The GCSE Thing!

Essential support for GCSE (9-1) mathematics

GO!ReaSdyE?T?
CROSSOVERThe Foundation & Higher
Rev(siusiitoabnlegfuoird: ebo/thWtieorsr3kboaoll kex/amStbouadryds guide

3)

Written by the teams at

We have been inundated with
requests for sample copies but, in
the interests of doing our bit towards
saving the planet, we have decided to

make this a digital sample.
Here you find the first section for
our ... thing, followed by 10 sample
double pages. These cover a range of
mathematics, and we hope that they

give a flavour of the full book.

Fize, Matt, Mel, Seager and Steve

Published by JustaRoo Limited, WR11 8SG, www.justaroo.co.uk
Copyright © JustaRoo Limited 2020
Copyright notice

All rights reserved. No part of this publication may be reproduced in any form or by any means (including
photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to
some other use of this publication) without the written permission of the copyright owner. Applications for the

copyright owner’s permission should be addressed to the publisher.
The rights of Faisal Khawaja, Matthew Nixon, Melanie Muldowney, Christian Seager and Steven Lomax to be
identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and

Patents Act 1988.
Every effort has been made to ensure that this publication is free of errors. We are, however, only human, and so
may have made a mistake or two (hopefully not!) and would ask that if you do spot an error you contact us at

[email protected] so we can make sure it is corrected in future versions.

HELLO Page 1 done!

You’ve probably picked this thing up, read the blurb on the back
cover several times, put it back down again etc. etc. ... so now
it’s time to get started!

But first, a few housekeeping rules about how to use this guide:
SAM
1 Make sure you have the correct equipment. You will need the usual maths stuff and we
also recommend you use a notebook/paper alongside this workbook to make notes, and
for more working out if needed.

2 Remind yourself of the key skills on the pages titled ‘Key Skills’ … clever title for
these pages huh? Make sure you identify any that you need to revise. Don’t just
ignore them ... do something about it!

3 Look through the command word glossary on the ‘Command Word’ page … we’re really
good at this titling stuff! These are the key words that start a question and give you
an instruction.
SAMPLE
4 Work through the ‘Ready’ section of a double page, making notes if needed. Then
have a go at the ‘Set’ and ‘Go’ questions (no cheating with the answers at the back ...
they’re for when you’ve finished).

5 Return to the same double page a couple of days later and check you still remember

the content. Then and only then are you allowed to tick the checklist and rip off the

corner of the page*. * if it’s not your workbook … check with the owner!

Also included ... a set of cut - out
flashcards (you can make more) to help you

remember the stuff you need to know

On each double-page there are three sections for you to work through:

Ready? Read the key information and work carefully through the examples.MPLE PMOALTHICS E
Sometimes highlighting is used to give extra guidance, and look out
for the maths police who point out common misconceptions!

SET? Dive right in and have a go at these questions. They are closely linked
to the worked examples (no curveballs yet). Check your answers.

GO! Here we go ... some more questions, but this time exam-style (curveballs
included). Again, don’t forget to check your answers.

You CAN do this ... now let’s get started!

You’re almost ready to get started … but first you must decide where you will work:

In bed under a duvet
On a sofa in front of the TV
At a well-lit desk with a proper chair
SAM
Turn to page ... Ready? Set? Go! Turn to page ... Ready? Set? Go!

Two-Way Tables 10 Interest and Growth 50

Frequency Trees 12 Depreciation and Decay 52

Rounding 14 Reverse Percentages 54

Error Intervals 16 Index Laws 56

Estimation 18 Expand and Simplify 58

Use of Calculator 20 Sequences 60
SAMPLE
Product of Prime Factors 22 Solving Equations 62

HCF & LCM 24 Forming and Solving 64

Real Life Multiples 26 Inequalities 66

Fractions 1 28 Factorising 1 68

Fractions 2 30 Factorising 2 70

Ratio 1 32 Changing the Subject 72

Ratio 2 34 Standard Form 1 74

Direct Proportion 36 Standard Form 2 76

Proportion: Best Value 38 Alternate/Corresponding Angles 78

Proportion: Recipes 40 Interior and Exterior Angles 80

Proportion: Exchange Rates 42 Plans and Elevations 82

Inverse Proportion 44 Constructions 84

Percentages 1 46 Bearings 86

Percentages 2 48 Pythagoras' Theorem 1 88

MPLE Mlaevkiasreibntlienhge...wheceh.a.n.ecachynkosdulie'svtcteictaiknobanto.ilhveeed Ace!

2 sCtilhdl enacaykisle.ld.a. tiaserfit?ew offNttohhwien..g.cytoernaerr

And now for a couple of non-sticky notes:
1) Try to do as much as possible without a calculator ... unless

it's completely obvious you need to use one
2) Diagrams are only drawn to scale when it says so

Checklist & Contents
(not just an index page!)

Aylow.u.a.ropynaaswngcseahwr1ed7ecsr0ks Krentomhwyeeomtufhlbnaeeesrfehodc(rcamturoutdlasoue)t SAMctSohloeisurrimosuatsnilcyyk.d.y.ifwnfhoetoreehsna?ts!

NIXON-3793 *Byoeulriesveelfin
417
139×3 yeoqucMuhoiparamrvkeeeecnattsllutrhee

SHIFT ALPHA S<=>D

. ×( n ) n x2 x3 xn
4 (–) sin cos tan
7 9 DEL AC
8 ÷
56
1 23 +–
Ans =
×10x

0 SAMPLE

10 11 12 13 14 15
cm
6just7aroo.8co.uk9
1 2 3 4 5

0

MULDOWNEY

Turn to page ... Ready? Set? Go! Turn to page ... Ready? Set? Go!

Pythagoras' Theorem 2 90 Straight Line Graphs 2 130

Trigonometry 1 92 Straight Line Graphs 3 132

Trigonometry 2 94 Non-Linear Graphs 134

Trigonometry 3 96 Speed, Distance, Time 136

Pythagoras with Trigonometry 98 Compound Measures 138

Circles 1 100 Real Life Graphs 140

Circles 2 102 Congruence 142

Arcs and Sectors 104 Similar Shapes 144

Surface Area and Volume 1 106 Reflections 146

Surface Area and Volume 2 108 Rotations 148

Sampling 110 Translations 150

Averages 112 Enlargements 152

Averages from a Table 114 Combined Transformations 154

Averages from Grouped Data 116 Vectors 156
MPLE
Frequency Diagrams 118 Probability 158

Scatter Graphs 120 Probability Tree Diagrams 1 160

Time Series 122 Probability Tree Diagrams 2 162

Pie Charts 124 Venn Diagrams 164

Coordinates 126 Simultaneous Equations 1 166

Straight Line Graphs 1 128 Simultaneous Equations 2 168

It's no good just owning a revision workbook ...

use it!

KeyMPLE Skills-9 -8 -7 -6 -5 -4 -3 -2 -1 0 GETTING LARGER Number
SAMPLE line
SAMRELLAMS GNITTEG 1 2345 67 8 9

Need to revise Nailed It!

