Download Mathematics revision booklet

Maths Department
Answering Scholarship Style Questions
1. Always read the question fully. If it’s an equation – write it out. If it’s a
worded question, note down the key bits of information before you start
working anything out.
2. You don’t have to do questions in order. Have a look through the paper
and pick the best ones for you! Skip out the tricky ones and then have a go
at the end.
3. Work quickly but don’t rush. It is better to do 8 questions fully and clearly
than 10 rushed questions.
4. If you find yourself doing some ridiculous working out, you’ve probably
gone wrong or there might be a different way. Check, rethink, or move on
quickly…
5. When there is an unknown quantity, try calling it “x” or “n” or some
suitable variable. ALWAYS make it clear at the start of the answer what
letter means what, (e.g. let x = number of ice creams).
6. Work down the page – keep to one equals sign per line and keep them
going vertically down the page.
7. Do not say things are equal if they are not – use therefore, (=>) or
because, (three dots in a triangle).
8. Where possible, start each part of an answer with a formula or rule. E.g.
for a circle area question, write out Area = πr2
9. Show full working out – there will be marks available for each step, even if
you don’t quite get to the solution.
10.
If you don’t know what to do, try drawing a diagram.
11.
Whatever you do, DON’T resort to trial and improvement, (unless
told to)…
12.
Start each new question on a new side. Underline your answer to
make it clear.
Most importantly, don’t panic. Maths scholarship papers are tough. Do as much
as you can. Don’t get stuck – move on!
Twelve Tips for Top Exam Performance!
1. Highlight – pick out key words / numbers you are going to use.
2. Simplify – if you have a fraction as your answer, always simplify to its
lowest terms!
3. BIDMAS – write BIDMAS at the top of questions where you need to do
different operations. Also write out the SDT triangle for speed, distance, time
questions.
4. Meanings – Check what maths words mean. Of means times; Product
means times; Sum means add!
5. Angles – Label angles as you go. Always work them out in alphabetical
order!
6. Probability – Always write probability as a fraction!
7. Converting – Show where you take your readings from a graph with a
ruler!
8. Accuracy – When drawing angles or lines, you must be within 1 degree or
1mm of the correct length
9. Don’t be Lazy – Label like crazy! Make sure that you have labelled
everything you measure – including bearings, angles, lengths, pie charts, lines,
etc.
10. Move on! – Don’t get stuck on a question. Give it a good go, but move on
if it’s too difficult. Finish the paper, and then go back to the question.
11. Working – SHOW YOUR WORKING. You get marks for working
things out.
1 mark – no working required
2 marks – some working probably needed
3 marks – working absolutely essential!!!
12. Maths
is tough - Learn from your mistakes – and avoid them next time.
Level 3: Algebra
Topic
Can I solve simple formulae questions?
Example
Algebra
When I double a number and add one, I get seventeen.
What is my number?
Can I substitute into equations using 1 or 2
operations?
If p = 2(a+b), find the value of p when a=4 and b=5
Can I simplify expressions, including with fractions?
Simplify: x + 5 + 2x + 4; Simplify 3c/c
Can I multiply out brackets involving x(x + n) and then
simplify?
Expand: x(x-2)
Can I factorise expressions, including quadratics?
Factorise: x² + 2x
Can I use algebraic convention?
Do I know what x + 2, 3x, x/2, x², 3(x + 2) mean?
Can I solve equations with whole numbers and
fractions?
Solve: 3x - 4 = 12 ; 4 - 3x = 9 ;
4) = 6
¼x - 4 = 2 ; ¾(3x -
Can I solve equations with trial and improvement?
Solve x(5x + 2) = 16 by constructing a table of values
Can I solve simultaneous equations, using a graph if
necessary?
Solve: 2x + y = 10 and 3x - 2y = 8
Can I solve simple inequalities?
½(x - 1) > 3 or 5 - 3x < 12
Can I plot graphs of linear and quadratic equations?
Can I forumlate linear equations?
Draw the line y = x - 1 and y = x² + 2x and find the
points of intersection
A rectangle has sides (2x+3) and (5x-1). If the
perimeter is 46cm, find x.
Level 3: Handling Data
Topic
Example
Averages
Calculate the mean, median, mode and range of the
Can I use the mean, median, mode and range?
following 5 numbers.
One class has a mean of 55%, and another has a mean of
Can I use the averages to compare to sets of data?
60%. Which class has done better?
10 boys have a mean mass of 52Kg. When another boy
Can I find the combined mean of two sets of data?
joins, the mean increases to 54Kg. How much does the
new boy weigh?
Charts
Do I understand tally charts, pictograms, bar charts,
Carroll diagrams and Venn diagrams?
Use a tally chart to collect data on favourite cars, and
Can I record data in tally charts and frequency tables?
display your results in a bar chart
Can I draw a simple line graph?
Can I interpret charts?
Use this pie chart to work out the most popular type of
car
Can I draw pie charts?
Construct a pie chart using angles or percentages
Can I say if a scatter graph has a negative or positive
correlation and explain what this means?
Can I draw a line of best fit on a scatter diagram?
Can I describe probability using words?
Can I describe probability using fractions?
Probability
Determine if the following events are certain, likely,
unlikely or impossible
What is the probability of picking a heart from a deck of
cards?
Can I determine if a die is fair or biased?
Do I know that different outcomes are possible from
repeating an experiment?
Do I know that all probabilities or all mutually exclusive
outcomes add up to one?
Can I list all the possible outcomes of two experiments?
List all the possible outcomes from rolling a die and pick
a card from a normal deck of cards
Level 3: Number
Topic
Example
4 Operations
Can I multiply and divide whole numbers and decimals There are 65 pages in a book. How many pages in 100
by 10, 100 or 1000?
books?
Can I recall multiplication facts up to 10x10 and use
5x7 = ?
60 ÷ 12 = ?
these for division?
Can I solve addition, subtractions, multiplications and
What is 34.2 times 3.6?
divisions with decimals up to two decimal places?
Can I use BIDMAS?
Solve 3 x 4 + 5.
Calculate 3 ( 4 + 5 )
Can I estimate to 1 significant figure and then multiply
Rewrite to 1 sig fig and solve: 63 x 87
or divide mentally?
.......................................... 29
Can I use a calculator to solve problems with
Calculate: 17.6 ÷ (47.92 - 41.4)
numbersof any size?
Fractions, Decimals and Percentages
Can I round to 1 or 2 decimal places?
Round 5.43156 to 2 decimal places
Can I use simple fractions and percentages to describe
amounts of a whole?
Find 60% of 500
Can I simplify a fraction to its simplest form?
Cancel 4/12 to its lowest terms.
Can I calculate fractions and percentages of amounts,
using a calculator if necessary?
Find 70% of £5. Calculate 3/4 of 60.
Can I express one amount as a percentage of another?
Can I work out a percentage increase or decrease?
Can I convert and order fractions, decimals and
percentages?
Can I use all four operations with fractions, including
mixed numbers?
A T-shirt marked at £18 is sold for £14.40. What
percentage discount was given?
A pair of trainers costing £24.50 are discounted by 40%.
How much do they now cost?
Put 2/7, 30% and 0.29 in order of size
3½ x 1¾ = ?
A car uses 40 litres to go 250 miles. How much fuel would
it need for a 300 mile journey?
A recipe for 6 people requires 150g of butter. How much
Can I use ratios to answer problems?
butter is needed for 8 people?
Number Properties
Can I calculate proportionate changes?
Do I understand multiple, factor, square & cube?
Write the next number in this sequence: 1, 4, 9, 16, ?
Do I know what prime numbers are?
Is 17 a prime number? What about 27?
Can I write a number as the product of prime factors?
Express 180 as a product of its prime factors
Can I use coordinates in all four quadrants?
Plot (3,-6) on a grid
Can I order, add and subtract negative numbers?
What is 3- (-2)?
Can I find the LCM and HCF of two numbers?
Find the largest number which exactly divides into 256
and 64
Can I substitute into a quadratic sequence?
Find the 5th term in the sequence: 3n² - 4n
Can I give the nth term of a sequence?
Write the nth term for the sequence: 3, 5, 7, ...
Level 3: Shape, Space and Measure
Topic
Example
Measurement
You should be able to draw and measure to the nearest
Can I use a ruler, compass, protractor and set square?
degree and millimetre
Can I roughly convert metric units into imperial units?
Approximately how many miles are equivalent to 40km?
Can I estimate sensibly?
Estimate the capacity of a teaspoon?
Can I calculate the distance between two co-ordinates?
Find the distance from (2,4) to (5, 8)
A train travels 200 miles at 80 miles per hour. How long
does the journey take?
