.
iA hendecagon is a regular polygon with 11 lines of
symmetry.
How many sides does it have?
iiA dodecagon is a regular polygon with order of rotational
symmetry 12.
How many sides does it have?
Look at rectangle ABCD.
A
Write true or false for each statement.
If the statement is false, write the correct statement. C
a
AC is the same length as BD.
b AB is parallel to AC.
c
BD is parallel to AB.
d All the angles are 90°.
d
2
176
B
D
8.1 Quadrilaterals and polygons
3
Copy and complete the side and angle
properties of these four quadrilaterals.
Choose from the words in the box.
a
opposite two all
b
A rectangle has:
pairs of sides the same length
pairs of parallel sides
angles are 90°.
A square has:
sides the same length
pairs of parallel sides
angles are 90°.
c
d
A rhombus has:
sides the same length
pairs of parallel sides
angles are equal.
A parallelogram has:
pairs of sides the same length
pairs of parallel sides
angles are equal.
Think like a mathematician
4
Work with a partner or in a small group to discuss these questions.
a
b
c
d
Is a square a rectangle? Is a rectangle a square?
Is a square a rhombus? Is a rhombus a square?
Is a parallelogram a rectangle? Is a rectangle a parallelogram?
Is a parallelogram a rhombus? Is a rhombus a parallelogram?
Discuss your answers with other groups in the class.
5
Zara is describing a square to Marcus.
Has Zara given Marcus enough
information for him to work out
that the quadrilateral is
a square? Explain your answer.
My quadrilateral
has two pairs of parallel
sides and all the angles
are 90º. What is the name
of my quadrilateral?
177
8 Shapes and symmetry
6
Look at isosceles trapezium ABCD.
This is angle CAB
Write true or false for each statement.
If the statement is false, write the correct statement.
a
AC is the same length as CD.
C
b AB is parallel to CD.
c
Angle CAB is the same size as angle ACD.
d Angle BDC is the same size as angle ACD.
Copy and complete the side and angle properties of these
three quadrilaterals.
The missing words are all numbers.
7
a
A
This is angle BDC
An isosceles trapezium has:
pair of sides the same length
pair of parallel sides
pairs of equal angles.
c
A kite has:
pairs of sides the same length
pair of equal angles.
Think like a mathematician
Work with a partner or in a small group to discuss these questions.
a
b
c
Is a trapezium always an isosceles trapezium?
Is an isosceles trapezium always a trapezium?
Is a kite a rhombus? Is a rhombus a kite?
Is a parallelogram a trapezium? Is a trapezium a parallelogram?
Discuss your answers with other groups in the class.
178
D
b
A trapezium has:
pair of parallel sides.
8
B
8.1 Quadrilaterals and polygons
9
Marcus is describing a kite to Zara.
My quadrilateral
has two pairs of sides the
same length and two pairs
of equal angles.
What is the name of my
quadrilateral?
Has Marcus given Zara the correct information for her to work
out that the quadrilateral is a kite? Explain your answer.
10 Follow this classification flow chart for each quadrilateral.
Write the letter where each shape comes out.
a
square
b rectangle
c
parallelogram
d kite
e
trapezium
f
rhombus
g isosceles trapezium
START
yes
yes
Two sides the
same length?
Only one pair of
parallel sides?
no
yes
All angles are
90º?
no
no
H
yes
I
yes
All sides the
same length?
Two pairs of
equal angles?
no
no
L
K
J
yes
M
All sides the
same length?
no
N
179
8 Shapes and symmetry
11 This diagram shows the hierarchy of
quadrilaterals.
In the diagram, a quadrilateral below
another is a special case of the one
above it.
For example, a square is a special
rectangle but a rectangle is not a
square.
Use the diagram to decide if these
statements are true or false.
a
A parallelogram is a special
trapezium.
b A kite is a special rhombus.
c
A trapezium is a special
quadrilateral.
Quadrilateral
Trapezium
Kite
Parallelogram
Isosceles
Trapezium
Rectangle
Rhombus
Square
Activity 8.1
On a piece of paper, write four statements like the ones in Question 11. Two of them
must be true and two of them must be false. Exchange statements with a partner.
Write if your partner’s statements are true or false. Exchange back and mark each
other’s work.
Discuss any mistakes.
Look back at this exercise.
a
How confident do you feel in your understanding of this section?
b What can you do to increase your confidence?
Summary checklist
I can identify the symmetry of regular polygons.
I can identify and describe the hierarchy of quadrilaterals.
180
8.2 The circumference of a circle
8.2 The circumference of a circle
In this section you will …
Key words
•
accurate
know and use the formula for the circumference of a circle.
You already know the names of the parts of a circle.
Did you know there is a link between the circumference of a circle
and the diameter of a circle?
This table shows the circumference and diameter measurements of
four circles.
Circle Circumference (cm)
A
B
C
D
9.1
19.8
25.1
37.1
Diameter (cm)
Circumference
÷ diameter
approximate
value
circumference
diameter
pi (π)
radius
semicircle
2.9
6.3
8
11.8
Copy the table and fill in the final column. Give your answers correct
to two decimal places. What do you notice?
You should notice that all the answers are 3.14 correct to 2 decimal
places.
This means that the ratio of the diameter to the circumference of
a circle is approximately 1 : 3.14
The number 3.14... has a special name, pi. It is written using the
symbol π.
π is the number 3.141 592 653 589…, but you will often use 3.14 or 3.142
as an approximate value for π.
You now know that circumference = π, so you can rearrange the formula
diameter
to get:
C = πd
where: C is the circumference of the circle
d is the diameter of the circle
Tip
C = πd means
C=π×d
181
8 Shapes and symmetry
Worked example 8.2
Work out the circumference of a circle with
a
diameter 3 cm b radius 4 m.
Use π = 3.14. Round your answers correct to 1 decimal place (1 d.p.).
Answer
a
Write the formula you are going to use.
Substitute π = 3.14 and d = 3 into the formula.
Work out the answer.
Round your answer to 1 d.p. and remember to write the
units, cm.
C = πd
= 3.14 × 3
= 9.42
= 9.4 cm
b d=2×r=2×4=8m
C = πd
= 3.14 × 8
= 25.12
= 25.1 m
You are given the radius, so work out the diameter first.
Write the formula you are going to use.
Substitute π = 3.14 and d = 8 into the formula.
Work out the answer.
Round your answer to 1 d.p. and remember to write the units, m.
Exercise 8.2
1
Copy and complete the workings to find the circumference of each circle.
Use π = 3.14. Round your answers correct to 1 decimal place (1 d.p.).
a
2
diameter = 6 cm
C = πd
diameter = 25 cm
C = πd
c
diameter = 4.25 m
C = πd
= 3.14 × 6
= 3.14 ×
=
=
=
=
=
cm (1 d.p.)
=
cm (1 d.p.)
Copy and complete the workings to find the circumference of each circle.
Use π = 3.142. Round your answers correct to 2 decimal places (2 d.p.).
a
radius = 7 cm
d=2×r
=2×7
= 14 cm
C = πd
= 3.142 × 14
=
=
182
b
cm (2 d.p.)
b
radius = 2.6 cm
d=2×r
c
=
×
m (1 d.p.)
radius = 0.9 m
d=2×r
=2×
=2×
=
cm
C = πd
=
m
C = πd
= 3.142 ×
=
=
=
=
cm (2 d.p.)
=
×
m (2 d.p.)
8.2 The circumference of a circle
Think like a mathematician
3
So far in this unit you have
used approximate values for π.
You have used π = 3.14 and
π = 3.142.
There is another approximate
value you can use: π = 22
7
A more accurate value for π is
stored on your calculator.
Can you find the button with
the π symbol on it?
a
b
c
d
4
Use the π button on your
calculator to work out
the accurate circumference of a circle with diameter 12 cm.
Write all the numbers on your calculator screen.
Now work out the circumference of the same circle using approximate
values for π of:
i
3.14
ii 3.142
iii 22
7
Compare your answers to parts a and b.
Which approximate value for π gives the closest answer to the accurate answer?
When you answer questions and you need to use π, which value of π
do you think it is best to use?
Explain why.
Work out the circumference of each circle. Use the π button on your calculator.
Round your answers correct to 2 decimal places (2 d.p.).
a
diameter = 9 cm
b diameter = 7.25 m
c
radius = 11 cm
d radius = 3.2 m
Think like a mathematician
5
Work with a partner or in a group to answer this question.
So far in this unit you have used the formula, C = πd
In questions 2 and 4, you found the circumference when you were given the radius.
Can you write a formula to find the circumference which uses r (radius) instead
of d (diameter)?
Test your formula on Question 4, parts c and d. Does it work?
Compare your formula with other groups in the class.
183
8 Shapes and symmetry
For questions 6 to 11, use the π button on your calculator.
6
Fu and Fern use different methods to work out the answer to
this question.
Work out the diameter of a circle with circumference 16.28 cm.
Give your answer correct to 3 significant figures.
This is what they write.
Fu
Step 2: Substitute in
Step 1: Make d
the subject of the the numbers.
formula.
d = 16.28
π
C=π×d
= 5.18208...
C
=
d
π
= 5.18 cm (3 s.f.)
d = πC
Fern
Step 1: Substitute Step 2: Solve the
in the numbers. equation.
C=π×d
16.28 = π × d
16.28 = d
16.28 = π × d
π
5.18208... = d
d = 5.18 cm (3 s.f.)
a
7
184
Look at Fu and Fern’s methods.
Do you understand both methods?
Do you think you would be able to use both methods?
b Which method do you prefer and why?
c
Use your preferred method to work out the diameter
of a circle with:
i circumference = 28 cm
ii circumference = 4.58 m
d Make r the subject of the formula C = 2πr
e
Use your formula from part d to work out the radius of
a circle with:
i circumference = 15 cm
ii circumference = 9.25 m
The circumference of a circular disc is 39 cm.
Work out the diameter of the disc.
Give your answer correct to the nearest millimetre.
Tip
Remember,
C = 2πr means
C = 2× π ×r
8.2 The circumference of a circle
8
9
A circular ring has a circumference of 5.65 cm.
Show that the radius of the ring is 9 mm, correct to the nearest
millimetre.
This is part of Ahmad’s homework.
Question
Work out the perimeter of this semicircle.
Answer:
perimeter = half of circumference + diameter
πd
P= +d
16 cm
2
π
= × 16 + 16
2
= 25.13 + 16
= 41.13 cm
a
Use Ahmad’s method to work out the perimeter of a
semicircle with:
i diameter = 20 cm
ii diameter = 15 m
iii radius = 8 cm
iv radius = 6.5 m
Round your answers correct to 2 d.p.
b Imagine you have a friend who does not know how to work
out the perimeter of a semicircle.
Are you confident you could explain to them how to
work it out?
Can you use your knowledge to explain how to work out
the perimeter of a quarter-circle?
Make a sketch of a quarter-circle to help you.
10 The diagram shows a semicircle and a quarter-circle.
Read what Zara says.
15 m
10 m
Is Zara correct? Show working
to support your answer.
I think the
perimeter of the
semicircle is greater
than the perimeter
of the quarter-circle.
185
8 Shapes and symmetry
11 Work out the perimeter of each compound shape.
Give your answers correct to two decimal places.
a
b
Tip
.4
c
m
Remember, the
perimeter is the
total distance
around the
outside of the
whole shape.
Make sure you
include all the
different parts of
the perimeter.
14
12 cm
4.5 m
8 cm
4.5 m
c
d
28 mm
28 mm
3.6 cm
4.5 cm
3.6 cm
4.5 cm
Summary checklist
I know that π is the ratio between the circumference and the diameter of a circle.
I can use the formula for the circumference of a circle.
8.3 3D shapes
In this section you will …
Key words
•
find the connection between the number of vertices, faces,
and edges of 3D shapes
front view, front
elevation
•
draw front, side, and top views of 3D shapes to scale.
side view, side
elevation
You already know how to describe a 3D shape using the
number of faces, vertices and edges.
You also know how to draw the top view, front view and
side view of a 3D shape.
The top view is the view from above the shape.
It is sometimes called the plan view.
The front view is the view from the front of the shape.
It is sometimes called the front elevation.
The side view is the view from the side of the shape.
It is sometimes called the side elevation.
You also need to be able to draw the top view, front view,
and side view of a 3D shape to scale.
186
top view, plan
view
Top
Side
Front
8.3 3D shapes
Worked example 8.3
The diagram shows a cuboid.
10 cm
35 cm
15 cm
a Write the number of faces, vertices and edges of the cuboid.
b Draw accurately the top view, front view and side view of the cuboid.
Use a scale of 1 : 5
Answer
a
6 faces, 8 vertices, 12 edges
b Top view
Front view
The faces are the flat surfaces, the vertices
are the corners, and the edges are where
two faces meet.
Use the scale to work out the dimensions
of the cuboid for the drawing.
The scale is 1 : 5, so 1 cm on the drawing
represents 5 cm in real life.
Length: 35 ÷ 5 = 7 cm
Height: 10 ÷ 5 = 2 cm
Width: 15 ÷ 5 = 3 cm
So, the top view is a rectangle 7 cm by 3 cm.
The front view is a rectangle 7 cm by 2 cm.
The side view is a rectangle 3 cm by 2 cm.
Side view
187
8 Shapes and symmetry
Exercise 8.3
Think like a mathematician
1
Work with a partner or in a small group to answer these questions.
a
Copy and complete this table showing the number of faces, edges,
and vertices of these 3D shapes.
3D shape
Number of Number of Number of
faces
vertices
edges
cube
cuboid
6
8
12
Tip
Make a sketch
of each shape to
help you.
tetrahedron
square-based
pyramid
triangular
prism
trapezoidal
prism
b
c
d
e
188
What is the connection between the number of faces,
vertices and edges for all of the 3D shapes?
Write a formula that connects the number of faces (F),
vertices (V) and edges (E).
Compare your formula with other groups in your class.
Do you have the same formula or a different formula?
Is your formula the same, just written in a different way?
Does your formula work for shapes with curved surfaces,
or does it only work for shapes with flat faces?
Explain your answer.
Tip
Remember, a
tetrahedron is a
triangular-based
pyramid.
Tip
You could start
your formula
E=…
8.3 3D shapes
2
Copy and complete the workings and scale drawings for
this question.
Draw the top view, front view, and side view of these shapes.
Use a scale of 1 : 2
a
cube
b
c
cuboid
cylinder
7 cm
3 cm
8 cm
10 cm
5 cm
14 cm
6 cm
Dimensions for
drawing:
6÷2=
cm
Dimensions for drawing:
8÷2=
cm
3÷2=
cm
5÷2=
cm
Dimensions for
drawing:
7÷2=
cm
10 ÷ 2 =
cm
14 ÷ 2 =
cm
Top view:
Top view:
Top view:
Tip
Tip
Tip
Draw a square of
side length 3 cm.
Draw a rectangle
4 cm by 2.5 cm.
Draw a circle of
radius 3.5 cm.
Front view:
Front view:
Front view:
Tip
Tip
Tip
Draw a square of
side length 3 cm.
Draw a rectangle
cm by
cm.
Draw a rectangle
cm by
cm.
Side view:
Side view:
Side view:
Tip
Tip
Tip
Draw a square of
side length 3 cm.
Draw a rectangle
cm by
cm.
Draw a rectangle
cm by
cm.
189
8 Shapes and symmetry
Think like a mathematician
3
Work with a partner or in a small group to answer these questions.
Li and Seb are drawing the plan view of this shape.
1 cm
2 cm
5 cm
4 cm
3 cm
This is what they draw.
Li Seb
1 cm
2 cm
4 cm
3 cm
a
b
c
d
e
f
190
4 cm
3 cm
Have either of them, or both of them, drawn the
correct plan view?
Have they drawn their plan views to scale?
Discuss your answers to parts a and b with other
groups in the class.
Draw the front elevation of the shape.
Draw the side elevation of the shape from
i
the left
ii the right.
Are your drawings for parts i and ii the same?
Discuss and compare your drawings in
parts d and e with other groups in the class.
Tip
Remember, the plan
view is the same as
the top view.
Tip
Remember, the side
elevation is the same
as the side view.
8.3 3D shapes
4
Draw the plan view, the front elevation and the side elevation
of this 3D shape.
Use a scale of 1 : 10
Tip
If the views from
the left side and
from the right
side are the
same, you only
need to draw one
side elevation.
30 cm
15 cm
35 cm
20 cm
25 cm
Tip
60 cm
A shipping
container is a very
large metal box
used to move
goods by lorry,
train or ship.
The diagram shows the dimensions of a shipping container.
5
2.6 m
6m
2.4 m
Ajani makes a house from three shipping containers.
The containers are arranged as shown in the diagram.
Draw the plan view, the front elevation, and
the side elevation of his house.
Use a scale of 1 : 100
3m
3m
Think like a mathematician
6
Work with a partner or in a small group to answer these questions.
The diagram shows two triangular prisms, A and B.
A
B
20 cm
5 cm
16 cm
4 cm
6 cm
3 cm
18 cm
24 cm
191
8 Shapes and symmetry
Continued
A is a right-angled triangular prism.
B is an isosceles triangular prism.
a
b
c
d
7
Will the side elevation of prism A be the same from the left side and
the right side?
Explain your answer.
Draw the plan view, the front elevation, and the side elevation of prism A.
Use the actual dimensions shown.
Discuss the methods you could use to accurately draw the triangle.
Which is the best method?
Will the side elevation of prism B be the same from the left side and
the right side?
Explain your answer.
Draw the plan view, the front elevation, and the side elevation of prism B.
Use a scale of 1 : 4
Discuss the methods you could use to accurately draw the triangle.
Which is the best method?
The diagram shows the dimensions of a village hall.
The roof is an isosceles triangular prism.
Draw the plan view, the front elevation, and the side elevation
of the village hall.
Use a scale of 1 : 200
3m
4m
20 m
12 m
192
Tip
Convert all the
dimensions
from metres
to centimetres
before using the
scale to work out
the dimensions
of the scale
drawings.
8.3 3D shapes
8
The diagram shows a shape drawn on dotty paper.
The shape is made from 1 cm cubes.
This diagram shows the plan view, front elevation and side
elevation for the shape.
Plan
A
B
C
Front
Side
9
The diagrams have been drawn accurately on 1 cm
squared paper.
a
Which diagram, A, B or C shows the
i plan view
ii front elevation
iii side elevation?
b Is it possible to have a shape made from a different number of
1 cm cubes which has the same plan view as the shape above?
Explain your answer.
c
Is it possible to have a shape made from a different number
of 1 cm cubes which has the same plan view, front elevation,
and side elevation as the shape above? Explain your answer.
The diagram shows four shapes drawn on
P
a
dotty paper.
The shapes are made from 1 cm cubes.
Draw accurately the plan view, front elevation
and side elevation for each of the shapes.
Use 1 cm squared paper.
S
F
The arrows in part a show the directions
from which you should look at the shapes
c
for the plan view (P), front elevation (F)
and side elevation (S).
b
d
193
8 Shapes and symmetry
10 This is part of Marcus’s homework.
Question
Accurately draw the outline of the plan view, front elevation and side elevation of
this shape. Do not include any internal lines.
P
30 cm
20 cm
F
S
Use a scale of 1 : 5 and use 1 cm squared paper.
Answer
Plan view
The length of the shape
is 20 cm, so it is made
from 10 cm cubes.
The scale is 1 : 5, so the
length of 20 cm needs
to be 20 ÷ 5 = 4 cm
The height of 30 cm
needs to be
30 ÷ 5 = 6 cm
The width of 10 cm
needs to be 10 ÷ 5 = 2 cm
Front elevation
Side elevation
Which of Marcus’s drawings is incorrect: the plan view, front elevation or side elevation?
Explain the mistake he has made.
194
8.3 3D shapes
11 The diagram shows a shape drawn on dotty paper.
The shape is made from cubes.
The measurements of the shape are shown in the diagram.
Accurately draw the plan view, front elevation, and side
elevation for this shape.
Use a scale of 1 : 2 and use 1 cm squared paper.
P
16 cm
4 cm
8 cm
F
S
12 The diagram shows a shape drawn on dotty paper.
The shape is made from cubes.
The width of the shape is shown in the diagram.
Accurately draw the plan view, front elevation and
side elevation for this shape.
Use a scale of 1 : 3 and use 1 cm squared paper.
Top
18 cm
Front
Side
Summary checklist
I can draw plan, front and side views of 3D shapes to scale.
195
8 Shapes and symmetry
Check your progress
Progress
Nimrah
1
Copy
thinks
and complete
of a number,
this sentence.
n.
Write
an expression
number
Nimrah
time.
A
regular
pentagon hasfor the
sides
of equal
length.gets
It haseachlines
of symmetry
a
She
multiplies
the
number
by
4.
b
She
subtracts
6
from
the number.
and rotational symmetry of order
.
cWriteShe
multiplies
by 3
d She divides the number by 6
2
True
or False the
for number
each statement.
then adds 5.
then subtracts 1.
a
A square is a special rectangle.
Copy the number line and show the inequality on the number line.
b A trapezium is a special parallelogram.
Write down the inequality that this number line shows.
c
A rhombus is a special parallelogram.
Work out the value of each expression.
d A rhombus is a special kite.
Loli lives with 3 friends. They share the electricity bill equally between the four of
3
Work out the circumference of these circles. Use the π button on your calculator.
them.
Round your answers correct to 2 decimal places (2 d.p.).
Write a formula to work out the amount they each pay, in:
a
diameter
b radius = 2.7 m
i words= 13 cm
ii letters
4
The
circle
27 cm.
Use circumference
your formulaof
in apart
a is
ii to
work out the
Workamount
out the diameter
of pay
the circle.
they each
when the electricity bill is $96.
Give
yourthese
answer
correct to the nearest millimetre.
6
Simplify
expressions.
5
Drawnthe
and side celevation
a
+ nplan
+ n view, bfront3celevation
+ 5c
9x − xof each shape.
Use a scale
of 1expressions
: 4 and useby
1 cm
squaredlike
paper.
7
Simplify
these
collecting
terms.
a
5c + 6c + 2d b 6c + 5k + 5c + k c b 3xy + 5yz − 2xy + 3yz
8
Work these out.
P
63cm
a
+ (x × 2)
b 6 (3 − w)
c 4 (3x + 2)
d 3 (7 − 4v)
9
Solve each of these equations12and
cm check your answers.
16 cm
a
n+3=8
b m − 4 = 12
c 3p = 24
d x=3
5
10 Shen has set a puzzle. Write an equation for
the puzzle. Solve the equation to find the
value of the unknown number.
24 cm
8 cm
F
196
S
Project 3
Quadrilateral tiling
This picture shows how you can tile an
area with rectangles.
They fit together with no gaps, because
at each point, there are four 90° angles,
which add up to 360°.
A tiling pattern like this is called a
tessellation.
Here is a kite, with a rectangle
drawn around it.
Draw a kite of your own, and a rectangle to
surround it in the same way. Then tessellate
your rectangle, keeping it in the same
orientation.
What do you notice?
Can you use this to prove that all kites
tessellate?
Next, let’s investigate parallelograms.
Draw a parallelogram, cut it out, then draw around it to make a tessellation pattern.
Can you use what you know about the angles in a parallelogram to prove that
they fit together without leaving any gaps?
Here is a trapezium.
Can you find a way to put two
identical trapezia together to make a
parallelogram?
Can you use this to prove that trapezia
will tessellate?
Can you find some irregular quadrilaterals
that tessellate?
You might like to use a dotty grid to explore
different options.
Look for ways to arrange your quadrilaterals
so the four angles that meet at each point
add up to 360°.
Are there any quadrilaterals that do not
tessellate?
197
9
Sequences and
functions
Getting started
1
2
3
For each of these sequences, work out
i
the term-to-term rule
ii the next two terms.
a
4, 7, 10, 13, ..., ...
b 28, 26, 24, 22, ..., ...
Write the first four terms of the sequence that has a first term
of 3 and a term-to-term rule of ‘Multiply by 2’.
This pattern is made from squares.
Pattern 1
a
b
Pattern 2
Draw the next pattern in the sequence.
Copy and complete the table to show the number of
squares in each pattern.
Pattern number
Number of squares
4
198
Pattern 3
1
3
2
5
3
4
5
c
Write the term-to-term rule.
d How many squares will there be in Pattern 10?
Work out the first four terms in each of these sequences.
a
nth term = 6n
b nth term = n − 1
Tip
Substitute n = 1,
2, 3 and 4 into
the nth term
formulae.
9 Sequences and functions
Continued
5
Copy these function machines and work out the missing inputs
and outputs.
a input
b input
output
output
2
5
__
+4
c input
4
5
__
× 10
__
__
16
7
__
15
output
d input
__
__
30
8
__
__
–7
__
5
__
output
÷4
__
6
9
Throughout history, mathematicians have been interested in number
patterns and sequences.
Look at this pattern of dots.
1 dot
3 dots
6 dots
10 dots
15 dots
The number of dots in each pattern forms the sequence 1, 3, 6, 10, 15, …
The numbers 1, 3, 6, 10, 15, … are called the triangular numbers,
because the dots can be arranged in the shape of a triangle as shown in
the pattern above.
You can see how the sequence is formed:
+2 +3 +4 +5
1
3
6
10
15
The next two terms in the sequence will be: 15 + 6 = 21
and 21 + 7 = 28
Can you work out the next three triangular numbers?
199
9 Sequences and functions
9.1 Generating sequences
In this section you will…
Key word
•
generate
use a term-to-term rule to make a sequence of numbers.
1 21
This is a sequence of numbers.
2
4 51
2
7
…
…
Each term is 1 1 more than the term before, so the term-to-term rule is
2
‘add 1 1 ’.
2
You can generate a sequence when you are given the first term and the
term-to-term rule.
For example, when the first term is 3 and the term-to-term rule is
‘multiply by 2 and add 5’ you get the sequence 3, 11, 27, 59, …
Tip
2nd term is
3 × 2 + 5 = 11
3rd term is
11 × 2 + 5 = 27
4th term is
27 × 2 + 5 = 59
Worked example 9.1
a
rite the term-to-term rule and the next two terms of this sequence.
W
6, 8 1 , 10 1 , 12 3 , …
4
2
4
b The first term of a sequence is 4.
The term-to-term rule of the sequence is: multiply by 3 and
then add 2.
Write the first three terms of the sequence.
Answer
a
Term-to-term rule is: add 2 1
You can see that the terms go up by 2 1 every time as
4
Next two terms are 15 and 17 1
4
b First three terms are 4, 14, 44
4
1
1 1
1
1
6 + 2 = 8 , 8 + 2 = 10 , etc.
4
4 4
4
2
1
You keep adding 2 to find the next two terms:
4
3
1
12 + 2 = 15 and 15 + 2 1 = 17 1
4
4
4
4
Write the first term, which is 4.
Then use the term-to-term rule to work out the
second and third terms:
second term = 3 × 4 + 2 = 14
third term = 3 × 14 + 2 = 44
200
9.1 Generating sequences
Exercise 9.1
1
Complete the workings to find the term-to-term rule and the next
two terms of each sequence.
a
7, 10 1 , 14, 17 1 , …
2
2
7 + 3 21 = 10 21 , 10 21 + ... = 14, 14 + ... = 17 21
The term-to-term rule is: add
The next two terms are:
17 21 + =
+ =
b
10, 9.8, 9.6, 9.4, …
10 − 0.2 = 9.8, 9.8 − = 9.6,
9.6 − = 9.4
The term-to-term rule is: subtract
The next two terms are:
9.4 − =
− =
2
3
For each of these sequences, write
i
the term-to-term rule
ii
a
5, 5 1 , 5 1 , 5 3 , …
b
c
5.4, 5.8, 6.2, 6.6, …
d
e
10, 9 3 , 9 1 , 8 4 , …
f
4
2
5
4
5
5
the next two terms.
7 1 , 8 2 , 10, 11 1 , …
3
3
3
1
1
9, 8 , 8, 7 , …
2
2
17, 16.2, 15.4, 14.6, …
Write the first three terms of each sequence. Show your working.
First Term-to-term rule
term
First Term-to-term rule
term
a
1
Add 1.4
b
6
Add 4 1
2
c
20
Subtract 2.5
d
40
Subtract 5 1
e
0.4 Multiply by 2
f
9
Divide by 2
3
Tip
For Question 3,
part a, work out
1 + 1.4 = 2.4, then
2.4 + 1.4 = …,
then … + 1.4 = …
201
9 Sequences and functions
4
Copy these sequences and fill in the missing terms.
a
2, 4 1 ,
, 8 3,
c
25, 24 3 ,
4
e
8,
5
,
5
,
,13, 15 1
5
, 23 1
2
, 24,
, 8.9, 9.2,
,
b
5, 8 3 , 11 6 ,
,
d
100, 89 1 ,
2
, 68 1 ,
2
f
,
7
7
, 24, 23.6,
, 22 1 ,
7
, 47 1 ,
2
,
, 22.4
Think like a mathematician
5
How can you answer these questions without working out more of the terms
in the sequences?
a
b
In the sequence 0.4, 0.8, 1.2, 1.6, 2, 2.4, …, what is the first term
greater than 10?
Is 45 a term in the sequence 5, 7 1 , 10, 12 1, 15, ...?
c
Is 5 1 a term in the sequence 30, 26 2 , 23 1, 20, ...?
2
3
Discuss your answers.
6
2
3
3
Write the first three terms of each of these sequences.
The first one has been started for you.
a
first term is 8
term-to-term rule is: multiply by 2 then subtract 5
first term = 8
second term = 8 × 2 − 5 = 16 − 5 = 11
third term = 11 × 2 − 5 = − 5 =
7
b first term is 15, term-to-term rule is: subtract 9 then multiply by 3
c
first term is 12, term-to-term rule is: divide by 2 then add 5
The first three terms of a sequence are 8, 10, 14, …
a
Which of these cards, A, B or C, shows the correct term-to-term rule?
b
202
A
multiply by 3 then subtract 14
B
divide by 2 then add 6
C
subtract 3 then multiply by 2
Which is the first term in this sequence greater than 50?
9.1 Generating sequences
8
Arun works out the terms in this sequence:
The first term in
First term is 10, term-to-term rule is subtract 6
this sequence
then multiply by 2.
which is a negative
Read what Arun says.
number is −4.
Is Arun correct? Show your working.
9
Work out the first three terms in each sequence.
a
first term is 4, term-to-term rule is: multiply by 3 then subtract 10
b first term is 10, term-to-term rule is: subtract 2 1 then multiply
2
by 2
c
first term is −6, term-to-term rule is: divide by 2 then add 5
10 Zara describes a sequence.
a
Work out the first three terms of the sequence.
My sequence has a
first term of 5, and
What do you notice?
the term-to-term rule
b Describe two different sequences that are
is ‘Multiply by 3 then
like Zara’s.
subtract 10’
Compare your answers with a partner’s.
Think like a mathematician
11 Sofia works out the terms in this sequence:
First term is 8, term-to-term rule is add
10 then divide by 2.
Read what Sofia says.
Is Sofia correct? Discuss your answers.
I will never have
a term in my
sequence which is
greater than 10.
12 This is part of Tania’s homework.
Question
The 10th term of a sequence is 50 2 . The term-to-term rule is
5
add 4 3
5
What is the 20th term of the sequence?
Answer
20th term = 2 × 10th term = 2 × 50 2 = 100 4
5
a
b
5
Explain why Tania’s method is wrong.
Work out the correct answer. Show all your working.
203
9 Sequences and functions
13 The 7th term of a sequence is 442.
The term-to-term rule is add 3 then multiply by 2.
What is the 4th term of the sequence?
Show all your working.
Look back at your answers to questions 12 and 13.
Write a short explanation of the method you used to solve these problems.
Discuss your method with a partner.
Did they use the same method?
Can you think of a better method?
Activity 9.1
On a piece of paper, write three questions similar to those in Question 2,
and three questions similar to those in Question 6.
Write the answers on a separate piece of paper. Make sure the questions
can be answered without using a calculator.
Exchange questions with a partner. Work out the answers to your
partner’s questions.
Exchange back and mark each other’s work.
If you think your partner has made a mistake, discuss with them where
they have gone wrong.
Summary checklist
I can find the term-to-term rule for a number sequence.
I can use the term-to-term rule for a number sequence.
204
9.2 Finding rules for sequences
9.2 Finding rules for sequences
In this section you will …
Key words
•
position number
make a sequence of numbers from patterns.
position-to-term
rule
This sequence of patterns is made from dots.
Pattern 1
Pattern 2
Pattern 3
5 dots
7 dots
9 dots
sequence of
patterns
The numbers of dots used to make the patterns form the sequence
5, 7, 9, …, …
As you go from one pattern to the next, two more dots are added each
time. The term-to-term rule is ‘add 2’.
You can use the term-to-term rule to work out the position-to-term rule.
The term-to-term rule for this sequence is ‘add 2’, so start by listing the
first three multiples of 2 and comparing them with the patterns of dots.
Multiples of 2:
Number of dots:
Position 1
1×2=2
2+3=5
Position 2
2×2=4
4+3=7
Position 3
3×2=6
6+3=9
The pattern is formed by adding multiples of 2, shown as red dots, to
the three blue dots at the start of each pattern.