Factors: Divide into Multiples: The result Prime numbers: 1=1 100%
50%
another number without a of multiplying a number by A number with exactly 0.5 = 1 33.3...%
2 25%
remainder, e.g. factors of an integer, e.g. multiples two factors 20%
0.33... = 1 10%
12 are 1, 2, 3, 4, 6 and 12 of 6 are 6, 12, 18, 24, ... 2, 3, 5, 7, 11, 13, 17, 19, etc. 3
conUvFesDresfPiuolns
0.25 = 1 Nailed It!
4

Integer: A positive or prNopuemrbteiers Even: Numbers that can 0.2 = 1
be divided exactly by 2 5
negative whole number or e.g. multiples of 2
zero, e.g. -3, 0 or 5 are 2, 4, 6, 8, etc. 0.1 = 1
10

Square number: Cube number: 1% = 1010 = 0.01
3% = 1030 = 0.03
The result of multiplying a The result of multiplying 30% = 130 = 0.3
whole number by itself Need to revise
a whole number by itself,
e.g. 3 × 3 = 9 Nailed It! Need to revise
then by itself again
e.g. 4 × 4 × 4 = 64

Place Millions Thousands Ones . Fractions
value . Tenths Hundredths Thousandths
Ten millions Millions Hundred Ten Thousands Hundreds Tens Ones
thousands thousands

The number 1,023,456 is one million, twenty three thousand, four Need to revise Nailed It!
hundred and fifty six

TTaimbeless 1×3=3 1×4=4 1×6=6 1×7=7 1 × 8 = 8 1 × 9 = 9 1 × 11 = 11 1 × 12 = 12
2×3=6 2×4=8 2 × 6 = 12 2 × 7 = 14 2 × 8 = 16 2 × 9 = 18 2 × 11 = 22 2 × 12 = 24
Know your 2×, 3×3=9 3 × 4 = 12 3 × 6 = 18 3 × 7 = 21 3 × 8 = 24 3 × 9 = 27 3 × 11 = 33 3 × 12 = 36
5× and 10× facts 4 × 3 = 12 4 × 4 = 16 4 × 6 = 24 4 × 7 = 28 4 × 8 = 32 4 × 9 = 36 4 × 11 = 44 4 × 12 =48
5 × 3 = 15 5 × 4 = 20 5 × 6 = 30 5 × 7 = 35 5 × 8 = 40 5 × 9 = 45 5 × 11 = 55 5 × 12 = 60
PLUS ... 6 × 3 = 18 6 × 4 = 24 6 × 6 = 36 6 × 7 = 42 6 × 8 = 48 6 × 9 = 54 6 × 11 = 66 6 × 12 = 72
7 × 3 = 21 7 × 4 = 28 7 × 6 = 42 7 × 7 = 49 7 × 8 = 56 7 × 9 = 63 7 × 11 = 77 7 × 12 = 84
8 × 3 = 24 8 × 4 = 32 8 × 6 = 48 8 × 7 = 56 8 × 8 = 64 8 × 9 = 72 8 × 11 = 88 8 × 12 = 96
9 × 3 = 27 9 × 4 = 36 9 × 6 = 54 9 × 7 = 63 9 × 8 = 72 9 × 9 = 81 9 × 11 = 99 9 × 12 = 108
10 × 3 = 30 10 × 4 = 40 10 × 6 = 60 10 × 7 = 70 10 × 8 = 80 10 × 9 = 90 10 × 11 = 110 10 × 12 = 120
11 × 3 = 33 11 × 4 = 44 11 × 6 = 66 11 × 7 = 77 11 × 8 = 88 11 × 9 = 99 11 × 11 = 121 11 × 12 = 132
12 × 3 = 36 12 × 4 = 48 12 × 6 = 72 12 × 7 = 84 12 × 8 = 96 12 × 9 = 108 12 × 11 = 132 12 × 12 = 144

Need to revise Nailed It!

Time Need to revise Nailed It! Metric
Units
Need to revise 1 year = 365 days 10 mm = 1 cm 1000 ml = 1 litre
Nailed It! 1 week = 7 days 100 cm = 1 m 1000 cm3 = 1 litre 1000 g = 1 kg
1 day = 24 hours 1000 m = 1 km
1 hour = 60 minutes
Twenty five past one A leap year 1 minute = 60 seconds Can you
has 366 days use?
or 1:25 a.m. or 01:25
or 1:25 p.m. or 13:25

January - 31 days July - 31 days A ruler A pair of compasses
February - 28 or 29 days August - 31 days A protractor
September - 30 days 10 11 12 13 14 15
March - 31 days October - 31 days cm
April - 30 days November - 30 days 6just7aroo.8co.uk9
May - 31 days December - 31 days 1 2 3 4 5
June - 30 days
0

SEAGER

Need to revise

Nailed It!

Addition 1 4 6 2 7 4 Subtraction 1 341567 14 8 Key Skills

+ 27653 – 26374
73927 1 9374

11 Calculations 2 20 045 - 9 999 = 10 046
(choose when to use a
2 20 045 + 9 999 = 30 044 written/mental method)

+ 10 000 Need to revise -10 000
Nailed It!
20 045 30 044 -130 045 10 045 10 046 20 045

+1
SAM
Multiplication 2 4 2 7 Division 1 5724 ÷ 4 1 43 1
4 517 12 4
× 38
1 94 1 6 523 2 4374 ÷ 12 0364.5
728 1 0 2 1 12 443 77 54 .60
92226

11

Need to revise Nailed It! X TTh Th 100s 10s 1s 111 ÷ TTh Th 100s 10s 1s 111
3 10 100 1000 25 2 10 100 1000
X10a0ndan÷db1y00100, 1 6 5
13 0 6 2 2
1 36 0 0 52
move digits13on.6e p×la1c0e move digits2o5ne2p÷la1c0e 2 252
move digits13t.w6o×pla10ce0s move digits2t5w2o ÷pla10ce0s
move digits2t5hr2ee÷ p10la0ce0s
move digits13th.6re×e p10la0ce0s 1 3 6 0

The Language of Algebra SAMPLE y
5
n + 2 means ‘n’ add 2 4 (4,3) Coordinates
3
n - 2 means ‘n’ subtract 2 (-4,2) 2
1
2n means 2 times ‘n’
n -5 -4 -3 -2 -1 O 1 2 3 4 5x
means ‘n’ divided by 2 -1 Need to revise
2 -2 (2,-4)
n2 means ‘n’ squared -3
-4 (x,y) Nailed It!
2 - n means 2 take away ‘n’ -5 (Left/Right, Up/Down)

equation

expression Algebra Eye Colour Tally Frequency Frequency
term term term
brown llll l 6 9

2x - 3 = 7 blue llll lll 8 8

green lll 3 7

grey llll 4 6

coefficient variable constant hazel llll 5 5

Constant: Within algebra, a number on its own is Tally chart: a way of 4

called a constant, e.g 3 is a constant in 2x - 3 collecting data 3

Variable: A quantity that varies in value and is Traffic Survey 2
usually represented by a letter ... often x or y Car
1
Coefficient: A number in front of a variable Motorbike
(multiplying it), e.g. 2 is the coefficient of 2x Van 0 Snail Racoon Snake Bear Ant Dog
MPLE Bus
Term: A term can be a constant, a variable, or either Pet
of these things multiplied together, e.g. 2, x, 2x Key = 2 people
Bar chart: a way of representing

data (can be horizontal or vertical)

Expression: A mathematical statement made by adding Pictogram: a way of DCiahaganrrdatms s

and/or subtracting terms, e.g. 2x - 3 representing data using symbols

Equation: A mathematical statement containing an 1 458 Key:
2 1 3699 2 1 = 21 kg
equals symbol, e.g. 2x - 3 = 7 3 057
44
Formula: An equation showing the relationship

between different quantities, e.g. C = πd

Identity: An equation that is always true, Stem and Leaf diagram: a way of Need to revise
e.g. 2x + 3x = 5x ... which should really be written as: Nailed It!
representing and ordering data. Each data value is
2x + 3x ≡ 5x split into a 'stem' and a 'leaf'. (MUST include a key)

Need to revise Nailed It!