Ben travels for 30 minutes at 60mph, and then for 45
Can I find the average speed of a multi-stage journey?
mins at 50mph. What is his average speed?
2D and 3D Shapes
Can I identify congruent shapes, orders of rotational
Tick a shape which is congruent to quadrilateral B
symmetry, and lines of symmetry?
Describe the properties of the following shapes: square,
Do I know the properties of quadrilaterals?
rectangle, trapezium, kite, parallelogram, rhombus
Can I use the SDT Triangle?
Can I find the perimeter and area of simple shapes?
Calculate the area and perimeter of a 5 by 9 rectangle.
Can I work out the area of a triangle and parallelogram?
What is the area of a triangle with a base of 4cm and a
height of 6cm?
Can I use Pythogoras' theorem with right-angled
triangles?
Calculate the missing side of this right angled triangle
Can I calculate volumes by using a formula?
Work out the volume of a cuboid with dimensions 3cm x
5cm x 8cm
Can I find the volume of a prism, including a cylinder?
Can I draw the net of a shape?
Can I draw 3D shapes on isometric paper?
Can I calculate the area and circumference of a circle?
Accurately draw a net of a 3cm x 2cm x 4cm cuboid
What is the area and circumference of a circle with a
diamter of 4cm
Can I find the radius of a circle with a given
Find the radius of a circle with circumference 20cm
circumference or area?
Transformations
Can I reflect shapes in a mirror line?
Reflect triangle ABC in mirror line m.
Can I rotate, and translate shapes?
Translate triangle A 3 squares left and 2 squares up.
Enlarge quadrilateral C, scale factor 3, about the center
Can I enlarge shapes by a number scale factor?
of enlargement C
Angles
Can I calculate missing angles in a triangle, on a straight
An angle in an isosceles triangle is 40 degrees. What
line, or around a point?
could the other two angles be?
Can I find missing angles on intersecting and parallel
lines?
Can I specify a direction by using compass points and 3
These questions will also involve scale drawings
digit bearings?
Can I calculate interior and exterior angles in a regular
Calculate the interior and exterior angles of a nonagon?
polygon?
Maths Department
Average Speed
What is it?
Average Speed is used when there is one of more parts of a journey. You
always hav e to consider the totals – i.e. the total distance and total time,
including stops!
Starting Out
Always note down the SDT triangle.
Now think of each measure – is it in the correct unit?
For km/ h, distance must be in km and time must be in hours!
For m/s, distance must be in m and time must be in seconds
Think about hours and minutes – how do you put that into a calculator?
(decimal or fraction)
If not using a calculator, leav ing time as a fraction is generally easier
Make sure you include any stops in your time
Calculating:
Div ide total distance by total time.
For fractions, conv ert into improper fractions, flip the second fraction, and
multiply
Maths Department
Example:
Ben runs three laps of a 400m running track. Each lap takes 2mins, 75sec, and 1 minute
45seconds respectively. Calculate his average speed in km/h.
Total Distance
= 3 x 400m
Total Time = 2 + 1 + 1
= 1200m
= 5 mins
= 1.2km
=
hours
= 1 km
=
hours
=1 ÷
= x
=
= 14.4km/h (or fraction equivalent)
Example 2 – no distance:
Daisy runs two laps of a different running track. She runs at 8km/h for the first lap and
12km/h for the second lap. Calculate her average speed in km/h.
Let d = distance of one lap
Total Distance = 2d
Total Time
= T (lap one) + T (lap two)
= d/s(lap one) + d/s(lap two)
=
=
=
+
+
hours
= 2d ÷
= 2d x
=
= 8 km/h (or decimal equivalent)
Maths Department
Remember:
The biggest mistake people make is finding the mean of the two
speeds, (e.g. Jimmy does a lap at 4km/h and a lap at 6km/h –
therefore his av erage speed is 5km/h).
Think about whether your answer makes sense! You walk at
approximately 5km/h, run at 12km/h, and a car driv es at 70km/h…
If you are giv en two speeds, the av erage speed will be between them.
If the same distance is cov ered at both speeds, the av erage will be
closer to the slower speed – this is because more time is spent at the
slower speed.
Sometimes a distance will cancel out when you divide – if you don’t
know a distance, giv e it a v ariable, (e.g. x or d)
Try using <shift > for a mixed number on your calculator. <Shift s↔d>
will giv e you a mixed number answer – useful for time.
Be VERY clear with your working out. Make it clear what your numbers
are and what you’re doing in each step.
FOIL
Maths Department
What is it?
FOIL is a way to remember how to multiply out two brackets.
F O I L
irst
utside
nside
ast
How to use FOIL?
When you have the product of two brackets, each with two terms inside.
For example:
(3x – 1)(2x + 3) = 6x2 + 9x - 2x – 3
= 6x2 + 7x – 3
Difference of two squares:
A useful result of using FOIL: a2 – b2 = (a + b)(a – b)
For example, 26 x 24 = (25 + 1)(25 – 1) = 252 – 12 = 624
Remember:
Be careful of negatives – a negative times negative = positive
Always collect like terms at the end – thinking about the number line!
When you square a bracket, you have to use foil;
e.g. (3x – 1)2 = (3x – 1)(3x – 1)
You don’t need to use FOIL for single brackets – e.g 5x(3x – 1) = 15x2 – 5x
Prisms
Maths Department
What are they?
Prisms are 3D shapes which have the same cross-sectional area. Common
examples include:
Cuboid
Cylinder
Triangular Prism, (Toblerone Box)
Cuboids
Volume of Cuboid = width x depth x height
Surface Area = sum of the areas of all the rectangular faces
Cylinders
Volume of Cylinder = πr2h
Surface Area = Area of top + Area of bottom + Area of Curved Surface
The curved surface is a rectangle – the width is the circumference of the
circle and the height is the height of the cylinder
Surface Area = 2πr2 + 2πrh
Other Prisms
Volume of Prism = Cross-sectional Area x Depth
The cross-sectional area is the area of a “slice” of the prism. It will be the
same at any point through the prism.
Maths Department
Example:
4cm
5cm
7cm
6cm
Volume
= Cross-sectional Area x depth
= ½(4 + 6)x 5 x 7
= 25 x 7
= 175cm3
Remember
Volume is a measure of 3D space – so units should be cubed!
A cube is a prism with all square sides! So the surface area will be
6 x the area of one face!
When you are dealing with π, make sure you give your answer in the
correct form
o In terms of π
o Approximate, (I.e. use π ≈ 3)
o To 3 significant figures if using a calculator
Show full working out and each step of your solution
Know the areas of 2D shapes – especially triangles, parallelograms, kites
and trapezia
Pythagoras’ Theorem
Maths Department
What is it?
Pythagoras’ Theorem links the hypotenuse of a triangle to the two shorter
sides.
c
a2 + b2 = c2
a
c2 - b2 = a2
hypoteneuse
b
c2 - a2 = b2
Example:
c2 = a2 + b2
x2 = 52 + 122
x
12cm
x2 = 25 + 144
x2 = 169
x=√
5cm
x = 13cm
Remember:
1. Always identify the hypotenuse – this is always opposite the right angle
2. Pythagoras’ Theorem can only be used in right angled triangles
3. Make sure you show full working out
4. Label lengths as you work through a question. If there is no diagram – draw
one
5. Know your square numbers up to 202. If using a calculator, check what you
should round to, (normally 3 significant figures)
6. Know the common triplets – i.e. 3,4,5
5,12,13
6,8,10
Number
BIDMAS
Brackets, Indices, Divide, Multiply,
Add, Subtract.
Order of Operations.
SHOW your working!!!
% of a Number
?%
x
Number
100
Look to cross-cancel top with bottom
50% = half of the number
25% = quarter of the number
10% = ÷ by 10
5% = half 10%
20% = double 10%
Fraction of a Number
Numerator x Number
Denominator
Cross-cancel where possible
1/2 = ÷ by 2
1/3 = ÷ by 3
1/4 = ÷ by 4
Product of Prime Factors
Use a Factor Tree, dividing by
primes, (2, 3, 5, 7, etc).
Product means TIMES
Don’t forget indices,
(e.g. 2 x 2 x 2 = 23)
Fraction to Decimal
Make fraction over 100
1/2 = 0.5
1/4 = 0.25
1/5 = 0.2
7/20 = 35/100 = 0.35
Fraction to %
Make fractions over 100
1/2 = 50%
1/4 = 25%
1/5 = 20%
9/25 = 36/100 = 36%
Decimal Places
Number of digits after the decimal
point.
0 – 4: round down
5 – 9: round up
Significant Figures
Number of digits from first nonzero digit.