Pattern 1
Pattern 2
Pattern 3
5 dots
7 dots
9 dots
Pattern 4
The position-to-term rule for this sequence is:
term = 2 × position number + 3
Draw the next pattern, to check.
Pattern 4: term = 2 × 4 + 3 = 11 ✓
11 dots
205
9 Sequences and functions
Worked example 9.2
This pattern is made from blue squares.
a
b
c
d
e
Write the sequence of the numbers of squares.
Write the term-to-term rule.
Draw the next pattern in the sequence.
Explain how the sequence is formed.
Work out the position-to-term rule.
Answer
a
There are 4 squares in the first pattern, 7 in the
second and 10 in the third.
4, 7, 10, …
b add 3
The term-to-term rule is ‘add 3’.
c
Pattern 4 will have 10 + 3 = 13 squares.
Pattern 4
d T
he pattern is formed by adding
multiples of 3, shown as red
squares, to the one blue square
at the start of each pattern.
e
Position
number
Term
3 × position
number
3 × position
number + 1
2
3
4
3
7
6
10 13
9 12
The term-to-term rule is ‘add 3’, so add a row to the
table which shows 3 × position number.
4
7
10 13
You can see that each number in this row is 1 less than
the equivalent number in the sequence. So if you add
1, you will get the terms of the sequence.
Position-to-term rule is:
term = 3 × position number + 1
206
4
Draw a table showing the first four position numbers
and terms.
1
(3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12)
(3 + 1 = 4, 6 + 1 = 7, 9 + 1 = 10, 12 + 1 = 13)
9.2 Finding rules for sequences
Exercise 9.2
1
This pattern is made from squares.
a
b
c
d
e
Write the sequence of the numbers of squares.
Write the term-to-term rule.
Draw the next pattern in the sequence.
Explain how the sequence is formed.
Copy and complete the table to find the position-to-term rule.
Position number
Term
2 × position number
1
3
2
2
5
4
3
4
2 × position number +
2
The position-to-term rule is: term = 2 × position number +
This pattern is made from dots.
a
b
c
d
e
Write the sequence of the numbers of dots.
Write the term-to-term rule.
Draw the next pattern in the sequence.
Explain how the sequence is formed.
Copy and complete the table to find the position-to-term rule.
Position number
Term
1
6
2
10
3
4
× position number
× position number +
The position-to-term rule is: term =
× position number +
207
9 Sequences and functions
3
This pattern is made from rectangles.
a
b
c
d
Write the sequence of the numbers of rectangles.
Write the term-to-term rule.
Draw the next pattern in the sequence.
Copy and complete the table to find the position-to-term rule.
Position number
Term
1
3
2
8
3
4
× position number
× position number −
The position-to-term rule is: term =
× position number −
Think like a mathematician
4
This pattern is made from squares.
Razi thinks that the position-to-term rule for the sequence of the numbers
of green squares is:
term = 2 × position number + 3
Is Razi correct? Explain the method you used to work out your answer.
Discuss the method you used with other learners. Did you use the same method
or a different method? What do you think is the best method to use?
208
9.2 Finding rules for sequences
5
This is part of Harsha’s homework.
Question
Work out the position-to-term rule for this sequence of
triangles.
Answer
The sequence starts with 4 and increases by 2 every time,
so the position-to-term rule is:
term = 4 × position number + 2
a
b
Explain the mistake Harsha has made.
Work out the correct answer.
Activity 9.2
a
b
6
Design your own sequence of patterns made from a shape of your choice.
Draw the first four patterns in your sequence.
Draw a table to show the number of shapes in each of your patterns.
Work out the position-to-term rule for your sequence.
Ask a partner to check that your work is correct.
Work out the position-to-term rule for each sequence.
a
10, 15, 20, 25, …
b 10, 30, 50, 70, …
Tip
Draw a table like the ones in
questions 1 to 3 to help you.
Think like a mathematician
7
This pattern is made from hexagons.
How many hexagons will there be in Pattern 20?
Show how you worked out your answer.
Discuss the method you used with other learners. Did you use the same method
or a different method? What do you think is the best method to use?
209
9 Sequences and functions
8
Mia is using trapezia to draw a sequence of patterns.
There are marks over the first and third patterns in her sequence.
Pattern 1
Pattern 2
Pattern 3
Pattern 4
How many trapezia will there be in Pattern 18?
Show how you worked out your answer.
Summary checklist
I can find and use the term-to-term rules for number sequences drawn as patterns.
I can find and use the position-to-term rules for number sequences drawn as patterns.
9.3 Using the nth term
In this section you will …
Key word
•
nth term
use algebra to describe the nth term of a sequence.
You already know how to work out the position-to-term rule of a linear
sequence.
Example: The sequence 5, 7, 9, 11, …, … has position-to-term rule;
term = 2 × position number + 3
You can also write the position-to-term rule as an nth term expression.
To do this, you replace the words ‘position number’ with the letter n.
So, in the example above, instead of writing:
term = 2 × position number + 3
you would write: nth term = 2 × n + 3
or, more simply: nth term = 2n + 3
210
Tip
2 × n is usually
written as 2n.
9.3 Using the nth term
Worked example 9.3
a
The nth term expression of a sequence is 2n − 1
Work out the first three terms and the tenth term of the sequence.
b Work out the nth term expression for the sequence 7, 10, 13, 16, …
Answer
a
1st term = 2 × 1 − 1 = 1
To find the first term, substitute n = 1 into the expression.
2nd term = 2 × 2 − 1 = 3
To find the second term, substitute n = 2 into the
expression.
3rd term = 2 × 3 − 1 = 5
To find the third term, substitute n = 3 into the expression.
10th term = 2 × 10 − 1 = 19
To find the tenth term, substitute n = 10 into the
expression.
b
Position
number (n)
Term
3×n
3×n+4
1
2
3
4
7 10 13 16
3 6 9 12
7 10 13 16
nth term = 3n + 4
Draw a table showing the position numbers and terms.
The term-to-term rule is ‘add 3’, so add a row to the table
which shows 3 × n.
(3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12)
You can see that if you work out 3 × n + 4, you will get the
terms of the sequence.
(3 + 4 = 7, 6 + 4 = 10, 9 + 4 = 13, 12 + 4 = 16)
Exercise 9.3
1
Copy and complete the workings to find the first four terms of each
sequence.
a
2
nth term = 2n + 1
1st term = 2 × 1 + 1 = 3
2nd term = 2 × 2 + 1 =
3rd term = 2 × 3 + 1 =
4th term = 2 × 4 + 1 =
b
nth term = 3n − 2
1st term = 3 × 1 − 2 = 1
2nd term = 3 × 2 − 2 =
3rd term = 3 × 3 − 2 =
4th term = 3 × 4 − 2 =
Work out the first three terms and the 10th term of the sequences
with the given nth term.
a
n+6
b n−3
c
9n
d 6n
e
2n + 5
f
3n − 1
g 5n + 3
h
4n − 3
211
9 Sequences and functions
3
Match each yellow sequence card with the correct blue nth term
expression card.
4
A
8, 9, 10, 11, …
i
n−4
B
4, 8, 12, 16, …
ii
2n + 4
C
−3, −2, −1, 0, …
iii
4n − 2
D
7, 14, 21, 28, …
iv
4n
E
6, 8, 10, 12, …
v
7n
F
2, 6, 10, 14, …
vi
n+7
The cards show one term from two different sequences.
A
B
12th term in the sequence
nth term is 8n − 4
7th term in the sequence.
nth term is 11n + 16
Which card has the greater value, A or B? Show your working.
5
Show that the first four terms of the sequence with nth term
1
n + 8 are 8 1 , 8 1 , 8 3 and 9.
4
4 2 4
6
Work out the first three terms and the 8th term of the sequences
with the given nth term.
1
n+6
2
a
c
0.2n + 1.5
b
5n − 2 1
d
4.5n − 0.25
2
Think like a mathematician
7
aWork out the first four terms of the sequences with the given nth term.
i
b
212
4n + 12
ii
1
n+1
4
1 − 1n
4
iii 12 − 4n
iv
Discuss with a partner the answers to these questions.
What is similar about the sequences in ai and aii?
What is similar about the sequences in aiii and aiv?
What is different about the sequences in ai and aiii?
What is different about the sequences in aii and aiv?
9.3 Using the nth term
Continued
c
The cards show the nth terms of some sequences.
Sort the cards into two groups.
Give a reason for your choice of groups.
7 − 1n
8 8
13 − n
15 − 2 n
3
9 − 5n
3n + 7
1 n − 19
4
1 n + 12
2
Discuss your choice of groups with other members of the class.
8
9
Look at this number sequence.
24, 18, 12, 6, 0, ...
Simply by looking at the numbers in the sequence, explain why you
can tell that the nth term expression for this sequence cannot be
2n + 22.
Ian and Lin use different methods to work out the answer to
this question.
The nth term expression for a sequence is 4n + 3.
Is the number 51 a term in this sequence?
Ian’s method
Lin’s method
Work out the terms in the Make an equation and
solve it to find n:
sequence:
n = 1, 4n + 3 = 7
4n + 3 = 51
n = 2, 4n + 3 = 11
4n = 51 − 3
n = 3, 4n + 3 = 15
4n = 48
Term-to-term rule is add 4,
n = 48
4 = 12
so sequence is
7, 11, 15, 19, 23, 27, 31, 35, Yes, 51 is the 12th term in
the sequence.
39, 43, 47, 51, …
Yes, 51 is in the sequence.
213
9 Sequences and functions
a
Use Ian’s method and Lin’s method to work out the answer to
this question.
The nth term expression for a sequence is 3n + 5.
Is the number 48 a term in this sequence?
b Write the advantages and disadvantages of Ian’s method and
Lin’s method.
c
Which method do you prefer? Explain why.
d Can you think of a better method? If you can, explain this
method.
e
Use your preferred method to work out the answers to these
questions.
iThe nth term expression for a sequence is 2n − 3.
Is the number 39 a term in this sequence?
iiThe nth term expression for a sequence is 6n + 7.
Is the number 60 a term in this sequence?
10 Copy and complete the workings to find the nth term expression
for the sequence 8, 10, 12, 14, …
Position number (n)
Term
2×n
2×n+
1
8
2
8
What do you
need to add to 2
to get 8? What do
you need to add
to 4 to get 10?
etc.
2 3 4
10 12 14
4
10 12 14
nth term = 2n +
11 Work out an expression for the nth term for each sequence.
Draw a table like the one in Question 10 to help you.
a
6, 8, 10, 12, …
b 5, 8, 11, 14, …
c
6, 11, 16, 21, …
d 3, 7, 11, 15, …
e
2, 10, 18, 26, …
f
2, 9, 16, 23, …
12 This pattern is made from rectangles.
Pattern 1
a
b
c
214
Pattern 2
Pattern 3
Tip
Pattern 4
Write the sequence of the numbers of rectangles.
Work out an expression for the nth term for the sequence.
Draw a table like the one in Question 10 to help you.
Use your nth term expression to find the number of rectangles
in the 20th pattern in the sequence.
9.3 Using the nth term
13 Sofia and Marcus are looking at the number sequence:
4, 4 1 , 5, 5 1 , 6, ...
2
2
Read what they say.
I think the
expression for the nth
term of this sequence
is 4n + 21
I think the
expression for the
nth term of this
sequence is 1n + 4
2
Is either of them correct? Explain your answer.
14 Work out an expression for the nth term for each sequence.
a
9 1 , 9 1 , 9 3 , 10, ...
b
4.6, 5.2, 5.8, 6.4, …
c
−1 1 , − 1, − 1 , 0, ...
d
−0.6, 0.8, 2.2, 3.6, …
4
2
2
4
2
Think like a mathematician
15 Zara and Arun are trying to work out the expression for the nth term of
the sequence 8, 6, 4, 2, …
They both start by drawing this table:
Position number (n)
Term
…×n
1
8
2
6
3
4
4
2
Read what they say next.
As the term-to-term
rule of the sequence
is subtract 2, I think
the next line in the
table is 2 × n
a
b
c
As the term-to-term
rule of the sequence
is subtract 2, I think
the next line in the
table is −2 × n
What do you think? Explain your answer.
Copy and complete the table. Use it to work out the expression
for the nth term of the sequence.
Compare your answers with those of other learners in the class.
215
9 Sequences and functions
16 Work out an expression for the nth term for each sequence.
a
18, 15, 12, 9, …
b 11, 7, 3, −1, …
c
7, 2, −3, −8, …
How well do you think you understand the nth term expressions?
Give yourself a score from 1: Still need lots of practice, to 5:
Feeling very confident.
Summary checklist
I can use the nth term expression for a number sequence.
I can work out the nth term expression for a number sequence.
9.4 Representing simple functions
In this section you will …
Key words
•
algebraically
work out input and output numbers from function machines.
A function is a relationship between two sets of numbers.
You can draw a one-step function as a function machine, like this.
input
1
1
22
1
44
+5
output
6
1
72
1
94
Tip
The numbers that go into the function machine are called
the input.
The numbers that come out of the function machine are called
the output.
216
function
function machine
input
inverse function
map
mapping diagram
one-step function
output
two-step function
9.4 Representing simple functions
You can also draw a function as a mapping diagram, like this.
input 0 1 2 3 4 5 6 7 8 9 10
output 0 1 2 3 4 5 6 7 8 9 10
The input numbers map to the output numbers.
You can also write a function algebraically as an equation.
Use the letter x to represent the input numbers.
Use the letter y to represent the output numbers.
You can then show the previous function machine like this:
x
y
+5
You can write the input (x) and output (y) numbers in a table.
x
1 2 21
y
6 7 21
41
4
91
4
You can also write the function as an equation like this: x + 5 = y
but it is more common to write the equation like this:
y=x+5
Tip
You usually
write a function
equation starting
with y = …
Worked example 9.4
a
Copy and complete the table of values for this two-step function machine.
x
×2
x
0
y
+1
2
3
412
y
b Draw a mapping diagram to show the function in part a.
c Write the function in part a as an equation.
Answer
a
x 0
2
3 41
y 1
5
7
2
10
To work out the y-values, multiply the
x-values by 2 then add 1.
0 × 2 + 1 = 1, 2 × 2 + 1 = 5, 3 × 2 + 1 = 7,
4 1 × 2 + 1 = 10
2
217
9 Sequences and functions
Continued
b
x
input
output
y
c
0
1
0
1
2
2
3
3
4
4
5
6
5
6
7
7
8
9 10
8
9 10
Draw a line connecting each x-value to its
y-value.
Draw an arrow on each line to show that 0
maps to 1, 2 maps to 5, 3 maps to 7 and 4 1
2
maps to 10.
Write the equation with ‘y =’ on the left.
Remember, you can write x × 2 + 1 more
simply as 2x + 1.
y = 2x + 1
Exercise 9.4
1
aCopy and complete the table of values for each one-step
function machine.
ii
i
x
x
2
x
2
y
–3
7 71 8 81
2
2
y
b Draw a mapping diagram for each function in part a.
c
Write each function in part a as an equation.
aCopy and complete the table of values for each two-step
function machine.
i
ii
x
×2
x
b
218
1 21 4 51
y
2
x
y
+3
+3
y
0 11 3 4 1
2
2
y
Write each function in part a as an equation.
x
÷2
x
y
–3
8 10 15 19
y
9.4 Representing simple functions
Think like a mathematician
3
Work out the missing values in the tables for these function machines.
a
i
ii
x
x
y
×2
x
y
b
4
6
9
12
+4
x
y
15
3
y
÷2
5
8 1 111
2
2
Compare your answers with those of other learners in the class.
Discuss the different methods you used.
What do you think is the best method?
Work out the missing values in the tables for these function machines.
ii
a
i
x
×3
51
x
b
y
12 1
2
8
y
–1
2
26
x
÷2
x
y
y
+5
4
10
81
2
111
2
Write each function in part a as an equation.
Activity 9.4
a
b
On a piece of paper, draw two function machines of your own, similar to those
in Question 4.
Draw a table for each function machine and give two x-values and two y-values.
On a different piece of paper, write the missing x-values and y-values.
Exchange function machines with a partner and work out their missing x-values
and y-values.
Exchange back and mark each other’s work.
Discuss any mistakes.
219
9 Sequences and functions
Think like a mathematician
5
Hannah works out the answer to this question.
Work out the missing values in the table for this function machine.
x
×2
x
y
y
+7
15
21
29
This is what she writes.
×2
+7
The function machine is: x
the equation is y = 2x + 7
If you reverse the function machine, you get:
÷2
x
–7
y
so
y
y−7
y−7
So the equation for the inverse function is 2 = x or x = 2
Use the inverse function to work out the missing values.
When y = 15, x = 152− 7 = 82 = 4
When y = 21, x = 212− 7 = 14
2 =7
When y = 29, x = 292− 7 = 22
2 = 11
Answer is:
x
y
a
b
4 7 11
15 21 29
What do you think of Hannah’s method?
What are the advantages and disadvantages of her method?
Discuss your answers.
Use Hannah’s method to answer this question.
Work out the missing values in the table for this function machine.
x
–4
÷3
y
x
y
2
5
8
Compare your equations and answers with those of other learners in the class.
220
9.4 Representing simple functions
6
Copy and complete these inverse function machines and equations.
a
x
+2
y
equation: y = x + 2
x
–2
y
reverse equation: x = …
x
÷4
y
equation: y = …
x
…
y
reverse equation: x = …
x
–3
×8
y
equation: y = 8(x – 3)
x
…
…
y
reverse equation: x = …
b
Tip
c
7
Match each function equation with its
inverse function equation.
The first one is done for you: A and v
You can draw function machines to help you
if you want to.
Ax
×6
y
y = 6x
vx
÷6
y
x=
A
y = 6x
i
x = 7y
B
y = x7
ii
x = 5y − 2
C
y=x+8
iii
x = 7(y + 3)
D
y = 2x + 4
iv
x=y−8
E
y = x7 − 3
v
x= 6
F
y= 5
vi
x=
x+2
In part c,
remember you
write (x − 3) × 8 as
8(x − 3).
y
6
y
y−4
2
221
9 Sequences and functions
8
aCopy and complete the function machine for each table of values.
i
ii
x
y
x
y
x
−3
0
1.5
6.2
x
−5
−1.5
2.5
4.25
y
6
9
10.5 15.2
y
−15
−4.5
7.5
12.75
b Write each function in part a as an equation.
Sofia and Arun are looking at this function machine and table of values.
9
x
y
x
4
5.5
7
y
13
19
25
I think the equation
for this function is
y = 4x − 3.
I think the equation
for this function is
y = 3x + 1.
Is either of them correct? Show all your working.
10 Work out the equation for this function machine and table of values.
x
y
x
1
2
3
y
5
8
11
Explain how you worked out your answer.
11 Marcus is putting numbers into a two-step
function machine.
Read what Marcus says.
Work out the equation for Marcus’s
function. Show all your working.
When x = 4, y = 11.
As my x-values
increase by 4, my
y-values increase by 2.
Summary checklist
I can work out output values of a function machine.
I can work out input values of a function machine.
I can write a function as an equation.
222
9 Sequences and functions
Check your progress
1
For each of these sequences, write
i
the term-to-term rule
a
2
2, 2 1 , 2 2 , 3, …
3
3
ii
the next two terms.
b
6.7, 6.4, 6.1, 5.8, …
This pattern is made from dots.
Pattern 1
a
b
c
d
Pattern 2
Pattern 3
Write the sequence of the numbers of dots.
Write the term-to-term rule.
Draw the next pattern in the sequence.
Copy and complete the table to find the position-to-term rule.
Position number
Term
1
3
2
5
3
4
× position number
× position number +
3
4
5
The position-to-term rule is: term =
× position number +
Work out the first three terms and the 10th term of a sequence with the given nth term.
1
a
n + 81
b 5n − 0.75
2
2
Work out an expression for the nth term for each sequence.
a
9, 11, 13, 15, …
b 15, 12, 9, 6, …
Work out the missing values in the tables for these function machines.
a
i
ii
x
÷4
x
y
b
8
y
–1
10
x
+9
x
4
61
2
y
−5
×2
y
22
29
−1
2
Write each function in part a as an equation.
223
10
Percentages
Getting started
1
2
3
4
5
There are 279 girls in a group of 450 children.
What percentage of the group are
a
girls
b boys?
Estimate, then work out
a
40% of 600
b 140% of 600
Explain the difference between 25% and 0.25%
Xavier earns $20 per hour. Sasha earns $25 per hour.
They are both given a pay increase of $2 per hour.
a
Write the increase as a percentage of Xavier’s pay.
b Write the increase as a percentage of Sasha’s pay.
Copy and complete this table.
100%
$850
4.50 m
10%
$85
60%
120%
c
0.5% of 600
350%
5.40 m
Percentages are often used, instead of actual values, in articles in newspapers,
in magazines, on the internet and on television.
224
10.1 Percentage increases and decreases
Look at these two sentences:
The population has increased from 3.25 million to 3.77 million
•
The population has increased from 3.25 million by 16%
•
The two sentences give the same information but in different ways.
The absolute change is 3.77 − 3.25 = 0.52 million.
The percentage change is an increase of 16%.
Percentages are easier to interpret than actual values if you want to:
describe one number as a fraction of another
•
describe an increase or decrease
•
compare two different increases or decreases.
•
In this unit, you will learn how to calculate percentage changes.
You will understand how useful percentages can be.
10.1 Percentage increases
and decreases
In this section you will …
Key words
•
learn to calculate percentage increases and decreases
absolute change
•
learn to write a change in value as a percentage.
percentage
decrease
The price of a train journey increases from $75 to $105
The price increase is $105 − $75 = $30
To find the percentage increase, you must write the increase as a
percentage of the original price.
percentage
increase
That is 30 × 100% = 0.4 × 100% = 40%
75
Suppose the price decreases from $75 to $60. The decrease is $15.
You can write this as a percentage of the original price in a similar way:
Tip
30 ÷ 75 = 0.4
15
× 100% = 0.2 × 100% = 20%
75
The percentage decrease is 20%.
For an increase or a decrease, 75 is the denominator of the fraction.
225
10 Percentages
Worked example 10.1
A library has 2800 books. Find the number of books if it
a increases by 84%
b decreases by 37%
Answer
a
84% = 0.84
84% of 2800 = 0.84 × 2800 = 2352
There are 2352 more books so the total is 2800 + 2352 = 5152
b 37% = 0.37
37% of 2800 = 0.37 × 2800 = 1036
There are 1036 fewer books so the total is 2800 − 1036 = 1764
Exercise 10.1
1
2
3
4
5
6
7
226
a
c
a
c
a
c
Find 15% of $70
Decrease $70 by 15%
Find 80% of 3200 people
Decrease 3200 by 80%
Find 2% of 19.00 kg
Decrease 19.00 kg by 2%
b
Increase $70 by 15%
b
Increase 3200 by 80%
b
Increase 19.00 kg by 2%
How much will she have if she increases her savings by
a
10%
b 50%
c
70%
I have saved $240.
d 100%
e
120%?
The population of a town is 45 000.
The population is expected to rise by 85% in the next ten years.
Estimate the population in ten years’ time.
Show that
a
81 is 135% of 60
b 60.8 is 190% of 32
c
308 is 220% of 140
a
What percentage of 950 is 380?
b What percentage of 380 is 950?
10.1 Percentage increases and decreases
8
a
What percentage of 40 years is 8 years?
b What percentage of 8 years is 40 years?
9
A metal bar is 1.80 metres long.
It is heated and the length increases by 0.5%.
a
What is the absolute increase in length?
b How long is the bar now?
10 Work out
a
20% of 60 km
b 90% of 60 km
c
170% of 60 km
d 260% of 60 km
11 Copy and complete this table.
Amount
$20
50 kg
90 m
40%
$8
140% 280% 420%
$84
126 m
12 The mass of a child is 22 kg. In the next 10 years, this mass
increases by 150%.
a
Find 150% of 22 kg.
b Find the mass after 10 years.
13 A shop lists its prices in a table.
Item
a
In a sale, all the prices are reduced by 30%. Calculate the
table
sale prices.
armchair
b How much would you save if you bought all three items in
bed
the sale?
Price
$280
$520
$1040
14 Electricity costs are rising by 8%.
The table shows the costs for one year for four customers.
Copy the table and fill in the last column to show the costs for
one year after the price rise.
Customer
A
B
C
Cost before the rise
$415
$629
$1390
Absolute change ($)
Cost after the rise
15 A garage is reducing the prices of cars. Calculate the new prices.
Model
Ace
Beta
Carro
Old price ($)
15 800
21 300
24 200
Decrease (%) Absolute change ($) New price ($)
2.0
12.0
0.5
227
10 Percentages
16 Mia sees this sign in a shop window:
She says: ‘The original price of a coat was $120 so the price
is now $84’
a
Explain the calculation that Mia has done and why her
statement is incorrect.
b What is the price of the coat now?
Think like a mathematician
17 There are 2000 people in a room.
The number increases by P%. Then the number decreases by P%.
a
b
c
d
How many people are in the room if P = 50?
Show how you calculated your answer.
What happens if P = 25?
Investigate other values of P.
Compare your answers with a partner’s.
18 A shop is selling a phone for $80. The shop increases the price
by 10%.
a
Find the new price.
After two weeks, the shop decreases the new price by 10%.
Read what Arun and Sofia say.
The price will go
back down to $80.
b
c
228
The price now will
be less than $80.
Explain why Arun is wrong and Sofia is correct.
Find the price of the phone after the decrease.
Prices
reduced
by 70%
10.1 Percentage increases and decreases
19 The same shop is selling a television for $400.
a
The shop increases the price by 20%. Find the new price.
b The shop increases the price by a further 20%.
Here are three statements:
• The new price is $560
• The new price is more than $560
• The new price is less than $560
Which statement is correct? Give a reason for your answer.
c
Show your answer to a partner.
Is he or she convinced by your explanation?
20 a
Sofia has savings of $500. She spends some money and says:
My savings have
decreased by 150%.
b
Is it possible for her savings to decrease by more than 100%?
Arun has 500 g of rice. He says:
I cooked some rice and
the amount I have has
decreased by 150%.
What can you say about this statement?
Summary checklist
I can write one number as a percentage of another value.
I can increase or decrease a value by a given percentage.
I can calculate the percentage change from one value to another.
229
10 Percentages
10.2 Using a multiplier
In this section you will …
Key word
•learn to use a multiplier to calculate a percentage increase
or decrease.
multiplier
Tip
65%
100%
100%
× 1.65
Tip
54%
100%
In this section, you will learn a more efficient way to calculate
percentage increases and decreases.
Suppose you want to increase $275 by 65%.
You start with $275 = 100%
Then 65% of $275 = $178.75 and the total is $453.75.
100% + 65% = 165%
You can find 165% of $275 in a single calculation.
165% = 1.65 and so 165% of $275 = 1.65 × $275 = $453.75
This is the value after the increase of 65%.
To increase the value by 65% you used a multiplier of 1.65.
Now suppose you want to decrease $275 by 54%.
Again $275 = 100%
So $275 − 54% = 100% − 54% = 46%
46% = 0.46 and so 46% of $275 = 0.46 × $275 = $126.50
This is the value after a decrease of 54%.
To decrease the value by 54% you used a multiplier of 0.46.
In general, original value × multiplier = new value
new value
You can also write this as multiplier = original value
× 0.46
46%
Worked example 10.2
The cost of a flight is $2300
Calculate the percentage change if
a
the cost is increased to $2850 b the cost is reduced to $1690
Answer
a
The multiplier for the increase is 2850 = 1.239 to 3 d.p.
2300
1.239 = 123.9% so the percentage increase is 23.9%
b The multiplier for the decrease is 1690 = 0.735 to 3 d.p.
2300
0.735 = 73.5% so the percentage decrease is 100% − 73.5% = 26.5%
230
The original value
is always the
denominator of
the fraction.
10.2 Using a multiplier
Exercise 10.2
In this exercise, always use a multiplier to calculate a percentage increase
or decrease.
1
What multiplier would you use to
a
increase a value by 63%
b decrease a value by 63%
c
increase a value by 103%
d decrease a value by 88%
2
Match each percentage change to the correct multiplier.
The first one is done for you: A and ii
A
B
C
D
E
F
3
i
ii
iii
iv
v
vi
50% increase
80% increase
80% decrease
120% increase
20% decrease
20% increase
× 0.2
× 1.5
× 1.8
× 1.2
× 0.8
× 2.2
8
Write the multiplier for
a
an increase of 45%
b an increase of 245%
c
a decrease of 45%
Here are some multipliers.
Write the percentage change in each case.
a
× 0.75
b × 1.22
c
× 3.33
d × 0.33
e
× 0.03
Increase each of these numbers by 85%
a
40
b 180
c
12
Find the value of 45 kg after the following changes.
a
an increase of 20%
b an increase of 170%
c
a decrease of 60%
a
The mass of a girl is 26.5 kg.
Several years later her mass has increased by 62%.
Calculate her new mass. Round your answer to 1 d.p.
b A man has a mass of 172.4 kg. He reduces his mass by 38%.
Calculate his new mass.
a
Increase 964 by 65%
b Increase 357 by 195%
c
Decrease 560 by 84%
9
Change each length by the percentage shown.
4
5
6
7
a
b
c
d
Length (mm)
90
240
660
320
Change
180% increase
12% increase
70% decrease
7% decrease
New length (mm)
231
10 Percentages
10 An athlete has a resting pulse rate of 60 beats per minute.
During a race, this increases to 160 beats per minute.
a
Calculate the percentage increase.
b Calculate the percentage decrease after the race, when his
pulse rate falls from 160 to 60 beats per minute.
11 a
Increase 96 by 25%
b Decrease 200 by 40%
c
Increase 60 by 100%
d Decrease 240 by 50%
Activity 10.2a
a
b
c
d
What do you notice about the answers to Question 11?
Write some more questions like this for a partner to answer.
Exchange questions with a partner and answer your partner’s questions.
Exchange back and check your partner’s answers. Discuss any mistakes.
12 a
The population of a town increases from 63 200 by 17%.
Calculate the new population.
The population of a city increases from 7.35 million to
12.82 million.
Calculate the percentage change.
The population of an island is 4120.
The population decreases by 16.5%.
Calculate the new population.
b
c
Think like a mathematician
13 Work with a partner on this question.
This table shows the changing population of China.
a
b
c
232
Year
1950
1960
1970
1980
1990
2000
2010
Population in millions
554
660
828
1000
1177
1291
1369
Calculate the percentage increase in the
population from
i
1950 to 1970
ii
1970 to 2000
iii 1960 to 1990
iv 1950 to 2010
Round your answers to 1 d.p.
In which decade was there the greatest percentage
increase in population?
Use the data to predict the population of China in 2020. Justify your answer.
10.2 Using a multiplier
Activity 10.2b
Find out how the population of your country has changed
from 1950 to 2010.
Are the percentage changes similar to or different from the
percentage changes in China?
Compare your answers with the answers of other learners.
Can you improve your answers?
Tip
A decade is a
period of 10 years.
14 Prices in a shop are reduced. Copy and complete this table.
Original price
Percentage reduction
$280
$420
$620
$750
20%
45%
Reduced price
$217
$705
15 The height of a tree is 3.65 m. Find the new height if the height
increases by
a
15%
b 132%
c
260%
16 The depth of water in a well is decreasing.
Calculate the percentage reduction from
a
Monday to Tuesday
b Tuesday to Thursday
c
Monday to Friday
Day
Monday
Tuesday
Wednesday
Thursday
Friday
17 Here are two sentences.
•
The population of India is 407% of the population of the USA.
•
The population of India is 307% more than the population of the USA.
a
Explain why both these sentences can be correct.
b Compare your explanation with a partner’s.
Can you improve your explanation or your partner’s?
18 Read what Marcus says:
When 650 is
a
Describe two different ways to
increased
by 184%
check that Marcus is correct.
the answer is 1846.
b Which way do you think is better?
Give a reason.
Depth
5.75 m
5.10 m
4.31 m
3.58 m
2.46 m
Summary checklist
I can use a multiplier to calculate a percentage increase or decrease.
I can identify a percentage increase by finding a multiplier.
233
10 Percentages
Check your progress
1
2
3
234
a
Write 32 as a percentage of 80.
b Write 80 as a percentage of 32.
Increase $240 by 35%
a
by first finding the increase in dollars
b by using a multiplier.
In 1960 the population of Indonesia was 88 million.
In 2010 the population of Indonesia was 242 million.
a
Calculate the percentage increase from 1960 to 2010.
b Estimate the population in 2060 if the rate of increase does not change.
11
Graphs
Getting started
1
2
3
A pair of shoes costs $25 less than a coat.
a
If the coat costs $110, find the cost of the shoes.
b The coat costs $x and the shoes cost $y
Write a function to show y in terms of x.
One Singapore Dollar can be exchanged for 80 Japanese Yen.
If d Singapore Dollars can be exchanged for y Japanese Yen, which of these
equations is correct?
y = d + 80 y = d − 80 y = 80d y = d
80
Here is a function: y = x + 2
a
Copy and complete this table of values.
x
y
0
1
2
4
3
b Use the table in part a to draw a graph of y = x + 2
This graph shows the temperature of some water.