Key SkillsNeed to revise 2D Isosceles: two equal sides Types of
Nailed It! Shapes Triangle
SAM (also has two equal angles)
Polygon
Equilateral: three equal sides
A shape with
straight sides (also has three equal angles ... each 60˚)

Name No. of sides Regular Scalene: no equal sides * any triangle with an
angle of 90˚ is called a
Triangle 3 All sides equal and all (and no equal angles)
Quadrilateral 4 angles equal right-angled triangle
5
Pentagon 6 Cube 3D Cuboid
Hexagon 7 Shapes
Heptagon 8 6 square faces 6 faces
Octagon 9 12 edges 12 edges
Nonagon 10 8 vertices 8 vertices
Decagon

Parallelogram Rectangle Triangular prism Square-based pyramid

2 pairs of parallel sides 4 right angles 5 faces 5 faces
9 edges 8 edges
Trapezium Rhombus Square 6 vertices 5 vertices
SAMPLE
Only 1 pair of parallel sides 4 equal sides

Kite Types of Cylinder Sphere Cone
Quadrilateral
2 pairs of adjacent equal sides

Need to revise Nailed It! Need to revise Nailed It!

= Equal to ≡ Identical to < Less than
≠ Not equal to
Symbols ≈ Approximately equal to ≥ Greater than OR equal to ≤ Less than OR equal to

> Greater than Need to revise Nailed It!

Right angle Acute angle Obtuse angle Reflex angle LAinengFlaec&ts

exactly 90˚ less than 90˚ more than 90˚ greater than 180˚
and less than 180˚

Angles meeting at aVerticalMPLE Angles meeting at a Vertically opposite Angles in a triangle
point add up to 360˚
meet at 90˚point on a straight lineangles are equaladd up to 180˚
add up to 180˚
Angles in a quadrilateral

add up to 360˚

Horizontal Parallel Perpendicular Need to revise
Parallel Nailed It!

Prime number Volume of a cuboid

Square number Volume of a prism

Cube number Volume of a cylinderSAM

Percentage change Pythagoras’ theorem

Area of a rectangle sin θ

Area of a triangle cos θ
SAMPLE
Area of a parallelogram tan θ

Area of a trapezium Exact values of sin θ

Area of a circle Exact values of cos θ

Circumference of a circle Exact values of tan θ

MPLE Speed Metric conversions: length

Density Metric conversions: mass

Pressure Metric conversions: capacity

height Volume = length × width × height has exactly two factors
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
length width V = lwh

Volume = the result of multiplying a whole number by itself
area of cross-section × length
1, 4, 9, 16, 25, 36, 49, ...
length
1 × 1 2 × 2 3 × 3 4 × 4 5 × 5 6 × 6 7 × 7 ...

the result of multiplying a whole number by itself, then by itself again

1, 8, 27, 64, 125, ...

1 × 1 × 1 2 × 2 × 2 3 × 3 × 3 4 × 4 × 4 5 × 5 × 5 ...
radius Volume = area of circular SAM
cross-section × height
height
V = πr2h

ac a2 + b2 = c2 actual change × 100%
original amount
b
sin θ = opposite length Area = length × width
opp hyp hypotenuse width
θ A = lw

adj adjacent Area = base × height ÷ 2
hypotenuse
opp hyp SAMPLEcos θ= height A= bh
θ base 2

adj

opp hyp tan θ = opposite height Area = base × height
θ adjacent
base A = bh
adj a
30˚ 45˚ 60˚ 90˚ height (h) Area = half the sum of the
q 0˚ parallel sides × height
1 23 b a+b
sin q 0 2 22 1 A = 2 ×h
radius
q 0˚ 30˚ 45˚ 60˚ 90˚ Area = π × radius × radius

cos q 1 3 21 0 A = πr2
222

q 0˚ 30˚ 45˚ 60˚ 90˚ diameter Circumference = π × diameter
tan q 0 C = πd
3 1 3 Doesn’t
3 exist

MPLE 1 kilometre = 1000 metres D Speed = distance
1 metre = 100 centimetres ST time
1 centimetre = 10 millimetres

1 tonne = 1000 kilograms M Density = mass
1 kilogram = 1000 grams DV volume
1 gram = 1000 milligrams

1 litre = 1000 millilitres F Pressure = force
1 litre = 100 centilitres PA area
1 centilitre = 10 millilitres

Remove brackets from MATHS Find the VALUE Use a rule to arrange
an algebraic expression
POLICE EVALUATE ORDER
EXPAND
In maths, Expand does Evaluate 43: Order from smallest to largest
5(2x + 3) = 10x + 15 not mean 4 × 4 × 4 = 64 5 , -2 , 0.24
Find the answer to Make a number simpler -2 , 0.24 , 5
Expand but keep its value close
a problem to what it was Give the answer
Make an algebraic without showing working
SOLVE expression simpler by ROUND
collecting like terms: out
Find the solution to an 3x + 4 + 2x = 5x + 4 74.26 rounded to ...
equation such as 2 significant figures is 74 WRITE
2x + 13 = 35 SIMPLIFY SAM
Give reasons to 1 decimal place is 74.3
Make a fraction simpler by
support the decision or Give a sensible
the answer cancelling common factors: approximate answer Write 3 as a decimal:
4
EXPLAIN 12 = 3 using rounding 0.75
16 4
ESTIMATE
Put brackets into an Perform one or more
algebraic expression steps to get an answer

FACTORISE CALCULATE
WORKor OUT

x2 + 6x + 8 = (x + 2)(x + 4) Calculate 15% of £40: MATHS
15y + 12 = 3(5y + 4) 10% –>£4 so 5% –>£2
POLICE

15% –>£6 Calculate doesn’t

wCorod‘mdsmo tashnoamdtewttoehrilnldygs’ouarteoSAMPLE drawCriengatuesainngatchceucraotrerecmt eanaycoaulchualvaetotro use

maths equipment

Estimate 21.7 × 6.3: CONSTRUCT
20 × 6 = 120

Create a neat drawing Command 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
that shows key features cm
LOMAX justaroo.co.uk

DRAW Words PROTRACTOR

Draw a plan of the 3D solid: Find the length or
size of an angle using ...

14 1c5m MEASURE

Create a rough Attach the correct 13
drawing that shows key name
12
features 11

10

96 jus7taroo8.co.uk

5 ... a ruler or
protractor
4
Mark a point on
PMOALTHICS E 3 a graph using a cross

No need SKETCH LABEL 2 0KH 1AWAJA
to use a ruler
Sketch a cylinder: Use correct maths Diameter
or a pair of vocabulary to explain
compasses
key features
MPLE Use reasons to Work out an answer PLOT
explain thinking, such as DESCRIBE to a problem
Plot the point (1,2):
GIVE Rotation 180˚centre (0,0) FIND y
JUSTor IFY y x
B Find the mode of
'the angles on a straight x 6,3,9,5,3: Fill in missing values
line add up to 180˚’ A
Give all working to Mode = 3 in a table such as
get to the answer Display information in Change from one
a chart or graph, such as form to another y=2x+1 x -2 -1 0 1 2
SHOW y 3
REPRESENT CONVERT
COMPLETE
a scatter graph
... and on a diagram such as
y
A A
x
0.2 B
A
0.1 0.5

B

B

Ready? SET?