Think about VALUE
Negatives when Multiplying and
Dividing
++=+
+-=-+=--=+
Same signs = +ve
Different signs = -ve
Number Properties
Multiple: in that number’s times
table
o Multiples of 7 = 7, 14, 21, 28...
Factor: something that goes into
the number
o Factors of 12 = 1, 2, 3, 4, 6, 12
Square: a number times itself
o 3 squared = 3 x 3 = 9
Cube: a number times itself, times
itself
o 2 cubed = 2 x 2 x 2 = 8
Prime: a number with two factors
o Primes = 2, 3, 5, 7, 11, 13, 17...
Ratio
Always multiply both sides of the
ratio by the same amount
Find the total by adding the
separate parts
Sequences
Always look at the difference
between each term, (number in the
sequence)
Look for a rule for the whole
sequence.
Fibonacci:
1, 1, 2, 3, 5, 8, 13, 21, 34...
Triangle Numbers:
1, 3, 6, 10, 15, 21, 28, 36...
Algebra
Words
Variable - letter
Coefficient – number in front of
letter
Index – floaty number, (plural
indices)
Term – sign, coefficient, variable
and index
Convention
Sum of x and y = x +y
Twice n = n + n = 2n
Product of p and q = pq
a + a = 2a
a x a = a2
a/a = 1
a4 = a x a x a x a
Simplification
When you add or subtract, the
indices never change
Collect like terms
o Same variable and index
o Include sign – think number line!
To multiply, think about:
o Sign: -ve or +ve?
o Multiply numbers
o Multiply variables
To divide:
o Put as a fraction
o Cancel coefficients
o Cancel same variables
Brackets
Multiply all the terms inside the
brackets by the term immediately
before the brackets
Be careful with negatives
Circle the term, including sign, in
front to help
E.g. 8 - 3 (2a – 4) = 8 – 6a + 12
Collect like terms at end
Factorise
Take out all common terms –
numbers and letters.
Must have highest common factor
E.g. 16y + 8 = 8(2y + 1)
Substitute
Replace the variable with its value,
(given in question).
Remember to use BIDMAS
A negative squared is positive
Equations
Use the inverses to get variables
on left and numbers on right
Check answer at end
Straight Line Graphs
Substitute into the equation to
generate co-ordinates
Your points should be on a
straight line
Shape and Space
Coordinates
(x,y) – along the corridor, up the
stairs!
Graphs
Types of Triangles
Equilateral – all sides and angles
equal
Isosceles – two sides and two
angles equal
Scalene – no sides or angles equal
Cuboid
x=y
Volume of Cuboid = b x w x h
Surface Area = sum of all
rectangular face areas.
Angles
x = -y
Transformations
Reflect = use mirror line
Rotate = turn around a point
Translate = move left, right, up or
down.
Enlargement = Use scale factor to
make a shape bigger.
o Draw lines from centre of enlargement
through each point on the shape
o Enlarged Perimeter = x scale factor
o Enlarged Area = x scale factor squared
180˚ = Straight line
180˚ in a triangle
360˚ around a point
Alternate Angles
Z-angles are equal
Corresponding Angles
F-angles are equal
Rectangle
Area = Base times height
Perimeter = distance all the way
around the edge
Area of Triangle
1/2 base x height
Polygon Angles
Exterior angles = 360/n
Interior angle =
180 – exterior angle
Bearings
Always start with 0˚ at North
Measure clockwise.
Think carefully about the scale –
what will fit on the page?
Circles
Circumference
C = π x diameter
Or C = 2π x radius
Area
A = π x radius x radius
Or A = πr2
Data Handling
Mean
Add all values and ÷ by number of
values.
Median
Middle value when in order.
Mode
Most common value.
Range
Difference between largest value
and smallest value.
Pie Charts
Do 360 ÷ total, and multiply all
your values by this.
1/2 = 50% = 180˚
1/3 = 33% = 120˚
1/4 = 25% = 90˚
1/5 = 20% = 72˚
1/10 = 10% = 36˚
Probability
Always express it as a fraction.
Always simplify.
Speed, Distance, Time
S = D/T
Correlation
Upwards sloping – Positive
correlation.
Downwards sloping – Negative
correlation.
No pattern = no correlation.
Introduction to Algebra
Maths Department
Key Words:
Variable – a letter used to represent any number
Term – can be a variable or number, or combination of variables
and numbers.
Expression – a collection of terms
Equation – Terms with an equals sign
Convention:
Variables follow the normal rules of arithmetic – just like numbers!
There are some things to make sure you remember:
2n means 2 lots of n; or n + n
n2 means n lots of n; or n x n
-4a means negative 4 lots of a; or –a –a –a –a
b + 5 means 5 more than b; or 5 + b
5 – b means b less than 5; or –b + 5
Like Terms:
Terms are like only if they have exactly the same variable.
Like terms can be added or subtracted, but remember that the
indices NEVER change.
4 Operations in Algebra
Maths Department
Addition and Subtraction:
1. Check if they are like terms. If they aren’t like terms, leave them alone.
2. The indices won’t change when you add or subtract.
3. Think about negative terms and their direction – use a number line!
To be like terms, the letters must be the same, including indices!
Multiplication:
1. Look at the signs; will the answer be positive or negative?
Negative times Positive = Negative
Negative times Negative = Positive
2. Multiply the coefficients.
3. Multiply the letters, remembering to add the indices.
3a2 x 4a = 12a3
-4b x 2b = -8b2
Division:
1. Look at the signs; will the answer be positive or negative?
a. Negative divided by Positive = Negative
b. Negative divided by Negative = Positive
2. Put the division into fraction.
3. If you can, divide the coefficients.
4. Cancel the letters, remembering to subtract the indices.
You can only cancel letters which are the same.
Brackets
Maths Department
Quite Simply:
1. Look at what you are multiplying by – is it positive or negative?
2. Multiply every term inside by the term in front of the brackets.
3. Be careful about the signs:
Negative times Positive = Negative
Negative times Negative = Positive
Remember to multiply the coefficients:
4a x 5b = 20ab
Don’t forget to add the indices when the letters are the same.
4a2 x a = 4a3
Examples:
1. 3(a+3) = 3a + 9
2. b(4c + 4) = 4bc + 4b
3. e2(4e3 + e) = 4e5 + e3
4. 3f(7f + 2g) = 21f2 + 6fg
5. -3(7h – 9) = -21h + 27
6. –(-i – 8) = i + 8
Remember to check if you can add any like terms at the end.
Factorising
Maths Department
Quite Simply:
1. If there is a common number factor, take that outside the brackets.
2. If there is a common letter factor, that that outside the brackets.
3. Remember to divide each term by what you’ve taken out:
Always take out the largest number factor:
8a + 12b = 4(2a + 3b)
[not 2(4a + 6b)]
Always take out all the common letters:
5bcd – 3bd = bd(5c – 3) [not b(5cd – 3d)]
Examples:
1. 3a + 9 = 3(a + 3)
2. 5m – 7mp = m(5 – 7p)
3. 15y + 20s = 5(3y + 4s)
4. 7h – 21hj = 7h(1 – 3j)
5. 6t2u + 7tu = tu(6t + 7)
6. 12a3b2 – 8a2b = 4a2b(3ab – 8)
Always factorise fully. Check that you have found the highest
common factor and taken out all the common letters.
Solving Equations
Maths Department
Quite Simply:
1. To solve an equation, you need to find out what the variable,
(letter), equals
2. Use inverse operations to get the variable on the left and the
numbers on the right.
Inverse Operations:
An inverse operation is the opposite
o The inverse of plus is minus
o The inverse of multiply is divide
How to solve Equations:
Multiply out any brackets – remember to multiply everything
inside the brackets by the term in front of the brackets
Simplify by collecting like terms
Isolate the variable by using inverse operations
Remember:
o 5a means 5 x a
o a means a ÷ 3
3
Maths Department
Examples:
1. 5a = 20 → a = 20 ÷ 5 → a = 4
2. b – 7 = 23
→ b = 23 + 7 → b = 30
3. c + 5 = 19
2
→ c = 19 – 5 → c = 14 x 2 → c = 28
2
4. 3(2d – 6) = 6 → 6d – 18 = 6 → 6d = 6 + 18 → 6d = 24
→ d = 24 ÷ 6 → d = 4
Remember:
ALWAYS multiply out brackets first
CHECK your answer using the original equation
Be careful with negatives
Use the inverse of plus and minus before multiplication or
division
Lay your work out neatly to avoid errors.
Straight Line Graphs
Maths Department
Quite Simply:
1. You will be given an equation, E.g. y = 2x + 4
2. You need to substitute values of x to find the value of y.
Remember the rules of algebra:
2x means 2 times x
½x means x divided by 2
Multiply or divide before adding or subtracting.
