Download 3239

Supplementary
Materials for
Mathematics
Audits
(Primary Teacher Training)
Euros Davies
Jan Morgan
The aim of the materials is to act as revision guides for trainees who are
undertaking a primary teacher training degree programme. It is not
intended that the trainees will necessarily work through every sheet, but
choose those most appropriate to their needs.
Worked examples and exercises are given, and answers can be found at
the end of the pack of materials. The Index is cross- referenced to
enable students to find the most appropriate material for them.
Individual tutors can use the materials as revision guides with a group of
students, or the materials can be worked through by individual students.
It should be stressed that the sheets are for revision purposes –
individual students may need further teaching or assistance if the topic is
a new one for them. It may be necessary for the students to be referred
to further examples or exercises for extension purposes. Further reading
may also be needed.
Other useful resources may be:
www.bbc.co.uk
GCSE Bitesize revision guide
Letts Revision Guide
CGP Revision Guide
This research and supplementary materials pack have been made possible
by financial support from ESCalate.
Contents
1. Averages
2. Brackets
3. Circles
4. Co-ordinates
5. Disproving a Hypothesis
6. Estimating results of calculations
7. Factorising algebraic expressions
8. Factors
9. Forming algebraic expressions
10. Fractions
11. Gradient
12. Grouped Data
13. Inequalities
14. Interpretation of a Graph
15. Nth terms
16. Order of arithmetic operations
17. Order of size / Rational – irrational numbers
18. Probability
19. Pythagoras’ Theorem
20. Ratio
21. Rounding
22. Straight lines
23. Surface area of 3D shapes
24. Tree diagrams
Topic: Averages
There are three types of average – mean, median, mode.

Consider the set of numbers:
7, 3, 4, 5, 4, 9, 2, 4, 6, 3
Mean – add up all the data and divide by the number of items = 47/10 = 4.7
Median – put the data in order and pick the middle one: 2, 3, 3, 4, 4, 4, 5, 6, 7, 9
(if there are two middle ones, add them and halve)
So median = 4
Mode – the most common data item = 4

Each has its advantages and disadvantages.
Median and mode are usually one of the data items; mean usually isn’t.
Median and mode are unaffected by extreme values.
Mean is best for further calculation, but adversely affected by extreme values.

Sometimes the data is given in table form – a frequency distribution:
Here, the number 3 occurs 4 times in the set, for example.
So totalling the 3s is easier by calculating 3x4 = 12.
These partial products are then added to give the grand total.
So Mean = ∑fx = 28 = 2.333
∑f
12
Median = 2,
Mode = 2,
x
1
2
3
4
f
2
5
4
1
12
fx
2
10
12
4
28
(there are 12 items, so middle one is between 6 th and 7th)
(1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4)
the most common, as there are five of them.
Exercises
1.
The hourly wages of employees in a firm are £15, £12, £ 20, £8, £15, £25. Work
out the average hourly rate (all three of them!)
2. If the employee earning £25 per hour has an increase to £100 per hour, what
effect does this now have on the three averages?
3. The mean age of ten boys is 9.6 years and of 8 girls is 10.2 years. What is their
combined average age?
4.
Work out all averages for the following data:
x
f
3
2
5
4
6
5
7
2
NB. x = item of data; f = frequency; ∑ = sum of
Topic: Brackets
See also: Factorising Algebraic Expressions sheet

Young children use these ideas when multiplying:
6 x 23 = 6 x (20 + 3) = 6 x 20 + 6 x 3 = 120 + 18 = 138

2a = a + a, so 2(y + z) = y + z + y + z = 2y + 2z
3(w – 2) = w – 2 + w – 2 + w – 2 = 3w – 6

This process is equivalent to multiplying each of the terms inside the bracket by the
term outside:
6(c – 3) = 6c – 18
3(p + q - 2r) = 3p + 3q – 6r
a(b + c) = ab + ac
x(x + y) = x2 + xy