40
Temperature (°C)
4
−3 −2 −1
0
30
y = −2x + 30
20
10
0
a
b
1
2
3
4
5
Time (minutes)
6
7
8
How does the graph show that the water is cooling?
How long does it take for the water to cool by 10 °C?
235
11 Graphs
In the 17th century, the Frenchman René Descartes showed how to plot
points on a grid and use this to draw lines and curves.
In his honour, we still call this method ‘Cartesian coordinates’.
y
=
x
y
3
(–2, 3)
(4, 2)
2
1
–3
–2
0
–1
1
2
3
4 x
–1
y = –2
–2
–3
René Descartes, 1596–1650
(1, –3)
Cartesian grid
You have used positive and negative numbers as coordinates to show points
on a Cartesian grid. You know that equations involving x and y can correspond
to lines and curves on such a coordinate grid.
In this unit, you will concentrate on straight-line graphs.
Two examples, y = −2 and y = x, are shown on the Cartesian grid above.
11.1 Functions
In this section you will …
•
represent situations in words and using functions.
The cost of hiring a hall is in two parts. There is
a booking fee of $15
•
a charge of $40 per hour.
•
The total cost of hiring the hall for 3 hours is $40 × 3 + $15 = $135
Suppose the hall is hired for n hours and the cost is $c.
Then c = 40n + 15
This function shows how to work out the cost for any number of hours.
236
Tip
The charge per
hour is multiplied
by the number
of hours, and this
is added to the
booking fee.
11.1 Functions
If you want to hire the hall for 3 hours, then n = 3 and
c = 40 × 3 + 15 = 135
The cost is $135.
If you want to hire the hall for 6 hours, then n = 6 and
c = 40 × 6 + 15 = 255
The cost is $255.
Worked example 11.1
The cost of hiring a digger is a fixed charge of $35 plus $10 per day.
a Find the cost of hiring the digger for 7 days.
b The cost of hiring the digger for n days is $y
Write a function to find the cost for any number of days.
Answer
a For 7 days, the cost is $35 + $10 × 7 = $105
b y = 10n + 35
Exercise 11.1
1
2
3
4
Arun buys some books online. The cost is $6 for each book plus
postage of $4.
a
Work out the total cost, including postage, of
i 3 books
ii 6 books
iii 12 books.
b Write a function to show the cost in dollars (c) of b books.
A plumber comes to a house to do a repair.
He charges a fixed fee of $45 plus $30 per hour.
a
Work out the total cost for a job that lasts
i one hour
ii 3 hours
iii 1.5 hours.
b Write a function to show the cost in dollars (c) of a job that
takes h hours.
Theatre tickets cost $12 each plus a booking fee of $3.
a
Work out the total cost, including the booking fee, of
i 4 tickets
ii 6 tickets
iii 10 tickets.
b If t tickets cost $d, write a function for d in terms of t.
The cost of scaffolding is $80 delivery plus $50 per week.
a
Work out the cost of hiring scaffolding for
i 2 weeks
ii 4 weeks
iii 7 weeks.
b The cost is $y for w weeks. Find an expression for y.
237
11 Graphs
5
The cost of printing photos is $2 per photo plus a fixed charge of $3.
a
Work out the cost of printing 40 photos.
b The cost of n photos is $c. Write a function for c.
c
Marcus pays $49.
How many photos were printed for Marcus?
The cost of hiring a car is a fixed fee of $25 plus $45 per day.
a
Show that the cost of hiring a car for 7 days is $340.
b Read what Sofia says:
If the cost of hiring
a car for 7 days is
Explain why Sofia is not correct.
$340, then the cost of
c
Write a function to show the cost in dollars
hiring it for 14 days is
(a) of hiring a car for n days.
2 × $340 = $680
A bamboo plant is 1.5 m tall. It grows 0.2 m every week.
a
Work out the height after
i 2 weeks
ii 4 weeks.
b How long will it take until the bamboo is 3.5 m tall?
Justify your answer.
c
Write a function to show the height in metres (h) after t weeks.
Read what Zara says:
a
How old is Zara if her father is
I am 2 years less
i 40
than half my
ii 52?
father’s age.
b Zara is z years old and her father is f years old.
Write a function for z in terms of f.
c
How old is Zara’s father if Zara is 30 years old?
6
7
8
Think like a mathematician
9
Here is a function: r = 18 − 3t
a
b
c
238
Work out the value of r when
i
t=2
ii
t=5
iii t = 0
iv
What happens when the value of t is more than 6?
A car is on a journey. The amount of fuel in the tank of the car after
t hours is r litres, where r = 18 − 3t
What can you say about the possible values of t?
t=6
11.1 Functions
10 Here is a shape. All the lengths are in cm.
The perimeter is p cm and the area is a cm2.
a
Show that p = 2L + 22
b Find a function for a in terms of L.
11 There are 108 litres of water in a tank. 9 litres flow out of
the tank every hour.
a
How much water is in the tank after
i 1 hour
ii 3 hours
iii 7 hours?
b How long will it be until the tank is empty?
c
There are l litres of water in the tank after h hours.
Complete this function: l = ...................
12 If x = 5 then y = 30
Which of these functions could be correct?
A y = 6x
B
y = 4 x + 10
C y = x + 30
D y = 40 − 2 x
E
y = 8x − 10
13 When x = 4, y = 6
a
Show that a possible function connecting y and x is y = 2 x − 2
b Show that a possible function connecting y and x is
y = 0.5x + 4
c
Find three more possible functions if y = 6 when x = 4
Write them in the form y = ...................
14 The cost of booking a room for a meeting is a fixed charge plus an
amount for each person.
The cost is $c for n people and c = 8n + 40
Explain what the numbers 40 and 8 show.
15 Here is a function: y = 20 x + 15
a
Describe a situation that this function could represent.
You must explain what x and y stand for.
You must explain what the numbers 20 and 15 tell you.
b Look at a partner’s answer to part a. Is the answer clear?
Can you improve it?
8
L
L
2
3
3
6
Someone says to you: ‘Why do you need to describe situations like the ones in this
exercise with a function when you can describe them in words?’ What would be
your reply?
239
11 Graphs
Summary checklist
I understand how a situation can be represented in words or as a function.
11.2 Plotting graphs
In this section you will …
Key word
•
construct a table of values for a function
plot
•
use the table to plot a graph.
Here is a function: y = 2 x − 1
You can substitute different values of x into the function to find
the y-value, for example:
if x = 3 then y = 2 × 3 − 1 = 5
•
•
if x = −2 then y = 2 × −2 − 1 = −4 − 1 = −5
•
and so on.
You can then complete a table of values like this.
x
y = 2x − 1
−2 −1
0
−5 −3 −1
1
1
2
3
3
5
4
7
The table gives you coordinates: (−2, −5), (−1, −3), (0, −1)
and so on.
You can use these coordinates to plot points on a grid.
You can draw a straight line through all the points.
240
y
8
7
6
5
4
3
2
1
–3 –2 –1–10
–2
–3
–4
–5
–6
y = 2x − 1
1 2 3 4 5 x
11.2 Plotting graphs
Worked example 11.2
Here is a function: y = 8x + 4
a
Copy and complete this table of values.
x
y = 8x + 4
−2 −1
0
4
1
2
3
b Use the table to draw the graph of y = 8x + 4
Answer
a
x
y
−2
−12
b
−1
−4
0
4
1
12
2
20
3
28
For example, if x = −2
then y = 8 × −2 + 4 =
−16 + 4 = −12
y
30
20
y = 8x + 4
Choose a scale so that you
can plot all the points.
10
–2
–1
0
1
3 x
2
–10
–20
Exercise 11.2
1
aCopy and complete this table of values for the function
y = 2x + 3
x
y
2
−2 −1
1
0
1
2
7
3
b Use the table to plot a graph of y = 2 x + 3
aCopy and complete this table of values for the function
y = 3x + 2
x
y
b
−2 −1
−4
0
1
2
3
11
Use the table to plot a graph of y = 3x + 2
241
11 Graphs
3
a
Copy and complete this table of values for the function y = 2 x + 6
x
y
b
c
d
4
a
5
0
1
8
2
Use the table to plot a graph of y = 2 x + 6
Extend the line on the graph to show that it goes through the
point (3, 12).
Show that the coordinates (3, 12) are correct for the
function y = 2 x + 6
Copy and complete this table of values for the function y = −2 x
x
y
b
a
−3 −2 −1
2
−2 −1
4
0
1
2
3
−6
If x = −2 then
y = −2 × −2 = 4
Use the table to plot a graph of y = −2 x
Copy and complete this table of values for the function y = 4 − x
x
y
−2 −1
5
0
1
2
3
1
4
5
Use the table to plot a graph of y = 4 − x
Where does the graph cross the x-axis?
Where does the graph cross the y-axis?
i Use the function to show that if x = 10 then y = −6
ii Is the point (10, −6) on your line?
aCopy and complete this table of values for the function
y = 10 x + 30
b
c
d
e
6
x
y
b
c
d
e
f
242
−2 −1
20
0
1
2
3
Tip
4
Use the table to plot a graph of y = 10 x + 30
Use a scale of 1 cm = 1 unit on the x-axis and 1 cm = 10 units
on the y-axis.
Where does the graph cross the y-axis?
i Use the function to find the value of y when x = 2.5
iiDoes this correspond to a point on
If the graph
your line?
is extended,
Read what Marcus says:
(10, 130) and (20,
Is Marcus correct? Justify your answer.
260) will be on
the line.
Compare your answers with a partner’s.
Is your partner correct?
11.2 Plotting graphs
7
Here is a function: y = 2 x + 40
a
Complete this table of values.
x
y
−10
0
10
20
30
100
40
Use the table of values to plot a graph of y = 2 x + 40
Where does the graph cross the y-axis?
Which of these points are on the line?
P (15, 70) Q (50, 140) R (37, 114)
S (−20, 0) T (100, 240)
e
Compare your answer to part d with a partner’s. Do you agree
about which points are on the line? Who is correct?
Here is a function: y = 5x − 15
a
Complete these coordinates of points on the graph of y = 5x − 15
i (4,
)
ii (7,
)
iii (0,
)
iv (20,
)
v (3,
)
b Where does the graph of y = 5x − 15 cross the y-axis?
c
Where does the graph of y = 5x − 15 cross the x-axis?
The cost of hiring a drill is in two parts. There is a delivery charge
of $5 plus $2 per day.
a
The cost of hire for n days is $c. Explain why c = 2 n + 5
b Copy and complete this table of values.
b
c
d
8
9
n
c
0
5
1
2
3
4
5
6
7
19
Tip
c is the subject
of the formula.
Put c on the
vertical axis.
Use the table to plot a graph of c = 2 n + 5. You only need
positive axes.
d Why do you only need positive axes?
10 A motor runs on diesel. Initially there are 40 litres of diesel in
the fuel tank.
The motor uses 5 litres per hour.
After h hours there are f litres of diesel remaining.
a
Explain why f = 40 − 5h
b Copy and complete this table of values.
c
h
f
0
1
2
30
3
4
5
6
243
11 Graphs
c
d
Use the table to draw a graph of f = 40 − 5h
How does the graph show that there were 40 litres of diesel
initially?
e
Use the graph to find the number of hours until the motor
runs out of diesel.
11 The cost of hiring a car is a fixed charge of $35 plus $15 per day.
a
Write a function to show the cost in dollars (y) of hiring a car
for d days.
b Copy and complete this table of values.
d
y
0
1
2
3
80
5
8
c
d
Use the table to plot a graph of the function.
Use the graph to find the cost of hiring a car for 7 days.
Use the function to check that your answer is correct.
e
How does the graph show the fixed charge?
12 The cost of hiring a van is $60 plus $20 per hour.
a
Work out a function to show the cost in dollars (c) of hiring
the van for n hours.
b Copy and complete this table of values.
n
c
c
d
0
1
2
100
3
4
5
6
Use your table to plot a graph to show
the cost.
Read what Zara says:
The cost for 12
hours is twice the
cost for 6 hours.
Is Zara correct? Give a reason for your answer.
244
11.2 Plotting graphs
Think like a mathematician
13 A plant is initially 10 cm high. It grows 2 cm a week.
After x weeks the height is y cm.
Write a function to show y in terms of x.
Copy and complete this table of values.
x 0
y 10
1
2
3
4
5
6
There are three lines on this graph, A, B and C.
c
d
e
Which line shows the growth of the plant?
The other two lines show the growth of two other
plants. Describe the growth of each of these plants.
Compare your answer to part d with a partner’s.
Do you agree on the answer?
Height
a
b
y
40
35
30
25
20
15
10
5
0
y = 10 + 5x
C
y = 10 + 2x
B
A
y = 10 + x
1 2 3 4 5 6 x
Weeks
14 A function is y = 2x + 5
a
Copy and complete this table of values.
x
y
b
c
d
e
−3 −2 −1
−1
0
1
2
3
11
Draw a graph of y = 2x + 5
Where does the graph cross
i the y-axis
ii the x-axis?
Is there a way to use the function to predict where the line will
cross the axes, before you make a table of values? Explain how
you can do this.
Test your method from part d to see if it gives the correct
answers for the function y = 2x + 2
Summary checklist
I can construct a table of values for a function of the form y = mx + c
I can use the table of values to draw a graph of the form y = mx + c
245
11 Graphs
11.3 Gradient and intercept
In this section you will …
Key words
•
coefficient
learn to interpret the values of m and c for a function of the
form y = mx + c
equation of a line
gradient
A function in the form y = mx + c where m and c are numbers is called
a linear function.
All the functions in Section 11.2 were linear functions.
Here are some linear functions and the values of m and c.
Function
y=x
y = 5x + 2
y = 3x − 4
y = 4x
m
1
5
3
4
c
0
2
−4
0
Function
y = −5x + 3
y = −2x − 4
y = −x + 10
y = 12
m
−5
−2
−1
0
linear function
x-intercept
y-intercept
c
3
−4
10
12
Tip
m is the
coefficient of x.
Tip
Here is a table of values and a graph for the linear
function y = 6x − 2
x
y
−3
−20
−2
−14
−1
−8
0
−2
1
4
2
10
y = −5x + 3 can
also be written as
y = 3 − 5x
3
16
y
20
y = 6x − 2 is the equation of the line.
The graph crosses the y-axis at (0, −2)
−2 is the y-intercept of this graph.
You can see from the graph that if x increases by 1
then y increases by 6.
+1
x
y
+1
−3
−20
−2
−14
+6
+6
+1
−1
−8
+1
0
−2
+6
+6
+1
1
4
15
10
+1
2
10
+6
6
5
3
16
–3
0
–1
–2
–5
+6
6
6 is the gradient of this graph.
In the following exercise, you will investigate
y-intercepts and gradients.
6
1
246
y = 6x − 2
1
1
–15
–20
6
1
6
1
6
1
1
2
3 x
11.3 Gradient and intercept
Exercise 11.3
1
a
Complete this table of values.
x
−3 −2 −1 0
1
2
3
2x
2x + 4
2x − 3
b Use the table to draw, on the same axes, graphs of
i y = 2x
ii y = 2x + 4
iii y = 2x − 3
c
Find the gradient and the y-intercept of each line.
The equations in part b are of the form y = 2x + c where c is
an integer.
d i Write another equation of this type.
iiWithout drawing the graph of this equation, make a
conjecture about what it looks like.
iii Draw the graph to test your conjecture.
Here are three equations.
y = 3x y = 3x + 3 y = 3x − 1
a
Draw a graph of each line. Use a table of values to help you.
b Find the gradient and the y-intercept of each line.
All the equations are of the form y = 3x + c where c is an integer.
c
Draw the graph of another line of the same type.
2
Think like a mathematician
3
4
Investigate the graphs of equations of the form y = 4x + c
where c is an integer.
What do you notice about all your graphs?
a
Copy and complete this table of values for y = −x + 5
x
y = −x + 5
b
c
d
e
−2
7
−1
0
1
4
2
3
4
1
5
Tip
Draw a number
of graphs on the
same set of axes.
Tip
y = −x + 5 is the
same as y = 5 − x
Add another row to this table to show values of y = −x + 2
Draw graphs of y = −x + 5 and y = −x + 2 on the same axes.
Write the gradient of each line.
Write the equation of another line parallel to these lines.
247
11 Graphs
5
a
Copy and complete this table of values for y = −2x + 9
x
y = −2x + 9
6
0
1
7
2
3
4
1
b Add another row to this table to show values for y = −2x + 6
c
Draw graphs of y = −2x + 9 and y = −2x + 6 on the same axes.
d Write the gradient of each line.
e
Write the equation of another line parallel to these lines.
Here are six equations of lines.
y = 4x + 6
7
−2 −1
13
y = 6x + 4
y = 4x + 2
y = 2x + 4
y = 6x + 2
a
Group together lines that are parallel.
b Write the equation of one more line for each group.
These three lines are parallel. One of the lines is y = 5x + 5
y
12
5
x
–6
8
a
b
a
Write the equations of the other two lines.
Write the equation of a parallel line that passes through the origin.
Copy and complete this table of values.
x
x+3
2x + 3
−x + 3
b
c
d
e
f
248
−2 −1
0
1
4
2
3
1
5
1
Use the table to plot graphs of y = x + 3, y = 2x + 3 and y = −x + 3
Plot all three graphs on the same axes.
On the same axes, draw the line y = 3
Write the gradient of each line.
Write the y-intercept of each line.
Write the equations of two more lines with the same y-intercept.
y = 2x + 6
11.3 Gradient and intercept
9
a
Copy and complete this table of values.
x
x−2
3x − 2
−2x − 2
−2 −1
−3
−5
0
0
1
−1
1
−4
2
3
b
Use the table to plot graphs of y = x − 2, y = 3x − 2 and
y = −2x − 2
Plot all three graphs on the same axes.
c
Write the gradient of each line.
d Write the y-intercept of each line.
e
Write the equations of two more lines with the same y-intercept.
10 The cost of a visit by an electrician is given by this function: y = 25x + 40
•
y is the total cost in dollars
Tip
•
x is the number of hours
You only need
•
there is a fixed charge and a charge per hour
positive axes.
a
Copy and complete this table for the function.
x
y
0
1
2
3
4
b
Use the table to plot a graph of y = 25x + 40
Use a scale of 2 cm to 1 unit on the x-axis and 2 cm to 50 units
on the y-axis.
c
Find the y-intercept and the gradient of the graph.
d What do the y-intercept and the gradient show about the
electrician’s charges?
11 The cost of a holiday abroad is in two parts:
•
the cost of the airline flights
•
a charge for each night in the hotel
The total cost is given by this function: y = 100x + 200, where
•
x is the number of nights
•
$y is the total cost.
a
Copy and complete this table of values.
x
y
b
c
d
0
1
2
3
4
5
6
7
900
Use your table to draw a graph of y = 100x + 200
Explain how the graph shows the cost of the flights.
What does the gradient of the graph show?
249
11 Graphs
12 Water is flowing out of a tank.
The amount of water is given by the function y = −3x + 40 where there
are y litres of water in the tank after x minutes.
a
Copy and complete this table of values.
x
y
0
1
2
3
4
28
5
6
7
8
b
Plot a graph to show how the amount of water in the tank changes
over time.
c
What does the y-intercept tell you about the water in the tank?
d What does the gradient tell you about the water in the tank?
13 Here is a function: y = 10x + 30
a
Construct a table of values with x going from 0 to 6.
b Use your table of values to draw a graph of y = 10x + 30
c
Describe a situation that the function y = 10x + 30 could represent.
iExplain what the variables x and y and the numbers 10 and
30 represent.
iiExplain how the numbers 10 and 30 are linked to the graph.
d Look at a partner’s answer to this question.
Is their explanation clear?
Think like a mathematician
14 Here are three functions.
y = x + 6 y = 2x + 6 y = 3x + 6
a
b
c
d
e
Plot a graph of each function. Plot them all on the same axes.
Find the y-intercept for each line.
The x-intercept is the x-coordinate where the line crosses the x-axis.
Find the x-intercept for each line.
Use your answers to part c to predict the x-intercept of the graph of y = 4x + 6
Draw the graph to see if you are correct.
Can you generalise your results and predict the x-intercept for the
graph of y = mx + 6 where m is a positive integer?
In this unit, you have looked at the graphs of linear functions.
These can be written as y = mx + c, where m and c are integers.
What general conclusions have you found?
250
11.4 Interpreting graphs
Summary checklist
I know that equations of the form y = mx + c correspond to straight-line graphs.
I know that m is the gradient and c is the y-intercept.
11.4 Interpreting graphs
In this section you will …
•
read and interpret graphs with several components
•
understand why graphs have specific shapes.
Graphs give information in a visual form. In real-life contexts, this can
help you to understand a situation. For example, a graph that shows
how the distance travelled by a car changes with time can help you to
see how fast the car is travelling.
Worked example 11.4
This graph shows the fares charged by two different taxis.
a
b
c
d
How much does each taxi charge for a journey of 7 km?
Find the fixed charge for each taxi.
Find the charge per kilometre for each taxi.
What distance will cost the same amount in either taxi?
Cost ($)
Each taxi has a fixed charge and a charge per kilometre.
y
40
35
30
25
20
15
10
5
0
Taxi B
Taxi A
1 2 3 4 5 6 7 8 x
Distance (km)
Answer
a
$26 for taxi A and $32 for taxi B
Find the y-coordinate on each line when the
x-coordinate is 7.
b $12 for taxi A and $4 for taxi B
This is the y-intercept of each line.
c
This is the gradient of each line.
$2 for taxi A and $4 for taxi B
d 4 km
This is the x-coordinate where the two lines cross.
251
11 Graphs
3
252
10:30
Ta
ne
sh
a
09:30
09:00
Distance (km)
Bibas 300
Lucas
200
Middja
100
Ackult
Razi and Jake are running laps of a running track.
a
How do you know from the graph that Jake
is running faster than Razi?
b For how long had Razi been running before
Jake started?
c
Where were the runners 9 minutes after
Razi started running?
This graph shows the journeys of a van and a car.
The van is travelling at a constant speed.
a
i What is the speed of the van?
iiFor how long does the van travel at that speed?
The speed of the car increases steadily
from 0 m/s for 20 seconds.
b What is the speed of the car after
20 seconds?
y
30
After the first 20 seconds, the car travels
25
at a constant speed for 20 seconds.
20
Then the speed steadily decreases
15
to 0 m/s in 10 seconds.
10
c
Copy the graph and plot the rest of
5
the journey of the car.
0
0
d Give your graph to a partner so they can
check your answer to part c.
Speed (m/s)
4
a
lik
Za
Time (24-hour clock)
Lucas is driving from Ackult to Bibas.
Simone is driving from Bibas to Ackult.
a
How long did Lucas take to get to Middja?
b For how long did Lucas stop in Middja?
c
How long did Simone take to get from
Bibas to Ackult?
d How far were the cars from Bibas when
they passed one another?
Distance from
Ackult (km)
2
Zalika and Tanesha are cycling on the same route.
The graph shows their journeys.
a
Zalika started at 09 : 00.
What time did Tanesha start?
b How far did Zalika travel in the first hour?
c
How long was Tanesha cycling before he
caught up with Zalika?
Laps run
1
35
30
25
20
15
10
5
0
10:00
Exercise 11.4
8
7
6
5
4
3
2
1
0
0
Simone
0 1 2 3 4 5
Time (hours)
i
z
Ra
ke
Ja
0 1 2 3 4 5 6 7 8 9 10
Time (minutes)
Car
Van
10
20
30
40
Time (seconds)
50
60 x
11.4 Interpreting graphs
e
f
7
1
2
230 260
3
4
200
d Find the fixed charge for plumber A.
e
Find the charge per hour for plumber A.
f
Find the fixed charge for plumber B.
g Find the charge per hour for plumber B.
There are two different tariffs for a
long-distance phone call.
Each tariff has a connection charge plus a
charge for each minute.
The tariffs are shown on this graph.
a
Which tariff is cheaper for a 5-minute call?
b A call costs 200 rupees on tariff B.
How long does it last?
c
What length of call costs the same on
both tariffs?
d Find the fixed charge for each tariff.
e
Work out the charge per minute for
each tariff.
This graph shows the growth of two plants over
a period of 6 weeks.
a
Which plant grew more quickly?
Explain how the graph shows this.
b Work out the initial height of plant X.
c
When were the plants the same height?
d Work out how many centimetres plant Y
grew each week.
Plumber B
100
0
5
Plumber A
1
0
2
3
4
Time (hours)
5
x
6
6
y
400
Cost (rupees)
6
0
300
300
Tariff B
Tariff A
200
100
0
0
1
2
3
4
Time (minutes)
y
40
Height (cm)
hours
cost ($)
y
400
Cost (dollars)
5
When is the car travelling at 15 m/s?
For how many seconds is the car
travelling faster than the van?
Two plumbers charge different rates.
Each plumber has a fixed charge and a
charge per hour.
a
Find the cheaper plumber for a job
that takes 2 hours.
b Plumber B charges $250 for a job.
Find the time for the job.
c
Copy and complete this table to show
the total cost for plumber A.
6 x
5
Plant Y
30
Plant X
20
10
0
0
1
2
3
4
Weeks
5
6 x
253
11 Graphs
8
Litres
A car and a van travel 70 km.
y
This graph shows the fuel in the tank of
20
each vehicle.
15
a
Work out how much fuel each vehicle
had at the start of the journey.
10
b Work out how much fuel each vehicle
used to travel 70 km.
5
c
The two lines cross at one point.
0
What does this indicate?
0
Arun and Marcus are walking along the
same path but in different directions.
The graph shows how far they are from home.
a
How far from home is each person when they
start walking?
b How far does Arun walk in 5 hours?
c
Arun is y km from home after x hours.
Write an equation for the line that shows
Arun’s journey.
d Describe Marcus’s journey.
Give as much detail as you can.
e
The lines cross. What does the point where
they cross indicate?
Van
10
Distance from home (km)
9
Car
20
y
50
60
70 x
2
3
4
Time (hours)
5 x
30
40
50
Kilometres
Marcus
40
30
20
Arun
10
0
0
1
Think like a mathematician
a
b
Describe how the temperature of each
liquid changes. Give as much detail
as you can.
When are the two liquids at the same
temperature?
Temperature (°C)
10 This graph shows the changing temperatures
of two liquids.
y
40
35
30
25
20
15
10
5
0
Liquid A
Liquid B
0 1 2 3 4 5 6 7 8 x
Time (minutes)
Summary checklist
I can interpret a real-life graph that shows a situation with several
distinct sections or shows more than one component.
254
11 Graphs
Check your progress
1
2
The cost of a holiday is $200 for travel plus $150 per night for a hotel.
a
Work out the cost of a 7-night holiday.
b A holiday that lasts n nights costs $c. Write a function for c in terms of n.
The perimeter of this shape is p cm.
w cm
a
Write a function for p in terms of w.
3 cm
b Copy and complete this table of values.
w
p
0
1
2
3
4
5
4 cm
6
3 cm
w cm
c
x
y = 3x + 6
4
5
−3 −2 −1
0
1
2
3
b Use the table to draw a graph of y = 3x + 6
c
Find the gradient and the y-intercept of the line y = 3x + 6
d Write the equation of a line parallel to y = 3x + 6 that passes through the origin.
Here is the equation of a line: y = 12 − 2x
a
Where does the line cross the y-axis?
b What is the gradient of the line?
The depth of water in two flasks is
y
changing. This graph shows the changes.
40
a
Describe how the depth in flask 1 is
changing.
30
b When do the two flasks have the
Flask 1
same depth of water?
20
c
The depth of water in flask 2 is
10
d cm after t minutes.
Choose the correct equation of the
Flask 2
0
line for flask 2.
1
2
3
4
5
0
6 x
d = 5t + 30 d = 6t + 30 Time (minutes)
d = 30 − 5t d = 30 − 6t
Depth (cm)
3
Use the table to draw a graph to show the
perimeter for different values of w.
Here is a function: y = 3x + 6
a
Copy and complete this table of values.
255
Project 4
Straight line mix-up
Here are nine function cards:
y
3
2
1
y
3
2
1
–4 –3 –2 –1 0
–1
–2
y = −2x − 4
–3
–4
–3 –2 –1 0
–1
–2
–3
–4
1 2 x
y = 4 − 2x
x
y
−1
2
0
0
1 2
−2 −4
y
5
4
3
2
1
y=x−1
1 2 3 x
–3 –2 –1 0
–1
–2
y = 3x − 4
x
y
−1 0
0 3
1
6
y = 2x
1 2 3 x
y=x
x
y
2
9
−1 0
3 4
1
5
2
6
You may wish to sketch the graphs, work out the equations, and work out a table
of values for each card.
Here are six property cards:
The gradient is positive
The gradient is negative
The y-intercept is negative
The line passes through
(0, 0)
The line is parallel to
y=x+2
The y-intercept is positive
Can you find a way to arrange the property cards and the function cards in a grid, so that
each function satisfies the property at the top of its column and at the left of its row?
For example, in the grid below, y = 3x − 4 has a positive gradient and a negative
y-intercept.
The y-intercept is
negative
The gradient is
positive
y = 3x − 4
Is there more than one way to arrange the cards?
256
12
Ratio and
proportion
Getting started
1
For each of these shapes, write the ratio of green squares to blue squares.
Write each ratio in its simplest form.
a
b
2
Write each of these ratios in its simplest form.
a
2:4
b 18 : 6
c
6:9
d
Share these amounts between Tim and Chan in the ratios given.
a
$18 in the ratio 1 : 2
b $25 in the ratio 2 : 3
Write the missing numbers in these conversions.
3
4
a
4m=
cm
b
6.5 cm =
d
0.8 kg =
g
e
2.3 l =
mm
ml
32 : 24
c
5t=
f
0.75 km =
kg
m
Ratios are used to compare two or more numbers or quantities.
Every day, ratios are used in all sorts of ways to work out all sorts
of things.
For example, builders use ratios to work out the amounts of
ingredients needed to mix together, to make concrete or mortar.
The ratio and ingredients vary, depending on what the builder will
do with the concrete or mortar.
257
12 Ratio and proportion
To make the mortar for laying brickwork or block pavements, a builder
would use cement and sand in the ratio 1 : 4. This means that every 1 kg
of cement must be mixed with 4 kg of sand. Builders often use a shovel or
bucket to measure their ingredients. For this mortar, they would need one
shovel (or bucket) of cement for every four shovels (or buckets) of sand.
To make a medium-strength concrete for a floor, a builder would use
three ingredients: cement, sand and gravel, mixed in the ratio 1 : 2 : 4.
This means that every 1 kg of cement must be mixed with 2 kg of sand
and 4 kg of gravel.
It is important that a builder uses the correct ratio of ingredients for
each job, otherwise walls may fall down or floors may crack.
12.1 Simplifying ratios
In this section you will …
Key words
•
adapt
simplify and compare ratios.
common factor
A ratio is a way of comparing two or more
highest common
Pastry recipe
quantities.
factor
In this pastry recipe, the ratio of flour to butter is
0.5 kg flour
ratio
0.5 kg : 250 g.
250 g butter
simplify
Before you simplify a ratio, you must write all
water to mix
quantities in the same units.
0.5 kg : 250 g is the same as 500 g : 250 g, which you
write as 500 : 250.
You can now simplify this ratio by dividing both numbers by the highest
500 : 250
common factor. In this case the highest common factor is 250.
÷250
÷250
2 : 1
Divide both numbers by 250 to simplify the ratio to 2 : 1.
If you cannot work out the highest common factor of the numbers in
a ratio, you can simplify the ratio in stages. Divide the numbers in the
ratio by common factors
until you cannot divide any more.
Tip
In the example above you could start by:
When the units
dividing by 10
•
500 : 250
are the same,
÷10
÷10
then dividing by 5
•
50 : 25
you do not need
÷5 10 : 5 ÷5
then dividing by 5 again
•
to write the units
÷5 2 : 1 ÷5
giving you the same answer of 2 : 1.
•
with the numbers.
258
12.1 Simplifying ratios
Worked example 12.1
Simplify these ratios.
a
12 : 20 b
12 : 30 : 24 c
2 m : 50 cm
÷4
The highest common factor of 12 and 20 is 4,
so divide both numbers by 4.
Answer
a
÷4
b ÷6
c
12 : 20
3 : 5
12 : 30 : 24
÷6
2 : 5 : 4
2 m : 50 cm
200 : 50
÷50
÷50
4 : 1
÷6
The highest common factor of 12, 30 and 24 is 6,
so divide all three numbers by 6.
First, change 2 metres into 200 centimetres.
The highest common factor of 200 and 50 is 50,
so divide both numbers by 50.
Exercise 12.1
1
2
3
Simplify these ratios.
a
2 : 10
b 3 : 18
e
36 : 12
f
180 : 20
i
10 : 35
j
75 : 10
Simplify these ratios.
a
5 : 10 : 15
b 8 : 10 : 12
d 18 : 15 : 3
e
27 : 9 : 45
This is part of Ben’s classwork.
a
Explain the mistake that Ben has made.
b Work out the correct answer.
c
g
k
5 : 25
4:6
72 : 20
c
f
20 : 15 : 25
72 : 16 : 32
d
h
l
30 : 5
9 : 15
140 : 112
Question
Simplify the ratio 6 : 12 : 3
Answer
6 ÷ 6 = 1 and 12 ÷ 6 = 2
So the ratio is 1 : 2 : 3
259
12 Ratio and proportion
Think like a mathematician
4
Arun and Sofia compare methods to simplify the ratio 4 mm : 6 cm.