3. This will then give you coordinates to plot on a graph.
the value of x tells you how far across
the value of y tells you how far up or down
Example:
Fill in the table for the equation y = 3x – 2
x
y
-2
-8
0
-2
3
7
Now you can plot the points on the graph to make a straight line:
(-2,-8)
(0,-2)
(3,7)
y
Maths Department
(3, 7)
x
(0, -2)
(-2, -8)
y = 3x - 2
y
(3, 7)
x
(0, -2)
(-2, -8)
Maths Department
Remember:
Always plot the points with x being along, and y being up or down
Draw your line right to the edge of the graph, not just between the
points.
Label your lines clearly
Show any working out – you might still get marks.
If the line isn’t straight, you will have made a mistake.
You might need to rearrange your equation so y is on the left.
Make sure you know what the lines y = x and y = -x look like.
Substitution
Maths Department
Quite Simply:
1. You will be given a number value for each letter.
2. Re-write the expression using numbers instead of letters.
3. Solve the expression, remembering BIDMAS:
Brackets
Indices
Division
Multiplication
Addition
Subtraction
Remember the rules of algebra:
ab means a times b
a/b means a divided by b
a2 means a times a
Always show your working out – you may still get marks.
Examples:
For all these examples, a = 4, b = 6, c = -1
1. 3a + b
= 3x4 + 6 = 18
2. b – a/2
= 6 - 4÷2
=4
3. 3a2
= 3x4x4
= 48
4. 5a + c
=5x4 + -1 = 19
5. 5b – c
= 5x6 - -1 = 31
Be careful with negatives; if you subtract a negative, you really
add it!
Writing Expressions
Maths Department
Quite Simply:
You will be given information about a number of something,
usually n.
Use this information to write, in algebra, how much of something
else there is
Ultimately, you will solve an equation to find the actual number of
something
Using algebraic convention:
Ben is n years old.
Daisy is 6 years younger than Ben
Daisy’s age = n – 6
John is twice as old as Ben
Twice means 2 times, hence 2n
John’s age = 2n
Sylvia is 3 years older than Ben
Sylvia’s age = n + 3
Hilda is 3 times older than Sylvia
Note the use of brackets!
Hilda’s age = 3(n + 3) = 3n +9
Multiply out brackets by
multiplying by 3
Solving Equations:
Maths Department
Together, John and Hilda have a combined age of 59. Find out how old
Hilda is.
John + Hilda = 59
so: 2n + 3n + 9 = 59
Collect like terms: 5n + 9 = 59
5n = 50
so n = 10
Hilda = 3n + 9.
Substitute for n: 3x10 + 9 = 39 years old
Remember:
Always read the question carefully!
Think carefully about how and when to use brackets.
Show all your working.
Probability
Maths Department
What is it?
A way of saying how likely something is to happen.
Using words:
Ranges from Certain to Impossible
Likely, 50:50, Unlikely
E.g. If I throw a die, I am unlikely to get a 6. I am likely to get a
number less than 6. It is 50:50 that I will get an odd number.
Using Fractions:
Probability of specific outcome = Number of ways that outcome can happen
Total number of possible outcomes
ALWAYS simplify if possible...
Examples:
Two tetra-dice are thrown and the results are multiplied:
1
2
3
4
1
1
2
3
4
2
2
4
6
8
3
3
6
9
12
4
4
8
12
16
Total number of outcomes = 16
Probability of even number = 12/16 = 3/4
Probability of prime = 4/16 = 1/4
Probability of greater than 10 = 3/16
Probability of square number = 6/16 = 3/8
Lines of Best Fit
Maths Department
What are they:
Show an approximate link between two values.
E.g. How much a bike is worth after a number of years.
Drawing a Line of Best Fit:
Use a ruler.
Rotate and move ruler until roughly half the plots are above and
roughly half are below.
There is never just one line of best fit – it is your interpretation.
Correlation:
Positive Correlation is upward sloping. Negative correlation is downward sloping
Using the Line:
Show clearly where you take your reading.
Anomaly
x
A “wrong” result or “anomaly” will be furthest away from the line.
Maths Department
Mean, Median, Mode and Range
Mean:
Add up all the values
Divide by the number of values
Remember that 4 lots of 2p is actually a total of 8p!
Median:
Put all the values in order of size
Find the middle value
If there are two middle values, find the half way point between
them
Mode:
Find the most common value
Range:
Find the largest and smallest value
Calculate the difference between them
Remember:
1. Frequency means the number of times something occurs.
2. When finding the mean, use long division, (or a calculator) when
dividing. Leave your answer as a decimal, unless told otherwise.
Maths Department
Pie Charts
What are they:
A visual way of displaying data
Working out the number of degrees:
Find the total of the values
Divide 360˚ by the total – this is the number of degrees for one
Now multiply all your amounts by this number
Drawing a Pie Chart:
Remember to use a protractor.
Line up the zero line on your protractor for each segment.
Don’t forget to label every section of your chart
Example:
Pet
Dog
Cat
Fish
Tiger
Total
Number
10
7
2
1
20
Pet
Dog
Cat
Fish
Tiger
Total
Number
10
7
2
1
20
First of all, divide 360 by 20 = 18.
So for each pet, there are 18 degrees...
Now multiply all your amounts by 18.
x18
180˚
126˚
36˚
18˚
360˚
Don’t forget to check that they add up to 360
Tiger
Fish
2
1
Dog
10
Cat
7
Don’t forget labels!
BIDMAS
Maths Department
Quite Simply:
Brackets
Indices
Division
Multiplication
Addition
Subtraction
Use of BIDMAS:
When given a calculation, BIDMAS tells us the order we must solve it in.
You must solve inside the brackets first
Then you must do indices next
Do any multiplication or division next
Finally, work out the addition or subtraction
Examples:
1. 12 – 3 x 2 = 12 – 6 = 6
2. 3 x (4 + 5) – 1 = 3 x 9 – 1 = 27 – 1 = 26
3. 19 – 42 + 2 x 5 = 19 – 16 + 2 x 5 = 19- 16 + 10 = 13
Remember:
Be careful of negatives. Positive times a negative is negative. Negative
times negative is positive.
Do addition and subtraction at the same time. Use a number line if
necessary
Converting Units
Maths Department
You must know:
1000m = 1km
100cm = 1m
10mm = 1cm
1000g = 1Kg
1000ml = 1 Litre
Converting Units:
Think about whether you need to multiply or divide by 10, 100, 1000 (or
other)
Examples:
4.5km = 4500m
519m = 0.519km
42.3cm = 0.423m
34100g = 34.1Kg
2.34 litres = 2340ml
1/4Kg = 250g
3/4km = 750m
Remember:
Check if your answer is sensible
Maths Department
Converting Fractions, Decimals and
Percentages
Fraction to Decimal:
1. Find an equivalent fraction with a denominator of 10, 100, 1000
E.g. 13/20 = 65/100 = 0.65
2. Use long division to divide the top number by the bottom number
3. Know and use certain conversions:
1/3 = 0.3333333
1/5 = 0.2
1/8 = 0.125
1/10 = 0.1
1/20 = 0.05
Decimal to Fraction:
Put the decimal as a fraction over 10, 100, or 1000 and then simplify
E.g. 0.12 = 12/100 = 6/50 = 3/25
Fraction to Percent:
1. Convert to decimal and multiply by 100 or
2. Find an equivalent fraction over 100
E.g. 17/25 = 68/100 = 68%
Percent to Fraction:
1. Put as a fraction over 100 and then simplify
E.g. 15% = 15/100 = 3/20
Maths Department
Percent to Decimal:
Divide by 100
Decimal to Percent:
Multiply by 100
Remember:
Always simplify any fractions if possible
Show your working out – even if you get the answer wrong, you
might still get marks
Check if your answer is sensible
Decimal Places
Maths Department
Decimal Places:
The first decimal place is the first digit after the decimal point
4.782
The second decimal place is the second digit after the decimal
point
Rounding to Decimal Places:
To round to two decimal places, take the complete number up
to the second decimal place.
Now look at the third decimal place. This will tell you to round
up or leave alone:
o 0 – 4: Leave alone
o 5 – 9: Round the second decimal place up
Examples:
5.21 = 5.2 (1 decimal place)
3.56 = 3.6 (1 decimal place)
0.0526 = 0.053 (3 decimal places)
523.565 = 523.57 (2 decimal places)
Remember:
Do not confuse with Significant Figures!!!
Tests of Divisibility
Maths Department
Quite Simply:
It is sometimes necessary to decide if a number is divisible by
another number
If a number is not divisible by anything other than 1 and itself, it
is prime.