A two-way table is a really useful way A. C omplete the table:
of organising information that includes
two (or more) categories. Finance Sales Ops Total
SAM
e.g. 1 The two-way table shows some information Male 17
about whether students in a class are left or
right-handed. Complete the table. Female 8 25 44

Left Right Total Total 32 90
8 43
Male Look for any row B. This table shows the languages that
Female 17 or column with only
Total 80 one missing piece of a group of students study at school.

information

Left Right Total 1 Which numbers can be

Male 8 35 43 calculated? French Spanish Other Total
80 - 43 = 37 Girls 9 16
Female 17 37 and Boys 6 32
43 - 8 = 35 Total 21 100
Total 80
Left Right Total
Male 8 35 43
Female 20 17 37
Total 52 80
2 Which numbers can now SAMPLE (i) How many girls study French?
(ii) How many students study Spanish?
be calculated?

37 - 17 = 20
and

35 + 17 = 52

Left Right Total 3 Calculate the final number C. Kelly asks 100 students if they like
Male 8 35 43 biology or chemistry or physics best.
Female 20 17 37 8 + 20 = 28
Total 28 52 80 or 38 of the students are girls.
21 of these girls like biology best.
80 - 52 = 28 18 boys like physics best.

Do both ways to check 7 out of the 23 students who like

Sometimes we need to create the table. chemistry best are girls.

e.g. 2 30 students were asked how they travel W ork out the number of students
to school. All students either walk, ride a
bike, or travel by car. who like biology best.

• 13 of the students are girls
• 4 of the boys walk
• 7 girls ride a bike
• 3 of the 5 students who travel by car

are boys
How many students walk to school?
MPLE
1 Decide on row and column titles

2 Fill in the given Walk Bike Car Total
Girls 4 7 2 13
information Boys 4 10 3 17
Total 8 17 5 30
3 Calculate the

missing values

PMOALTHICS E ‘3 out of 5’ gives Highlight the key
you two values information like in
Don't forget to
8 students walk to school for the table e.g. 2
answer the question

10

GO! Two-Way Exceptional!

Tables

1. 50 students were asked to choose an activity Female Netball Tennis RoundersSAMTotal
Male 9 12 1 22
from netball, tennis or rounders. Total 0 14 14 28
9 26 15 50
15 chose rounders
28 of the students were female
9 of the females chose netball
No males chose netball
14 males chose tennis

Marcus has completed the two-way table using
this information. Do you agree with Marcus?
Explain why.

2. Students were asked to choose an activity

from origami, metalwork or pottery.
SAMPLE
There were 257 students
84 chose pottery
154 of the students were girls
79 of the girls chose origami
No boys chose origami
Equal numbers of boys and girls chose
metalwork

Work out the difference between the number
of boys who chose metalwork and the number
of boys who chose pottery.

3. Orange cordial is sold in various size bottles:

Demi (375 ml)
Standard (750 ml)
Magnum (1.5 litres)
One weekend a shop sold 123 bottles of cordial.
MPLE
84 of the bottles were sold on Sunday
19 of the bottles sold on Sunday were
1.5 litres
17 of the bottles sold on Saturday were
Demi bottles
29 of the bottles sold were Magnum bottles
36 of the bottles sold were Standard size
bottles

How many Demi size bottles were sold on
Sunday?

Ready? NIXON-3793 Use the same calculator at home
139×3 that you use in class. Different models label
some of the same functions differently;
417 e.g. xn and ^ both find powers

SHIFT

() S-D

n x2 x3 xn

n (–) sin cos tan

7 8 9 DEL AC
4 5 6×÷

1 2 3+–

.0 ×10x Ans =

Here are some of the most useful calculator functions to know. e.g. 1 Calculate 52

x2 Squares a number Ans Uses the previous answer in 5 x2 = 25
the next calculation

n Finds the positive square root (–) Inputs a negative number e.g. 2 C alculate 3 64
SAM
x3 Cubes a number Enters a fraction 3n 6 4 = 4
e.g. 3 Calculate 46
3n Finds the cube root n Enters a mixed number 34n xn 6 = 4096
(you might need to press SHIFT) e.g. 4 C alculate 4 16
34n n 1 6 = 2
xn Finds any power of a number Converts between a fraction
S<=>D and a decimal

n Finds any root of a number ( ) Two separate buttons
(you might need to press SHIFT) (one for each bracket)

When doing complex calculations, try to make your display look identical
to the calculation, (or you can work out different parts separately).

e.g. 5 Use your calculator to work out the value of 12.7 × 0.8 as a decimal. Write down all the figures on the display.
9.78 + 6.4

Use the buttons on your calculator to make a copy of the calculation:

12 . 7×0 . 8 9 . 7 8 + 6 . 4 = S<=>D

SAMPLE is the 'down' key 0.6279357231

1 Work out the 'top' part 2 Then work out the 'bottom' part
OR 1 2 . 7 × 0 . 8 = 10.16 9 . 7 8 + 6 . 4 = 16.18

3 Carry out the division You might need this S<=>D
button throughout
1 0 . 1 6 ÷ 1 6 . 1 8 = 0.6279357231

e.g. 6 Work out 8 –π 22 . Give your answer to 3 significant figures. 1 8 – 2 x2 = 4
SHIFT
SHIFT
n 8 – 2 x2 ×10x = OR
2 Ans ÷ ×10x = 1.273239545
1.128379167 = 1.13 to 3 s.f. πMost calculators don't
3 n Ans = 1.128379167 = 1.13 to 3 s.f.
have a button

W ork out these calculations. Write D. W ork out 4.65 × 0.8 SET?
15.4 - 9.76
your answer as a decimal.

A. 18.7 Give your answer to 2 decimal places.
13.8 + 4.36
MPLE
B. 18.7 + 4.36 E. U se your calculator to work out: F . Use your calculator to work out:
13.82 (2 + 5 14 )3
28.7 × 9.8
C. 18.72 15.4 + 9.76 (i) W rite down all the figures on your
13.8 + 4.36 calculator display.
(i) W rite down all the figures on your
calculator display. (ii) W rite the answer to 2 decimal
places.
(ii) W rite the answer to 3 significant
figures.

14

GO! CaUlsceuOlfatAor Incredible!

1. U se your calculator to work out:SAM

3 33.3 - 3π

2.01
a) Write down all the figures on your

calculator display.

b) Write your answer to part a) to two
significant figures.

2. Sue is using her calculator to work out:
19.73

11.2 - 6.58
Her answer is:

676.0425893

Do you agree? Explain why.
SAMPLE
3. W ork out:

3 15. 76 - 4.63 ÷ 2.5
3

a) Write down all the figures on your
calculator display.

MPLE b) Write your answer to part a) to 2 decimal
places.

4. Use your calculator to work out:

28.7 + 4sin60˚
0.4 + π

Write down all the figures on your
calculator display.