My first step is to
change 4 mm into
0.4 cm.
a
b
5
My first step
is to change 6 cm
into 60 mm.
Who do you think has the better first step? Explain why.
Discuss your answer with other learners in the class.
Simplify these ratios.
a
500 m : 1 km
b 36 seconds : 1 minute
c
800 ml : 2.4 l
d 1.6 kg : 800 g
e
3 cm : 6 mm
f
2 days : 18 hours
g 2 hours : 48 minutes
h
8 months : 1 year
Zara uses this recipe for orange preserve.
6
Orange preserve
750 g oranges
1.5 kg sugar
Tip
Remember that
both quantities
must be in the
same units before
you simplify.
The ratio of
oranges to sugar
is 2:1.
juice of one lemon
Is Zara correct? Explain your answer.
Simplify these ratios.
a
600 m : 1 km : 20 m
b 75 cm : 1 m : 1.5 m
d 3.2 kg : 1600 g : 0.8 kg
e
$1.08 : 90 cents : $9
7
c
f
300 ml : 2.1 l : 900 ml
4 cm : 8 mm : 0.2 m
Activity 12.1
On a piece of paper, write two ratios similar to those in Question 5 and write
two ratios similar to those in Question 7.
Make sure each ratio can be simplified.
Write the answers on a separate piece of paper.
Exchange questions with a partner. Work out the answers to your partner’s questions.
Exchange back and mark each other’s work.
Discuss any mistakes that have been made.
260
12.1 Simplifying ratios
8
Marcus and Sofia are mixing paint. They mix 250 ml of white paint
with 750 ml of red paint and 1.2 litres of yellow paint.
The ratio of white
to red to yellow
paint is 1:3:5.
9
The ratio of white
to red to yellow
paint is 25:75:12.
Is either of them correct? Explain your answer.
Preety answers this question.
Five cups hold 1.2 litres and three mugs hold 900 ml.
Which holds more liquid, one cup or one mug?
This is what she writes.
1.2 litres = 1200 ml
Ratios:
÷5
5 cups : 1200 ml
1 cup : 240 ml
÷5
÷3
A mug holds 60 ml more than a cup.
3 mugs : 900 ml
1 mug : 300 ml
÷3
Use Preety’s method to answer these questions.
a
Four bags of sugar have a mass of 1.3 kg and three bags
of flour have a mass of 960 g.
Which has a greater mass, one bag of sugar or one bag
of flour?
b Eight pens have a total length of 1.2 m.
Five pencils have a total length of 90 cm.
Which is longer, a pen or a pencil?
261
12 Ratio and proportion
Think like a mathematician
10 Work with a partner or in a small group to
answer this question.
This is part of Jed’s homework.
a
b
c
d
Explain why Jed’s first step is to multiply
both of the numbers in the ratio by 10.
What are the advantages of Jed’s
method? Can you think of any
disadvantages?
How could you adapt Jed’s method to
simplify the ratio 0.03 : 0.15?
Discuss your answers with other groups
in the class.
Question
Simplify these ratios.
a 1.5 : 2 b 0.8 : 3.6
Answer
a
1.5 : 2
×10
×10
15 : 20
÷5 3 : 4 ÷5
b
0.8 : 3.6
×10
×10
8 : 36
÷4 2 : 9 ÷4
11 Use Jed’s method to simplify these ratios.
a
0.5 : 2
b 1.5 : 3
c
1.2 : 2.4
d 3.6 : 0.6
e
7.5 : 1.5
f
2.4 : 4
g 1.8 : 6.3
h
2.1 : 0.7 : 1.4
12 Oditi goes for a run three times a week.
Her notebook shows the time she took for each run
one week.
a
Oditi thinks the ratio of her times for Monday to
Monday
1 hour 40 mins
Wednesday to Friday is 1 : 2 : 3.
Wednesday
50 mins
Without doing any calculations, explain how you
Friday
2½ hours
know Oditi is wrong.
b Oditi’s mum uses this method to work out the ratio of Oditi’s times.
×10
÷5
c
262
Monday :
1 hour 40 mins :
1.4 :
14 :
14 :
Wednesday
50 mins
0.5
5
1
: Friday
: 2.5 hours
: 2.5
: 25
: 5
Explain the mistakes Oditi’s mum has made.
Work out the correct ratio of Oditi’s times. Show all your working.
12.2 Sharing in a ratio
In this exercise you have answered questions on:
•
simplifying ratios with two or three numbers
•
simplifying ratios with quantities in different units
•
simplifying ratios with decimal numbers.
a
Which questions have you found
i
b
the easiest ii
the hardest? Explain why.
How can you improve your skills in simplifying ratios?
Summary checklist
I can simplify ratios when the quantities have different units.
I can compare ratios when the quantities have different units.
I can simplify a ratio with more than two parts.
12.2 Sharing in a ratio
In this section you will …
Key words
•
profit
divide an amount into two or more parts in a given ratio.
share
Sometimes you need to share an amount in a given ratio.
For example, Zara, Sofia and Marcus buy a painting for $600.
Zara pays $200, Sofia pays $300 and Marcus pays $100.
You can write the amounts they pay as a
ratio like this:
Simplify the ratio by dividing by 100 to give:
Zara : Sofia : Marcus
200 :
2 :
300
3
: 100
: 1
You can see that Zara paid twice as much as Marcus, and Sofia paid
three times as much as Marcus. There are now 6 equal parts in total
(2 + 3 + 1 = 6). When they sell the painting, they need to share the six
parts of the money fairly between them. They can do this by using the
same ratio of 2 : 3 : 1.
263
12 Ratio and proportion
Follow these steps to share an amount in a given ratio.
1
Add all the numbers in the ratio to find the total number of parts.
2
Divide the amount to be shared by the total number of parts to
find the value of one part.
3
Use multiplication to work out the value of each share.
Worked example 12.2
Share $840 between Alan, Bob and Chris in the ratio 2 : 3 : 1
Answer
2+3+1=6
First, add the numbers in the ratio to find the total number
of parts.
840 ÷ 6 = 140
1 part = $140
Then divide the amount to be shared by the total number of
parts to find the value of one part.
Alan gets 2 × 140 = $280
Bob gets 3 × 140 = $420
Chris gets 1 × 140 = $140
Finally, work out the value of each share using multiplication.
Make sure you write the name of the person with each
amount.
Exercise 12.2
1
Copy and complete the workings to share $80 between So, Luana and
Kyra in the ratio 3 : 2 : 5.
Total number of parts: 3 + 2 + 5 =
Value of one part: $80 ÷
So gets: 3 ×
=
Luana gets: 2 ×
2
264
=
=
Kyra gets: 5 ×
=
Share these amounts between Mia, Beth and Fen in the given ratios.
a
$90 in the ratio 1 : 2 : 3
b $225 in the ratio 2 : 3 : 4
c
$432 in the ratio 3 : 5 : 1
d $396 in the ratio 4 : 2 : 5
12.2 Sharing in a ratio
Think like a mathematician
3
Look again at your answers to questions 1 and 2.
a
b
4
5
6
7
Think about a method you can use to check you have shared each
amount correctly.
Discuss your method with a partner. Do they have the same method?
If they have a different method, which method do you think is better?
Explain your answer.
Dave, Ella and Jia share their electricity bills in the ratio 3 : 4 : 5.
a
How much does each of them pay when their electricity bill is
i $168
ii $192
iii $234?
b Show how to check your answers to part a.
A choir is made up of men, women and children in the ratio 5 : 7 : 3
Altogether, there are 285 members of the choir.
a
How many members of the choir are
i men
ii women
iii children?
b How many more women than men are there in the choir?
c
How many more men than children are there in the choir?
A box of fruits contains oranges, apples and peaches in the ratio
4 : 2 : 3.
The box contains 72 fruits altogether.
a
How many fruits in the box are
i oranges
ii apples
iii peaches?
b The ratio of the number of oranges, apples and peaches is
changed to 3 : 1 : 4.
There are still 72 fruits in the box.
How many fruits in this box are
i oranges
ii apples
iii peaches?
Aden, Eli, Lily and Ziva run their own business.
They share the money they earn from a project in
Project earnings: $450
the ratio of the number of hours they put into
Time spent working on project:
the project.
Aden: 6 hours Eli: 4 hours
On the right is the time-sheet for one of
their projects.
Lily: 3 hours
Ziva: 5 hours
How much does each of them earn from
this project?
265
12 Ratio and proportion
8
Here is a set of ratio cards.
$50
Share $150 …
$25
… in the ratio 2 : 3 : 1
… in the ratio 3 : 1 : 4
$55
Share $120 …
$84
… in the ratio 2 : 6 : 1
$45
$11
$15
Share $132 …
9
$60
$66
Tip
The angles in a triangle add up to
degrees.
Think like a mathematician
10 In pairs or groups, look at the following question and answer.
This is part of Zara’s homework.
Question
A grandmother leaves $2520 in her will, to be shared
among her grandchildren in the ratio of their ages.
The grandchildren are 6, 9 and 15 years old.
How much does each child receive?
Answer
Ratio for grandchildren is 6 : 9 : 15
Total number of parts = 6 + 9 + 15 = 30
Value of one part = $2520 ÷ 30 = 84
6-year-old child gets: $84 × 6 = $504; 9-year-old child gets
$84 × 9 = $756, 15-year-old gets: $84 × 15 = $1260
Check: $504 + $756 + $1260 = $2520 ✓
266
$14
… in the ratio 1 : 5 : 6
$28
Share $126 …
Sort the cards into their correct groups.
Each group must have one pink, one yellow
and three blue cards.
The angles in a triangle are in the ratio 2 : 3 : 5.
Work out the size of the angles.
$75
12.2 Sharing in a ratio
Continued
Zara has got the answer correct.
However, some of her calculations were difficult and she had to use a calculator.
a
b
c
d
How can she make the calculations easier?
Rewrite the solution for her.
Do not use a calculator.
Compare you answers to parts a and b with other groups in your class.
Did you come up with the same idea or different ideas?
What extra step could she add to simplify her solution?
11 Every year, on his birthday, David shares $300
among his children in the ratio of their ages.
This year the children are aged 4, 9 and 11.
Show that, in two years time, the oldest child will receive
$7.50 less than he receives this year.
12 Zhi, Zhen and Lin buy a house for $180 000.
Zhi pays $60 000, Zhen pays $90 000 and Lin
pays the rest.
Five years later, they sell the house for $228 000.
They share the money in the same ratio that they
bought the house.
Lin thinks he will make $9000 profit on the sale of
the house.
Is Lin correct? Show all your working.
13 Akello, Bishara and Cora are going to share $960, either
in the ratio of their ages or in the ratio of their heights.
Akello is 22 years old and has a height of 168 cm.
Bishara is 25 years old and has a height of 152 cm.
Cora is 33 years old and has a height of 160 cm.
a
Without working out the answer, which ratio do you think will
be better for Bishara, age or height? Explain your decision.
b Work out whether your decision was correct.
If it was not, explain why you think you made the
wrong decision.
Summary checklist
I can share an amount in a given ratio with two or more parts.
267
12 Ratio and proportion
12.3 Ratio and direct proportion
In this section you will …
Key words
•
comparison
use the relationship between ratio and direct proportion.
justify
You can see ratios in a variety of situations, such as mixing ingredients
in a recipe or sharing an amount among several people.
Ratios can also be used to make comparisons.
For example, suppose you wanted to compare two mixes of paint.
Pink paint is made from red and white paint in a certain ratio (red : white).
If two shades of pink paint have been mixed from red and white paint,
how do you decide which shade is darker?
The shade which is darker is the shade with the greater proportion of red
paint. You can change the ratios into fractions, decimals or percentages
to compare the proportions of red paint in each shade.
proportion
shades
Pink
Pink
Worked example 12.3
Pablo mixes two shades of pink paint in the ratios of red : white paint shown below.
Perfect pink
3:4
Rose pink 2 : 3
a What fraction of each shade of pink paint is red?
b Which shade is darker? Justify your choice.
Answer
Perfect pink: 3 + 4 = 7
3
Fraction red = 7
Add together the numbers in the ratio to find the total number
of parts.
Three parts out of seven are red. Four out of seven are white.
Rose pink 2 + 3 = 5
Fraction red = 52
Add together the numbers in the ratio to find the total number
of parts.
Two parts out of five are red. Three out of five are white.
b Perfect pink: 3 × 5 = 15
For each shade, write the fraction that is red as an equivalent
fraction with common denominator 35.
For perfect pink, 15 parts out of 35 parts are red.
For rose pink, 14 parts out of 35 parts are red.
There are more parts of red in perfect pink, so this shade
is darker.
a
7×5
35
2
×
7
14
Rose pink:
=
5×7
35
Perfect pink is darker,
because it contains more
parts of red.
268
12.3 Ratio and direct proportion
Exercise 12.3
1
Copy and complete the workings to change each ratio into a fraction.
a
A bag of nuts contains cashew nuts and peanuts in the ratio 2 : 7.
What fraction of the nuts are
i cashew nuts
ii peanuts?
Total number of parts = 2 + 7 =
i fraction that are cashew nuts = 2
ii fraction that are peanuts = 7
b
A box of toys has plastic and paper toys in the ratio 3 : 5.
What fraction of the toys are
i plastic
ii paper?
Total number of parts = 3 + 5 =
i fraction that are plastic = 3
ii fraction that are paper = 5
c
A basket of fruit has apples and bananas in the ratio 3 : 1.
What fraction of the fruit are
i apples
ii bananas?
Total number of parts = 3 + 1 =
i fraction that are apples =
ii fraction that are bananas =
269
12 Ratio and proportion
Think like a mathematician
2
Tio and Kai work out the answer to this question.
A school choir is made up of girls and boys in the ratio 2 : 1.
There are 36 students in the choir altogether.
How many of the students are girls?
Tio uses the ‘sharing in a ratio’ method. Kai uses the ‘fraction of an amount’ method.
Tio
Total number of parts = 2 + 1 = 3
Value of one part = 36 ÷ 3 = 12
Number of girls = 2 × 12 = 24
Kai
Fraction of the choir that are
girls = 2 = 32
2+1
Number of girls = 32 × 36
= 36 ÷ 3 × 2 = 24
Work with a partner or in a small group to discuss these questions.
a
b
3
Compare Tio and Kai’s methods. How are they similar? How are they different?
Whose method do you prefer? Explain why.
A tin of biscuits contains coconut and ginger biscuits in the ratio 3 : 7.
The tin contains 50 biscuits.
a
What fraction of the biscuits in the tin are coconut biscuits?
b How many coconut biscuits are in the tin?
A school tennis club has 35 members. The ratio of boys to girls is 4 : 3.
a
What fraction of the club members are girls?
b How many girls are in the club?
4
Activity 12.3
On a piece of paper, write two questions similar to questions 3 and 4 in this exercise.
Write the answers on a separate piece of paper.
Make sure the questions can be answered without using a calculator.
Exchange questions with a partner. Work out the answers to your partner’s questions.
Exchange back and mark each other’s work.
Discuss any mistakes that have been made.
270
12.3 Ratio and direct proportion
Think like a mathematician
5
Work with a partner or in a small group to discuss this question.
The ratio of red to blue counters in a bag is 5 : 4.
Read what Sofia and Zara say.
It is possible
that there are 62
counters in the
bag.
It is possible
that there are 72
counters in the
bag.
Is either of them correct? Explain how you know.
Discuss your answers with other groups in the class.
6
The ratio of boys to girls in class 8C is 5 : 7.
Which of these cards shows the number of learners that could be in
class 8C?
A
7
8
25
B
C
28
32
D
36
E
38
Justify your choice.
The ratio of men to women in a book club is 3 : 5.
The number of adults in the book club is greater than 20 but fewer
than 30.
How many adults are in the book club?
This is part of Jan’s classwork.
Question
A bag contains blue and yellow cubes. 3 of the cubes
11
are blue.
What is the ratio of blue to yellow cubes?
Answer
3 are blue, so 1 – 3 = 11 – 3 = 8 are yellow.
11
11
11
11
11
So the ratio of blue to yellow is 3 : 8 = 3 : 8.
11 11
271
12 Ratio and proportion
9
Use Jan’s method to work out the following.
a
A bag contains green and red counters. 23 of the counters
are green.
What is the ratio of green to red counters?
b A box of books contains history and science books. 3 of the
7
books are science books.
What is the ratio of history to science books?
c
A café sells sandwiches and cakes. 94 of the items they sell
are cakes.
What is the ratio of sandwiches to cakes that the café sells?
Shani mixes two shades of blue paint in the
following ratios of blue : white.
Sky blue
3:2
Sea blue
7:3
Blue
Blue
a
What fraction of each shade of blue
paint is white?
b Which shade of blue paint is lighter?
Show all your working. Justify your choice.
10 Angelica mixes a fruit drink using mango juice and orange juice
in the ratio 3 : 5.
Sanjay mixes a fruit drink using mango juice and orange juice in
the ratio 5 : 11.
a
What fraction of each fruit drink is orange juice?
b Whose fruit drink, Angelica’s or Sanjay’s, has the higher
proportion of orange juice?
Show all your working. Justify your choice.
11 In the Seals swimming club there are 13 girls and 17 boys.
a
What fraction of the children are boys?
In the Sharks swimming club there are 17 girls and 23 boys.
b What fraction of the children are boys?
c
Which swimming club has the higher proportion of boys?
Show all your working. Justify your choice.
272
Tip
The paint
which is lighter
has a greater
proportion
of white.
12.3 Ratio and direct proportion
Think like a mathematician
12 aWork with a partner or in a small group to discuss the different methods
you could use to answer this question.
b
Lin has black and white counters in the ratio 40 : 840
Ian has black and white counters in the ratio 25 : 535
Who has the greater proportion of black counters, Lin or Ian?
Compare and discuss the different methods with other groups in the class.
What do you think is the best method? Explain why.
13 Liam and Hannah collect coins and stamps.
Liam has 20 coins and 320 stamps.
Hannah has 15 coins and 270 stamps.
Use your favourite method from Question 12 to decide who has the
greater proportion of stamps. Justify your choice.
14 Two jewellery shops sell watches and rings.
Bright Jewellery has 12 watches and 180 rings for sale.
Mega-Jewellery has 30 watches and 438 rings for sale.
Which shop has the greater proportion of watches? Justify
your choice.
Summary checklist
I can use the relationship between
ratio and direct proportion.
273
12 Ratio and proportion
Check your progress
Progress
Nimrah
1
Simplify
thinks
these
of aratios.
number, n .
Write6 cm
an :expression
for bthe number
Nimrah
gets each time.
a
5 mm
12 seconds
: 1 minute
c
400 ml : 1.6 l
a
She
multiplies
the
number
by
4.
b
She
subtracts
6
from
2
Five bags of peanuts have a mass of 1.375 kg and two bags of walnutsthe number.
c
have
She
a mass
multiplies
of 540 g.
the number by 3
d She divides the number by 6
then
5.
subtracts
1.
Which
hasadds
a greater
mass, one bag of peanuts or then
one bag
of walnuts?
Copy
the number
line and
the women
inequality
the number
3
A running
club is made
upshow
of men,
andon
children
in theline.
ratio 8 : 5 : 7.
WriteAltogether,
down thethere
inequality
that
this
number
line
shows.
are 260 members of the running club.
WorkHow
out many
the value
of
expression.
memberseach
of the
running club are
Loli lives
with 3 friends. They share
electricity bill equally between
the four of
a
men
b the
women
c
children?
them.
4
A school quiz club has 45 members. The ratio of boys to girls is 4 : 5.
Write
a formula
out members
the amount
they each pay, in:
a
What
fractiontoofwork
the club
are boys?
i words
ii letters
b How many boys are in the club?
Use your formula in part a ii to work out the
5
Ellenamount
mixes two
shades
ofpay
greywhen
paint the
in the
followingbill
ratios
of black : white.
they
each
electricity
is $96.
6
Simplify these expressions.
Silver grey 2 : 5
Stone grey 3 : 8
a
n+n+n
b 3c + 5c
c 9x − x
7
Simplify
these
expressions
collecting
like
terms.
a
What
fraction
of eachbyshade
of grey
paint
is white?
a
5c
+
6c
+
2d
b
6c
+
5k
+
5c
+
k
c
3xy + 5yz − 2xy + 3yz
b Which shade of grey paint is lighter?
your working. Justify your choice.
8
WorkShow
theseall
out.
a
3 + (x × 2)
b 6 (3 − w)
c 4 (3x + 2)
d 3 (7 − 4v)
9
Solve each of these equations and check your answers.
a
n+3=8
b m − 4 = 12
c 3p = 24
d x=3
5
10 Shen has set a puzzle. Write an equation for
the puzzle. Solve the equation to find the
value of the unknown number.
274
13
Probability
Getting started
1
2
3
4
This is a spinner.
Each colour is equally likely.
a
Find the probability of green.
b Find the probability of blue or yellow.
An unbiased 6-sided dice is thrown.
Work out the probability of getting
a
3
b 6
c
an even number
d less than 5.
Tomorrow at 11:00 it will be sunny, cloudy or wet.
The probability it will be sunny is 25% and
the probability it will be cloudy is 40%.
Find the probability it will be wet.
A large number of drawing pins are dropped on the floor.
87 land point up and 135 land point down.
Work out the experimental probability of landing point up.
Do you know the game ‘rock, paper, scissors’? It is a very
old game and is known by other names as well.
Two people simultaneously show either a fist (rock), the
first two fingers pointing forwards (scissors) or an open
hand (paper).
Scissors beats paper, paper beats rock and rock beats
scissors. This is because scissors cut paper, paper wraps
rock and rock blunts scissors.
If both players choose the same thing it is a draw
(neither wins) and they play again.
Red
White
Yellow
Blue
Green
scissors
beats paper
paper
beats rock
rock
beats scissors
275
13 Probability
This may seem a trivial game but in 2005 the Maspro Denkoh
electronics corporation used it to decide whether to give the contract
to auction its $20 million collection of paintings to Sotheby’s or to
Christie’s auction houses.
Christie’s won with paper, after taking the advice of Flora and Alice,
the 11-year-old daughters of one of the directors of the company.
Their argument was that for beginners, rock seems strongest, so they tend
to start with that. Playing against a beginner, you should start with paper.
This game illustrates two methods of finding probabilities.
One method is to say that each different play – rock, scissors, paper – is
equally likely. If the three outcomes are equally likely, each one has a
probability of 1 .
3
Flora and Alice realised that, for less experienced players, the outcomes
are not equally likely. The probability of starting with rock is more than 1 .
3
13.1 Calculating probabilities
In this section you will …
Key word
•
find the probability of complementary events
•
use lists and diagrams to show equally likely outcomes
complementary
event
•
use lists and diagrams of outcomes to calculate probabilities.
This is a spinner.
The probability that it points to red is 0.2.
The probability that it points to blue is 0.15.
We can write those probabilities as P(red) = 0.2 and P(blue) = 0.15
The sum of the probabilities for all six colours is 1.
This means the probability the spinner does not point to red,
P(not red) = 1 − 0.2 = 0.8
The probability the spinner does not point to blue,
P(not blue) = 1 − 0.15 = 0.85
Getting blue and not getting blue are complementary events.
One of them must happen and they cannot both happen.
If A is an event and A′ is the complementary event,
then P(A′ ) = 1 − P(A)
276
Red
Blue
13.1 Calculating probabilities
Worked example 13.1a
The probability that it will be sunny tomorrow is 40%.
The probability it will not rain tomorrow is 95%.
Find the probability that tomorrow
a will not be sunny
b it will rain.
Answer
a
P(not sunny) = 1 − P(sunny) = 100% − 40% = 60%
b P(rain) = 1 − P(not rain) = 100% − 95% = 5%
Worked example 13.1b
Two unbiased 6-sided dice are thrown.
Find the probability of getting
a the same number on both dice
b a total of 6
c a total of 9 or more.
a
he diagram shows all
T
possible outcomes.
There are 36 outcomes
altogether.
The loop shows the
outcomes with the same
number: (1, 1), (2, 2) and
so on.
There are 6 of them.
6
which is
The probability is 36
Second dice
Answer
6
× × × × × ×
5
× × × × × ×
4
× × × × × ×
3
× × × × × ×
2
× × × × × ×
1
× × × × × ×
1
2 3 4
First dice
5
6
equivalent to 61
277
13 Probability
b T
his table shows the total for
each outcome.
Five outcomes give a total of
6 (shown by a blue loop).
5
The probability is 36
c
sing the same table as for
U
part b, ten outcomes give
a total of 9, 10, 11 or 12
(shown by the red loop).
Second dice
Continued
6
5
4
3
2
1
7
6
5
4
3
2
1
8 9 10 11 12
7 8 9 10 11
6 7 8 9 10
5 6 7 8 9
4 5 6 7 8
3 4 5 6 7
2 3 4 5 6
First dice
For example, 5 on
the first dice and
3 on the second
gives a total of 8.
The probability is 10 = 5
36
18
Exercise 13.1
1
2
3
278
The probability that a football team will win a match is 0.3.
The probability that the team will draw is 0.1.
Work out the probability that the team will
a
not win
b not draw
c
lose
d not lose.
Tomorrow must be hotter, colder or the same temperature as today.
The probability it will be hotter is 55%.
The probability it will be colder is 25%.
Work out the probability that it will
a
not be hotter
b not be colder
c
not be the same temperature.
A spinner has five colours on it.
The probability it shows green is 0.32.
The probability it shows purple is 0.17.
Find the probability that the colour is
a
not green
b not purple.
Tip
All dice in this
exercise are
unbiased, 6-sided
dice.
13.1 Calculating probabilities
4
There are lots of coloured toys in a box. Here are the percentages
of some of the colours.
Colour
Percentage
5
6
yellow
15%
8
red
30%
green
10%
a
Why do the percentages add up to less than 100%?
A child takes a toy at random.
b Find the probability that the toy is
i not orange
ii
not green
iii not red
iv not yellow.
Two dice are thrown. Find the probability that
a
both dice show 5
b one dice shows a 5 and the other does not
c
neither dice shows a 5.
Two dice are thrown. The numbers are added together.
a
Draw a table to show all the possible outcomes.
b Find the probability that the total is
i 3
ii 7
iii 12
c
Copy and complete this table of probabilities.
Total
Probability
7
orange
25%
2
3
4
5
6
7
8
9
Tip
Use the diagram
from part a of
Worked Example
13.1b.
iv 9
10 11 12
Two dice are thrown. The numbers are added together.
a
Find the probability that the total is
i 5 or less
ii more than 5
iii 10 or more
iv less than 10
v a prime number.
7
b Find an event with a probability of 36
c
Give your answer to part b to a partner to check it is correct.
A fair coin and a fair dice are thrown. This table shows the possible
outcomes.
Dice
Coin
a
b
H
T
1
H1
2
3
4
5
6
Tip
T3 stands for a
tail on the coin
and 3 on the dice.
T3
Copy and complete the table.
How many outcomes are there? Are they all equally likely?
279
13 Probability
c
d
Find the probability of
i 6 and a tail
ii
4 and a head
iii a head and an even number
iv a tail and a number less than 3.
Find the probability of each of the events in part c not happening.
5
e
Describe an event with a probability of 12
f
Give your answer to part e to a partner to check.
9
Here are two spinners.
a
The two spinners are spun. Draw a diagram to
4
1
show all the outcomes.
b Work out the probability that
3
2
i both spinners show a 1
ii neither spinner shows a 1
iii both spinners show the same number
iv the spinners do not show the same number.
c
The two scores are added together.
Draw a table to show the possible totals.
d Find the probability that the total is
i 4
ii
5
iii not 7
iv a multiple of 3
v a factor of 12.
e
Now the scores on the spinners are multiplied.
Draw a table to show the possible products.
f
Find the probability of each of the different possible products.
g Find the probability that the product is
i 6 or more
ii less than 6
iii an odd number
iv an even number.
10 aTwo fair coins are flipped.
Copy and complete this table to show the outcomes.
First coin
280
H
T
Second coin
H
T
HT
3
1
2
Tip
Use a table like
the one in part
b of Worked
example 13.1b.
Tip
The product
is the result of
multiplying two
numbers.
13.1 Calculating probabilities
b
Read what Arun says:
When you throw
Explain why Arun is not correct.
two coins there are three
Find the probability of
outcomes. They are 2 heads,
2 tails or a head and a tail.
i 2 heads
ii
2 tails
So the probability of 2 heads
iii a head and a tail.
is 31
Another way to show the outcomes
when two fair coins are thrown is a tree diagram.
Copy the tree diagram and fill in the missing outcomes.
First Second Outcome
Explain how the table in part a and the tree diagram in part d
coin
coin
show the same outcomes.
Three fair coins are thrown. One possible outcome is HHH, a
H
...
head on all three coins. List all the possible outcomes in this way.
Draw a tree diagram to show the results of throwing three fair
H
T
HT
coins. Use it to check your answer to part f.
When three fair coins are thrown, find the probability of
H
...
T
i 3 heads
ii 3 tails
iii not getting 3 heads
iv 2 heads and 1 tail
T
...
v 1 head and 2 tails.
c
d
e
f
g
h
Think like a mathematician
11 Investigate the possible outcomes when 4 fair coins are thrown.
You should find all the possible outcomes and find probabilities of different events.
Use your experience from Question 10 to help you.
12 Zara has three cards with numbers on them.
2
4
5
She puts the cards side by side in a random order to make a 3-digit number.
a
List all the possible numbers. Make sure you have found them all.
b Find the probability that the number formed is
i an odd number
ii
an even number
iii more than 400.
Zara adds an extra card. Now she has four cards.
2
4
5
8
Zara takes two cards at random and places them side by side to make a 2-digit number.
c
List all the possible numbers she can make. Make you sure you have found them all.
d Find the probability that the 2-digit number
i is 48
ii
is not 48
iii is an odd number
iv is an even number
v
includes the digit 2.
281
13 Probability
Now Zara takes three cards at random and places them side
by side to make a 3-digit number.
e
List all the possible numbers she can make.
f
Find the probability that the 3-digit number is
i an odd number
ii
an even number
iii
less than 500.
In this exercise you have used different methods to find
outcomes. What are they? Which do you prefer and why?
Summary checklist
I can find the probability of a complementary event.
I can use a chart, a table or a list to find all possible outcomes.
I can use lists and diagrams of outcomes to calculate probabilities.
13.2 Experimental and
theoretical probabilities
In this section you will …
Key words
•
experimental
probability
calculate experimental probabilities and compare them to
theoretical probabilities.
You can use equally likely outcomes to calculate probabilities.
When this is not possible you can do an experiment.
A spreadsheet is used to simulate throwing a dice 200 times.
Here are the results of the experiment.
Score
Frequency
theoretical
probability
1 2 3 4 5 6
30 36 37 33 35 29
From the information in the table, we can work out the experimental probabilities:
•
30
The experimental probability of 1 is 200
= 0.15
•
36
The experimental probability of 2 is 200
= 0.18
•
282
33  29  98
The experimental probability of an even number is 36 200
= 0.49
200
13.2 Experimental and theoretical probabilities
We know that each number is equally likely with a fair dice so we can also
calculate the theoretical probabilities:
The theoretical probability of 1 is 61 = 0.167 to 3 d.p.
•
The theoretical probability of 2 is 61 = 0.167 to 3 d.p.
•
The theoretical probability of an even number is 63 = 12 = 0.5
•
The experimental probabilities and the theoretical probabilities are very
similar. This shows that the spreadsheet simulation is reliable.
Worked example 13.2
Read what Marcus says.
Event
2 heads 2 tails
Frequency 17
14
1 head and 1 tail
19
a
I have thrown
Calculate the experimental probability of
2 coins 50 times.
each outcome.
The results are in
b Calculate the theoretical probability of
this table.
each outcome.
c Marcus’s teacher thinks Marcus has made up his results.
What do you think? Give a reason for your answer.
Answer
a
The experimental probability of 2 heads is 17
= 0.34
50
The experimental probability of 2 tails is 14
= 0.28
50
The experimental probability of 1 head and 1 tail is 19
= 0.38
50
b There are four equally likely outcomes: HH, HT, TH, TT
The theoretical probability of 2 heads is 14 = 0.25
The theoretical probability of 2 tails is also 14 = 0.25
There are two ways to get 1 head and 1 tail: HT or TH
The theoretical probability is 24 = 12 = 0.5
c
The experimental and theoretical probabilities are not similar.
It looks as if Marcus may have made up his results.
283
13 Probability
Exercise 13.2
1
A learner throws a coin 50 times. This table shows the results.
T
H
H
T
T
H
T
H
T
H
T
T
T
T
H
T
H
H
T
T
T
H
H
T
T
H
T
H
H
T
H
H
T
T
H
T
H
H
T
T
H
H
T
T
H
T
T
H
T
T
a
Use the first row of the table to calculate the experimental
probability of a head based on the first 10 throws.
b Use the first two rows of the table to calculate the
experimental probability of a head based on 20 throws.
c
In the same way, find the experimental probability of a head
based on
i 30 throws
ii
40 throws
iii 50 throws.
d Compare the experimental probabilities you have found so
far with the theoretical probability of a head.