Some examples of when you might need a test of divisibility
o Testing if a number is prime
o Simplifying fractions
o Finding prime factors
o Simplifying ratios
Divisible by 2:
Any even number is divisible by two, (ends in 0, 2, 4, 6, or 8)
Divisible by 3:
If the sum of the digits is a multiple of 3, then the number is
divisible by 3
o E.g. 2139 = 2 + 1 + 3 + 9 = 15 = 5x3 => 2139 is divisible by 3
Divisible by 4:
If the last two digits are a multiple of 4, then the number is
divisible by 4
o E.g. 23124 is divisible by 4
o E.g. 23126 is not divisible by 4
Maths Department
Divisible by 5:
Any number ending in 5 or 0 is divisible by 5
Divisible by 6:
Anything divisible by 3 which is even can be divided by 6
Divisible by 7:
Very tricky – use long division, (or times table knowledge)
Divisible by 8:
If you can halve the number 3 times, it is divisible by 8
Divisible by 9:
If the sum of the digits is a multiple of 9, then it is divisible by 9
o E.g. 42336 = 4+2+3+3+6 = 18 = 9x2 => 42336 is divisible by 9
Divisible by 10:
Any number ending in 0 is divisible by 10
Remember:
Divisible means you can divide it by a certain number
28, 35, and 49 are all divisible by 7 because 7 goes into them
Fraction Operations
Maths Department
Adding Fractions:
If two fractions have the same denominator, add the numerators
E.g.
If they have different denominators, find equivalent fractions which have the
same denominator
E.g.
Subtracting Fractions:
Similar to adding fractions, but make sure you take away!
E.g.
Multiplying Fractions:
Multiply the top numbers and multiply the bottom numbers.
E.g.
Dividing Fractions:
Turn the 2nd fraction upside down, and then multiply tops and bottoms
E.g.
=
=5
Remember:
Always check if you can simplify!!!
When adding or subtracting, only add or subtract the
numerators!
You can cross-cancel top and bottom when multiplying, (look
for common factors!)
Maths Department
Fractions and Percentages of Amounts
Fractions of Amount:
1.
2.
3.
4.
Write out using a x sign
Look to simplify or cross-cancel
Multiply top numbers
Divide by bottom numbers
E.g. Find 3/5 of 25:
3 x 25
5
→
3 x 25
5
5
→
3 x 5 = 15
→
(3 x 11)÷2 = 16.5
1
E.g. Find 3/14 of 77:
3 x 7
14
→
3 x 77
14
11
2
Percentages of Amounts:
1.
2.
3.
4.
5.
Write out as a fraction, using a x sign
Change £ to pence or metres to cm
Look to simplify or cross-cancel
Multiply top numbers
Divide by bottom numbers
E.g. Find 30% of 80:
30 x 80 →
100
30 x 80 →
3 x 8 = 24
100
E.g. Find 15% of £4:
15 x 400 →
100
15 x 400 →
100
15 x 4 = 60p
Maths Department
Express two amounts as a percentage:
1. Make sure they are the same units – both in pence or cm
2. Write as a fraction and simplify
3. Make the denominator into 100 and the numerator is the percentage
E.g. Express 42 as a percentage of 70:
42 →
42
6
70
70
10
→
6
10
=
60 = 60%
100
E.g. Express 15cm as a percentage of 3m:
15 →
300
15
300
5
100
→
5 =
100
5%
Remember:
Percentage means out of 100
Check your answer – does it seem sensible?
100cm = 1m
1000m = 1km
Indices
Maths Department
What are they?
An index or power tells us how many times something is
multiplied by itself
Indices can be used with numbers or variables
Squared means “to the power of 2”
Cubed means “to the power of 3”
Examples:
32 = 3 x 3 = 9
43 = 4 x 4 x 4 = 64
y5 = y x y x y x y x y (NOT 5 x y)
Uses:
Indices are used to express numbers as products of prime factors.
E.g. 24 = 2 x 2 x 2 x 3 = 23 x 3
They are used to simplify expressions in algebra.
E.g. 3 x a x a x b x b x b = 3a2b3
Remember:
Indices come second in BIDMAS
c2 means c x c (NOT c + c or c x 2)
Number Properties
Maths Department
Factors:
A factor is a whole number which will divide exactly into a number
The factors of 20 are 1, 2, 4, 5, 10 and 20 because they all go
into 20
Multiples:
A multiple of a number is something in that number’s times table.
The multiples of 7 are 7, 14, 21, 28, 35, 42, etc...
Prime Number:
A number is prime if it has exactly two factors.
The first few primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc...
One is NOT a prime number
Squares:
A number multiplied by itself gives a square number.
The first few squares are: 1, 4, 9, 16, 25, 36, 49, 64, etc...
The square root is the number you multiplied by itself
o E.g. The square root of 25 is 5, or √25 = 5
Maths Department
Cubes:
A number multiplied by itself, and then by itself again, gives a cube number.
The cubes you should know are: 1, 8, 27, 64, etc...
The cube root is the number you multiplied by itself and then
itself again
o E.g. The cube root of 27 is 3, or 3√27 = 3
1
2
3
4
5
6
7
8
9
10
1
1,2
1,3
1,2,4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
1, 2, 3...
2, 4,
6...
3, 6,
9...
4, 8,
12...
5, 10,
15...
6, 12,
18...
7, 14,
21...
8. 16.
24...
9, 18,
27...
10, 20, 30...
Prime?
No
Yes Yes No
Square?
1x1
No No 2x2
Cube?
1x1x1 No No No
Square root of:
1
4
9
16
Yes
No
No
25
No
No
No
36
Yes
No
No
49
No
No
2x2x2
64
No
3x3
No
81
No
No
No
100
Factors:
Multiples:
Maths Department
Prime Factors
Prime Numbers:
A Prime Number is a number which can only be divided by 1 and itself
The first few prime numbers are: 2, 3, 5, 7, 11, 13,17, 23, 29, 31, 37,...
Finding Prime Factors:
Is it divisible by 2?
Yes
No
2 is a Prime Factor.
Is it a Prime Number?
Write down the
number divided by 2
Yes
Indices:
No
It is a Prime Factor
Divide it by 3, 5, 7, 11, etc
Stop!
This is a Prime Factor
1. When you have found all the prime factors, put them in order of size.
2. Use indices if possible
Examples:
1. 24 = 23 x 3
2. 15 = 3 x 5
3. 20 = 22 x 5
4. 100 = 22 x 52
5. 26 = 2 x 13
6. 42 = 2 x 3 x 7
Ratios
Maths Department
What are they?
A ratio tells you how something is shared.
3:4 means that for every 3 units of something, there are 4 units of
something else
Totals:
Think about what the parts add up to – that is the total.
1:5 means that there are 6 units in total
Using Ratios:
If you multiply one side of a ratio, you must multiply the other side by the
same amount.
2:7 (total 9) = 4:14, (total18) = 6:21, (total 27)
Note that the total is multiplied by the same amount too.
Maths Department
Example:
Bill and Ted share £36 in the ratio 2:7. How much does each person get?
Bill : Ted
Total
2 : 7
9
times 4
36
Bill : Ted
Total
2 : 7
9
8 : 28
36
times 4
Bill gets £8, Ted gets £28
Remember:
You must multiply both sides of the ratio by the same amount
Always check your answer, (E.g. check that £8 + £28 = £36)
Read the questions very carefully
To simplify a ratio, divide both sides by the same amount.
Maths Department
Speed, Distance and Time
What are they:
Speed, distance and time can all be linked by simple formulae.
SDT Triangle:
Speed = Distance ÷ Time
Distance = Speed x Time
Time = Distance ÷ Speed
D
S
T
Time amounts:
Always use time as a fraction or decimal.
Learn these:
30 mins = 0.5 hours or 1/2 hour
20 mins = 0.3333 hours or 1/3 hour
15 mins = 0.25 hours or 1/4 hour
40 mins = 0.6666 hours or 2/3 hour
45 mins = 0.75 hours or 3/4 hour
Remember:
Be very careful about your units for time – there are 60 minutes in
an hour, and 60 seconds in a minute!
Check that you can convert from kilometres per hour to metres
per second
Read the question very carefully – what are they really asking you
to do?
Significant Figures
Maths Department
Significant Figures:
The first significant figure is the first non-zero digit
78203
The second significant figure is the second digit, (including zeroes)
Rounding to Significant Figures:
To round to two significant figures, take the complete number up to the
second significant figure.
Now look at the third significant figure. This will tell you to round up or leave
alone:
o 0 – 4: Leave alone
o 5 – 9: Round the second significant figure up
Always keep the place value of the digits the same
Examples:
521 = 500 (1 significant figure)
356 = 360 (2 sig figs)
0.05126 = 0.0513 (3 sig figs)
500923.565 = 501000 (3 sig figs)
Remember:
Do not confuse with Decimal Places!!!