HINT: An open bracket needs a closed bracket

Ready? REMEMBER:
When dealing with percentages, sometimes
you will be given the final value and SAM The “original” is
have to work out the original amount. always 100%
It's a good idea to draw a picture (not a
SET?
masterpiece ... just a sketch).
A. A bike is reduced by 40% in a sale.
e.g. 1 The normal price of a watch is reduced by 20% in The selling price is now £1500
a sale. The sale price of the watch is £200. Work out What was the original cost of the bike?
the normal price of the watch.
B. The price of bread increases by 5%.
Before After Sale price A loaf of bread now costs £1.26
What did it cost before the increase?
100% if 20% is 80% 20%
taken off C. A packet of cereal is advertised as 20%
extra free. There are 900 grams of
Sale price: ÷8 80% = £200 ÷8 £200 cereal in the pack.
10% = £25 How much cereal was in the original pack?
x10 100% = £250 x10 Original price = £250
D. An electronics store is advertising
Sometimes you’ll be given the reduction 30% off everything.
or increase (the amount that is taken
off or added on) and have to work out A TV is advertised as “£210 off”.
either the original or the new value. What was the cost before the reduction?
SAMPLE
e.g. 2 In a sale, the normal price of a T-shirt is reduced by E. A pair of shoes cost £60 in a sale.
24%. The reduction is £3.66 They are advertised as 25% off.
Work out the sale price. What was the original price of the shoes?

Before After Reduction

100% if 24% is 76% 24%
taken off

Reduction: 24% = £3.66 £3.66
÷24 1% = £0.1525 ÷24
x76 76% = £11.59 x76 Sale price = £11.59

e.g. 3 M al’s wages are increased by 3%. After the increase, his
annual salary is £29,355

What was his original salary?

Before After

100% if 3% is 100% 3%
MPLE added on

103% is £29,355

(new salary)

New salary: 103% = £29,355 ÷ 103
÷ 103
x100 1% = £285 x100
100% = £28,500 Original salary
= £28,500

Annual means per year
(so does ‘per annum’)

??

GO! SEGEASTRNEEVCERREP* Genius!
*reverse percentages!

1. A clothes shop is advertising 20% offSAM

everything in a sale.

Gareth buys a pair of trousers for £40
He works out the original price as follows:
20% of £40 = £8 so £40 + £8 = £48
Gareth is wrong. Explain why.

2. D uring editing, the running time of a film is

reduced by 16% to 2 hours and 6 minutes.
How long was the film before editing
took place?
SAMPLE
A. The necklace originally cost less than £145.60 3. T he price of a necklace is reduced by 30%.
B. The necklace originally cost exactly £145.60
C. The necklace originally cost more than £145.60 One week later, the price is then increased
by 30%.

The necklace is now on sale for £145.60
Which of these statements is correct?
You must give reasons for your answer.

MPLE4. M att’s annual salary has increased by 4%.

The increase is £1553.60
Matt says that this increase will mean he

now earns more than £40,000 per annum.
Is Matt correct?

Ready? It is REALLY USEFUL to know:

* the symbol : is used to describe ratios
* how to simplify ratios ... 6 : 4 is the same as 3 : 2

and 15 : 6 is the same as 5 : 2

A ratio describes how the size of one quantity e.g. 3 M el and Chris share some money in the ratio 3 : 5
Chris receives £14 more than Mel.
compares to the size of another quantity.
How much money did Mel receive?
e.g. 1 HM oewl amnducChhmrisonsehyardeoe£s8e0acinh the ratio 3 : 5
person receive? Mel Chris
M3e l : Ch5r is 3 : 5
SAM
÷ 3+5=8 £14 is the amount So £14 ÷ 2 = £7 £7
80 8 = 10£ ££80 is the whole
more than Mel each part is worth

amount being shared So each part is worth £10 £7 £7 £7 £7 £7 £7 £7 £7

£10 £10 £10 £10 £10 £10 £10 £10 Mel = 3 × £7 = £21 Chris = 5 × £7 = £35

Mel = 3 × £10 = £30 Chris = 5 × £10 = £50 Mel receives £21

e.g. 2 Mel and Chris share some money in the ratio 3 : 5 e.g. 4 M el, Chris and Fize share some sweets.
If Chris receives £40, how much money did they share? Chris gets three times as many sweets as Mel.
Fize gets twice as many sweets as Chris.
Mel Chris
3 : 5 Mel's sweets : Chris' sweets : Fize's sweets = 1 : n : m
Work out the values of n and m.

Mel Chris Fize
1 : 3 × 1 : 2 × 3 = 6
£40 is the amount £40 ÷ 5 = £8TheM1elgoesSAMPLwith E

Chris receives So each part is worth £8

£8 £8 £8 £8 £8 £8 £8 £8 3 x Mel's swee ts x Chris' sweets
Mel = 3 × £8 = £24 2
So ... 1 : 3 : 6
Total amount shared = £24 + £40 = £64 n = 3 and m = 6

e.g. 5 Mel and Chris share some money in the Mel Chris Mel receives
ratio 1 : 3 1 : 3 1 Drawing a picture really
4 does help ... it's common to
What fraction of the total does Mel
receive? think the answer 1
is 31PMOALTHICS E 4
instead of

4 parts in total

A. D ivide £60 in the ratio: C. Evie and Steve share some money in SET?

(i) 1 : 4 the ratio 2 : 7
(ii) 3 : 7 Steve receives £45 more than Evie.
(iii) 4 : 11 How much money do Evie and Steve have
(iv) 3 : 17
(v) 1 : 2 : 3 in total?

MPLE D. Tom is sending some letters. F. Sweets are shared in the ratio n : 3,

3 out of every 4 letters are sent where n is a positive integer.
first class. There are 24 sweets in total.
Find 3 possible values for n.
If 12 letters are sent first class, how
B. Ezra and Isla are sharing money in many letters are sent altogether?

the ratio 2 : 5

(i) If Ezra receives £12, how much G. John, Kam and Liam share some money in
money does Isla receive? the ratio 5 : 6 : 9
In total, John and Liam receive £56.
(ii) ImfoInselayrdeocethiveeys s£h1a2r.5e 0in, how much E. Elijah and Mia share some money in the Work out the amount of money Kam
total?
ratio 1 : 4

(i) What fraction does Elijah get? receives.

(ii) What fraction does Mia get?

61

GO! Ratio 1 Good work!

1. T he ratio of the angles in a triangle is:

1 : 5 : 6
Fatima thinks the triangle is right-angled.
Do you agree? Explain your answer.
SAM

2. Harley and Lucy shared some sweets in theSAMPLE

MPLE ratio 7 : 4
Harley got 21 more sweets than Lucy.
Maya thinks Lucy got 12 sweets because:
21 ÷ 7 = 3 and 3 × 4 = 12
Do you agree? Explain your answer.

3. There are between 20 and 38 students in a

class. The ratio of girls to boys is 5 : 7
How many students could there be in the

class?

4. There are 60 people in a club.

Half of the people in the club are women.
The number of women in the club is three times
the number of men in the club.
The rest of the people are children.
The ratio of the number of children in the club
to the number of men in club is n : 1
Work out the value of n.