The learner throws the coin another 50 times. Here are the results.
H
T
T
H
H
H
T
T
T
T
H
T
H
H
T
H
T
T
H
T
T
T
H
H
H
T
H
H
H
H
T
H
T
T
H
H
T
T
H
H
T
T
T
H
T
H
T
T
T
T
e
2
Use the two sets of results to find the experimental probability
of a head based on 100 throws. How close is it to the
theoretical probability?
This spinner has 3 sectors.
The probability of red, P(red) = 0.6
Blue
The probability of white, P(white) = 0.3
The probability of blue, P(blue) = 0.1
Here are the results of 50 spins.
White
R
W
R
R
R
284
W
R
W
W
R
R
W
R
R
R
R
R
W
R
R
B
W
R
R
W
B
R
R
R
R
B
R
R
R
R
B
R
R
B
W
R
R
R
B
R
R
W
R
R
R
Red
13.2 Experimental and theoretical probabilities
a
b
c
d
Use each row to find an experimental probability of red based
on 10 spins.
Find two different sets of 25 spins and use them to find the
experimental probability of red.
Use all 50 spins to find experimental probabilities of red,
white and blue.
Here are the results of 800 spins.
Colour
Frequency
e
red
489
white blue
218
93
Use these results to find experimental probabilities for
each colour.
Read what Marcus says:
It is better to use a large
number of spins to work out
experimental probabilities.
3
Do you agree? Give a reason for your answer.
This question is about throwing six dice together
and seeing if there is at least one 6.
Four learners each threw six dice together a
number of times. Here are their results.
Name
Number of throws
Frequency of at least one 6
a
b
c
Arun
10
7
Sofia
20
9
Marcus
40
31
Zara
50
36
Work out the experimental probability of at least one 6 for
each learner.
Combine the four sets of results to get another experimental
probability.
A computer simulated 500 throws. There was at least one six
333 times.
Work out an experimental probability from this data.
285
13 Probability
d
In fact, the theoretical probability of throwing at least one 6 is
0.6651.
Compare the experimental probabilities with the theoretical
probability.
Activity 13.2
Work with another learner on this question. Each pair will need a dice.
Design and carry out an experiment to answer this question:
Is your dice unbiased?
Before you start, you need to decide:
how many times to throw a dice
•
•
how to record your data
•
how to compare experimental probabilities and theoretical probabilities.
Write your plan before you start. Give reasons for your conclusion based on your data.
4
Work with one or more other learners on this question.
You learnt about the number π (pi) in Unit 8. It is the ratio of the
circumference of a circle to its diameter.
The value of π is a decimal that does not terminate and has no
pattern to its digits.
Here are the first 200 decimal places of π:
3.141 592 653 589 793 238 462 643 383 279 502 884 197
169 399 375 105 820 974 944 592 307 816 406 286 208 998
628 034 825 342 117 067 982 148 086 513 282 306 647 093
844 609 550 582 231 725 359 408 128 481 117 450 284 102
701 938 521 105 559 644 622 948 954 930 381 96
Look at this statement:
All the digits from 0 to 9 are equally likely.
a
b
c
286
Devise and carry out an experiment to test this statement.
Use experimental probabilities and compare them with
theoretical probabilities.
Describe your experiment and your result.
Give a reason for your conclusion.
Look at the results of another pair.
How do they compare with yours?
13.2 Experimental and theoretical probabilities
5
You need a spreadsheet to answer this question. You also need to
know how to use it to generate random numbers.
a
Carry out a simulation to model throwing a coin 50 times.
Find the experimental probability of throwing a head and
compare it with the theoretical probability.
b Repeat part a another 5 times. How much do the experimental
probabilities vary?
c
You now have the results of 300 simulated throws. Use them
all to find an experimental probability of throwing a head.
d Experiment with larger numbers of throws, finding an
experimental probability of throwing a head each time.
Comment on your results.
In some situations, you can find theoretical probabilities based on equal likelihood
and you can also find experimental probabilities. What is the connection between
the two?
Summary checklist
I can use the results of an experiment to find experimental probabilities
and compare them to the theoretical probability.
287
13 Probability
Check your progress
1
2
4
3
A spinner has a yellow section. The probability of landing on yellow is 0.27.
Work out the probability of getting a different colour.
Zara writes the digits 3, 6 and 9 in a random order to make a 3-digit number.
a
List all the possible 3-digit numbers she could make.
b Find the probability that Zara’s number is
i less than 500
ii an odd number
iii a multiple of 3.
An unbiased tetrahedral dice has four faces
showing the numbers 1, 2, 3 and 4.
a
Two unbiased tetrahedral dice are thrown.
Copy and complete this table to show the
possible totals.
First dice
1
4
3
2
1
2
5
1
2
3
4
Second dice
b
4
Work out the probability of
i a total of 6
ii a total of less than 6
iii the same number on each dice.
Here are the results of a computer simulation of throwing a dice 40 times.
3
2
a
b
288
5
6
4
2
3
2
1
2
2
2
2
2
6
6
1
6
6
5
4
1
5
3
3
3
1
6
6
3
1
1
4
6
2
3
Find the experimental probability of getting 3.
Compare the experimental and theoretical probabilities of getting a 3.
6
1
3
6
Project 5
High fives
For this problem, you need to be able to generate random numbers between 1 and 5.
You could do this by:
•
using the random number function on a calculator or spreadsheet
•
putting five counters in a bag and picking one out
•
making a five-sided spinner
•
rolling a ten-sided dice and subtracting 5 if you get an answer greater than 5.
Imagine spinning two 1 to 5 spinners and writing down the higher of the two numbers.
If you did this lots of times, how often would you expect to write each number?
Let’s try an experiment to find out.
Step 1: Generate some pairs of random numbers.
Step 2: Write the larger number in each pair. For example, if you get the numbers
2 and 4, write 4; if you get 3 and 3, write 3.
Step 3: Do this 100 times. (You may want to work in a group to do this, or use a
spreadsheet to generate the numbers automatically.)
Step 4: Display your results. You could use a bar chart to do this.
Are your results what you expected?
Sometimes, mathematicians predict the results of an experiment using theoretical
distributions.
Here is a sample space diagram that you could complete to identify the higher
number for every possible combination:
Spinner B
1
1
2
3
4
5
Spinner A
2
3
4
5
3
In 25 trials, how many times would you expect to write each number?
How could you scale this up to work out how often each number would occur in 100 trials?
Can you picture the sample space diagram if the spinners went from 1 to 6 instead of
1 to 5? How many times would you expect to write each number in this case?
If your spinners went from 1 to 7, how many times would you expect to write each
number?
Can you work out how many times you would write the largest number if the spinners
went from 1 to 10?
Or from 1 to 100?
289
14
Position and
transformation
Getting started
1
2
3
4
5
6
290
Use a protractor to
a
measure the size of this angle
b draw an angle of 125°.
Work out the distance between these coordinates.
a
(9, 2) and (9, 11)
b (3, 8) and (5, 8)
A point P has coordinates (6, 1). P is translated 3 squares right
and 4 squares up to point P′.
Work out the coordinates of P′.
Make a copy of this diagram.
a
Reflect shape A in the x-axis. Label the shape B.
b Reflect shape A in the y-axis. Label the shape C.
Copy the diagram.
Rotate the shape about the centre C
a
90° anticlockwise
b 180°
Copy this shape onto squared paper.
Enlarge the shape using scale factor 2.
y
4
3
2
1
–4 –3 –2 –1 0
–1
–2
–3
A
–4
C
1 2 3 4 x
14 Position and transformation
In Stage 7 you learned how to transform 2D shapes by reflecting,
translating or rotating them.
Here is a summary of the key points:
The shape before any transformation is called the object.
•
The shape after the transformation is the image.
•
You need a mirror line to reflect a shape.
•
When you translate a shape, you move it a given distance,
•
right or left and up or down.
When you rotate a shape, you turn it through a given number
•
of degrees.
You turn it about a fixed point, called the centre of rotation.
You turn it either clockwise or anticlockwise.
With any of these three transformations, only the position of the shape
is changed. The shape and size of the shape are not changed. An object
and its image are always identical. They are congruent.
In stage 7 you also learned about enlargements. An enlargement of a
shape is a copy of the object, but it is bigger. You can use a microscope
to look at enlarged images of very small objects. In this picture, you
can see a dust mite. These mites are about 0.04 mm long so they cannot
usually be seen without the use of a microscope. A typical mattress on
a bed may have from 100 000 to 10 million mites inside it. This is not a
very nice thought as you go to bed at night!
291
14 Position and transformation
14.1 Bearings
In this section you will ...
Key word
•
bearing
use bearings as a measure of direction.
A bearing describes the direction of one object from another.
It is an angle measured from north in a clockwise direction.
A bearing can have any value from 0° to 360°. It is always written with three figures.
N
N
A 120°
B A
In this diagram, the bearing
from A to B is 120°.
65°
B
In this diagram, the
bearing from A to B is 065°.
Worked example 14.1
The diagram shows three towns, A, B and C.
a Write the bearing of B from A.
b Write the bearing from A to C.
c Write the bearing of B from C.
A
B
C
Answer
a
raw a north arrow
D
from A, and a line
joining A to B. Measure
the angle from the north
arrow clockwise to the
line joining A to B.
b D
raw a north arrow from
A, and a line joining A to C.
Measure the angle from the
north arrow clockwise to the line
joining A to C.
N
N
c
raw a north arrow
D
from C, and a line
joining C to B.
Measure the angle
from the north arrow
clockwise to the line
joining C to B.
N
A
A 130°
B
The bearing is 130°.
292
210°
80°
B
C
C
The bearing is 210°.
The bearing is 080°.
14.1 Bearings
Exercise 14.1
1
For each diagram, write the bearing of B from A.
Use a protractor to measure the angle from north in a clockwise direction.
a
b
N
c
N
N
N
B
B
A
d
A
A
A
B
B
Draw diagrams similar to those in Question 1, to show these
bearings of B from A.
a
025°
b 110°
c
195°
d 330°
This is part of Freya’s homework.
2
3
Question
Write the bearing of B from A in this diagram.
Answer:
The angle is 32°, so the bearing of B from
A is 32°.
N
B
A
Is Freya correct? Explain your answer.
Activity 14.1
a
b
c
Draw four diagrams similar to those in Question 1, to show different bearings
of B from A.
On a different piece of paper, write the bearings you have drawn.
Exchange diagrams with a partner. Measure the bearings they have drawn.
Exchange back and check each other’s answers. Discuss any mistakes.
293
14 Position and transformation
4
The diagram shows the positions of a shop and a school.
Tip
N
To find the
bearing of the
shop from the
school (part a) you
need to measure
the angle at the
school. To find
the bearing of the
school from the
shop (part b) you
need to measure
the angle at
the shop.
Shop
N
School
a
b
Write the bearing of the shop from the school.
Write the bearing of the school from the shop.
Think like a mathematician
5
The diagram shows the position of a tree and a lake.
Seren, Taylor and Ros are standing at the tree.
Seren walks straight from the tree to the lake.
N
N
a
On what bearing must she walk?
Lake
Taylor walks north from the tree. After a short distance
she then walks to the lake.
Tree
b Is the bearing she walks on to the lake, larger or smaller
than the bearing from the tree to the lake?
Explain your answer.
Ros walks south from the tree. After a short distance she then walks to the lake.
Is the bearing she walks on to the lake, larger or smaller than the bearing
c
from the tree to the lake? Explain your answer.
d Discuss your answers to parts b and c with other learners in the class.
What can you say about how bearings change as you move north or south
from the original position before turning to walk towards another object?
294
14.1 Bearings
6
Arun goes for a walk.
The diagram shows Arun’s initial
position (A), a farm (F), a pond (P),
a tree (T) and a bridge (B).
Write the bearing Arun follows
to walk from
a
A to F
b F to P
c
P to T
d T to B
e
B to A.
N
N
F
N
B
N
N
A
P
T
Think like a mathematician
7
Work with a partner or in a small group to answer these questions.
a
For each diagram, write the bearing of Y from X and X from Y.
i
ii
N
iii
N
N
Y
X
Y
b
c
d
e
N
N
N
X
Y
X
Draw two different diagrams of your own, plotting two points X and Y.
In each diagram, the bearing of Y from X must be less than 180°.
For each of your diagrams, write the bearing of Y from X and of X from Y.
What do you notice about each pair of answers in parts a and b?
Copy and complete this rule for two points X and Y, when the bearing of
Y from X is less than 180°.
When the bearing of Y from X is m°, the bearing of X from Y is ................°.
Discuss your answers to parts c and d with other groups in your class.
295
14 Position and transformation
8
This is part of Marcus’s homework.
Question
iWrite the bearing of B
from A.
iiWork out the bearing
of A from B.
Answer
iBearing of B from
A is 127°
iiBearing of A from B is
180° + 127° = 307°
N
N
N
N
A 127 °
A 127 °
180 °
B
127 °
B
Marcus uses alternate angles to work out the bearing of A from B.
For each diagram
i write the bearing of B from A
iiuse Marcus’s method to work out the bearing of A
from B.
a
c
N b
N
N
N
N
77°
B
A
N
118°
B
16°
A
B
A
296
14.1 Bearings
9
This is part of Sofia’s homework.
Question
iWrite the bearing of P from Q.
iiWork out the bearing of Q from P.
N
N
Q
223 °
P
N
Answer
iBearing of P from Q is 223°
ii Bearing of Q from P is
223° – 180° = 043°
N
Q
180 °
43 ° 43 °
P
Sofia uses alternate angles to work out the bearing of Q from P.
For each diagram
i
write the bearing of P from Q
ii
use Sofia’s method to work out the bearing of Q from P.
a
N
b
c
N
N
N
N
N
Q
P
244°
P
Q
Q
348°
204°
P
In this exercise, you have learned these three skills:
•
measuring bearings in diagrams
•
drawing bearing diagrams
•
working out bearings using alternate angles.
a
Which of these did you find the easiest? Explain why.
b Which of these did you find the hardest? Explain why.
c
What could you do to improve these skills?
297
14 Position and transformation
Summary checklist
I can use bearings as a measure of direction.
14.2 The midpoint of a line segment
In this section you will ...
Key words
•
line segment
work out the coordinates of the midpoint of a line segment.
midpoint
The diagram shows two line segments, AB and CD.
The midpoint of AB is halfway between A and B.
You can see from the diagram that the midpoint of AB is (3, 3).
You can see from the diagram that the midpoint of CD is (1, 0).
y
4
(–1, 3) C
3
2
1
–4 –3 –2 –1 0
–1
–2
–3
–4
Worked example 14.2a
The diagram shows two line segments, LM and PQ.
y
4
3
2
1
–4 –3 –2 –1 0
–1
–2
P
–3
–4
L
M
Q
1 2 3 4 5 x
a Write the coordinates of the midpoint of LM.
b Work out the coordinates of the midpoint of PQ.
298
A (1, 3)
B (5, 3)
1 2 3 4 5 x
D (3, –3)
14.2 The midpoint of a line segment
Continued
Answer
a
You can see that the y-coordinate of the midpoint
is 3, because all the points on the line LM have a
y-coordinate of 3.
(3, 3)
You can see that the x-coordinate of the midpoint is
3, because it is exactly halfway along the line LM.
b (1, 0)
y
4
3
2
1
Q
4 squares up
–3 –2 –1 0 1 2 3 4 5 x
–1
2 squares up
P
–2
4 squares across 8 squares across
–3
To go from P to Q, you go 8 squares across and 4
squares up (shown by the red line).
To go from P to the midpoint, you do half of this,
so you go 4 squares across and 2 squares up
(shown by the blue line).
Exercise 14.2
1
Write the coordinates of the midpoint of each line segment.
y
5
4
3
2
A
B
1
0
0 1 2 3 4 5 x
y
5
4
3
2
1
0
C
D
0 1 2 3 4 5 x
y
5
4
3
2
1
0
E
F
0 1 2 3 4 5 x
y
5
4
3
2
1
0
G
H
0 1 2 3 4 5 x
299
14 Position and transformation
2
Match each line segment with the correct
midpoint.
An example is done for you.
Line segment AB and iii.
i
(1, −2)
ii
(−1, −6)
iii (2, 3)
iv (−5, 4)
(−3 1 , −3)
v
vi
2
vii
viii
(−3, −1)
(2 1 , 5)
2
1
(5, −2 )
2
C
D
y
6
5
4
A
3
2
1
–6 –5 –4 –3 –2 –1 0
–1
E
F
–2
P
Q
–3
–4
–5
–6
M
I
J
B
G
K
1 2 3 4 5 6 x
H
N
Think like a mathematician
3
Discuss the answers to these questions with a partner or in a small group.
Zalika and Maha use different methods to find the midpoint of the line segment
AB where A is (3, 4) and B is (11, 4).
Zalika’s method
Draw a diagram.
y
5
4
3
2
1
0
A
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x
a
b
c
d
e
f
300
×
midpoint at (7, 4)
Maha’s method
The y-coordinates of A and B
are the same, so the y-coordinate of the midpoint is 4.
To find the x-coordinate:
11 – 3 = 8, 8 ÷ 2 = 4, 3 + 4 = 7
The midpoint is at (7, 4)
Write the advantages and disadvantages of Zalika’s method.
Explain how Maha’s method works.
Write the advantages and disadvantages of Maha’s method.
Whose method do you prefer? Explain why.
Can you think of a better method?
Discuss your answers with other groups in your class.
L
14.2 The midpoint of a line segment
4
Work out the midpoint of the line segment joining each pair
of points.
Write whether A, B or C is the correct answer.
Use your preferred method.
a
(7, 1) and (7, 7)
A (7, 6)
B (7, 3)
C (7, 4)
b (4, 2) and (10, 2)
A (7, 2)
B (6, 2)
C (5, 2)
c
d
A (4, 9)
(4, 11) and (4, 2)
A
(8, 15) and (15, 15)
B (4, 6 1 )
C (4, 4 1 )
B (7, 15)
C
2
(11 1 , 15)
2
2
1
(12 , 15)
2
Worked example 14.2b
You can calculate the midpoint of a
line segment by finding the means
of the x-coordinates and the
y-coordinates of the end points.
y
6
4
2
The diagram shows the line
segment PQ.
–8 –6 –4 –2 0
–2
–4
–6
Q (–4, –6)
Calculate the coordinates of the
midpoint of PQ.
P (10, 4)
2 4 6 8 10 x
The mean of a and
b is a + b
2
Answer
10 + −4
= 6 =3
2
2
Add the x-coordinates of P and Q and divide the
result by 2.
4 + −6
= −2 = −1
2
2
Add the y-coordinates of P and Q and divide the
result by 2.
The midpoint of PQ is (3, −1).
5
Copy and complete the workings to calculate the midpoint of the
line segment joining each pair of points.
a
b
c
d
(
(8, 0) and (12, 6) (
(5, 2) and (8, 10) (
(2, 3) and (6, 7)
(0, 4) and (7, 11)
(
) ( ) ( )
)=( , )=( , )
) = , = (6 , )
2+6 3+7
,
= 8 , 10 = 4,
2
2
2 2
8 + 12 0 + 6
,
2
2
20 6
2 2
5 + 8 2 + 10
,
2
2

13

 2
)


2 

 
0 + 7 4 + 11
,
=
,
 =
2
2
2  
 2
1
2
,



301
14 Position and transformation
6
aE is the point (6, 0), F is the point (14, 8) and G is the point
(3, 15).
Work out the midpoint of the line segments
i EF
ii EG
iii FG
b Draw a coordinate grid. Plot the points E, F and G.
Check your answers to part a by finding the midpoints on
your diagram.
Think like a mathematician
7
Discuss the answers to these questions with a partner or in a small group.
Shen and Hassan calculate the midpoint of the line joining the points (−5, −8)
and (−1, 9).
This is what they write.
Shen
x-coordinate: –5 + –1 = –4 = –2
2
y-coordinate: –8 + 9 = –1 = – 1
2
2
2


1
Midpoint is at –2, – 
2
2
a
b
c
8
9
302
Hassan
x-coordinate: –1 + –5 = –6 = –3
2
2
y-coordinate: 9 + –8 = 1
2
2


1
Midpoint is at –3, 

2
Who, out of Shen and Hassan, has the correct midpoint?
Explain the mistake the other student has made.
Look again at their methods.
Shen added the x and y coordinates of (−5, −8) to (−1, 9).
Hassan added the x and y coordinates of (−1, 9) to (−5, −8).
Does it matter in which order you add the x and y coordinates?
Explain your answer.
Discuss your answers with other groups in your class.
Calculate the midpoint of the line segment between
a
(5, −2) and (2, −6)
b (−4, 5) and (3, 0)
c
(−7, 5) and (−10, 10).
A parallelogram has vertices at P (2, 5), Q (−2, 3), R (2, −1) and S (6, 1).
The diagonals are PR and QS. Show that the diagonals have the same midpoint.
14.2 The midpoint of a line segment
10 Calculate the coordinates of the midpoint of each side of this triangle.
E
y
40
30
20
10
–4 –3 –2 –1 0
–10
–20
–30
–40
D
1 2 3 4x
F
11 A quadrilateral has vertices at (−2, 1), (0, 4), (5, 2) and (1, −1).
Do the diagonals have the same midpoint? Justify your answer.
Think like a mathematician
12 The midpoint of a line segment is (4, 1). One end of the line segment is (2, 5).
a
Work out the coordinates of the other end of the line segment.
b Compare the method you used to answer part a with a partner’s method.
Did you both use the same method? Did you use different methods?
c
Discuss the methods you used with other learners in the class.
Which do you think is the best method to use to answer this type of question?
Explain why.
13 The midpoint of a line segment is (7, 2). One end of the line
segment is (−1, 6).
Work out the coordinates of the other end of the line segment.
14 Here are six cards showing the coordinates of the points A to F.
A
(2, 0)
B
(−3, −2)
C
(−7, 5)
D
(1, 4)
E
(5, −3)
F
(−4, 2)
Three line segments are made using the six cards.
The midpoint of all three line segments is (−1, 1).
What are the three line segments? Show how you worked out
your answers.
Summary checklist
I can work out the coordinates of the midpoint of a line segment.
I can work out the coordinates of the end of a line segment when
I know the coordinates of the other end and the midpoint.
303
14 Position and transformation
14.3 Translating 2D shapes
In this section you will ...
Key words
•
column vector
translate shapes on a coordinate grid using vectors.
congruent
You already know that when you translate a 2D shape on a coordinate
grid, you move it up or down and right or left.
You can describe this movement with a column vector.
This is an example of a column vector:  
2
 5
The top number tells you how many units to move the shape right
(positive number) or left (negative number).
The bottom number tells you how many units to move the shape up
(positive number) or down (negative number).
 2
means ‘move the shape 2 units right and 5 units up’.
 5
 −2
 −3 means ‘move the shape 2 units left and 3 units
For example:  
down’.
If the scale on the grid is one square to one unit, the numbers tell you
how many squares to move the object up/down and across.
When a shape is translated, only its position changes. Its shape
and size stay the same. This means that the object and its image are
always congruent.
304
image
object
translate
14.3 Translating 2D shapes
Worked example 14.3
y
4
3
2
1
T
–4 –3 –2 –1 0
–1
–2
–3
–4
The diagram shows triangle T on a coordinate grid.
Draw the image of triangle T after each translation.
a
 3
 2
c
 −3
 1 
b
 2
 −1
d
 −1
 −3
1 2 3 4 x
Answer
a
b
c
d
y
4
3
2
c
1
T
–4 –3 –2 –1 0
–1
–2
d –3
–4
Move triangle T 3 squares right and 2 squares up.
Move triangle T 2 squares right and 1 square down.
Move triangle T 3 squares left and 1 square up.
Move triangle T 1 square left and 3 squares down.
a
1b 2 3 4 x
Exercise 14.3
1
The yellow cards show translations.
The blue cards show column vectors.
Match each yellow card with the correct blue card.
The first one is done for you: A and iii
A
4 squares left, 1 square up
B
4 squares right, 1 square down
C
4 squares left, 1 square down
D
4 squares right, 1 square up
i
 4
 −1
ii
 4
 1
iii
 −4
 1 
iv
 −4
 −1
305
14 Position and transformation
2
The diagram shows triangle P on a coordinate grid.
Copy the grid, then draw the image of triangle P after
each translation.
a
3
 3
 2
b
 2
 −2
The diagram shows shape A on
a coordinate grid.
Copy the grid, then draw the
image of shape A after
each translation.
4
a
 3
 2
b
 4
 −2
c
 −2
 2 
d
 −1
 −2
This is part of Fin’s homework.
c
 −1
 3 
d
 −2
 −1
y
4
3
2
1
A
–4 –3 –2 –1 0
–1
–2
–3
–4
y
8
7
6
5
4
3
2
1
0
P
0 1 2 3 4 5 6 7 8 x
1 2 3 4 x
Question
A triangle ABC is translated using the column vector  3 
–2
The image of ABC is A'B'C'.
Write the column vector that translates A'B'C' back to ABC.
Answer
2 
 
–3
a
b
Is Fin’s answer correct? Explain your answer.
How could Fin check whether his answer is correct?
Think like a mathematician
5
Look at this question in pairs or groups, then discuss the answers to parts a,
b and c.
Read what Zara says.
If I translate a shape
 2
 3
using the column vector   , I can translate
the shape back to its original position using
 −2
the column vector  
 −3
306
14.3 Translating 2D shapes
Continued
a
b
Show that Zara is correct.
Write the column vectors that translate a shape back to its original position
after these translations.
i
c
6
 −4
 7 
ii
 3
 −5
When a shape is translated using a column vector, it can be translated back
to its original position.
Write a general rule for finding the column vector that will translate a shape
back to its original position.
The diagram shows triangle DEF.
∠DEF = 90°, ∠DFE = 45° and ∠EDF = 45°
DF has a length of 4 units.
a
Copy the grid, then draw the image of the triangle
 3
after the translation  −2
7
 −2
 −8
iii
Label the triangle D′E′F′.
b Copy and complete these statements.
∠D′E′F′ = ......°, ∠D′F′E′ = ......° and ∠E′D′F′ = ......°.
D′F′ has a length of ...... units.
c
Copy and complete these statements.
Choose from the words in the box.
When you compare an object and its image after
a translation:
• corresponding lengths are ...............
• corresponding angles are ...............
• the object and the image are ................
The diagram shows two shapes, P and Q.
Choose the column vector (A, B or C) that translates
y
E
5
4
F
3 D
2
1
0
0 1 2 3 4 5 6 7 8 x
different
equal
congruent
not congruent
a
shape P to shape Q
A
 2
 3
B
 −2
 3 
C
 2
 −3
b
shape Q to shape P
A
 2
 3
B
 −2
 3 
C
 2
 −3
y
5
4
3
2
1
0
shorter
longer
smaller
bigger
P
Q
0 1 2 3 4 5 6 x
307
14 Position and transformation
8
The diagram shows shapes L, M, N, P and Q on a coordinate
grid. Write the column vector that translates
a
shape N to shape L
b shape N to shape P
c
shape N to shape Q
d shape N to shape M
e
shape L to shape P
f
shape P to shape M.
y
4
3
2
N
1
M
0
–4 –3 –2 –1
–1
Q
–2
L
1P 2 3 4
Think like a mathematician
9
The diagram shows triangle JKL.
Marcus and Arun translate triangle JKL using the
J
 5
column vector  
 −4
They label the image J′K′L′.
Read what Marcus and Arun say.
K
L
y
4
3
2
1
–3 –2 –1 0
–1
–2
–3
I can calculate the
coordinates of J′ like this:
 5
( −3, 4 ) +  −4 = ( −3 + 5, 4 + −4 )
= (2, 0 )
I think you should write:
 5
 −3 + 5 
( −3, 4 ) +  −4 =  4 + −4
a
b
c
308
 2
 0
= 
Explain why Marcus is correct and Arun is incorrect.
Use Marcus’s method to calculate the coordinates of K′ and L′.
Use the diagram to check your answers are correct.
Discuss your methods and answers to parts a and b with other
learners in your class.
J9
1 2 3 4 x
K9
L9
x
14.3 Translating 2D shapes
10 A rectangle ABCD has vertices at the points A (−2, 3), B (4, 3),
C (4, −2) and D (−2, −2).
 8
ABCD is translated using the column vector  5
 
a
b
Calculate the coordinates of A′, B′, C′ and D′.
Check your answers are correct by drawing a diagram and
translating rectangle ABCD.
c
Compare and discuss your working for part a with that of a
partner.
Have you used the same methods? Are both sets of working
easy to understand?
11 This is part of Joule’s classwork.
She has spilt some juice on her work.
Question
A square EFGH has vertices at the points
E (–5, –1), F (3, –1), G (3, 7) and H (–5, 7)
EFGH is translated using column vector  8  to E'F'G'H'.
5
Work out the coordinates of the vertices of E'F'G'H'.
Answer
E' (–8, 6), F' (0, 6), G' (0, 14), H' (–8, 14)
a
b
Work out the coordinates of vertices
i F′
ii G
iii H
Explain how you worked out the answers to part a.
Summary checklist
I can translate shapes on a coordinate grid using vectors.
I can work out the vector of a translation given the object and the image.
I can work out the coordinates of the image of a shape given the vector.
309
14 Position and transformation
14.4 Reflecting shapes
In this section you will ...
Key words
•
mirror line
•
reflect shapes on a coordinate grid given the equation
of the mirror line
reflect
identify a reflection and its mirror line.
You already know how to reflect a shape when you use the x-axis or
y-axis as the mirror line.
You must also be able to reflect a shape on a coordinate grid in other
mirror lines.
To do this, you need to know the equation of the mirror line.
Some examples are shown on the grid on the right.
All vertical lines are parallel to the y-axis and have the equation
x = ‘a number’.
All horizontal lines are parallel to the x-axis and have the equation
y = ‘a number’.
x = –1 y
3
2
1
x=2
0
–3 –2 –1
–1
1 2 3x
y = –2
y=1
–2
–3
Worked example 14.4
Draw a reflection of this triangle in the lines
a
b
x=4
y=4
y
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
Answer
a
y
6
5
4
3
2
1
0
310
First draw the mirror line x = 4 on the grid.
object
Take each vertex of the object in turn and plot its
reflection in the mirror line.
Use a ruler to join the reflected points with straight
lines to make the image.
image
x=4
0 1 2 3 4 5 6 7 x
14.4 Reflecting shapes
Continued
b
y
6
5
4
3
2
1
0
First draw the mirror line y = 4 on the grid.
Take each vertex of the object in turn and plot its
reflection in the mirror line.
object
y=4
Use a ruler to join the reflected points with straight
lines to make the image.
image
Notice that the vertices at (5, 4) and (7, 4) are the
same on the object and the image.
0 1 2 3 4 5 6 7 x
Exercise 14.4
1
Copy each diagram and reflect the shape in the mirror line with the given equation.
a mirror line x = 3
b mirror line x = 4
c mirror line x = 2.5
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
d mirror line y = 4
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
e mirror line y = 3
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
f mirror line y = 3.5
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
311
14 Position and transformation
2
Copy each diagram and reflect the shape in the mirror line with the given equation.
a
mirror line x = 4
b mirror line x = 3
c mirror line x = 2
y
7
6
5
4
3
2
1
0
d
3
0 1 2 3 4 5 6 7 x
y
7
6
5
4
3
2
1
0
mirror line y = 3
e
y
7
6
5
4
3
2
1
0
y
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
0 1 2 3 4 5 6 7 x
mirror line y = 2
0 1 2 3 4 5 6 7 x
This is part of Gille’s homework.
Question
Reflect shape A in the line y = –1. Label the shape A’.
Answer
Mirror line y = −1
y
3
2
A9
A
1
0
–5 –4 –3 –2 –1
–1
1 2 3 4 x
–2
–3
y = –1
–4
a
b
312
y
7
6
5
4
3
2
1
0
Explain the mistake Gille has made.
Copy the diagram of shape A and draw the
correct reflection.
f
0 1 2 3 4 5 6 7 x
mirror line y = 4
y
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 x
14.4 Reflecting shapes
4
5
The diagram shows shape B on a coordinate grid.
Draw the image of shape B after reflection in the line
a
x = −1
b y = −2
c
x = 0.5
d y = −0.5
B
y
4
3
2
1
–4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
This is part of Oditi’s homework.
1 2 3 4 5 x
Question
y
x=4
Draw a reflection of the orange 4
3
triangle in the line x = 4.
2
Explain your method.
1
Answer
0
0 1 2 3 4 5 6 7 x
Reflected triangle drawn on
grid in green.
I reflected each corner of the triangle in the line x = 4,
then I joined the three corners together.
a
b
Make a copy of this grid.
Use Oditi’s method to draw these reflections.
i Reflect the triangle in the line x = 4
ii Reflect the parallelogram in the line y = 5
iii Reflect the kite in the line x = 8
What do you think of Oditi’s method?