Angle Chasing
Maths Department
What does this mean?
Sometimes we need to find certain angles using facts given.
We may be given parallel lines, isosceles triangles, quadrilaterals,
and perpendicular lines.
We need to learn certain facts about angles to help us find the
missing angles
Angles in Isosceles Triangles:
Angles in a triangle add up to 180˚
Two angles in an isosceles triangle are always the same.
Two lines are the same length, this is shown with a dash on those sides.
a
SAME
If a = 30˚, the other two angles add up to 180˚ – 30˚ = 150˚
Therefore, the three angles in this triangle are 30˚, 75˚, and 75˚
Maths Department
Angles in Parallel Lines:
This means they
are parallel
All these angles
are EQUAL
All these angles
are EQUAL
Remember:
1. Use what you know to find the missing angles – work in alphabetical order.
2. Look at the angle; although the drawings are not to scale, (so a protractor
is no good), they are roughly correct.
3. Obtuse angles must be more than 90˚; Acute angles are less than 90˚
3. If in doubt, have a guess! The angle might the same as another one
you’ve already found!
4. Angles on a straight line add up to 180˚
Maths Department
Angles
What are they?
An angle is the amount of turn between two lines.
They are measured in degrees.
An Acute angle is smaller than 90˚
An Obtuse angle is between 90˚ and 180˚
A Reflex angle is between 180˚ and 360˚
A Right Angle is equal to 90˚
Angles in Triangles and Quadrilaterals:
Angles in a triangle add up to 180˚
Angles in a quadrilateral, (4 sided shape) add up to 360˚
Interior and Exterior Angles:
Interior Angle
Exterior Angle
Exterior Angles add up to 360˚
Exterior Angles = 360 ÷ number of sides
Interior Angle plus Exterior Angle = 180˚
Interior Angle = 180 – Exterior Angle
Maths Department
Example:
E.g. An Octogon has 8 sides
Exterior angle = 360 ÷ 8 = 45˚
Interior angle = 180 – 45 = 135˚
Sum of Interior Angles = 135 x number of sides, (8) = 1040˚
Sum of Interior Angles:
There are two ways to find the sum of interior angles.
Sum of Interior Angles = Interior Angle x Number of Sides
Or,
Sum of Interior Angles = (n – 2) x 180
(n = number of sides)
Remember:
1. Perpendicular means at right angles. Parallel means that the lines will
never meet, (like train tracks).
2. In an isosceles triangle, 2 angles are the same. This is a very useful fact to
remember!
3. The exterior angle of a regular polygon is always equal to the angle
formed by two adjacent vertices and the centre.
These two angles are equal
Area and Perimeter
Maths Department
What are they?
The area of a shape is the amount of space enclosed within a shape
The perimeter is the length around the outside of a shape.
Area of a rectangle, (A):
The area of a rectangle is its base multiplied by its height
A=bxh
Perimeter of a rectangle, (P):
The Perimeter is the total length of all the sides
P=b+b+h+h
P = 2b + 2h
Rectangle Example:
Area = 8 x 12 = 96cm2
12cm
Perimeter = 8 + 8 + 12 + 12 = 40cm
8cm
Maths Department
Area of a triangle, (A):
The area of a triangle is half its base multiplied by its perpendicular height
A=½xbxh
Triangle Example:
Area = ½ x 8 x 6 = 12cm2
Perimeter = 10 + 10 + 8 = 28cm
10cm
6cm
8cm
Remember:
1. Perpendicular means at right angles. The height of the triangle must be at
right angles to the base.
2. Remember your units – area is cm2 , m2 , etc. but perimeter is cm, m, etc.
Maths Department
Bearings
What are they?
A bearing is a direction
It can either be a compass point or a three-digit angle
Bearings are always measured from North
Compass Points:
Bearings:
000
N
NW
NE
315
˚
˚
W
E
SW
045
˚
270
090
˚
˚
SE
S
225
˚
135
180
˚
˚
Remember:
1. Bearings ALWAYS have 3 digits, (E.g. 030˚)
2. Always start measuring from North!
3. When finding a bearing, if there isn’t a north line, draw one!
4. If you know a distance but not a bearing, you might have to use a
compass to draw an arc.
5. Read the question carefully – sometimes a sketch will help. Think carefully
about the scale of the drawing.
Circles
Maths Department
What is π?
π, (the greek letter Pi), is 3.1412... It is an irrational number, which means the
decimals go on for every without recurring.
You can use the π button on your calculator when solving circle problems.
Radius, (r):
Diameter, (d):
Circumference, (C):
The circumference of a circle is the distance around the outside.
It can be calculated by:
C=πxd
(remember, the diameter is twice the radius)
Maths Department
Area, (A):
The area of a circle is the amount of space inside the circumference
A=πxrxr
A = π x r2
Parts of a circle:
Area = ½ x π x r x r
r = 5cm
Perimeter = ½ x π x d (+10)
=½xπx5x5
= ½ x π x 10 (+ 10)
= 39.26990817cm
= 25.70796327cm
= 39.3cm (3 sig figs)
= 25.7cm (3 sig figs)
Remember:
1. Always use 3 significant figures unless told otherwise.
2. Read worded questions very carefully – what are they asking?
Maths Department
Nets
What are they?
A net is a “flat” version of a 3D shape. When folded, it would make the 3D
shape.
They can be useful to work out the surface area of a cuboid.
Net of a Cuboid:
3cm
4cm
10cm
3cm
4cm
10cm
Remember:
1. Always use a ruler to draw your net.
2. You calculate the surface area of a cuboid by adding all the areas of all
the faces, (areas of back, front, side, side, top, and bottom).
3. Label the lengths of your net.
5. The volume of a cuboid is the length times width times depth.
Maths Department
Surface Area and Volume
What are they?
The surface area of a 3D shape is the sum of the areas of each of its faces
The volume is the amount of space contained in the shape.
Area of a rectangle, (A):
The area of a rectangle is its base multiplied by its height
A=bxh
Surface Area of a Cuboid:
You need to find the area of each of its 6 faces
Surface Area = Area of Top + Area of Bottom + Area of Front
+ Area of Back + Area of Right Side + Area of Left Side
Volume of a Cuboid (V):
The volume of a cuboid is the width times the height times the depth, (or
length)
V=wxhxd
Maths Department
Cuboid Example:
2cm
4cm
10cm
m
Surface Area
= 2x10 + 2x10 + 4x10 + 4x10 + 2x4 + 2x4
= 20 + 20 + 40 + 40 + 8 + 8
= 136cm2
Volume
= 10 x 4 x 2
= 80cm3
Remember:
1. Remember your units – area is cm2 , m2 , etc. but volume is cm3, m3, etc.
2. If you have a net of the cuboid, you can count squares to find the surface
area.
Transformations
Maths Department
What are they?
A transformation is a way of changing a shape’s position or orientation.
There are 4 main types of transformation:
Rotation
Reflection
Translation
Enlargement
Rotation:
Rotation means turning a shape around a fixed point.
We need to know the angle of rotation, the point to rotate about, and the
direction; clockwise or anti-clockwise.
E.g. Rotate triangle A 90˚ clockwise about the point (1,0)
These dashed lines are the
same length.
A
The angle between
the dashed lines is 90˚
B
This point is (1,0)
Note that the shape of the triangle
hasn’t changed, but the orientation has!
Maths Department
Reflection:
Reflection means “flipping” a shape across a mirror line.
We need to know where the mirror line is.
Each point in the reflection will the same distance from the mirror line as its
equivalent point in the original image.
E.g. Reflect triangle A in the line x = 2
Mirror Line
B
A
All these distances to and
from the mirror line are equal.
x = 2 because at every point on
this line, x is equal to 2
Maths Department
Translation:
This means moving a shape in a straight line left, right, up or down
It can be a combination of two or more movements.
Carefully count squares and then redraw the shape
E.g. Translate triangle A 3 squares left and 5 squares down.
Count 3 squares
left
A
Then count 5 squares down
Then do the same for each point
in the shape. Note the size and
orientation haven’t changed
B
Maths Department
Enlargement:
Enlargement means making a shape bigger.
We need to know the center of enlargement.
We also need to know how many times bigger our new shape will be. This is
called the scale factor.
o If the scale factor is 3, all the edges of the new shape will be three
times bigger.
o The perimeter will also be three times bigger
o The area will be 32 = 9 times bigger.
E.g. Enlarge triangle A, scale factor 2, center (1,1)
Draw these lines – they show
where the new points will be.