Ready? It is REALLY USEFUL to remember how to convert
between metric measures using the facts:
1 kg = 1000 g 1 litre = 1000 ml

We can solve proportion problems by scaling up e.g. 2 A drink is made by mixing 25 ml of cordial with 200 ml of
and/or scaling down: i.e. if we know a recipe for lemonade.
3 people, we can work out the recipe for 1 person Ross has 200 ml of cordial and 1 —21 1500 ml
(divide by 3) and then multiply by 4, 5, 6, 7, etc
to work out the recipe for 4, 5, 6, 7, etc people. litres of lemonade.

e.g. 1 T he recipe for Victoria Sponge Cake is shown here: What is the maximum number of drinks he can make?
SAM
Consider each
ingredient
Serves 6 people Cordial: separately Lemonade:
200 g butter
200 g sugar 25 ml –>1 drink 200 ml –>1 drink
200 ml –>8 drinks x8
4 eggs 100 ml –> —21 drink ÷2
200 g flour 1500 ml –>7—21 drinks x15
250 ml cream
6 raspberries He does not have enough lemonade for 8 drinks

The maximum number of drinks he can make is 7

a) How much flour is needed to make the cake for 12 people? e.g. 3 Here are the ingredients needed to make 12 cakes.

From the recipe ... Oliver has 500 g of sugar, 1000 g of butter,
Makes 12 cakes 1000 g of flour and 500 ml of milk.
6 people –> 200 g flour 50 g sugar
x2 12 people –> 400 g flour x2 400 g of flour is needed 200 g butter
200 g flour
b) How many eggs are needed to make the cake for 3 people? Work out the greatest number of cakes Oliver
can make.
From the recipe ... SAMPLE
2 eggs are needed 10 ml milk Sugar:
6 people –> 4 eggs
÷2 3 people –> 2 eggs ÷2 x10 50 g –> 12 cakes x10
500 g –> 120 cakes
c) How much cream is needed to make the cake for 15 people?

From the recipe ... Butter or Flour: Milk:

÷2 6 people –> 250 ml cream x5 200 g –> 12 cakes x5 x50 10 ml –> 12 cakes x50
x5 1000 g –> 60 cakes 500 ml –> 600 cakes
3 people –> 125 ml cream
15 people –> 625 ml cream There are other The butter and flour limit how many cakes he can make
ways of scaling
The greatest number of cakes he can make is 60
from 6 to 15

625 ml cream is needed

A. The recipe for Corned Beef Casserole B. The recipe for Amanda’s Lush Cake SET?

(serves 6 people) is: (serves 4 people) is: C. T he recipe to make 800 ml of Mango
600 g corned beef 1 onion
300 ml of gravy stock 3 carrots 2 eggs 80 g of sugar Smoothie is:
(i) W ork out the recipe to serve 18 people. 150 g of flour 60 g of margarine 1 —21 cups mango juice

MPLE(ii) Work out the recipe to serve 3 people. (i) How much flour is needed to make the 1 banana
cake for 8 people? 1 cup of mango chunks
(iii) Work out the recipe to serve 9 people. 200 ml Greek yogurt
(ii) How many eggs are needed to make the Amber has 500 ml of Greek yoghurt,
cake for 20 people? 2 bananas and plenty of mango juice
and mango chunks.
(iii) How much sugar is needed to make the What is the maximum amount of
cake for 10 people? Mango Smoothie she can make?

(iv) Kai has 150 g of margarine and plenty
of the other ingredients. How many
people can he make cake for?

(v) Write the recipe for 6 people.

61

GO! PrRoepcoirpteison: Cracking!

1. T he recipe for Grandma Margaret’s famous

Butterfly Cakes is:

2 eggs 150 g margarineSAM
100 g sugar 120 g flour

......... eggs The recipe makes 12 cakes.
......... sugar
......... margarine Paige has 10 eggs, 500 g each of sugar,
......... flour margarine and flour. She thinks that the
maximum number of cakes she can make is 40.
2. T he recipe for Lizzie’s Jam Cake (serves Is Paige correct? Explain why.

6 people) is:
1 egg 1 cup of sugar
2 cups of flour 60 g of margarine
Jam to spread
Keanu thinks he needs 1 kg of margarine to
make Jam Cake for 10 people.
Do you agree? Explain your answer.

SAMPLE 3. Carrot soup (serves 4)

400 g carrots
600 ml vegetable stock
Salt and pepper
Maisie has 1.5 kg of carrots and 3 litres
of vegetable stock.
What is the maximum number of people
she can make carrot soup for?

4. The recipe for Evan’s Fabulous Tuna Pasta

Bake (serves 4 people) is:
MPLE People Ingredient
6 oz pasta 2 oz mushrooms 6
1 tin of tuna —21 pint of tomato soup 3 ..... oz pasta
1 onion 1 oz margarine 1
2 oz cheese Crisps to sprinkle on the top 10 100% more onions
..... 1—41 pints of tomato soup
Match the people to ingredients (one has been
done for you) and find the missing values. —21 oz cheese
50% more tuna

Ready? SET?

An important skill in algebra is simplification. You A. S implify:
will often be asked to put something in its simplest
form. One way to do this is collecting like terms. (i) 3a + 4 + 5a

7x 8x x 5x 6y 2xy 2x2 7 SAM

Like terms Not like terms (ii) 5x + 2y + 2x + 3y

e.g. 1 Simplify 2 + 5a + 4a2 - 3a + 1 (iii) 6x2 + 2x - 2x2

1 Identify the like terms: 2 + 5a + 4a2 - 3a + 1 (iv) 3a2 + 2b – 2b2 + 5b – 3b2
2 Collect the like terms: 3 + 2a + 4a2 + 5a - 3a
B. E xpand:
= + 2a
(i) 3(a + 2)
Another important skill in algebra is expanding

brackets. We multiply everything inside the brackets

by the term outside the brackets.

e.g. 2 E xpand 4(2m - 3n) PMOALTHICS E

4 x 2m 4 x -3n Multiply both terms:
4(2m - 3n) is not 8m - 3n
= 8m - 12n SAMPLE (ii) 4(2x + 3)
(iii) y(y - 6)
We need to be able to combine both of the above skills. (iv) h(7h + 3) Reymxeym=beyr2:
e.g. 3 Expand and simplify 3(2a + 5) + 5(a – 2)

1 Expand the brackets: 3(2a + 5) = 6a + 15 (v) 4k(10 + k)

and

+ 5(a - 2) = + 5a - 10

2 Identify the like terms: 6a + 15 + 5a - 10 (vi) 6m(2m – 1)
3 Collect the like terms: 11a + 5
C. Expand and simplify:
When expanding double brackets like this: (x + 2)(x + 3),
there are 4 different multiplications that need to be (i) 4(x + 2) + 3(x + 3)
completed first.

e.g. 4 E xpand and simplify (y + 5)(y - 7)

1 mWurlittipelidcaowtniontsh:e y × y y × -7 5 × y 5 × -7

2 Identify the like terms: y2 - 7y + 5y - 35 (ii) 3(2a + 3) – 5(a + 4) Bbteheetecwsxaeuprebreentfsrutaslhciweotni titsohwn o
(iii) 2(4p + 3) – 6(2p – 3)
3 Collect the like terms: y2 - 2y - 35 - 7y + 5y
= - 2y
MPLE
You could also use a grid method.

e.g. 5 E xpand and simplify (y - 3)(y - 6) D. Expand and simplify:

y x y = y2 × y -3 -3 x y = -3y (i) (x + 4)(x + 5)
y y2 -3y

-6 x y = -6y -6 -6y +18 -6 x -3 = 18 (ii) (y - 3)(y + 6)

Like terms: (iii) (p – 6)(p – 10)
-6y - 3y = -9y

Collect the like terms: y2 - 9y + 18

01

GO! Expand And Legend!
Simplify

1. F ind the value of p:

3(2x + p) + 4(x - 2) = 10x + 1
SAM
2. Eve is expanding and simplifying this expression:
SAMPLE3. Cian is expanding and simplifying:
4(3x – 2) – 5(4x + 1)
She writes: (y + 2)(y - 3)
He writes:
4(3x – 2) – 5(4x + 1) y2 + 5y – 6
= 12x – 8 – 20x + 5
= 32x - 3 Do you agree? Explain your answer.
She is wrong. Explain why.