Is it easy to follow? Can you think of a better
method to use to reflect shapes when the mirror
line goes through the shape? Explain your answer.
y
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 x
313
14 Position and transformation
Think like a mathematician
6
Work with a partner, or in a small group, to answer
this question.
The diagram shows a rectangle.
It also shows the line y = x.
a
Reflect the rectangle in the line y = x.
b Compare your answer with other groups in your
class and discuss the methods used.
c
Use your preferred method to reflect these shapes
in the line y = x.
i
ii
y
6
5
4
3
2
1
0
y
5
4
3
2
1
0
y=x
y
6
5
4
3
2
1
0
Mirror line y = x
y=x
0 1 2 3 4 5 6 x
iii
y
6
5
4
3
2
1
0
y=x
0 1 2 3 4 5 x
0 1 2 3 4 5 6 x
0 1 2 3 4 5 x
Think like a mathematician
7
Work with a partner, or in a small group, to answer
this question.
Alicia reflects trapezium ABCD in the line y = x.
The diagram shows the object, ABCD, and its image, A’B’C’D’.
a
The table shows the coordinates of the vertices of the
object and its image.
Copy and complete the table.
Object A (3, 6)
Image
b
c
d
314
A’ (
,
B (3, 4)
) B’ (
,
y
A
6
5
y=x
B
4
B9 A9
3
D
C
2
C9
1
D9
0
0 1 2 3 4 5 6 x
C(
,
) D(
,
)
) C’ (
,
) D’ (
,
)
What do you notice about the coordinates of ABCD and its image A’B’C’D’?
Write a rule you can use to work out the coordinates of the image of
a shape when it is reflected in the line y = x.
Does your rule in part c work for any shape reflected in the line y = x?
Explain your answer.
14.4 Reflecting shapes
8
9
The diagram shows shape ABCD on a coordinate grid.
It also shows the line y = x.
a
Write the coordinates of the points A, B, C and D.
When shape ABCD is reflected in the line y = x, the image
is A′B′C′D′.
b Use your rule from Question 7, part c to write the
coordinates of the points A′, B′, C′ and D′.
c
Copy the diagram. Reflect shape ABCD in the line y = x.
d Check the coordinates of the points A′, B′, C′ and D’ you
worked out in part b are correct.
If any of the coordinates are incorrect,
check your answers with a partner.
The diagram shows shape ABCD on a coordinate grid.
It also shows the line y = −x.
a
Make a copy of the diagram. Reflect ABCD in the
line y = −x and label the image A′B′C′D′.
b The table shows the coordinates of the vertices of the
object and its image.
Copy and complete the table.
Object A (−1, 2)
Image A’ (
,
B (−1, 4)
) B’ (
,
y
y=x
6
A
B
5
4
3 D
2
1
C
0
0 1 2 3 4 5 6 7 x
y
B 4
3
A 2
1
–5 –4 –3 –2 –1 0
–1
–2
–3
C(
,
)
D(
,
)
) C’ (
,
) D’ (
,
)
C
D
1 2 3 x
y = –x
c
What do you notice about the coordinates of ABCD and its
image A′B′C′D′?
d Write a rule you can use to work out the coordinates of the image
of a shape when it is reflected in the line y = −x.
e
Does your rule in part d work for any shape reflected in the line y = −x?
Explain your answer.
10 The diagram shows triangle PQR on a coordinate grid.
y
It also shows the line y = −x.
4
a
Write the coordinates of the points P, Q and R.
3
P
When shape PQR is reflected in the line y = −x, the image is
2
P′Q′R′.
1
b Use your rule from Question 9, part d to write the coordinates
–4 –3 –2 –1 0 1 2 x
–1
of the points P′, Q′ and R′.
–2
c
Copy the diagram. Reflect shape PQR in the line y = −x.
Q
R
y = –x
d Check the coordinates of the points P′, Q′ and R′ you worked
out in part b are correct.
If any of the coordinates are incorrect, check your answers
with a partner.
315
14 Position and transformation
Activity 14.4
Work with a partner for this question.
Make a copy of these coordinate axes on a piece of
squared paper.
Draw a rectangle inside the shaded region.
Exchange your diagram with a partner.
Reflect their rectangle in the line y = x. Label it A.
Reflect their rectangle in the line y = −x. Label it B.
Exchange back and mark each other’s work.
Discuss any mistakes.
y
5
4
3
2
1
–5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
1 2 3 4 5 x
11 The diagram shows shapes J, K, L, M, N and P.
y
Choose the correct equation of the mirror line for each
6
of these reflections.
5
J
M
a
J and K
A x=1
B y=3
C x=3
4
3
b J and M
A x=4
B y=4
C x=5
N
2
c
M and N
A y=3
B y=4
C y = 3.5
K
L
P
1
d K and L
A x=3
B y=1
C y=2
0
0 1 2 3 4 5 6 7 8 9x
e
L and P
A x = 5.5
B y = 5.5
C x=8
12 The diagram shows eight triangles, labelled A to H.
y
Identify which of the following are reflections.
8
For each one that is a reflection,
7
A
B
write the equation of the mirror line.
6
a
triangle A to triangle B
b triangle A to triangle C 5
c
triangle B to triangle F
d triangle B to triangle E 4
F G
C
3
e
triangle D to triangle E
f
triangle G to triangle E
2
D
E
g triangle C to triangle E
h
triangle F to triangle G 1
H
i
triangle D to triangle H
j
triangle E to triangle H 0
0 1 2 3 4 5 6 7 8 x
316
14.5 Rotating shapes
Look back at Question 1a.
a
Write the steps you took to draw the reflection of the shape.
You might begin with:
Step 1: Draw the mirror line.
Step 2:
b
Look back at Question 6ciii. Write the steps you took to draw
the reflection of this shape.
c
Which steps were the same or different for these two
questions? Explain why.
Summary checklist
I can reflect shapes on a coordinate grid in the lines x = ‘a number’
and y = ‘a number’.
I can reflect shapes on a coordinate grid in the lines y = x and y = −x.
I can identify a reflection and its mirror line.
14.5 Rotating shapes
In this section you will ...
Key words
•
anticlockwise
rotate shapes on a coordinate grid and describe a rotation.
centre of rotation
When you carry out a rotation, or describe a rotation, you need three
pieces of information:
the angle of the rotation
•
the direction of the rotation (clockwise or anticlockwise)
•
the coordinates of the centre of rotation.
•
clockwise
317
14 Position and transformation
Worked example 14.5
a
Draw the image of this shape after a
rotation 90° clockwise about the centre
of rotation (−2, −1).
b
Describe the rotation that takes
shape A to shape B.
y
6
5
4
3
2
1
0
y
4
3
2
1
–5 –4 –3 –2 –1 0
–1
–2
1 2 3 x
A
B
0 1 2 3 4 5 6x
Answer
a
Start by tracing the shape, then put the point of your
pencil on the centre of rotation.
y
4
3
2
1
–5 –4 –3 –2 –1 0
× –1
(–2, –1)
–2
b
Turn the tracing paper 90° clockwise, then make a note
of where the image is.
Draw the image onto the grid.
1 2 3 x
Rotation is 180°
The centre of rotation is
at (3, 3).
When you describe a rotation, give the number of degrees
and the coordinates of the centre of rotation. Note that
when the rotation is 180° you do not need to say clockwise
or anticlockwise as both give the same result.
Exercise 14.5
1
Copy each diagram and rotate the shape using the given information.
a
90° clockwise
centre (2,1)
y
4
3
2
1
0
318
b
90° anticlockwise
centre (−2, 2)
y
4
3
2
1
0 1 2 3 4 5x
–4 –3 –2 –1 0 1 x
c
180°
centre (−1, −2)
y
2
1
–4 –3 –2 –1 0
–1
–2
1 2 x
14.5 Rotating shapes
Think like a mathematician
2
This is how Milosh rotates a shape when the centre of rotation is not on
the shape and he doesn’t have tracing paper.
Question
Rotate the shape 90° clockwise about centre (1, 2). 5y
4
3
2
1
0
Answer
Draw a vertical line from the
shape to the centre of rotation.
Rotate the line 90° clockwise and
draw in the new shape.
y
5
4
3
2
1
0
y
5
4
3
2
1
0
×
0 1 2 3 4 x
×
0 1 2 3 4 x
90° clockwise
0 1 2 3 4 x
Discuss the answers to these questions in pairs or groups.
a
What do you think of Milosh’s method?
Do you think it makes it easier to rotate a shape?
Do you think you could use this method?
Do you think it would work for all rotations?
b Use Milosh’s method or your own method to rotate each shape
90° clockwise about centre (2, 4).
Remember, you must not use tracing paper.
y
5
4
3
2
1
0
0 1 2 3 4 5 x
y
5
4
3
2
1
0
0 1 2 3 4 5 6 x
319
14 Position and transformation
3
aCopy each diagram and rotate the shape using the given information.
Do not use tracing paper.
i
4
y
5
4
3
2
1
0
ii
0 1 2 3 4 5x
iii
y
5
4
3
2
1
–3 –2 –1 0
–1
–2
1 2x
90° anticlockwise
90° clockwise
centre (1, 1)
centre (2, 2)
b Use tracing paper to check your answers to part a.
This is part of Rohan’s classwork.
Question
Rotate this parallelogram
90° anticlockwise about
­centre (2, 3).
y
4
3
2
1
–4 –3 –2 –1 0
–1
–2
–3
a
b
320
–3 –2 –1 0
–1
–2
–3
–4
–5
180°
centre (1, 0)
y
4
3
2
1
–4 –3 –2 –1 0
–1
–2
–3
Answer
I have used a dotted line to
show the image.
y
5
4
3
2
1
1 2 3 4x
×
1 2 3 4x
What is wrong with Rohan’s answer?
Copy the object onto squared paper and draw the correct image.
1 2 3 4 5 x
14.5 Rotating shapes
Think like a mathematician
5
This is part of Marcus’s homework.
Question
Describe the rotations that take shape A to shape B.
a
b
y
5
4
3
A
2
1
–1 0
B
1 2 3 4 5 x
y
5
4
3
2
1
0
A
B
0 1 2 3 4 5 6 x
Answer
a Rotation 90° clockwise
b Rotation 180° anticlockwise, centre (3, 2)
Read what Sofia and Zara say to Marcus.
You haven’t
given enough
information in
part a.
You have given
too much
information in
part b.
Discuss the answers to these questions in pairs or groups.
a
Are Sofia and Zara correct? Explain your answers.
b Look again at part a of Marcus’s homework.
You can see that the centre of rotation is at (1, 1)
because this point is the same on both the image
and the object.
How can you work out the centre of rotation when no
point is the same on both the image and the object
(for example, in part b of Marcus’s homework)?
Tip
Try drawing lines
to corresponding
vertices on the
object and
the image.
321
14 Position and transformation
Continued
c
Use your answer to part b to work out the centre of rotation in each diagram.
Notice that both rotations are 180°.
i
y
4
3
2
1
−4 −3 −2 −1 0
−1
A
−2
d
e
6
322
B
1 2 3 4 x
y
5
4
3
2
1
0
A
B
0 1 2 3 4 5 6 7 8 9 x
Complete this sentence: ’I can find the centre of a 180° rotation by ...................’
Does your method for finding the centre of a 180° rotation work
for a 90° rotation?
Test your answer on these two diagrams.
i
f
ii
y
5
4
3
2
1
0
ii
B
A
0 1 2 3 4 5 x
y
5
4
3
2
1
0
B
A
0 1 2 3 4 5 6 7 x
Describe a method you can use to work out the centre of rotation
for a 90° rotation.
The diagram shows seven triangles.
Match each rotation with the correct description.
a
A to B
i
90° clockwise, centre (3, 5)
b B to C
ii
180°, centre (4, 1)
c
C to D
iii 180°, centre (6, 5)
d C to E
iv 90° anticlockwise, centre (3, 8)
e
F to G
v
180°, centre (4, 4)
y
8
A
7
B
6
5
4
3
2
1
0
E
C
D
F
G
0
1 2 3
4
5 6
7 8 x
14.5 Rotating shapes
7
8
The diagram shows triangles R, S, T, U, V and W
on a coordinate grid.
Describe the rotation that transforms
a
triangle R to triangle S
b triangle S to triangle T
c
triangle T to triangle U
d triangle U to triangle V
e
triangle V to triangle W.
The diagram shows seven shapes labelled A to G.
Here are seven cards labelled i to vii.
Each card shows a rotation of one shape to another.
For example, card i means rotate shape A to shape B.
i
A to B
ii
A to C
iv
B to D
v
E to B
vi
G to A
vii
T
0
–4 –3 –2 –1
–1
W
–2
–3
–4
y
7
6
5
4
3
2
1
0
C to B
iii
D to F
A
B
y
5
4
3
2
1
0
b
T
T
S
0 1 2 3 4 5 6 x
S
–5 –4 –3 –2 –1 0
1
G
D
Tip
You could use
the angle, the
direction, or the
centre of rotation.
S
x
V
0 1 2 3 4 5 6 7 8 9 10 11 x
c
y
5
4
3
2
1
1 2 3 4 x
E
C
Put the cards into groups using one property of the rotations.
Describe the property of each group.
b Sort the cards into different groups using a different property
of the rotations.
Describe the property of each group.
Describe the rotation that transforms S to T in each diagram.
a
U
F
a
9
y
4
R3
2
S
1
y
4
3
2
1
–5 –4 –3 –2 –1 0
T
1 2 3 4x
Summary checklist
I can rotate shapes on a coordinate grid.
I can describe rotations on a coordinate grid.
323
14 Position and transformation
14.6 Enlarging shapes
In this section you will …
Key words
•
centre of
enlargement
enlarge shapes using a positive whole number scale factor
from a centre of enlargement.
enlargement
An enlargement of a shape is a copy of the shape that changes the
lengths but keeps the same proportions. In an enlargement, all angles
stay the same size.
Look at these two rectangles.
scale factor
image
object
4 cm
2 cm
1 cm
2 cm
The image is an enlargement of the object.
Every length on the image is twice as long as the corresponding length
on the object.
The scale factor is 2.
The centre of enlargement tells you where to draw
the image on a grid. In this case, as the scale factor
3 cm
is 2, not only must the image be twice the size of the
centre of
object, it must also be twice the distance from
enlargement
6 cm
the centre of enlargement.
You can check you have drawn an enlargement
correctly by drawing lines through the
corresponding vertices of the object and image.
All the lines should meet at the centre of
the enlargement.
This is also a useful way to find the centre of
enlargement if you are only given the object and
centre of
the image.
enlargement
324
14.6 Enlarging shapes
Worked example 14.6
Draw enlargements of the following triangles using the given scale factors and
centres of enlargement, marked with a red dot.
a
scale factor 2
b
scale factor 3
Answer
a
Start by looking at the corner of the triangle that is
closest to the centre of enlargement (COE).
This corner is 1 square to the right of the COE so, with a
scale factor of 2, the image will be 2 squares to the right
of the COE.
Plot this point on the grid, then complete the triangle.
Remember to double all the lengths.
b
One of the corners of this triangle is on the centre of
enlargement, so this corner doesn’t move.
Look at the bottom right corner of the triangle. This
corner is 1 square to the right and 1 square down from the
COE. With a scale factor of 3, the image will be 3 squares
to the right and 3 squares down from the COE.
Plot this point on the grid, then complete the triangle.
Remember to multiply all the other lengths by 3.
325
14 Position and transformation
Exercise 14.6
1
2
Copy each shape onto squared paper.
Enlarge each shape using the given scale factor and centre
of enlargement.
a
scale factor 2
b
scale factor 3
c
scale factor 4
d
scale factor 2
e
scale factor 3
f
scale factor 4
This is part of Geraint’s homework.
Question
Enlarge this triangle using a scale
factor of 2 and the centre of
enlargement shown.
Answer
a
b
326
Explain Geraint’s mistake.
Make a copy of the triangle on squared paper.
Draw the correct enlargement.
Tip
Make sure you
leave enough
space around
your shape to
complete the
enlargement.
14.6 Enlarging shapes
3
The vertices of this triangle are at (2, 2), (2, 3) and (4, 2).
a
Copy the diagram on squared paper.
Mark with a dot the centre of enlargement at (1, 1).
Enlarge the triangle with scale factor 3 from the
centre of enlargement.
b Write the coordinates of the vertices of the image.
y
8
7
6
5
4
3
2
1
0
x
0 1 2 3 4 5 6 7 8 9 10 11
Think like a mathematician
4
Marcus and Arun enlarge this square using scale factor 3.
Marcus uses a centre of enlargement at (1, 1).
Arun uses a centre of enlargement at (0, 1).
Read what Marcus and Arun say.
y
5
4
3
2
1
0
x
0 1 2 3 4 5 6 7
There are no
invariant points on
my object and
image.
There is one
invariant point on
my object and
image.
Work with a partner or in a small group to answer these questions.
a
Make two copies of the grid above and enlarge the square using
scale factor 3 with
i Marcus’s centre of enlargement
ii Arun’s centre of enlargement.
b Look at the diagrams you draw for part a.
What do you think Marcus and Arun mean by ’invariant points’?
c
Describe where a centre of enlargement must be, for you to have
one invariant point.
d Describe where a centre of enlargement must be, for you to have
no invariant points.
Discuss your answers with other groups in your class.
327
14 Position and transformation
5
The vertices of this trapezium are at (3, 2), (7, 2), (5, 4)
y
and (4, 4).
6
a
Copy the diagram onto squared paper.
5
Mark with a dot the centre of enlargement at (5, 2). 4
Enlarge the trapezium with scale factor 2 from the 3
centre of enlargement.
2
b Write the coordinates of the vertices of the image. 1
0
c
Write the coordinates of the invariant point.
0 1 2 3 4 5 6 7 8 9 10
x
Activity 14.6
Work with a partner for this question. Read the instructions before you start.
a
On a coordinate grid, draw a quadrilateral of your choice.
b Ask your partner to enlarge your quadrilateral by a scale factor of your choice.
Give them the coordinates of the centre of enlargement, which must be
somewhere on the perimeter of the quadrilateral.
You must make sure the enlarged shape will fit on the coordinate grid.
6
a
328
Check each other’s work and discuss any mistakes.
Each diagram shows an object and its image after an enlargement.
For each part, write down the scale factor of the enlargement.
b
y
8
7
6
image
5
4
object
3
2
1
0
x
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
y
8
7
6
5
image
4
3
2
1
0
x
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
object
c
14.6 Enlarging shapes
7
The diagram shows shape ABCD and its image A′B′C′D′.
a
Write the scale factor of the enlargement.
Read what Marcus and Zara say:
I think the centre of
enlargement is at (−3, −4).
I think the centre of
enlargement is at (−4, −3).
b
y
C9
5 B9
4
3
2
1
A9
0 1 2 3 4 5x
–5 –4 –3 –2 –1
B
C –1
–2
A
D –3
D9
–4
–5
Who is correct? Explain how you worked out your answer.
Think like a mathematician
8
Work with a partner or in a small group to answer these questions.
Zara drew a triangle with vertices at (1, 1), (2, 1) and (1, 3).
She enlarged the shape by a
If I multiply the
scale factor of 3, centre (0, 0).
coordinates
of each vertex by 3 it will
Read what Zara said.
give me the coordinates of the enlarged
triangle, which are at (3, 3), (6, 3)
and (3, 9).
a
Show, by drawing, that in this case Zara is correct.
Read what Arun said.
This means that,
for any enlargement, with any
scale factor and centre of enlargement,
I can multiply the coordinates of each
vertex by the scale factor to work out
the coordinates of the
enlarged shape.
b
c
Tip
A counterexample is just
one example
that shows a
statement is
not true.
Use a counter-example to show that Arun is wrong.
What are the only coordinates of the centre of an enlargement
where you can multiply the coordinates of the vertices of the object
to get the coordinates of the vertices of the image?
Summary checklist
I can enlarge shapes using a positive whole number scale factor from
a centre of enlargement.
329
14 Position and transformation
Check your progress
1
2
3
4
Make two copies of
this diagram.
a
On one copy, reflect
the shape in the
line x = 7.
b On the other copy,
reflect the shape in
the line y = 3.
5
The diagram shows shapes A, B and C.
Describe the rotation that takes
a
A to B
b A to C.
Copy this shape onto squared paper.
6
N
Draw diagrams to show these bearings of B from A.
For each one, start with the diagram on the right.
a
045°
b 170°
c
215°
d 340°
Work out the midpoint of the line segment joining each pair of coordinates
A
a
(3, 7) and (9, 7)
b (2, 8) and (11, 6)
The diagram shows shapes M, N, P and Q on a coordinate grid.
y
Write the column vector that translates:
4
3
a
shape M to shape N
N
2
b shape N to shape Q
M
1
c
shape Q to shape P
x
–3 –2 –1–10 1 2 3
d shape P to shape M.
y
6
5
4
3
2
1
0
x
0 1 2 3 4 5 6 7 8 9 10
P
A
Q
–2
–3
y
4
3
B
2
1
0
–4 –3 –2 –1–1
C
–2
–3
–4
1 2 3
Enlarge the shape using scale factor 3 and the centre of enlargement shown.
330
x
15
Distance, area
and volume
Getting started
1
Work out these calculations.
a
2
1
× 32
8
b
5
× 32
8
c
Work out the area of each shape.
a
1
× 15
5
d
8
× 15
5
b
5 cm
7m
8m
12 cm
3
The diagram shows a cuboid.
Work out
a
the volume of the cuboid
b the surface area of the cuboid.
3 cm
4 cm
6 cm
The metric system that is used today was developed in France,
in the late 18th century, by Antoine Lavoisier.
At that time, different countries used different units for
measuring, which was very confusing.
The modern metric system is called the International System of
Units (SI) and is now used by about 95% of the world’s population.
However, some countries that use the metric system still use some
of their old units as well. For example, in the UK, Liberia and the USA,
distances and speeds on road signs are shown using miles, not
kilometres. Rulers are often marked in both inches and centimetres.
Antoine Lavoisier, 1743–1794.
331
15 Distance, area and volume
15.1 Converting between miles
and kilometres
In this section you will …
Key words
•
kilometre
convert between miles and kilometres.
mile
In some countries, such as the USA, Liberia and the UK, distances are
measured in miles rather than kilometres.
A kilometre is a shorter unit of measurement than a mile.
One kilometre is about 5 of a mile.
8
If the blue line below represents a distance of 1 mile, then the red line
represents a distance of 1 kilometre.
1 mile
1 kilometre
To convert a distance in kilometres to a distance in miles, multiply by 5 .
8
To convert a distance in miles to a distance in kilometres, multiply by 8 .
5
Worked example 15.1
a
b
c
d
Which is greater, 20 miles or 20 km?
Convert 72 kilometres into miles.
Convert 50 miles into kilometres.
Which is further, 200 km or 120 miles?
Answer
20 miles
1 mile is greater than 1 km, so 20 miles is greater than 20 km.
b 72 ÷ 8 = 9
9 × 5 = 45 miles
To multiply 72 by 5 , first divide 72 by 8, then multiply the
8
answer by 5.
c
To multiply 50 by 8 , first divide 50 by 5, then multiply the
5
answer by 8.
Convert 200 km into miles (or 120 miles into km) so the units
are the same.
a
50 ÷ 5 = 10
10 × 8 = 80 km
d 200 ÷ 8 = 25
25 × 5 = 125 miles
200 km is further
332
125 miles is greater than 120 miles, so 200 km is further than
120 miles.
15.1 Converting between miles and kilometres
Exercise 15.1
1
Write true (T) or false (F) for each statement.
a
15 miles is further than 15 km.
b 100 km is exactly the same distance as 100 miles.
c
2.5 km is further than 2.5 miles.
d 6 km is not as far as 6 miles.
e
In one hour, a car travelling at 70 miles per hour will travel a
shorter distance than a car travelling at 70 kilometres per hour.
Read what Zara says.
My brother lives
Is Zara correct?
35 km from my house.
Explain your answer.
My sister lives 35 miles from
2
my house. I live closer
to my brother than to
my sister.
3
Copy and complete these conversions from kilometres to miles.
a
64 km
64 ÷ 8 = 8
8×5=
miles
b
40 km
40 ÷ 8 =
×5=
miles
c
56 km
56 ÷
=
×
=
miles
Copy and complete these conversions from miles to kilometres.
a
55 miles 55 ÷ 5 = 11
11 × 8 =
km
4
b
20 miles
20 ÷ 5 =
c
85 miles
85 ÷
×8=
=
km
×
=
km
Think like a mathematician
5
Read what Sofia says.
Discuss a strategy Sofia
could use to help her
decide when she should
multiply by 5 and when she
8
should multiply by 8
5
6
7
Convert each distance to miles.
a
24 km
b 48 km
Convert each distance to kilometres.
a
10 miles
b 100 miles
When I convert between
miles and kilometres, I
never know whether to
multiply by 5 or 8
8
5
c
96 km
d
176 km
c
125 miles
d
180 miles
333
15 Distance, area and volume
Think like a mathematician
8
Look at this question:
Which is further, 107 km or 70 miles?
Discuss with a partner or in a small group:
a
b
c
d
Do you think it is easier to change 107 km into miles or 70 miles into km
without using a calculator? Explain why.
If you could use a calculator, would this change your answer to part a? Explain why.
When you compare a number of km and a number of miles, explain how
you would decide which unit to convert.
Test your answer to part c on these questions:
i
ii
9
Which is further, 90 miles or 150 km?
Which is further, 51 miles or 80 km?
Use only the numbers from the cloud to complete these statements.
a
120 km =
miles
b 105 miles =
km
115 140
75
c
224 184
168
km =
miles
d
miles =
km
Activity 15.1
Hamza and Inaya use different methods to convert 23 miles into kilometres.
This is what they write.
Hamza
23 × 8 = 23 ÷ 5 × 8
5
23 ÷ 5 = 4 35
4 35 × 8 = 4 × 8 + 35 × 8
= 32 + 24
5
= 32 +4 4 = 36 4 km
5
a
b
c
334
5
Inaya
23 × 85 = 23 × 1 35
= 23 × 1 + 23 × 35
= 23 + 69
5
= 23 + 13 54
= 36 54 km
Whose method do you prefer, Hamza’s or Inaya’s? Explain why.
Can you think of a better method?
Discuss your answers to parts a and b with other learners in your class.
15.1 Converting between miles and kilometres
10 Work out the missing numbers in these conversions.
Use your preferred method.
a
17 miles =
km
b 33 miles =
Tip
km
c
54 miles =
km
d
28 km =
miles
e
42 km =
miles
f
75 km =
miles
Give each answer
as a mixed
number in its
simplest form.
11 Every car in the USA is fitted with a milometer.
The milometer shows the total distance a car has travelled.
Evan is a salesman.
This is the reading on his car’s milometer at the start of one week.
125 465 miles
This is the reading on his car’s milometer at the end of the week.
126 335 miles
a
b
How many kilometres has Evan travelled in this week?
Evan is paid 20 cents for each kilometre he travels.
This is to pay for the fuel he uses.
Evan works out that, in this week, he will be paid more than
$250 for the fuel he uses.
Is Evan correct? Explain your answer.
In this section you have learned to convert between miles and kilometres.
a
b
Match each statement to the correct method.
A
Convert from miles to km
i
×5
B
Convert from km to miles
ii
×8
8
5
Explain to a partner how you remember these methods.
Summary checklist
I can convert between miles and kilometres.
335
15 Distance, area and volume
15.2 The area of a parallelogram and
a trapezium
In this section you will …
Key word
•
trapezia
derive and use the formulae for the area of a parallelogram
and a trapezium.
Look at this parallelogram.
Imagine you cut off a triangle from the left
end of the parallelogram and move it to the
right end.
You can see that you have made a rectangle.
So the area of the parallelogram is the same
as the area of the rectangle with the same
base and perpendicular height.
height
base
You can write the formula for the area of a
parallelogram as:
area = base × height
or simply A = bh
a
Now look at this trapezium.
The lengths of its parallel sides are a and b.
h
Its perpendicular height is h.
b
Two trapezia can be put together like this to
make a parallelogram with a base length of
(a + b) and a height h.
The area of the parallelogram is:
area = base × height = (a + b) × h
The area of one trapezium is half the area
of the parallelogram.
So, the area of a trapezium is:
A = 12 × (a + b) × h
336
h
a+b
15.2 The area of a parallelogram and a trapezium
Worked example 15.2
Work out the area of each shape.
12 mm
a
b
5 cm
8 mm
7 cm
18 mm
A = bh = 7 × 5
= 35 cm2
Write the formula, then substitute the values of b and h.
Answer
a
Work out the answer. Remember to include the units (cm2).
b A = 12 × (a + b ) × h
Write the formula.
Substitute the values of a, b and h.
= 1 × (12 + 18) × 8
2
= 1 × 30 × 8
2
Work out 12 + 18 = 30 first.
Then work out 1 of 30 = 15
= 15 × 8
= 120 mm 2
2
Finally work out 15 × 8
Remember to include the units (mm2) with your answer.
Exercise 15.2
1
Copy and complete the workings to find the area of each
parallelogram.
a
b
4 cm
1.5 m
8 cm
2
6m
A = bh = 8 × 4 =
cm2A = bh =
×
=
m2
Copy and complete the workings to find the area of each trapezium.
6 cm
a
b
4 mm
5 cm
7 mm
8 cm
A = 1 × (a + b) × h = 1 × (6 + 8) × 5
2
2
= 1×
2
=
×5=
cm2
×5
12 mm
1
A = × (a + b) × h = 1 × (
2
2
= 1×
2
=
×7=
+
)×7
×7
mm2
337
15 Distance, area and volume
3
This is part of Bembe’s homework.
Question
Work out the area of
this parallelogram.
Answer Area = bh
= 7×5
= 35 cm 2
4 cm
5 cm
7 cm
a
Explain the mistake Bembe has made.
b Work out the correct answer.
Sofia, Marcus and Zara are discussing the methods
they use to find the area of a trapezium.
4
I work out
a + b, then divide
by 2, then multiply
by h.
Will they all get the same answer?
Explain why.
I work out half of h,
then work out a + b,
then multiply my two
answers together.
I work out a + b,
then multiply by h,
then divide by 2.
Think like a mathematician
5
Work with a partner to answer this question.
Look back at the methods used by Sofia, Marcus and Zara in Question 4.
Whose method would it be best to use to find the areas of these trapezia?
Explain why.
a
b
c
d
338
a = 6 cm, b = 4 cm, h = 3 cm
a = 7 cm, b = 4 cm, h = 6 cm
a = 2 m, b = 3 m, h = 5 m
a = 16 mm, b = 14 mm, h = 12 mm
15.2 The area of a parallelogram and a trapezium
6
Work out the area of each trapezium using the method shown.
Look back at Question 4 to check the method.
a
Sofia’s method
b
Marcus’s method
Zara’s method
3m
8 cm
6 mm
5m
6 cm
5 mm
7m
7
c
9 cm
9 mm
This is part of Zalika’s homework.
Question
What is the difference in area between these two shapes?
B
A
12 mm
6 cm
9 cm
15 mm
10 cm
Answer
Area A = b × h = 15 × 12 = 180
Area B = 1 × (a + b) × h = 1 × (6 + 10) × 9
2
2
1
= × 16 × 9 = 8 × 9 = 72
2
Difference = 180 – 72 = 108
a
b
Explain the mistake Zalika has made.
Work out the correct answer.
Think like a mathematician
8
Work with a partner or in a
small group to discuss this
question.
Read what Zara says.
Is Zara correct? Explain
your answer.
If you double the base
length of a parallelogram
and double the height of the
parallelogram, the area of
the parallelogram will
be doubled.
339
15 Distance, area and volume
9
The diagram shows a trapezium.
a
Work out an estimate of the area
of the trapezium.
b Use a calculator to work out the
accurate area of the trapezium.
10 Here are four shapes, A, B, C and D.
A
2.3 cm
3.8 cm
To work out an
estimate, round all
the numbers to one
significant figure.
4.6 cm
9.8 cm
B
4.5 cm
Tip
3.7 cm
4.2 cm
5.4 cm
C
D
2.9 cm
2.7 cm
8.2 cm
3.4 cm
Here are five area cards.
i
9.86 cm2
ii
18.81 cm2
iv
15.54 cm2
v
11.07 cm2
iii
24.48 cm2
Using estimation only, match each shape with the correct area card.
Use a calculator to check your answers to part a.
Sketch a shape that has an area equal to the area on the card
you have not matched.
Tip
11 A parallelogram has an area of 832 mm2.
It has a perpendicular height of 2.6 cm.
Be careful with the units.
What is the length of the base of the parallelogram?
a
b
c
Summary checklist
I can derive and use the formula for the area of a parallelogram.
I can derive and use the formula for the area of a trapezium.
340
15.3 Calculating the volume of triangular prisms
15.3 Calculating the volume of
triangular prisms
In this section you will …
Key words
•
cross-section
derive and use the formula for the volume of a triangular
prism.
prism
You already know how to work out the volume of a
cuboid by multiplying the length (l) by the base (b) by the
height (h).
You can multiply the dimensions in any order.
A cuboid is a rectangular prism.
h
l
A prism is a 3D shape which has the same 2D shape throughout
its length.
b
This 2D shape is called the cross-section of the prism.