The distance from C to B is 2
times the distance from C to A
B
A
C
Maths Department
Remember:
1. Always use a ruler to draw straight lines.
2. Read the question very carefully!
3. Make sure you know what the lines y=2, x=1, y=x look like.
4. Always draw accurately – make sure you use the squares in the grid.
5. Make sure you label your new shapes correctly.
6. Check that you know the difference between clockwise and anticlockwise.
7. Congruent means that the shapes are the same shape, same size.
8. Similar means that the shapes are the same shape, different sizes.
Practice Makes Perfect 13+ Level 1
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 20145 to nearest
thousand
7x9
3+4x5
Simplify 4/10
Convert 0.3 into fraction
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 50% of £48?
Two square numbers less
than 10
Prime factors of 30
5 - 12 =
1, 2, 4, 8, 16, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
4n - 2n + 5n
Multiply out: 4(2a - 1)
Solve: 4b = 20
Area =
Volume =
3cm
1cm
5cm
5cm
2cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Area of circle with r = 2 (Use
pi = 3)
5 miles = ? Km
Probability of even number
on a fair die?
Mode of: 3, 5, 2, 1, 2, 7, 2
Score:
/20
Practice Makes Perfect 13+ Level 1
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 13732 to nearest
thousand
8x4
5-3x2
Simplify 6/15
Convert 0.8 into a fraction
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 50% of $34
Two prime numbers less than
10
Prime factors of 12
6-9=
4, 5, 7, 10, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
3a + 8a - a + 4a
Multiply out 5(3a - 1)
Solve 3b = 9
Area =
Volume =
2cm
2cm
3cm
7cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Area of circle with r = 1 (Use
pi = 3)
? miles = 8 Km
Probability of odd number on
a fair die?
Mode of: 6, 4, 9, 6, 2, 1, 6
Score:
/20
Practice Makes Perfect 13+ Level 1
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 9503 to nearest
thousand
6x9
5-6+9
Simplify 18/20
Convert 0.2 into a fraction
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 50% of £82?
A square number between 20
and 30
Prime factors of 18
6 - 14 =
32, 16, 8, 4, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
6n - n - 5n + 2n
Multiply out 2(5a - 2b)
Solve: 6b = 12
Area =
Volume =
3cm
1cm
4cm
6cm
1cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Area of circle with r = 3 (Use
pi = 3)
10 miles = ? Km
Probability of zero on a fair
die?
Mode of: 5, 4, 0, 9, 0, 2, 4, 0
Score:
/20
Practice Makes Perfect 13+ Level 2
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 1952 to nearest 100
8x6
12 ÷ 3 + 1
Simplify 24 / 30
Convert 7/10 into a
percentage
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 25% of £20
A cube number greater than
20
Prime factors of 45
(-3) - 9
13, 9, 5, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
2n + 4 - 5n
Factorise 4a - 6
Solve b - 5 = 2
Perimeter =
Volume =
3cm
5cm
4cm
5cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Circumference of circle with r
= 6 (Use pi = 3)
1 KG = ? Lbs
Probability of prime number
on a fair die?
Median of: 3, 5, 2, 1, 3, 7, 2
Score:
/20
Practice Makes Perfect 13+ Level 2
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 1809 to nearest 100
7x6
15 ÷ (5 - 2)
Simplify 21/24
Convert 0.42 into a
percentage
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 25% of £32
A cube number less than 10
Prime factors of 60
(-7) + 5
12, 5, -2, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
4a - 3 - 5a
Factorise 8a - 2
Solve b + 7 = 10
Perimeter =
Volume =
5cm
8cm
3cm
9cm
2cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Lines of Symmetry
Circumference of circle with r
= 5 (Use pi = 3)
? KG = 2.2 Lbs
Probability of cube number
on a fair die?
Median of: 5, 3, 9, 1, 1, 6, 3
Score:
/20
Practice Makes Perfect 13+ Level 2
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 4971 to nearest 100
9x4
(12 + 8) ÷ (4 + 1)
Simplify 10/15
Convert 3/10 into a
percentage
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 25% of £40
Two cube numbers
Prime factors of 32
(-7) - 12
8, 7, 5, 2, ___, ___
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
8 - 3n - 4n
Factorise 10a - 15
Solve b - 3 = 8
Perimeter =
Volume =
4cm
3cm
5cm
7cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Circumference of circle with r
= 1 (Use pi = 3)
2 KG = ? Lbs
Probability of square number
on a fair die?
Median of: 3, 5, 3, 3, 6, 1, 8
Score:
/20
Practice Makes Perfect 13+ Level 3
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 3.371 to 1 decimal
place
7x8
3 + 5 - 3²
Find 1/5 of 35
Convert 32% into a fraction
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 15% of £12
Two prime numbers that have
a product of 65
Prime factors of 80
(-3) x (- 7)
1, 3, 6, 10, 15, __, __,
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
4ab x 3a²
Multiply out and simplify: 2a
+ 3(a - 1)
Solve 3b - 1 = 20
Area =
Surface Area =
2.5cm
5cm
4cm
10cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Area of circle with r = 6 (Use
pi = 3)
40km = ? Miles
Probability of picture card in
deck of cards? (no jokers)
Range of: 3, 5, 2, 1, 3, 7, 9
Score:
/20
Practice Makes Perfect 13+ Level 3
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 2.603 to 1 decimal
place
6x8
3² - 2³ + 6
Find 1/9 of 27
Convert 88% into a fraction
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 45% of £5
Two prime numbers that have
a product of 21
Prime factors of 56
(-9) x (- 5)
1, 4, 9, 16, ___, ___
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
5a³ x 2ab
Multiply out and simplify: 5
+ 2(5a - 4)
Solve 5b + 2 = 37
Area =
Surface Area =
0.8cm
2cm
5cm
5cm
2cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Area of circle with r = 5 (Use
pi = 3)
20km = ? Miles
Probability of even number in
deck of cards? (no jokers)
Range of: 7, 3, 9, 0, 8, 8, 4,
Score:
/20
Practice Makes Perfect 13+ Level 3
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 4.082 to 1 decimal
place
8x9
4 - 3² - 5
Find 1/8 of 16
Convert 44% into a fraction
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
What is 35% of £4
Two prime numbers that have
a sum of 24
Prime factors of 96
(-6) x (- 9)
2, 4, 7, 11, 16, ___, ___
11. Simplify
12. Brackets
.
14. Area and Perimeter
15. Volume
Solve 4b - 5 = 7
Area =
Surface Area =
3ab x 5b
Multiply out and simplify:
+ 2(3a - 2)
5
3.2cm
2cm
3cm
6cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Rotational Symmetry?
Area of circle with r = 4 (Use
pi = 3)
15km = ? Miles
Probability of prime number
in deck of cards? (no jokers)
Range of: -4, 0, 6, 3, -1, 4, 0
Score:
/20
Practice Makes Perfect 13+ Level 4
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 4280 to 2 significant
figures
12 x 7
12 - (3 + 1³)
-4
How many days will 4 bottles
of milk last if I drink 1/4 of a
bottle a day?
Convert 4/25 into a decimal
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
Increase £30 by 20%
Cube root of 27
Prime factors of 300
(-72) ÷ 8
0, 3, 8, 15, 24, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
12a³ ÷ 3ab
Factorise fully: 14a - 35
Solve 2(3a - 4) = 10
Perimeter =
Volume =
1.3cm
1.5cm
3cm
7.1cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Circumference of circle with r
= 50cm (Use pi = 3)
? km = 100 Miles
Probability of a letter with 1
line of symmetry in
MATHEMATICS
Mean of: 3, 5, 2, 1, 3, 7, 9, 1,
3, 2
Score:
/20
Practice Makes Perfect 13+ Level 4
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 691 to 1 significant
figure
5 x 14
4 x (1 + 5).
-3
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
Increase £60 by 30%
Cube root of 8
Prime factors of 660
81 ÷ -9
1, 3, 6, 10, ___, ___
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
5a³ ÷ 10a²
Factorise fully: 24a - 60
Solve 4c - 2 = 3c +9
Perimeter =
Volume =
How many days will 8 bottles Convert 17/20 into a decimal
of milk last if I drink 2/7 of a
bottle a day?
1.9cm
2.5cm
4cm
3.1cm
2cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Circumference of circle with r
= 17cm (Use pi = 3)
? km = 55 Miles
Probability of a letter with
rotational symmetry in
MATHEMATICS
Mean of: 4, 1, 0, 0, 5, 0, 4, 3,
2, 5
Score:
/20
Practice Makes Perfect 13+ Level 4
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 4280 to 2 significant
figures
12 x 7
12 - (3 + 1³)
-4
How many days will 4 bottles
of milk last if I drink 1/4 of a
bottle a day?