4. E xpand and simplify:

a) (2y + 8)(y + 6)
b) (x - 6)(x + 6)
c) (3a – 8)(4a - 5)
d) (2p + q)(p - 3q)
MPLE
5. ABGH is a square. C DE
5 cm GF
BCDG is a rectangle.
DEFG is a square. B

AH
(x + 2) cm

Show that the total area of the shape
is x2 + 9x + 39 cm2

Ready? In maths, lines of best fit are:
* straight lines that follow the trend (pattern) of the points

* straight lines with roughly the same number of crosses
either side of the line

Scatter graphs (also called a scatter diagrams) are used for data that contains pairs of values. They illustrate if

there is a relationship between these values - this relationship is called the correlation.

e.g. Dom collects information about the maximum temperature and Visitors
number of vistors to an outside splash pool over 10 days and
shows this on a scatter graph. 100 SAM

One of the points is an outlier.

a) Circle this outlier. It's the cross away from the trend 80
b) Give a possible reason for this outlier.

It may have been a rainy day. There are several possible reasons LINE OF BEST FIT

c) For all the other points write down the type of correlation 60
shown by the graph.

Positive correlation. The trend is bottom left to top right 50-
46-
d) Draw a line of best fit. In the direction of the crosses
e) On the 11th day the maximum temperature was 21˚C, estimate 40

the number of visitors to the pool on this day.

46 visitors. Read up from 21˚C, to the line of 20 OUTLIER
best fit, then read across
SAMPLE
f) Explain why it would not be sensible to use the scatter graph

to predict the number of visitors on a day when the maximum

temperature was 16˚C. of the given data. 015 20 21 25
Temperature (°C)
16˚C is outside the range 16˚C is away from
the group of crosses

Type s of Corre lation The closer the points are to the line of best fit, the stronger the correlation.

Positive Strong Positive Negative Strong Negative No correlation

Weight of book (g) The table shows the number of pages and SET?
260 weights, in grams, for each of 10 books.

240
MPLE
Pages 80 130 100 140 115 90 154 140 105 70

Weight (g) 168 252 180 258 230 208 256 246 210 160
220

a) Complete the scatter graph to show the information in the table. The
first 6 points have been plotted for you.

200 b) For these books, describe the relationship between the number of
pages and the weight of a book.

c) Draw a line of best fit.

180

d) Use your line of best fit to estimate

(i) The number of pages in a book of weight to 220 g.

16060 80 100 120 140 160 (ii) The weight of a book with 134 pages.

Number of pages

61

GO! Wonderful!
SGcraatpthesr
Drinks sold
40 1. T he graph shows information about the number

30 of hot drinks sold from a vending machine and the
outside temperature for a period of 10 days.
a) One of the points is an outlier. Circle this point.
b) What could this point represent? Give an example.
SAM
20

0 0 5 10 15 20
Temperature (°C)

2. Rhys is constructing a scatter graph to show theSAMPLEScience mark
40
results for 9 students in a science test and a maths
test. 30

Science 3 5 7 8 9 13 14 17 18 20
mark 15 23 30 29 27 31 25 34 32
Maths
mark

His scatter graph is incorrect.
Describe two mistakes Rhys has made.

100 5 10 15 20
Maths mark

MPLEPositive Outside Shoe 3. Match one statement from each column to create a
correct relationship.
correlation temperature size

Negative Monthly Heating
correlation pay cost

No Number Hours
correlation of pets worked

Ready? 1. Mahe sure your calculator is in degrees mode
2. Know where the sin, cos, tan buttons are on

your calculator

Trigonometry is the study of the relationship between SET?
side lengths and angles of triangles.
A. L abel each of the below triangles with
The sides of a right-angled triangle have special names. opposite, adjacent and hypotenuse.
The hypotenuse is opposite the right angle. The other
two sides are called opposite or adjacent depending on
which angle is being used:
Opposite
AdjacentHypotenuse50˚ Hypotenuse

SAM
These two triangles are the
same, but have different
angles labelled

40˚ Opposite
Adjacent

With a side length and a given angle (not the right angle) (ii)
we can use trigonometry to find lengths of missing sides
using sine (sin), cosine (cos) and tangent (tan). 81˚

sin q = H0 cos q = HA tan q = A0 (i) (iii)

sthyqemibsaontlghfleeor O SAMPLE A O 68˚ 27˚
SH CH TA

e.g. 1 F ind the value of y on this 12 cm B. Each of these triangles has a side
triangle to 1 decimal place. labelled x. Decide whether you would

use sin, cos or tan to find this length.

1 Label the sides 40˚ 71˚ 21 cm
(i)
H y
x
O 12 cm 2 Make a decision ... cross off O as
4 cm x
cos ithas h it. Aq =a nHAd H are
(ii)
in A 61˚
40˚ CH not hing wit
so we use 10 cm
Ay 38˚
= 3 Su bstitu te (q = angle = 40˚, A = y, H = 12 cm): cos 40 12 y
(iii)
4 S olve: A y = cos 40 × 12
C×H y = 9.19253... x
Cover up
what we are so y = 9.2 cm (l d.p.) C. Find the value of x in each triangle in
question B. Give your answers to one
finding decimal place.

e.g. 2 F ind the length of PQ to Q

MPLE2 decimal places. 8 cm

let’s call 1 Label the sides P 38˚ R
PQ ‘y’ Q

yA 8 cm 2 Make a decision ... cross off H as
P 38˚ H O it has
O nothing wit ht aitn. qO =a nAOd A are
R in TA so we use

8 cm, A = y): tan 38 8y
= 3 Su bstitu te (q = angle = 38˚, O =
y = tan838
4 Solve: T÷O A 8 ÷ tan 38

y = 10.2395... so PQ = 10.24 cm (2 d.p.)

10

GO! Trigonometry 1 Smokin’!

tan 4 8 = 8k 1. Mackenzie is finding the value of k inSAM
this triangle. Here is his working.
tan 48 × 8 = 8.884900119
k = 8.9 cm (1 d.p.) k 8 cm

48˚

Mackenzie is wrong. Explain why.

2. T he diagram shows two right-angled

triangles. D

B SAMPLE

12 cm 25˚
A 32˚ C

Find the length of DC to a suitable HINT: Find length BC
in triangle ABC first
degree of accuracy.
3. Find the length of k in this triangle.

k

57˚

33˚ 12 cm

Explain any decisions you make.