When you work out the volume of a cuboid, if you start with
b × h, you find the area of the rectangular cross-section of the
cuboid.
h
l
When you multiply this area by the length l, you get the volume
of the cuboid.
b
The diagram shows a triangular prism.
You can see that the cross-section of the prism is a triangle.
You can work out the volume of the prism using the formula:
h
l
b
volume = area of cross-section × length
341
15 Distance, area and volume
Worked example 15.3
Work out the volume of this triangular prism.
6 cm
15 cm
4 cm
Answer
First, work out the area of the triangular cross-section.
area = 1 × b × h
2
= 1 ×4×6
2
Substitute in the values.
The lengths are all in cm, so the area is in cm2.
= 12 cm 2
volume = area × length
= 12 × 15
= 180 cm3
Multiply the area of the cross-section by the length.
Substitute in the values.
The area is in cm2 and the length is in cm, so the volume is in cm3.
Exercise 15.3
1
Copy and complete the workings to find the volume of each
triangular prism.
a
b
4m
8 cm
10 cm
6 cm
7m
3m
Area of cross-section = 12 × b × h
Area of cross-section = 12 × b × h
= 1 ×6×8
= 1×
=
=
cm2
Volume = area of cross-section × length
2
× 10
=
×7
=
cm3
=
m3
Work out the volume of each triangular prism.
a
b
2m
6m
30 cm
9 cm
342
m2
Volume = area of cross-section × length
=
12 cm
×
2
2
7m
15.3 Calculating the volume of triangular prisms
Think like a mathematician
3
Yari and Mike use different methods to work out
the volume of this triangular prism.
This is what they write.
7 mm
20 mm
5 mm
Yari
Area of cross-section = 21 × b × h
Mike
Volume of cuboid = length × base × height
= 20 × 5 × 7 = 700 mm 2
= 21 × 5 × 7 = 17.5 mm 2
Volume of prism = area × length Volume of prism = volume of cuboid ÷ 2
= 17.5 × 20
= 700 ÷ 2
3
= 350 mm
= 350 mm 3
Discuss the answers to these questions with a partner or in a small group.
a
b
c
4
How does Mike’s method work? Why does it give the same answer
as Yari’s method?
Which method do you prefer? Explain why.
Can you think of a different method you can use to work out the volume
of a triangular prism?
Discuss your answers with other groups in the class.
This is part of Vin’s homework.
Question
Work out the volume of
7 cm
this triangular prism.
Answer
Area of cross-section = 21 × b × h
8 cm
120 mm
= 21 × 7 × 8 = 28 cm2
Volume = area × length = 28 × 120 = 3360 cm3
Vin has got the answer wrong.
Explain Vin’s mistake and work out the correct answer.
343
15 Distance, area and volume
5
The table shows the base, perpendicular height and length of
four triangular prisms.
Copy and complete the table.
Base
Height
Length
Volume
a
4 cm
8 cm
5 mm
cm3
b
2 cm
15 mm
8 mm
mm3
c
7m
9m
10 cm
m3
d
30 mm
6 cm
200 mm
cm3
Tip
Make sure the
length, width and
height are all in
the same units
before you work
out the volume.
Think like a mathematician
6
The diagram shows a compound prism.
The compound prism is made of a triangular prism and
a cuboid.
a
b
c
7
10 cm
Show that the volume of the compound prism is
1920 cm3.
7 cm
Discuss with other learners in the class the method
you used to work out the volume.
What do you think is the easiest method to use to work out
the volume of a compound prism? Explain why.
20 cm
8 cm
Work out the volume of each compound prism.
a
b
8 mm
3m
6m
2.5 m
344
4m
7 mm
15 mm
12 mm
15.3 Calculating the volume of triangular prisms
Activity 15.3
Work with a partner to answer this question.
On a piece of paper, draw two triangular prisms like those in question 1.
Make sure you write all the dimensions on your prisms.
a
On a different piece of paper, work out the volume of each prism.
b Exchange pieces of paper with your partner and work out the volume
of each of their prisms.
c
Exchange back and mark each other’s work. Discuss any mistakes.
8
The diagram shows a triangular prism.
The volume of the prism is 96 cm3.
a
Work out the area of the shaded triangle.
b Copy and complete these possible dimensions for
the shaded triangle:
Option 1: base =
cm and height =
cm
height
8 cm
base
Option 2: base =
cm and height =
cm
Tip
c
Compare your answers to part b with those of
Choose your own
a partner.
values for the base and
Did you have the same base and height measurements,
height of the triangle
or were they different?
that will give the area
Discuss the number of different possible combinations.
you found in part a.
9
The diagram shows a triangular prism.
The volume of the prism is 168 m3.
a
Work out the height of the triangle.
b Compare the method you used to answer part a
with other learners in the class. Which method do
height
you think is best to use to answer this type of question?
12 m
Explain why.
4
m
10 A triangular prism has a base of 10 cm, a height of 6 cm
and a length of 15 cm.
a
Work out the volume of the triangular prism.
b Work out the dimensions of three other triangular prisms
with the same volume.
345
15 Distance, area and volume
11 The diagram shows a triangular prism made from silver.
Jan is going to melt the prism and make the silver
into cubes.
25 mm
The side length of each cube is 8 mm.
Jan thinks he can make nine cubes from this prism.
Is Jan correct? Explain your answer. Show all your working.
12 mm
30 mm
In this lesson you have looked at volumes of triangular prisms.
hat do you think is the most important thing to remember
W
when working out volumes of triangular prisms?
Summary checklist
I can derive and use the formula for the volume of a triangular prism.
15.4 Calculating the surface area
of triangular prisms and pyramids
In this section you will …
Key words
•
net
calculate the surface area of triangular prisms and pyramids.
surface area
You already know how to draw the net of a cube or cuboid to help you
work out the surface area of the shape. You can use the same method to
help you work out the surface area of triangular prisms and pyramids.
346
15.4 Calculating the surface area of triangular prisms and pyramids
Worked example 15.4
5 cm
The diagram shows a triangular prism.
5 cm
4 cm
a Sketch a net of the prism.
b Work out the surface area of the prism.
6 cm
8 cm
Answer
a
4 cm
The prism has a rectangular base (A), measuring 8 cm
by 6 cm.
D
A
6 cm
B
5 cm
E
C
It has two rectangular faces (B and C) that measure
8 cm by 5 cm.
8 cm
It has two triangular faces (D and E), each with base
length 6 cm and perpendicular height 4 cm.
5 cm
Work out the area of rectangle A.
b Area A = l × w = 8 × 6
= 48 cm2
Area B = l × w = 8 × 5
= 40 cm2
1
Work out the area of rectangle B.
Note that C has the same area as B.
1
Area D = 2 bh = 2 × 6 × 4
= 12 cm2
Work out the area of triangle D.
Note that E has the same area as D.
Surface area = 48 + 40 × 2 + 12 × 2
= 48 + 80 + 24
= 152 cm2
Add the areas together.
Remember to include 40 × 2 and 12 × 2.
Remember the units (cm2).
Exercise 15.4
1
Copy and complete the workings to find the surface area of each shape.
a
10 cm
b
14 cm
8 cm
12 cm
6 cm
10 cm
347
15 Distance, area and volume
12 cm
8 cm
8 cm
D 6 cm
10 cm
A
10 cm
B
12 cm
E
E
C
Area of D = 12 × 6 ×
=
cm2
Area of E = Area of D
Surface area = 96 +
+
+2×
1
Area of B = 2 × 10 ×
=
cm2
=
cm2
Area of C, D and E = Area of B
Surface area =
+4×
=
cm2
=
cm2
For each of these solids
i
sketch a net
ii
work out the surface area.
a
triangular prism (isosceles)
b triangular prism (right-angled triangle)
13 cm
5 cm
10 cm
6 cm
30 cm
8 cm
24 cm
d
15 cm
18 cm
The diagram shows a triangular prism and a cube.
Which shape has the greater surface area?
Show your working.
5 cm
4 cm
15 cm
6 cm
7 cm
9 cm
triangular-based pyramid
(all triangles equal size)
13 cm
square-based pyramid
(all triangles equal size)
12 cm
c
348
C
Area of A = 10 ×
Area of A = 8 × 12 = 96 cm2
Area of B =
×
=
cm2
Area of C =
×
=
cm2
3
A
10 cm
D
12 cm
2
14 cm
B
4
The diagram shows a triangular-based pyramid
and a cuboid.
In the triangular-based pyramid, all triangles are the
same size.
Show that the surface area of the triangular-based
pyramid is 8 m2 more than the surface area of the cuboid.
3.5 m
15.4 Calculating the surface area of triangular prisms and pyramids
2m
1.5 m
2m
4m
Think like a mathematician
This square-based pyramid has a base side length of x cm.
a
Write an expression for the area of the base of the pyramid.
The perpendicular height of each triangular face is double the
base side length.
b
c
d
e
height
5
x
Write an expression for the area of one of the triangular
faces of the pyramid.
Write a formula for the surface area of the pyramid.
In Pyramid A x = 5. In Pyramid B x = 7.
Use your formula to work out the difference in surface area between
the two pyramids.
Compare and discuss your answers to parts c and d with the rest of the class.
Activity 15.4
When you have answered this question, you will swap your solution with a partner.
They will follow the method you have used and check your working.
Make sure you set out your solution so it is easy for your partner to follow.
Once you have checked each other’s solutions, discuss each other’s work and
give feedback on the methods used.
The surface area of this triangular-based pyramid is 249.6 cm2.
height
Work out the height of the triangular face (all triangles are the
same size).
12 cm
349
15 Distance, area and volume
6
The base of a triangular pyramid is an equilateral triangle
with base length 6 cm and perpendicular height 5.2 cm.
The sides of the triangular pyramid are isosceles triangles
with base length 6 cm and perpendicular height 8.7 cm.
Work out the surface area of the pyramid.
This triangular prism has a volume of 180 cm3.
The area of the triangular cross-section of the prism is A.
Use the information given to work out the surface area
of the triangular prism.
7
l = 5 cm h = A b = 2h x = 2 1 × l y = 1.4 × l
5
Summary checklist
I can calculate the surface area of a triangular prism.
I can calculate the surface area of a pyramid.
350
Tip
Draw a diagram
to help you.
x
h
l
b
y
15 Distance, area and volume
Check your progress
1
2
3
Write true (T) or false (F) for each statement.
a
22 miles is further than 22 km.
b 50 km is exactly the same distance as 50 miles.
c
200 km is not as far as 200 miles.
a
Convert 112 km into miles.
b Convert 205 miles into km.
Work out the area of each shape.
a
4
b
16 m
4 cm
8m
7 cm
24 m
Work out the volume of this triangular prism.
4 cm
9 cm
3 cm
5
a
b
Sketch the net of this shape.
Use your net to work out the surface area of the shape.
13 cm
5 cm
10 cm
12 cm
6
aSketch a net of this square-based pyramid. All the triangular faces of
the pyramid are the same size.
b Use your net to work out the surface area of the pyramid.
7 cm
10 cm
351
Project 6
Biggest cuboid
Start with a 12 cm by 12 cm square of paper.
Draw six rectangles that can be cut out and fitted together to make a cuboid.
For example, these six rectangles could be joined to make this 2 cm by 3 cm
by 5 cm cuboid:
There are lots of gaps between the rectangles, so perhaps we could have
made a cuboid with a bigger surface area and a bigger volume.
Can you find a cuboid that uses more of the paper?
What is the volume of your cuboid?
What different volumes of cuboid can you make from a 12 cm by 12 cm square?
Can you find any cuboids that use the whole square of paper?
What is the biggest volume of cuboid you can make?
352
16
Interpreting and
discussing results
Getting started
The frequency diagram shows the masses of the
Mass of family members
members of one family.
8
a
How many members of this family have a mass in
the 80–120 kg group?
6
b How many members are in this family altogether?
c
What fraction of the family members have a mass
4
in the 0–40 kg group?
Look at the following sets of data. Which type of
2
diagram, graph or chart do you think is best to use to
display each set of data? Justify your choice.
0
120 160
40
80
a
The proportion of different flavour potato chips
Mass (kg)
sold in a shop one day.
b The sales of coats each month for a year.
Transport to work
c
The number of girls and boys going to an
after-school club each day for one week.
d The heights of 200 students in a college.
Car
120 people were asked how they travel to work.
Bus
108 ° 90 °
The pie chart shows the results.
Train
a
What percentage of the people travel to work by
30 ° 132 °
car?
Bicycle
b What fraction of the people travel to work by
train? Write your answer in its simplest form.
c
How many of the people travel to work by bus?
These are the weekly wages, in dollars, of the workers in an office.
Frequency
1
2
3
4
500
a
b
c
525
650
510
500
495
740
630
450
500
Work out the mode, median and mean weekly wages.
Which average weekly wage best represents this data?
Give a reason for your choice of average.
Work out the range in the weekly wages.
353
16 Interpreting and discussing results
Month
When you study statistics, you need to be able to draw and understand
charts, graphs, tables and diagrams. A ‘picture’ of the data can make it
easier to understand the information.
For example, look at the table on the
Number of boxes of breakfast cereal sold
right. It shows the number of boxes of
Jan Feb Mar Apr May Jun
breakfast cereal sold at a grocery store
Top shelf
30
33
28
23
44
22
each month from January to June. It also
shows which shelf the boxes were on in
Middle shelf
32
52
46
40
65
51
the store. There is a lot of information in
Bottom shelf 26
10
20
35
24
14
the table and it is difficult to understand
all this information just by looking at
the table.
Now look at this bar chart, which shows the total
Monthly breakfast cereal sales at a grocery store
monthly sales. You can easily see that May had
the largest number of sales, by quite a long way,
Jun
while the total sales in the other months were all
very similar.
May
Apr
Mar
Feb
Jan
0
20
40
60
80 100 120
Number of boxes sold
Number of boxes sold
When the data is put into a line graph, showing the Monthly breakfast cereals sales at a grocery store
from May to June
monthly sales and the positions on the shelf, you
can see that sales from the middle shelf were always
Top shelf
Middle shelf
Bottom shelf
greater than sales from the other shelves. The sales
from the top and bottom shelves were quite close
70
to each other on some occasions.
60
All this information could be important to a
50
grocery store when it is planning where to place
40
30
items to maximise sales. It could also help the store
20
to identify the months in which it needs to order
10
extra stock.
354
0
Jan
Feb Mar Apr May Jun
Month
140
16.1 Interpreting and drawing frequency diagrams
16.1 Interpreting and drawing
frequency diagrams
In this section you will …
Key words
•
class interval
draw and interpret frequency diagrams for discrete and
continuous data.
A frequency diagram shows how often particular values occur in a set of
data. One example of a frequency diagram is a bar chart. In a bar chart,
the bars are used to represent the frequency.
When you draw a bar chart for grouped data, you must use suitable
classes and have equal class intervals.
When you draw a bar chart for discrete data, you should
make sure:
the bars are all the same width
•
the gaps between the bars are equal
•
you label each bar with the relevant data group
•
you give the frequency diagram a title and label the axes
•
you use a sensible scale on the vertical axis.
•
When you draw a bar chart for continuous data, you should
make sure:
the class intervals are all the same width
•
there are no gaps between the bars
•
you use a sensible scale on the horizontal axis
•
you give the frequency diagram a title and label the axes
•
you use a sensible scale on the vertical axis.
•
classes
continuous data
discrete data
frequency
diagram
grouped data
355
16 Interpreting and discussing results
Worked example 16.1
he frequency diagram shows how many
T
pieces of fruit the students in class 8T ate
in one week.
i How many students ate 4–7 pieces of fruit?
iiHow many more students ate 8–11 pieces
of fruit than ate 12–15 pieces?
iii How many students are there in class 8T?
b The frequency table shows the masses of
20 teachers.
Draw a frequency diagram to show the data.
Mass, m (kg)
60 < m ≤ 70
70 < m ≤ 80
80 < m ≤ 90
90 < m ≤ 100
Frequency
3
8
6
4
Number of pieces of fruit eaten by 8T in one week
10
8
Frequency
a
6
4
2
0
8–11
12–15
0–3
4–7
Number of pieces of fruit
Answer
a
i
6 students
ii
9 − 4 = 5 students
The bar for 4–7 has a height of 6 on the
frequency axis.
The frequency for 8–11 is 9 and the frequency
for 12–15 is 4.
Subtract one from the other to find the
difference.
Add together the frequencies for all the
groups.
All the bars are the same width and, as the
data is continuous, there are no gaps between
them.
iii7 + 6 + 9 + 4 = 26 students
b
Mass of 20 teachers
Frequency
8
Both the horizontal and vertical axes have
sensible scales.
6
The frequency diagram has a title and the axes
are labelled.
4
2
50
356
60
70
90
80
Mass (kg)
100
110
16.1 Interpreting and drawing frequency diagrams
Exercise 16.1
1
Number of phone calls made by the employees
of a company on one day
Number of phone calls
The frequency diagram shows the number of
30–39
phone calls made by all the employees of a
company on one day.
20–29
a
How many employees made 10–19 phone
calls?
10–19
b How many more employees made 30–39
phone calls than made 0–9 phone calls?
0–9
c
How many employees are there in the
0
8
10
2
4
6
company?
Frequency
Explain how you worked out your answer.
The frequency table shows the number of cups of
coffee sold each day in a coffee shop during one month.
a
Draw a frequency diagram to show the data.
Number of cups Frequency
b Which month do you think your frequency
of coffee sold
diagram represents? Explain your answer.
0–19
2
c
Read what Marcus says.
20–39
3
40–59
6
60–79
12
The frequency diagram
80–99
5
shows that the most cups
2
of coffee sold was 99.
Is he correct? Explain your answer.
Think like a mathematician
3
Work with a partner or in a small group to answer this question.
Ryan recorded the number of text messages he sent each day for one month.
Here are his results.
23
14
a
b
c
17
4
19
12
0
20
16
9
18
13
7
20
17
11
15
19
18
1
12
20
10
20
18
24
14
2
Record this information in a frequency table.
Choose suitable classes. Make sure you have equal class intervals.
Draw a frequency diagram to show the data.
Compare your frequency table and diagram with those of other groups.
Discuss the classes used.
Which classes do you think are best to show this data? Explain why.
357
16 Interpreting and discussing results
4
Erin recorded the number of emails she sent each day for one month.
Here are her results.
31 17 37 11 35 34 36 15 33 22 31 18 34 12 28
14 30 21 39 16 13 38 34 29 10 19 39 32 38 15
Marcus, Arun and Zara discuss what classes to use.
Read what they say.
I would use the
classes 0–4, 5–9,
10–14, etc.
I would use the
classes 10–14,
15–19, 20–24, etc.
a
b
c
d
I would use the
classes 10–19,
20–29, 30–39, etc.
Who do you think has chosen the most suitable classes, Marcus,
Arun or Zara?
Explain why.
Explain why you think the classes chosen by the other two are
not suitable.
Record the information in a frequency table.
Draw a frequency diagram to show the data.
Think like a mathematician
5
358
Work with a partner or in a small group to answer this question.
The frequency table shows the ages of the members
Age, a years Frequency
of a choir.
10 ≤ a < 20
12
a
Explain what you think the class 10 ≤ a < 20 means.
20 ≤ a < 30
8
b Explain why you cannot use the classes 10–19,
30 ≤ a < 40
15
20–29, etc.
40 ≤ a < 50
6
c
In which class would you include someone aged
exactly 30 years?
d Draw a frequency diagram to show the data.
e
Discuss and compare your answers to parts a–d with other groups in your class.
16.1 Interpreting and drawing frequency diagrams
6
The frequency table shows the speeds of cars
passing a speed camera on one day.
The speeds are recorded in kilometres per hour (km/h).
a
Draw a frequency diagram to show the data.
b The speed limit is 80 km/h. How many cars
are travelling over the speed limit?
c
Read what Sofia says.
Speed of car,
s (km/h)
50 < s ≤ 60
60 < s ≤ 70
70 < s ≤ 80
80 < s ≤ 90
90 < s ≤ 100
Frequency
2
3
6
12
5
The frequency
diagram shows that the
slowest car was travelling
at 50 km/h.
7
Is she correct? Explain your answer.
Here are the heights, in centimetres, of some plants.
25
39
a
b
c
32
20
30
27
26
33
34
37
22
32
33
25
34
24
Record this information in a frequency table.
Use the classes 20 ≤ h < 25, 25 ≤ h < 30, 30 ≤ h < 35
and 35 ≤ h < 40.
Draw a frequency diagram to show the data.
How many of the plants are at least 25 cm high?
Explain how you worked out your answer.
31
30
28
29
Tip
At least 25 cm
means 25 cm or
more.
359
16 Interpreting and discussing results
8
The frequency diagrams show the population of a village by age
group in 1960 and 2010.
a
b
90
80
70
60
50
40
30
20
10
0
Population of a village by age
group, 2010
Frequency
Frequency
Population of a village by age
group, 1960
0
60
20
40
Age (years)
80
100
90
80
70
60
50
40
30
20
10
0
0
20
40
60
Age (years)
80
100
Look at the graphs.
Write two sentences to compare the age groups in the
population of the village in 1960 and 2010.
Read what Marcus says.
Approximately 25% of
the population were over
the age of 40 in 1960,
compared with approximately
60% in 2010.
Is Marcus correct? Show your working to support your answer.
In this section, you have recorded information in a frequency table.
You have had to choose your own classes.
a
Explain to a partner the method you use to decide what classes to use.
b
Does your partner use the same method as you or a different one?
c
Discuss your methods and decide which is better/easier to use.
Summary checklist
I can draw and interpret frequency diagrams for discrete and continuous data.
360
16.2 Time series graphs
16.2 Time series graphs
In this section you will …
Key words
•
time series graph
draw and interpret time series graphs.
trend
A time series graph is a series of points, plotted at regular time intervals
and joined by straight lines.
Time series graphs are used to show trends, which tell you how the data
changes over a period of time.
When you draw a time series graph, make sure you:
put time on the horizontal axis
•
use an appropriate scale on the vertical axis
•
plot each point accurately
•
join the points with straight lines
•
give the time series graph a title and label the axes.
•
Worked example 16.2
The table shows the value of a car over a period of five years.
a
b
c
d
Draw a time series graph to show the data.
During which year did the car lose the most value?
Describe the trend in the value of the car.
Use the graph to estimate the value of the car after 2 1 years.
2
Age of car
(years)
0
1
2
3
4
Value of
car ($)
25 000
20 000
17 000
14 900
13 400
Answer
a
Age of car (years) goes on the
horizontal axis.
Value of car over five years
30 000
Value of car ($) goes on the
vertical axis.
Value of car ($)
25 000
20 000
The vertical axis has a sensible
scale that is easy to read.
15 000
All the points are plotted
accurately and joined with
straight lines.
10 000
5 000
0
0
1
2
3
Age of car (years)
4
The graph has a title and the axes
are labelled.
361
16 Interpreting and discussing results
Continued
b During the first year
The greatest loss is $5000, in the first year.
c
The losses are $5000, $3000, $2000 and $1500 so each
year, the loss is less than in the year before.
he value of the car decreases
T
every year, but the loss each year
is less than in the year before.
d $16 000 (see red line on graph)
Read up from 2 1 on the horizontal axis to the line,
2
then across to the vertical axis to read off the value.
Exercise 16.2
2
362
Company profit
Profit ($ million)
The time series graph shows the profit made by a
company each year for a six-year period.
a
How much profit did the company make in
i 2006
ii 2007?
b In which year did the company make the largest
profit?
c
Between which two years was the greatest
increase in profit?
d Between which two years was the greatest
decrease in profit?
e
Describe the trend in the company profits over
the six-year period.
The time series graph shows the value of a
house over a ten-year period.
a
What was the value of the house in
200
180
i 2000
ii 2010?
160
b In which year did the house reach its
140
greatest value?
120
c
Between which two years was the
100
greatest increase in the value of the
0
house?
d Describe the trend in the value of the
house over the ten-year period.
e
Use the graph to estimate the value of the house in
i 2003
ii 2009.
Value of house ($ thousands)
1
3
2
1
0
2006 2007 2008 2009 2010 2011
Year
House value 2000–2010
2000 2002 2004 2006 2008 2010
Year
16.2 Time series graphs
Think like a mathematician
The time series graph shows the average price of crude oil, per barrel,
every ten years since 1965.
Price per barrel ($)
3
55
50
45
40
35
30
25
20
15
10
5
0
1965
Average price of crude oil (to the nearest dollar)
1975
1985
Year
1995
2005
2015
Marcus and Sofia are trying to work out in which year the average price
of crude oil was at its highest.
Read what they say.
I think the
average price
of crude oil was
at its highest in
2005.
I don’t think you can
tell from this graph in
which year the average
price of crude oil was
at its highest.
What do you think? Explain why.
Discuss your answers and explanations with other learners in your class.
For questions 4 and 5, work in groups of three or four.
i
On your own, draw a time series graph to show the data.
Then describe the trend in the data and answer the question.
ii
Compare your time series graph, description and answers
with those of the other members of your group. Discuss the
different scales you have used in your graphs and compare
the descriptions you have given. Decide which are best.
Check your answers to the questions are the same.
363
16 Interpreting and discussing results
4
The table shows the number of people staying in a guest house each month
for one year.
Month
Jan Feb Mar Apr May Jun
Number of people 8
6
11 15
17 20
Jul Aug Sep Oct Nov Dec
24 26 18 14
8
7
Between which two months did the number of people at the guest house
change the most?
The table shows the average price of silver and copper, per ounce, every
four years since 1990. The prices are rounded to the nearest $0.10.
5
Year
Average price of silver $
Average price of copper $
a
b
c
d
1990
4.80
1.20
1994
5.30
1.10
1998
5.50
0.80
2002 2006 2010 2014 2018
4.60 11.60 20.20 19.10 15.70
0.70 3.10 3.10 3.20 3.10
Draw a time-series graph to show both sets of data.
Write true (T) or false (F) for each statement
i Between 1990 and 2002 the prices did not change very much.
ii Silver increased by the greatest amount between 2002 and 2006.
iii
We can use the graph to predict an accurate price of copper and
silver in 2022.
Use your graphs to estimate the price in 2008 of:
i silver
ii
copper.
Marcus says ‘The price of both silver and copper went up in 2002.’
Explain why Marcus may not be correct.
Think like a mathematician
6
Zara goes to a one-hour cycling class every Monday, Wednesday
and Friday evening. She records the distance she cycles during each class.
Zara wants to draw a time series graph and look at the trend in
her data over one year.
Read what Zara and Sofia say.
I think I will
plot every
distance I have
recorded over
the year.
If you do that, you’ll
have more than
150 points to plot!
I think I would plot
fewer points.
What do you think? Explain why.
Discuss your answers and explanations with other learners in your class.
364
16.2 Time series graphs
A sports shop sells the rugby shirts of two teams,
Number of rugby shirts in stock
Scarlets and Dragons.
over an 8-week period
80
The time series graph shows the number of rugby
Scarlets
Dragons
70
shirts the shop has in stock each week over an
60
8-week period.
50
a
Describe the trend in the sales of
40
i Scarlets rugby shirts
30
ii Dragons rugby shirts.
20
b Do you think the shop has enough Scarlets
10
rugby shirts in stock for week 9?
Explain your answer.
0
1 2 3 4 5 6 7 8 9 10
Week number
c
Do you think the shop has enough Dragons
rugby shirts in stock for week 9?
Explain your answer.
The time series graph shows the number of hotel rooms booked in a seaside town.
It shows the number booked in spring, summer, autumn and winter from 2018 to 2020.
Number of rugby shirts in stock
7
Number of hotel rooms booked in seaside town from 2018 to 2020
90
80
70
60
50
40
30
20
10
0
Sp
rin
Su g
m
m
A er
ut
um
W n
in
te
Sp r
rin
Su g
m
m
A er
ut
um
W n
in
te
Sp r
rin
Su g
m
m
A er
ut
um
W n
in
te
r
Number of hotel rooms
booked (100s)
8
2018
a
b
c
d
e
2019
Year and season
2020
Describe how the number of hotel rooms booked changes over the seasons during 2018.
Do similar changes over the seasons that you have noticed in 2018, also happen in 2019
and 2020? Explain your answer.
Describe the yearly trend in the number of hotel rooms booked.
Use your graph to predict the number of hotel rooms that will be booked in Autumn 2021.
Explain why your answer to part d may be incorrect.
Summary checklist
I can draw and interpret time series graphs.
365
16 Interpreting and discussing results
16.3 Stem-and-leaf diagrams
In this section you will …
Key words
•
mean
draw and interpret stem-and-leaf diagrams.
median
A stem-and-leaf diagram is a way of showing
Key: 2 3 means 23 cm
data in order of size.
2 3 4 7 8
When you draw a stem-and-leaf diagram,
3 0 6 7 7 8
make sure:
4 1 9 9
you write the numbers in order of size from
•
stem
leaves
smallest to largest
you write a key to explain the numbers
•
you keep all the numbers in line vertically and horizontally.
•
mode
range
stem-and-leaf
diagram
Worked example 16.3
Here are the temperatures, in °C, recorded in 20 cities on one day.
9
14
19
16
26
18
35
29
6
27
17
8
32
25
21
32
30
21
16
32
a Draw an ordered stem-and-leaf diagram to show this data.
b How many cities had a temperature over 28 °C?
c Use the stem-and-leaf diagram to work out
i the mode
ii the median
iii the range of the data.
‘Ordered’ means you
write the numbers in
order of size from
smallest to largest.
Answer
a
366
Key: 1 9 means 19 °C
0
9 6 8
1
9 7 6 4 6 8
2
3
6 1 9 7 5 1
5 2 0 2 2
First, draw an unordered stem-and-leaf diagram.
Start by writing a key. You can use any of the numbers to
explain the key.
Write in the stem numbers. In this case, they are the tens digits,
0, 1, 2 and 3.
Now write in the leaves, taking the numbers from the table above.
For example, 9 has 0 as the stem and 9 as the leaf; 19 has 1 as the
stem and 9 as the leaf; 26 has 2 as the stem and 6 as the leaf;
and so on.
Complete for all 20 temperatures.
16.3 Stem-and-leaf diagrams
Continued
Key: 1 9 means 19 °C
0
6 8 9
1
4 6 6 7 8 9
2
3
1 1 5 6 7 9
0 2 2 2 5
Now draw an ordered stem-and-leaf diagram.
Rewrite the diagram with all the leaves in order, from the smallest
to the biggest.
Make sure the stem numbers are in line vertically and the leaves
are in line vertically and horizontally.
b 6 cities
You can see from the stem-and-leaf diagram that 29 °C, 30 °C,
32 °C, 32 °C, 32 °C and 35 °C are all over 28 °C.
c
mode = 32 °C
You can see that 32 °C is the temperature that appears most often.
ii median = 21 °C
There are 20 temperatures, so the median temperature is the average
of the 10th and 11th values. These are both 21 °C, so the median
is 21 °C.
iii range = 35 − 6
= 29 °C
You can easily see from the diagram that the highest temperature
is 35 °C and the lowest is 6 °C. Subtract to find the range.
i
Exercise 16.3
1
Shen listed the playing times, to the nearest minute, of some CDs.
He recorded the results in a stem-and-leaf diagram.
a
How many CDs did Shen list?
b What is the shortest playing time?
c
How many of the CDs had a playing time longer than
60 minutes?
d Work out
i the mode
ii the median
iii the range of the data.
Key: 4 5 means 45 minutes
4
5 5 7 9
5
0 2 5 6 8 9
6
1 2 4 6 7
Think like a mathematician
2
Work with a partner to answer this question.
a
11
2 4 6 8
12 0 0 1 2 4 5 6 9
A cafe keeps a record of the number of cups of coffee
13 1 4 4 5 8 9
sold each day.
14 2 7 7 7
The stem-and-leaf diagram shows the number of cups
15 0 1 1 3 6 8
of coffee sold each day for one month.
iWhich month does the stem-and-leaf diagram represent?
ii What is the largest number of cups of coffee sold on one day?
iii What is the modal number of cups of coffee sold on one day?
367
16 Interpreting and discussing results
Continued
b
3
Discuss these questions with other learners in your class.
i Were you able to answer all of the questions in part a?
ii What is missing from the stem-and-leaf diagram?
iiiCan you work out the answers to part a, even though something
is missing from the stem-and-leaf diagram?
These are the file sizes, in kilobytes (kB), of 30 files on Greg’s computer.
101 128 117 109 154 139 166 155 117 145 135 162 117 168 125
131 140 160 151 125 152 108 139 130 165 158 103 130 110 148
a
b
c
d
Draw an ordered stem-and-leaf diagram to show this data.
How many of the files are larger than 150 kB?
Which average, the mode or the median, better represents this data? Explain why.
Greg works out that the range in his file sizes is 65 kB.
Is Greg correct? Explain your answer.
Think like a mathematician
4
Work with a partner to answer this question.
Key: 0 4 means 4 birds
Opaline counted the number of birds in her garden each day
0 4 8 9
for 10 days.
1 2 4 9
The stem-and-leaf diagram shows her results.
2 0 2 3 7
This is how Opaline works out the mean number of birds in her garden each day.