Convert 4/25 into a decimal
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
Increase £30 by 20%
Cube root of 27
Prime factors of 300
(-72) ÷ 8
0, 3, 8, 15, 24, __, __
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
12a³ ÷ 3ab
Factorise fully: 14a - 35
Solve 2(3a - 4) = 10
Perimeter =
Volume =
1.3cm
1.5cm
3cm
7.1cm
3cm
16. Quadrilaterals
17. Circles
18. Metric / Imperial
19. Probability
20. Averages
Name?
Circumference of circle with r
= 50cm (Use pi = 3)
? km = 100 Miles
Probability of a letter with 1
line of symmetry in
MATHEMATICS
Mean of: 3, 5, 2, 1, 3, 7, 9, 1,
3, 2
Score:
/20
Practice Makes Perfect 13+ Level 5
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 0.04039 to 3
significant figures
13 x 6
(4 - 6)³ + (12 ÷ 4)
How many bottles of coke do
I drink in 25 days if I drink 2/5
of a bottle a day?
Change 9/15 into a
percentage
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
Price of a shirt, normally £45,
if reduced by 40%?
Square root of 81
Prime factors of 32 x 90
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
(-2ab)³
Multiply out and simplify
5a - (5 - 3a)
Solve 3c + 4 = 2c - 1
Area =
Surface Area =
8 + (7 - 10)
(- -1.6, -0.8, -0.4, -0.2, ___, ___
5 + 3)
5cm
3cm
8cm
16. Quadrilaterals
17. Circles
How many lines of symmetry? Perimeter of semi circle with
radius 4cm (Use pi = 3)
1.5cm
3cm
18. Metric / Imperial
19. Probability
20. Averages
44 Lbs = ? Kg
Probability of a letter with
rotational symmetry in
MATHEMATICS
Mean of: 3, 5, 2, 1, 4 minus
mode of 7, 9, 1, 3, 2, 3
Score:
/20
Practice Makes Perfect 13+ Level 5
1. Rounding
2. Multiplication
3. BIDMAS
4. Fractions
5. FDP
Round 0.04039 to 3
significant figures
13 x 6
(4 - 6)³ + (12 ÷ 4)
How many bottles of coke do
I drink in 25 days if I drink 2/5
of a bottle a day?
Change 9/15 into a
percentage
6. Percentage
7. Number Props
8. Prime Factors
9. Negatives
10. Sequence
Price of a shirt, normally £45,
if reduced by 40%?
Square root of 81
Prime factors of 32 x 90
11. Simplify
12. Brackets
13. Equations
14. Area and Perimeter
15. Volume
(-2ab)³
Multiply out and simplify
5a - (5 - 3a)
Solve 3c + 4 = 2c - 1
Area =
Surface Area =
8 + (7 - 10)
(- -1.6, -0.8, -0.4, -0.2, ___, ___
5 + 3)
5cm
3cm
8cm
16. Quadrilaterals
17. Circles
How many lines of symmetry? Perimeter of semi circle with
radius 4cm (Use pi = 3)
1.5cm
3cm
18. Metric / Imperial
19. Probability
20. Averages
44 Lbs = ? Kg
Probability of a letter with
rotational symmetry in
MATHEMATICS
Mean of: 3, 5, 2, 1, 4 minus
mode of 7, 9, 1, 3, 2, 3
Score:
/20
Practice Makes Perfect 13+ NUMBER
1. Significant Figures
2. Decimal Places
3. 10, 100, 1000
4. BIDMAS
5. % of Amounts
Round 3082 to 2 significant
figures
Round 4.592 to 1 decimal
place
3.02 x 1000
3² - 4 x 2 + 7
Find 35% of £3
6. Fractions of Amounts
7. FDPs
8. FDP
9. % increase
10. Fractions
Find 3/8 of 24Kg
Convert 3/25 into a
percentage
Which is larger, 0.35, 34%, or
2/5?
Increase £12 by 15%
Find the sum of 3/5 and 1/8
11. Ratios
12. Express a percentage
13. Prime Number
14. Prime Factors
15. Negatives
Share £35 in the ratio 3:4
Express 12cm as a % of 3m
Write all the prime numbers
between 20 and 30
Write 48 as the product of
prime factors, using indices
Calculate 3 - 5 - 2
16. Sequences
17. Multiplication
18. Division
19. Estimation
20. Proportion
What is the 7th term in this
sequence: 1, 4, 7, 10...
Evaluate 0.7 x 0.8
Calculate 36 ÷ 0.9
Estimate £39.30 x 9.03
A recipe for 4 people need
200g of sugar. How much
sugar is needed for 6 people?
Score:
/20
Practice Makes Perfect 13+ ALGEBRA
1. Simple Formulaie
2. Substitution
3. Substitution with
indices
I think of a number, multiply
by 3 and add 4 to get 19.
What number did I think of?
a = 2, b = -3
3a - b =
6. Brackets
4. Collecting Like Terms
5. Multiplying Terms
a = 4, b = -1
a² - b²
4a - 2b + a - 3b
3ab x -2b
7. Factorise
8. Convention
9. One step equation
10. Two Step Equations
Multiply out and simplify
3a - 2(a - 4)
Factorise fully:
12a - 18b
Write the product of x and y
4a = 36
3b - 3 = -9
11. Equation with
Brackets
12. Equation with
variables on both sides
13. Equation with
Fractions
14. Forming Expressions
15. Writing Expressions
2(3c - 1) = 22
4d - 2 = 3d + 9
Perimeter =
Bob has n sweets. Jimmy has
twice as many as Bob. Write
the number of sweets Jimmy
has in terms of n.
1e=8
4
3a
5a
16. Writing Expressions
Bobina has n + 3 sweets.
Jimmette has 3 times as
many. How many does
Jimmette have?
17. Straight Line Graphs
If y = 2x + 3, fill in the table:
x = -1 0 3
y=
18. Straight Line Graphs
If y = 4 - 2x, fill in the table:
x = -1 0 3
y=
19. Graphs
20. Graphs
Draw x = 3
Draw y = x
Score:
/20
Practice Makes Perfect 13+ SHAPE
1. Units
2. Units
3. Units
4. SDT
5. SDT
Convert 3.502km into metres
Convert 320g into Kg
Estimate 25miles in Km
A man runs at 8km/h for 30
minutes. How far does he go?
A woman runs 12km at
8km/h. How long, in hours
and minutes, does it take?
6. Congruent
7. Symmetry
8. Symmetry
9. Quadrilaterals
11. Polygons
Order of rotational
symmetry?
Name
Interior angle =
What does congruent mean? How many lines of symmetry?
11. Area
12. Area
13. Perimeter
14. Nets
15. Volume
Area =
Area =
Perimeter =
Sketch the net of a cube
Volume =
2cm
4cm
16. Circles
4cm
3cm
7cm
17. Circles
3cm
3cm
2cm
5cm
5cm
18. Reflection
19. Angles
Area of a circle with r = 4 (use Circumference of circle with
pi = 3)
r = 3 (Use pi = 3)
Calculate x
2cm
20. Angles
Calculate x
x
30
x
50
Score:
/20
Practice Makes Perfect 13+ DATA
1. Mode
2. Median
3. Mean
4. Range
5. Averages
Calculate the mode of
4, 2, 7, 2, 5, 2
Find the median of
4, 5, 1, 2, 9
Find the mean of
4, 2, 6, 9, 4
Find the range of
4, 2, 7, 5, 0
Three numbers have a mode
of 4 and a mean of 5. What
are they?
6. Venn Diagram
7. Carroll diagram
8. Frequency Table
9. Frequency Table
10. Pie Charts
20 people, 12 like rounders,
15 like cricket, 7 like both
Complete:
Girl
8AG
Boy
7
8S
Total
17
9
Total
34
Calculate Median
Cars Frequency
1
5
2
4
3
1
4
1
Calculate Mean
Pets Frequency
0
4
1
3
2
1
3
2
20 people. How many degrees
per person?
11. Pie Charts
12. Pie Charts
13. Scatter Plot
14. Scatter Plots
1/3 of people like Bond
movies. How many degrees
on a pie chart?
Estimate percentage:
Describe correlation
Finish sentence:
As
people run faster, they
.............................................
15. Conversion
$
$50 =
50
speed
10 20
weight
16. Probability
Probability of prime number
on fair die?
17. Probability
18. Probability
Probability of picking blue
Probability of violin player =
sweet = 1/6. In total, 24
2/13. Probability of non-violin
sweets. How many are blue?
player?
£
30 40 50
19. Outcome Table
20. Probability
Multiply two dice
Probability of square number
x
1
2
3
2
3
4
Score:
/20