4. N ewtown is 16 kilometres from Oldtown
on a bearing of 075˚

How far east of Oldtown is Newtown?
Give your answer to three significant
figures.
MPLE

75˚ Newtown

Oldtown 16 km HINT: Make a right-angled triangle

Ready? It is REALLY USEFUL to remember:

* Parallel lines are lines that will
never meet (like train tracks)

* The notation for parallel lines is > or >>

ALTERNATE * The sum of the angles in a triangle is 180˚
angles are equal * T he sum of the angles at a point is 360˚
AB (look for the “Z” shape) and on a straight line is 180˚ A
BA
BB
A
* Vertically opposite anglesSAM
are equal

Angles labelled A are the C CORRESPONDING
same size D
Angles labelled B are the angles are “eFq”usahlape)
same size (look for the

CO-INTERIOR or ALLIED C
angles add up to 180˚
(look for the "C" shape) D

These two Angles labelled C are the same size
angles add up Angles labelled D are the same size

to 180˚ Don’t measure angles when you are PMOALTHICS E
asked to work them out

e.g. 1 Calculate the value of x. SAMPLE x e.g. 2 Giving reasons, calculate the value of y.
Give reasons for your answer.
39˚ * The sum of the angles y
* The sum of the angles on a straight 141˚ in a triangle = 180˚
line = 180˚ 180˚ - 141˚ = 39˚ 56˚ 1
* Alternate angles are equal
* Corresponding angles are equal so ... y = 54˚ 56˚ + 70˚
so ... x = 39˚ = 126˚
2 180˚ - 126˚ = 54˚ 54˚ 70˚
Can you spot
‘F’ shape? the Ca‘Zn y’ osuhsappoet?the

Put reasons as
bulle t points

Give reasons for all your answers C. Calculate the value of y. SET?

A. Calculate the value of x. E. Find the value of y.
x˚
42˚
MPLE 59˚
y˚
119˚ y˚ 51˚

B. Calculate the values of m and n. D. Calculate the value of x.

56˚

n˚ 155˚ x˚
m˚ 75˚
65˚ 42˚

10

GO! CAolrterArensngpalotenesdAiNnDg Quality!

1. Jane says: 132˚ SAM
f
“Angle f = 48˚ because alternate angles
are equal“

Do you agree with Jane?

Explain your answer.

2. F ind the value of x, giving reasons for each

stage of your working.

SAMPLE50˚ x˚
30˚
60˚

MPLE3. All angles are in degrees. 4.

Find the value of y. 80˚
(2x + 5)˚
3(y - 2)
2y + 5 Giving reasons, find the value of x.

HINT: Try to work out as many
angles as possible ...

write them on the diagram

Ready? You REALLY NEED to know the language of enlargements:

r s* The scale factor tells you how much large (or maller)

the new shape will be

* The centre of enlargement tells you where the rays* meet

Enlargements are one of the 4 transformations:

tcelfer, rotate , translate, enlarge e.g. 2 Enlarge the triangle by a y

We enlarge a shape using a centre of enlargement scale factor of 1 , centre 6
(1,3). 3 4
and a scale factor. 2SAM

e.g. 1 Enlarge the rectangle by y y
7

a scale factor of 2 using 6 6
5

(0,2) as the centre of 4 4 O 2 4 6 8 10 x

enlargement. 3 2 y
2

1 P lot the centre of enlargement, 1 O 2 4 6 8 10 x 6
4
and draw rays from it through the O 1 2 3 4 5 6 7 8 9 10 x original corner –> 9 right, 3 up 2
corners of the shape
2 C hoose a corner of the shape, scale factor –> x 1 the same O
3 as ÷ 3
y and count and from the
7 centre to that corner new corner –> 3 right, 1 up

6 y 2 4 6 8 10 x
7
5 6 e.g. 3 D escribe fully the single transformation
5 that maps shape A onto shape B.
4 4
3
3 *rays 2
2 1

1 O 1 2 3 4 5 6 7 8 9 10 x y
7
O 1 2 3 4 5 6 7 8 9 10 x 4 R epeat for the other cornersSAMPLE 1 D raw rays through the

3 M ultiply the movement by the and join them up to create the 6 matching corners
new shape
scale factor and plot the new 5 2 F ind the coordinates where
corner (3 right, 1 down becomes y
6 right, 2 down) 7 4A Centre they meet (this is the
6 3 4 5x centre of enlargement)
y 5 3
7 4 3 Work out the scale factor
6 3 1
5 2 using matching sides:
4 2B SF = 3 ÷ 1 = 3
3
2 1 It’s an enlargement, with a scale
1 factor of 3 and a centre of
3 enlargement of (2,4)

-2 -1 O 12

-1

O 1 2 3 4 5 6 7 8 9 10x 1 You must include the word
O 1 2 3 4 5 6 7 8 9 10x ‘enlargement’ AND state the

centre and scale factor

A. E nlarge the triangle by a scale factor SET?
of 2 using (0,0) as the centre of
enlargement.

y B. Enlarge the triangle using a scale
9 factor of 3, centre (2,3).

8 C. Enlarge the shape using a scale factor
MPLE
7 of 1 and (-3,4) as the centre of enlargement.
y 2

64 y
53
42 6
31
5

4

3

2 -4 -3 -2 -1 O 1 2 3 4x 2
1 -1
1
O 1 2 3 4 5 6 7x -2
-5 -4 -3 -2 -1 O 1 2 3 4 5x
-3 -1

-4 -2

10 -3

-4

GO! Enlargements Lush!

y 1. Describe the single transformation that maps
8
7 shape B onto shape A.
6
5 To ‘map’ means
to ‘move’
4 AB SAM

3
2
1

O 1 2 3 4 5 6 7 8 9x

2. E nlarge shape Q by a scale factor of 2.5 SAMPLE y
6
using (0,5) as the centre of enlargement. 5
Label the image R.
Q
y
8 4
7 3
6 2
5 1

4B O 1 2 3 4 5 6 7 8x

3 3. Martha is asked to enlarge rectangle A

A using a scale factor of 2 and centre (5,1).
She has labelled her enlargement B.
2 Do you agree with Martha?
1
O 1 2 3 4 5 6 7 8x Explain your answer.

MPLE4. The diagram shows an enlargement of PA Not drawn
to scale
triangle ABC. L
CB
The centre of enlargement is at P.
The enlargement is labelled LMN.

PA = 10 cm
AL = 15 cm

What is the scale factor of the enlargement?

NM

Ready? SET? GO! ... Crossover

This crossover revision guide / workbook / study guide is unique. It is
designed to support all students throughout their mathematics GCSE (9 - 1)
journey. For many students it's a revision guide; for others it's a workbook
to kickstart their GCSE course; for some it could be an ongoing study
guide throughout the course. Focused on the content common to both

Foundation and Higher tiers, each double-page spread has three sections:

Ready? Key information, vocabulary, misconceptions and carefully
crafted worked examples

SET? A selection of questions closely linked to the examples to
provide deliberate practice of the essential skills

GO! Exam-style practice with a wider selection of questions
including reasoning and problem-solving

Also included are a set of cut-out revision flashcards, a command word
glossary, and a key skills checklist. Not forgetting a full set of answers.

Meet the team

Fize, Matt, Mel, Seager and Steve

Not only have they been teachers for decades, their websites, JustMaths and
Kangaroo Maths, have helped support teachers and students since 2003. Bringing
these experiences together for a new project, they were determined to get it right
... no matter how long it took. After 3 years in the making, they are delighted to
bring you their idea of the ultimate revision guide workbook study guide ... thing.

THE ISBN SECRET: (Sum of odd-numbered digits) + (Sum of even-numbered digits × 3) is always a multiple of 10
CHECK: 9 + 3 × 7 + 8 + 3 × 1 + 8 + 3 × 3 + 8 + 3 × 0 + 6 + 3 × 5 + 0 + 3 × 0 + 3

9 + 21 + 8 + 3 + 8 + 9 + 8 + 0 + 6 + 15 + 0 + 0 + 3

390 and 90 is a multiple of 10

www.justaroo.co.uk