From my stem-and-leaf diagram:
1st line
mean = 4 + 83 + 9 = 21
= 7, mean = 7
3
2nd line
mean = 2 + 4 + 9 = 15
= 5, mean = 15
3
3
3rd line mean = 0 + 2 +4 3 + 7 = 12
= 3, mean = 23
4
Overall mean = 7 + 153+ 23 = 45
= 15
3
a
b
368
Is Opaline’s method a correct method? Explain your answer.
What do you think is the best method to use to work out the mean
from a stem-and-leaf diagram?
16.3 Stem-and-leaf diagrams
5
6
Ashish counted the number of cars passing his school between
8.30 a.m. and 9 a.m. each day for 12 days.
The stem-and-leaf diagram shows his results.
a
Use the stem-and-leaf diagram to work out
i the mode
ii the median
iii the mean of the data.
b Which average, the mode, median or mean, best represents
this data? Explain why.
The students in class 8B took a test.
The stem-and-leaf diagram shows their scores out of 40.
Key: 0 1 means 1 car
0
1 2 3 6 7 7 9
1
7
2
0 4 4 4
Key: 1 8 means 18
a
b
c
0
6 8 8 9 9
1
6
2
5 6 8 8 9
3
0 1 2 3 3 5 6 7 8 8
4
0 0 0 0
What percentage of the students had a score greater than 32?
What fraction of the students had a score less than 25%?
Any student scoring less than 40% must re-sit the test.
How many students do not have to re-sit the test?
Discuss the answers to these questions with a partner.
a
Do you think stem-and-leaf diagrams are a good way to
represent data? Explain why.
b
Do you think stem-and-leaf diagrams are easy to understand?
Explain why.
c
What are the advantages and disadvantages of drawing and
interpreting stem-and-leaf diagrams?
Summary checklist
I can draw and interpret stem-and-leaf diagrams.
369
16 Interpreting and discussing results
16.4 Pie charts
In this section you will …
Key words
•
categorical data
compare pie charts.
You already know how to draw and interpret a pie chart.
You can use pie charts to compare different sets of categorical data data,
but remember that a pie chart shows proportions, not actual amounts.
pie chart
proportions
sector
Worked example 16.4
The pie charts show the proportion of male and female teachers in two schools.
Gender of teachers in Elm School
Gender of teachers in Oak School
Male
108 °
252 °
144 °
Female
216 °
There are 20 teachers in Oak School. There are 45 teachers in Elm School.
a Which school has the greater proportion of female teachers?
b Which school has the greater number of female teachers? Show your working.
Answer
a
ak School has the greater
O
proportion of female
teachers.
b Oak: 252 × 20 = 7 × 20
360
10
= 14 female teachers
Elm: 144 × 45 = 2 × 45
360
5
= 18 female teachers
Elm School has the greater
number of female teachers.
370
The blue sector (female teachers) in the pie chart for
Oak School is larger than the blue sector in the pie chart
for Elm School.
Work out the number of female teachers in each school.
Start by writing the fraction of female teachers, then
simplify this fraction and multiply by the number of
teachers.
Although the proportion of female teachers is greater in
Oak School, there are more female teachers in Elm School.
16.4 Pie charts
Exercise 16.4
1
The pie charts show the proportion of boys and girls in two swimming clubs.
Gender of children in the Dolphins
swimming club
210 ° 150 °
Gender of children in the Seals
swimming club
Girls
Boys
120 °
240 °
There are 120 children in the Dolphins swimming club.
There are 72 children in the Seals swimming club.
a
Which swimming club has the greater proportion of girls?
b Which swimming club has the greater number of girls?
Copy and complete the working.
240
× 120 =
Dolphins: 150
girls Seals: 360 × 72 =
girls
360
The ................... swimming club has the greater number of girls.
2
The pie charts show the proportion of Ivan’s income that he
made from gardening, washing windows and painting houses in
2009 and 2019.
Ivan’s business in 2019
Ivan’s business in 2009
Gardening
135 °
45 °
a
180 °
Washing windows
Painting houses
90 °
135 °
135 °
What fraction of Ivan’s income came from gardening in
i 2009
ii 2019?
371
16 Interpreting and discussing results
b
Copy and complete these sentences. Choose from the words
in the rectangle.
doubled
i
3
372
stayed the same
tripled halved
more than tripled
In 2019 the proportion of Ivan’s income that came from
gardening had ...................... compared to 2009.
ii
In 2019 the proportion of Ivan’s income that came from
painting houses had ...................... compared to 2009.
iii In 2019 the proportion of Ivan’s income that came from
washing windows had ...................... compared to 2009.
In 2009, Ivan’s total income was $12 000.
In 2019, Ivan’s total income was $24 000.
c
Show that Ivan earned the same amount of money from
Tip
gardening in 2009 and 2019.
Use the fractions
d Show that Ivan earned six times as much money from
you found in
washing windows in 2019 as in 2009.
part a.
e
How much more money did Ivan earn from painting houses
in 2019 than in 2009?
The pie charts show the results of a
Men’s favourite chocolate
Women’s favourite chocolate
survey about the types of chocolate
preferred by men and by women.
480 men took part in the survey.
Plain
Caramel
Caramel
Plain
600 women took part in the survey.
81 °
a
How many men chose plain
60
°
84 ° 120 °
White
135 °
chocolate?
White
Milk
Milk
b How many women chose plain
chocolate?
c
Hassan thinks that more
men than women like milk
chocolate. Is Hassan correct?
Show how you worked out your answer.
d The ‘Caramel’ sector is the same size for men and for women.
Without doing any calculations, explain how you know that more
women than men chose ‘Caramel’.
16.4 Pie charts
4
The pie charts show the favourite sports of the students in two schools.
Castlehill School
2%
Riverside School
Rugby
18%
25%
13%
10%
32%
14%
Football
Tennis
12%
Hockey
7%
Cricket
38%
15%
14%
Other
There are 1600 students in Castlehill School.
There are 1100 students in Riverside School.
Which school had the larger number of students who chose tennis as
their favourite sport?
Show your working.
Think like a mathematician
5
Work with a partner or in a small group to answer these questions.
The pie charts show the proportions of different makes of car sold by two
garages in 2019.
Make of car sold in Kabir’s garage
Make of car sold in Ekta’s garage
30 °
Kia
90 °
120 °
120 °
Ford
45 °
40 °
120 °
155 °
Seat
Nissan
In 2019, Kabir sold 600 cars in total.
a
How many cars does Ekta sell in total if she sells
i the same number of Kia cars as Kabir
ii the same number of Ford cars as Kabir
iii the same number of Nissan cars as Kabir?
373
16 Interpreting and discussing results
Continued
b
c
6
Explain how you worked out your answers to part a.
Can you describe a rule to follow to answer this type of question?
Discuss your answers and methods with other groups in your class.
The pie charts show the proportions of different sizes of T-shirt
sold in a shop on two days.
Sizes of T-shirts sold on Tuesday
Sizes of T-shirts sold on Monday
60 ° 80 °
220 °
Small
40 °
Medium
180 ° 140 °
Large
7
On Monday, the shop sold 144 T-shirts.
On Tuesday, the shop sold the same number of small T-shirts as
on Monday.
a
How many small T-shirts did the shop sell on Tuesday?
b How many T-shirts did the shop sell altogether on Tuesday?
The pie charts show the proportions of types of rice sold by two
shops in May.
Type of rice sold in Shop B
Type of rice sold in Shop A
30 °
150 ° 120 °
60 °
20 °
Black
Brown
Red
30 °
White
In May, Shop A sold 6 kg of black rice.
374
130 °
180 °
16.4 Pie charts
a
Copy and complete this table to show the amounts of rice sold
by Shop A.
Amounts of rice sold by Shop A
Type of rice Degrees in pie chart
Kilograms sold
6 kg
black
30º
brown
120º
red
60º
white
150º
Total:
360º
Tip
Use the fact that
6 kg of rice is
represented by
30º in the pie
chart.
In May, Shop B sold the same number of kilograms of red rice as
shop A.
b Copy and complete this table showing the amounts of rice sold
by Shop B.
Amounts of rice sold by Shop B
Type of rice Degrees in pie chart
Kilograms sold
black
20º
brown
180º
red
30º
white
130º
Total:
360º
c
Tip
Use the fact that
the number of
kilograms of red
rice sold is the
same in both
tables.
Read what Sofia says.
In Shop A, 60° of the
pie chart represents red rice.
In Shop B, 30° of the pie chart represents
red rice. This means that, without doing
any calculations, I can say that the total
amount of rice sold in Shop A is double
the total amount of rice sold
in Shop B.
Explain why Sofia is incorrect.
Summary checklist
I can compare pie charts.
375
16 Interpreting and discussing results
16.5 Representing data
In this section you will …
Key word
•
justify
choose how to represent data.
When you represent data using a diagram, graph or chart, you need to
decide which type is best to use.
This table will help you to decide.
What type of diagram /
graph / chart?
Venn or Carroll diagram
When do I use it?
Bar chart
When you want to
compare discrete data
Dual bar chart
When you want to
compare two sets of
discrete data
Compound bar chart
When you want to
combine two or more
quantities into one bar, to
look at individual amounts
and total amounts
When you want to
compare continuous data
Frequency diagram
Time series graph
376
When you want to
sort data or objects
into groups with some
common features
When you want to see
how data changes over
time
What does it look like?
16.5 Representing data
What type of diagram /
graph / chart?
Scatter graph
When do I use it?
Pie chart
When you want to
compare the proportions
of each sector with the
whole amount
When you want to show
some information in a
quick way that is easy to
understand
Infographic
Stem-and-leaf diagram
What does it look like?
When you want to
compare two sets of data
points
When you want to
compare data that is
grouped, but you still want
to see the actual values
40% delivered
on time
60% delivered
late
Key: 2 3 means 23 cm
2
3 4 7 8
3
0 6 7 7 8
4
1 9 9
Worked example 16.5
Look at the following sets of data.
Which type of diagram, graph or chart do you think is best to use to display the data?
Justify your choice.
a The value of gold every Monday morning
b The ingredients in 100 ml of two different salad dressings
c The ages of the people on a bus
Answer
a Time series graph – so you can see how the value of gold changes over time.
b Compound bar chart – so you can easily compare the amounts of each ingredient.
c Stem-and-leaf diagram – so you can see the data as grouped data but also see
the exact values.
377
16 Interpreting and discussing results
Exercise 16.5
1
2
3
Look at the following sets of data.
Which type of diagram, graph or chart do you think is best to use
to display the data? Justify your choice.
a
The percentage of the members of two running clubs that are
men, women, girls and boys
b The ages and heights of the horses at a riding school
c
The scores, out of 50, of 30 students in a spelling test
d The mass of a baby chimpanzee each week
A group of 30 students study science at advanced level.
Four students study physics, biology and chemistry.
Five students study only chemistry and biology, three study only
chemistry and physics, and two study only physics and biology.
Six students study only physics, seven study only biology and three
study only chemistry.
a
Draw a diagram, graph or chart to represent this data.
b Justify your choice of diagram, graph or chart.
c
Make one comment about what your diagram, graph or chart
shows you.
The table shows the monthly average mass of a baby girl from
newborn to one year old.
Month
Mass (kg)
a
b
c
378
0
1
2
3
4
5
6
7
8
9
10 11 12
3.2 4.2 5.1 5.8 6.4 6.9 7.3 7.6 7.9 8.2 8.5 8.7 8.9
Draw a diagram, graph or chart to represent this data.
Justify your choice of diagram, graph or chart.
Make one comment about what your diagram, graph or chart
shows you.
16.5 Representing data
Think like a mathematician
4
The table shows the ingredients of two different cans of beans.
Ingredient
Can A
Can B
beans
48 g
67 g
water
32 g
30 g
tomato paste
17 g
18 g
sugar
2g
8g
salt
1g
2g
Read what Arun and Zara say.
I think it is
best to use a
compound bar
chart to represent
this data.
I think it is best
to use a pie chart
to represent this
data.
Discuss the answers to these questions in a small group, and then with
other groups in the class.
a
b
c
5
If you represented this data in a compound bar chart
i what parts of the data would it be easier to compare?
ii what parts of the data would it be more difficult to compare?
If you represented this data in a pie chart
i what parts of the data would it be easier to compare?
ii what parts of the data would it be more difficult to compare?
Complete each statement with either ‘compound bar chart’ or ‘pie chart’.
iWhen you are comparing individual and total amounts, it is better
to use a ...............
ii When you are comparing proportions, it is better to use a ...............
These are the numbers of pages of a book that Daylen reads each day for four
weeks.
25
30
a
b
c
d
5
11
18
22
34
19
16
7
35
27
12
10
12
32
20
27
14
33
8
11
27
24
39
17
9
22
Draw a diagram, graph or chart to represent this data.
Justify your choice of diagram, graph or chart.
Make one comment about what your diagram, graph or chart shows you.
Work out
i the mode ii the median iii the range of the data.
379
16 Interpreting and discussing results
6
Zara recorded the number of minutes she spent
doing homework each evening for one month.
The frequency table shows her results.
Time,
t (minutes)
0 ≤ d < 20
20 ≤ d < 40
40 ≤ d < 60
60 ≤ d < 80
80 ≤ d < 100
Frequency
1
6
2
8
14
a
Draw a diagram, graph or chart to
represent this data.
b Justify your choice of diagram, graph or chart.
c
Make one comment about what your diagram, graph or chart
shows you.
A scientist measured the length and mass of 12 sea turtles.
The table shows her results.
7
Length (cm) 87 99 92 84 108 105 109 94 85 95 100 90
Mass (kg)
125 150 135 112 175 163 188 132 115 144 158 128
a
b
c
Draw a diagram, graph or chart to represent this data.
Justify your choice of diagram, graph or chart.
Make one comment about what your diagram, graph or chart
shows you.
Summary checklist
I can choose how to represent data.
380
16.6 Using statistics
16.6 Using statistics
In this section you will …
Key words
•
mean
use mode, median, mean and range to compare sets of data.
median
You can use an average to summarise a set of data. This could be the
mode, median or mean.
You can use the range to measure the spread of the data.
The larger the range, the more varied the data.
You already know how to work out the mode, median, mean and range.
Here is a reminder:
•
The mode is the most common value or number.
•
The median is the middle value, when they are listed in order.
•
The mean is the sum of all the values divided by the number
of values.
•
The range is the largest value minus the smallest value.
You can use these statistics to compare two or more sets of data.
mode
range
Worked example 16.6
A health club recorded the masses (in kilograms) of eight men and six women.
Men:
65, 79, 68, 72, 77, 77, 81, 67
Women: 68, 52, 47, 49, 50, 58
Calculate the mean and the range of each set of data and use these values to
compare the two sets.
Answer
The mean for the men is 73.25 kg.
The mean for the women is 54 kg.
65 + 79 + 68 + 72 + 77 + 77 + 81 + 67 586
= 8 = 73.25 kg
8
68 + 52 + 47 + 49 + 50 + 58 324
= 6 = 54 kg
6
On average, the men are 19.25 kg
heavier than the women.
73.25 − 54 = 19.25 kg
The range for the men is 16 kg.
81 − 65 = 16 kg
The range for the women is 21 kg.
68 − 47 = 21 kg
The women’s masses are more
varied than the men’s.
21 kg is greater than 16 kg.
381
16 Interpreting and discussing results
Exercise 16.6
1
2
3
4
382
In the 2010 football World Cup, Spain won and Brazil was knocked out
in the quarter finals.
Spain: 0, 2, 2, 1, 1, 1, 1
Brazil: 2, 3, 0, 3, 1
The numbers of goals they scored in their matches are shown.
a
Work out the mean score for each team.
b Use the means to state which team scored more goals, on average,
per match.
c
Work out the range for each team.
d Use the ranges to state which team’s scores were more varied.
A teacher measured the heights of two groups of children.
Here are the results.
Group A: 84 cm, 73 cm, 89 cm, 80 cm, 77 cm
Group B: 77 cm, 85 cm, 75 cm, 69 cm, 82 cm, 67 cm, 72 cm
a
For each group
i write the heights in order of size
ii write the median height
iii work out the range in heights.
b Use the medians to state which group is taller, on average.
c
Use the ranges to state which group’s heights are less varied.
The maximum daytime temperature (ºC) was recorded in Madrid and
Cartagena during one week in August.
Here are the results.
Madrid:
38, 34, 36, 32, 35, 37, 36
Cartagena: 30, 32, 29, 30, 28, 30, 33
a
For each city
i write the temperatures in order of size
ii write the modal temperature
iii work out the range in temperatures.
b Use the modes to state which city is hotter, on average.
c
Use the ranges to state which city’s temperatures are more varied.
A nurse measured the total mass of 20 baby boys as 64 kg.
The total mass of 15 baby girls was 51 kg.
Which babies were heavier on average, the boys or the girls?
Give a reason for your answer.
16.6 Using statistics
Think like a mathematician
5
The test marks of two groups of students are shown.
Maths:
77, 89, 75, 80, 80, 91, 78, 76, 76, 76
Science:
a
72, 79, 77, 87, 81, 62, 75, 87
Copy and complete this table.
Mean
Median Mode
Range
Maths
Science
b
c
d
e
6
In which group, Maths or Science, do you think the students did better on
average?
In which group, Maths or Science, do you think the students had more
consistent scores?
Compare your answers to parts b and c with those of other learners in the class.
Discuss these questions.
iWhich average did you use to compare the scores? Why did you use this
average? Why did you not use the other averages?
iiWhat does ‘more consistent’ mean? What statistic did you use to decide
which group had more consistent scores?
Now you have discussed the answers of other learners in your class, which
average do you think is the best to use to compare these scores? Explain why.
Nialls recorded the temperatures in two experiments.
First experiment (ºC)
Second experiment (ºC)
a
b
c
29, 28, 21, 33, 30
28, 29, 28, 33, 32, 31, 32, 29
Work out the mean, median and range for each experiment.
State whether each of these statements is True (T) or
False (F). Justify your answers.
i
The temperatures in the first experiment are higher, on
average, than the temperatures in the second experiment.
ii
The temperatures in the first experiment are more varied
than the temperatures in the second experiment.
Is it possible to work out the modal temperature for each
experiment? Explain your answer.
383
16 Interpreting and discussing results
Think like a mathematician
7
Work with a partner or in a small group to answer this question.
You are going to roll two dice and subtract the numbers on the dice to give a score.
Always subtract the numbers to give you a positive, or zero, score.
a
b
What is the smallest score you can get?
What is the largest score you can get?
You are going to roll the dice 40 times.
c
d
e
f
g
384
Tip
For example, if
you roll these
numbers you get
a score of 5.
Draw a table to record the scores you get.
Your table needs a ‘Tally’ column and a ‘Frequency’
column.
Now roll the dice 40 times and record all your scores.
When you have finished, make sure the frequency
column adds up to 40.
Work out
i the mode
ii the median
iii the mean score for your data.
Which average best represents your data?
Give a reason for your choice of average.
Compare your data and averages with other learners in the class.
Do you have different averages? Do you have the same averages?
Have you chosen the same average to represent your data?
Discuss your answers.
16.6 Using statistics
8
The frequency tables show the number of goals scored in each
match by two hockey teams in 20 matches.
Team A
Number of goals 0 1 2 3 4 5
Frequency
4 1 4 2 4 5
Team B
Number of goals 0 1 2 3 4 5
Frequency
0 6 1 5 4 4
Read what Marcus, Zara and Arun say.
I think that, on
average, team A
scored more goals
per match.
I think that, on
average, team B
scored more goals
per match.
I think that, on
average, they scored
the same number of
goals per match.
a
b
Show that Marcus, Zara and Arun could all be correct.
Which average do you think best represents the data in the
tables. Explain why.
Who do you agree with, Marcus, Zara or Arun?
Summary checklist
I can use mode, median, mean and range to compare sets of data.
I can decide when it is more appropriate to use mean, mode or median to compare
sets of data
385
16 Interpreting and discussing results
Check your progress
1
Here are the ages, to the nearest year, of the members of a bowling club.
15
29
12
23
16
17
20
14
24
27
28
22
23
25
14
24
10
20
22
29
a
2
Record this information in a frequency table.
Use the classes 10 ≤ a < 15, 15 ≤ a < 20, 20 ≤ a < 25 and 25 ≤ a < 30.
b Draw a frequency diagram to show the data.
c
How many of the members are at least 20 years old?
Explain how you worked out your answer.
The students in class 8T took a test. These are the results, marked out of 50.
28
45
a
b
c
d
e
12
34
50
43
28
8
24
28
39
36
46
37
27
18
50
29
49
38
28
9
36
Draw an ordered stem-and-leaf diagram to show the scores out of 50.
What percentage of the students had a score greater than 35?
What fraction of the students had a score less than 25?
Any student scoring less than 40% must re-sit the test.
How many students do not have to re-sit the test?
Read what Zara says.
I had the average
score, because
my mark was 28 out
of 50.
i
ii
386
18
39
Show that Zara could be correct.
Show that Zara could be wrong.
16 Interpreting and discussing results
3
The pie charts show the favourite animals of the students in two schools.
Haywood School
Ryefield School
2%
15%
10%
35%
18%
5%
Elephant
4
30%
25%
15%
25%
Lion
8%
Giraffe
Zebra
12%
Warthog
Other
There are 1200 students in Haywood School.
There are 700 students in Ryefield School.
Which school had the larger number of students choosing lion as their favourite animal?
Show your working.
The test marks, out of 20, of two groups of students are shown.
Art:
9, 16, 8, 10, 18, 16, 12, 8, 16, 9
Music: 12, 20, 6, 18, 6, 16, 11, 19
a
Copy and complete this table.
Mean
Median
Mode
Range
Art
Music
b
c
In which group, Art or Music, do you think the students did better on average?
Justify your answer.
In which group, Art or Music, do you think the students had more
consistent scores? Justify your answer.
387
Glossary and index
absolute change when a numerical value increases or decreases, the absolute
change is the positive difference between the old and new
values
accurate
exact or correct to a given level of accuracy such as 1 d.p.
or 3 s.f.
adapt
change to make suitable for a new situation
advantages
good points
algebraically
using algebra
alternate
two equal angles between two parallel lines on opposite sides
angles
of a transversal
anticlockwise turning in the opposite direction to the hands of a clock
approximate
a number rounded to a suitable degree of accuracy
value
arc
part of the circumference of a circle
bearing
an angle measured clockwise from north
bisector
a line that divides a line segment or an angle into two equal
parts
brackets
used to enclose items that are to be seen as a single
expression
categorical
data that can be divided into specific groups such as eye
data
colour and favourite sport
centre of
the fixed point of an enlargement
enlargement
(COE)
the point that remains still when a shape is turned
centre of
rotation
changing the
rearranging a formula or equation to get a different letter on
subject
its own
circumference the perimeter of a circle
class interval
the width of a group in a grouped frequency table/diagram
classes
groups used for recording data in a grouped frequency table/
diagram
clockwise
turning in the same direction as the hands of a clock
closed interval a range of numbers that does include its endpoints
coefficient
a number in front of a variable in an algebraic expression or
equation; the coefficient multiplies the variable
388
227
183
262
146
217
107
317
181
117
292
118
17
370
324
317
37
181
355
355
317
57
30
column vector
two numbers, placed vertically, that describe a translation
of a point or shape; the top number describes the horizontal
movement, while the bottom number describes the vertical
movement
common factor a number that is a factor of two or more different numbers
compare
look at and see what is the same or different
comparison
to look at how things are the same or different
complementary two events are complementary if one of them must happen
event
and they cannot both happen
congruent
identical in shape and size
conjecture
a possible value based on what you know
constant
a number on its own (with no variable)
construct
use given information to write an equation
(algebra)
construct
use given information to draw shapes, angles or lines in
(geometry)
diagrams using compasses and a ruler
continuous data data that can take any value within a given range
corresponding two equal angles formed by parallel lines and a transversal
angles
cross-section
the 2D face formed by slicing through a solid shape
cube root
the cube root of a number produces the given number when
it is cubed; for example, the cube root of 125 is 5, because 53
is 125
decimal number a number in the counting system based on 10; a number with
at least one digit after the decimal point; the part before the
decimal point is a whole number, while the part after the
decimal point is a decimal fraction
decimal places the number of digits after the decimal point
(d.p.)
degree of
the level of accuracy in any rounding
accuracy
derive
construct a formula
diameter
a straight line between two points on the circumference of a
circle (or surface of a sphere) that passes through the centre
of the circle (or sphere)
disadvantages bad points
discrete data
data that can only take exact values
enlargement
a transformation that increases or decreases the size of a
shape to produce a mathematically similar shape
equation of a when a function is used to draw a line, the function is called
line
the equation of the line
304
258
82
268
276
304
19
30
51
115
355
107
341
20
68
73
73
37
181
146
355
324
246
389
equivalent
equivalent
calculation
equivalent
decimal
estimation
expand
brackets
experimental
probability
expression
exterior angle
of a triangle
factor
factor tree
factorisations
factorise
formula
equal in value
a calculation which uses a different method but gives exactly
the same answer
a decimal number that has the same value as a fraction
approximation of an answer, based on a calculation with
rounded numbers
multiply all parts of an expression inside the brackets by the
term directly in front of the bracket
a probability resulting from an experiment involving a large
number of trials
a collection of symbols representing numbers and
mathematical operations, but not including an equals sign (=)
the angle formed by extending one of the sides of a triangle
a whole number that divides exactly into another whole number
a method of finding prime factors
expressions that have been factorised
write an algebraic expression as a product of factors
an equation that shows the relationship between two or more
quantities
plural of formula
any diagram that shows frequencies
formulae
frequency
diagram
front view, front a 2D drawing of a solid shape seen from the front
elevation
function
a relationship between two sets of numbers
function
a method of showing a function
machine
generalise
use a set of results to make a general rule
generate
make or form
geometric
an adjective meaning that something relates to geometry
gradient
the slope of a line; a positive gradient slopes up from left to
right; a negative gradient slopes down from left to right
grouped data where data values are recorded in groups, rather than as
individual values
HCF
an abbreviation for highest common factor
hierarchy
a system in which things are arranged in order of their
importance
highest common the largest number, and/or letters, that is a factor of two or
factor (HCF) more other numbers or expressions
390
33
70
138
88
43
282
30
111
98
11
48
47
37
37
355
186
216
216
25
200
106
246
355
11
180
11
hypotenuse
the longest side of a right-angled triangle, opposite the right
angle
image
a shape after a transformation
improper
a fraction in which the numerator is larger than the
fraction
denominator
improve
make better/easier
index
the number of times a number is multiplied; 73 = 7 × 7 × 7 =
243, so the index is 3; the plural of index is indices
inequality
a relationship between two expressions that are not equal
input
a number that goes into a function machine
integer
a whole number that can be positive or negative or zero, with
no fractional parts
inverse
the operation that has the opposite effect to another operation;
for example, the inverse of ‘multiply by 5’ is ‘divide by 5’
inverse function the equation that reverses a function
inverse
the operation that reverses the effect of another operation
operation
investigate
explore an idea or method
justify
give a reason to support your decision
kilometre
measure of distance, approximately 58 of a mile
LCM
an abbreviation for lowest common multiple
line segment
a part of a straight line between two points
linear
an expression with at least one variable, where the highest
expression
power of any variable is 1
linear function a function of the form y = mx + c where m and c are numbers
lines of
a line of symmetry divides a shape into two parts, where each
symmetry
part is the mirror image of the other
lowest common the smallest positive integer that is a multiple of two or more
multiple (LCM) positive integers
map
the process of changing an input number to an output
number using a function
mapping
a type of diagram that represents a function
diagram
mean
an average of a set of values, found by adding all the values
and dividing the total by the number of values in the set
median
the middle number when a set of numbers is put in order
mentally
a mental method, worked out in your head
midpoint
the centre point of a line segment
118
304
142
146
11
57
216
11
19
220
38
15
268
332
11
298
31
246
174
11
217
217
156
366
88
298
mile
measure of distance, approximately 8 of a kilometre
332
mirror line
a line dividing a diagram into two parts, each part being a
mirror image of the other
310
5
391
mixed number
a number which is the sum of a whole number and a proper
fraction
mode
the most common number in a set of numbers
multiplier
a number by which you multiply a given value
natural numbers 0, 1, 2, 3, 4, 5, ...
net
a flat diagram that can be folded to form a 3D shape
nth term
the general term of a sequence where n represents the
position number of the term
object
a shape before a transformation
one-step
a function that has only one mathematical operation
function
order
arrange in a particular pattern, for example, from the
smallest to the biggest
output
a number that comes out of a function machine
partitioning
a method of multiplying two numbers where the units,
tens, hundreds, etc. in one of the numbers are multiplied
separately by the other number
percentage
the decrease in the value of a number, written as a percentage
decrease
of the original value
percentage
the increase in the value of a number, written as a percentage
increase
of the original value
pi (π)
the ratio of the circumference of a circle to the diameter of
the circle
pie chart
a circle divided into sectors, where each sector represents its
share of the whole
place value
the value of a digit in a number based on its position in
relation to the decimal point
plot
put points on a graph in order to draw a line
population
all the possible people from whom you choose a sample
position number the position of a term in a sequence of numbers
position-toa rule that allows you to calculate any term in a sequence,
term rule
given its position number
power
2 × 2 × 2 = 23 is called ‘2 cubed’ or ‘2 to the power 3’
prime factor
a factor that is a prime number
prism
a solid 3D shape that has the same cross-section along its length
profit
a gain in money; you make a profit if you sell something for
more than you paid for it
proportion
amount compared to the whole thing, usually written as a
fraction, decimal or percentage
proportions
fractions (or percentages) of the whole
quadrilateral
a flat shape with four straight sides
392
142
366
230
20
346
210
304
216
82
216
154
225
225
181
370
89
240
132
205
205
24
11
341
267
268
370
174
radius
a straight line from the circumference of a circle (or surface
of a sphere) to the centre of a circle (or sphere)
the difference between the largest and smallest numbers
in a set
an amount compared to another amount, using the symbol :
numbers that can be written as fractions; this includes all
integers
the multiplier of a number that gives 1 as the result; for
example, the reciprocal of 2 is 1 because 1 × 2 = 1
182
recurring
in a recurring decimal, a digit or group of digits is repeated
decimal
forever
reflect
draw the image of a shape as seen in a mirror
regular polygon a 2D shape with three or more straight sides, where all the sides
are equal in length and all internal angles are the same size
reverse
a method of checking your answer by working backwards
calculation
through the calculation using inverse operations
rotational
a shape has rotational symmetry if, in one full turn, it fits
symmetry
exactly onto its original position at least twice
round
make an approximation of a number, to a given degree of
accuracy
scale factor
the ratio by which a length is increased (or decreased)
sector
a portion of a circle made from two radii and the part of the
circumference that joins them
semicircle
half a circle
sequence of
patterns made from shapes; the number of shapes in each
patterns
pattern forms a sequence of numbers
shades
a colour which is lighter or darker than a similar colour
share
to split up into parts
short division a method of division where remainders are simply placed in
front of the following digit
side view, side a 2D drawing of a solid shape seen from the side
elevation
significant
the first significant figure is the first non-zero digit in a
figures (s.f.)
number; for example, 2 is the first significant figure in both
2146 and 0.000 024
simplified
written in its simplest form
simplify
(a ratio) divide all parts of the ratio by a common factor
solve
find the value (or values) of the unknown letter (or letters) in
an equation
square root
the square root of a number multiplied by itself gives that
number; for example, the square root of 36 is 6 or −6
139
range
ratio
rational
numbers
reciprocal
2
2
366
258
23
160
310
174
96
174
73
324
370
185
205
268
263
93
186
73
158
258
37
20
393
stem-and-leaf
diagram
strategies
subject of a
formula
substitute
a way of displaying data, similar to a horizontal bar graph
but with sets of digits, in order of size, forming the bars
methods
the letter on its own on one side (usually the left) of the
formula
replace part of an expression, usually a letter, with another
value, usually a number
the total area of the faces of a 3D shape
a single number or variable, or numbers and variables
multiplied together
a decimal number that does not go on forever
366
a rule to find a term of a sequence, given the previous term
86
probability calculated on the basis of equally likely
outcomes. It is equal to the number of favourable outcomes
divided by the total number of possible outcomes.
a series of points, plotted at regular time intervals, joined by
straight lines
a 2D drawing of a solid shape seen from above
282
transform a shape by moving each part of the shape the same
distance in the same direction
a line that crosses two or more parallel lines
plural of trapezium
a pattern in a set of data
a function that has two mathematical operations
304
a fraction that has a numerator of 1
a letter in an equation for which the value is yet to be found
when you turn a fraction upside down you swap the
numerator and the denominator, so 2 turned upside
3
down is 3
139
30
160
variable
a symbol, usually a letter, that can represent different values
vertically
two equal angles formed where two straight lines cross
opposite angles
written method worked out on paper
x-intercept
the value of x where the graph of a line crosses the x-axis
y-intercept
the value of y where the graph of a line crosses the y-axis
30
106
surface area
term
terminating
decimal
term-to-term
rule
theoretical
probability
time series
graph
top view, plan
view
translate
transversal
trapezia
trend
two-step
function
unit fraction
unknown
upside down
2
394
165
37
37
346
30
138
361
186
107
336
361
217
88
250
246