Using a grid:

y
2 2y
z
2z
x
x
x2
y
xy
If we have two sets of brackets, the approach is the same:
(x + 1)(x + 2) = (x + 1)x + (x + 1)2
x
= x2 + x + 2x + 2
x
x2
= x2 + 3x + 2
2 2x
Exercises
Multiply out:
2(3 + 2x) =
4(a + 2b + 3c) =
x(x – 1) =
a(b + c – a) =
3y(x + z) =
x(x2 – ab) =
2w(w – 3) =
4(2x –y + 3z) =
(x + 3)(x + 4) =
(x – 1)(x + 2) =
1
x
2
(p + 2)(p –2) =
(m +2)(2m + 1) =
Topic: The Circle

People often mix up the circumference (or perimeter) and the area of a circle.
The circumference is a distance – all the way round the circle - 2πr or πd
where r = radius, d = diameter.
It is measured in ‘length units’ (centimetres or metres, for example)
The area is the space enclosed by the circumference; area = πr 2
It is measured in ‘area units’ (cm2 or m2, for example)

Example:
r
r
A semicircle has diameter 20 cm.
What is its perimeter?
What is its area?
20 cm.
It has half the circumference of a full circle
For the curved part = ½ x πd = ½ x 3.14159 x 20 = 31.4159 cm.
Remember to add the diameter here!
So total perimeter = 31.4159 + 20 = 51.4159 cm.
Area of semicircle = half area of whole circle
= ½πr2 = ½ x 3.14159 x 102
= 157.08 cm2
(NB. d = 20, so r = 10)
Exercise.
1. Calculate the perimeter and area of the given shape
(Semicircle and rectangle)
6
14
2.
A bicycle wheel has diameter 80 cm.
How far would I cycle if it made 40 revolutions?
d
3. A washer looks like two concentric circles of
diameters 10 and 20 mm. respectively.
What is the area of the washer ring?
Topic: Coordinates
Vocab: Axis, axes, coordinate, origin
See also: Interpretation of a Graph sheet
When position is to be identified on a piece of paper, it has to be referenced to a starting point (origin).

On a straight line, we just need to know the distance
along the line from the origin, O.
4eg. A is 3 units from O
B is –2 units from O
B
-2
-3
-1 O
1
2
A
3
y

In two dimensions, we need to use two reference
lines – called axes. These are usually labelled
the x-axis and y-axis.
• P(4,3)
Q
The point P has coordinates (4,3). This means
It is 4 units in the x-direction and 3 units in the
y-direction. Points are marked with a cross.
•
0
• R
x
The ‘horizontal’ coordinate is given first.
Q is (-2,1); R is (0, -1)


Axes do not have to be labelled x and y.
A car is travelling along a straight road and is 1km away
after 5 mins and 4km away after 10 mins.
We plot Time (t) horizontally and Distance (d) vertically:
d
4
1
+ (10, 4)
+ (5, 1)
5
10
t
Geographers use this system for map references.
(613206: 613
, 206
; walk in the house, then go up the stairs!)
Exercises
1.
The following coordinates are three corners of a rectangle. Plot them and find the coordinates of
the remaining corner: (3, 4), (5, 4) , (5, -1) .
2.
The following points are reflected in the y-axis. Plot them and give the coordinates of their images:
(2, 3) , (1, 1) , (3, 0) , (0, 2) , (-2, -1)
3.
Which point is furthest from the origin: (3, 5)
(6, 0)
(-2, 5.5)
(-1, -6) ?
Topic: Disproving a hypothesis

If a given statement has to be disproved, all you have to do is to find one example to
contradict the statement.
Eg. Hypothesis:
all even numbers are divisible by 4
Try 14. This is even. 14 = 3 rem 2;
4
So 14 is not divisible by 4.
Thus statement is incorrect.
Eg. Hypothesis:
1 + 1 = 2
m n
m+n
Try m =2 and n = 4;
statement give s 1 + 1 = 2 = 1
2
4
6
3
But we know that ½+ ¼ = ¾
So hypothesis is incorrect.

You cannot use this approach to prove a hypothesis is correct.
Eg. Hypothesis: if m is a whole number, 2m is always even.
Try m = 3:
m = 4:
2m = 2x3 = 6; even
2m = 2x4 = 8; even
This only ‘proves’ the hypothesis is true when m = 3 and m = 4.
It confirms nothing else!
Exercise.
1.
Show that the following statements are incorrect:
- the sum of any three consecutive whole numbers is always odd
- the square of any number is larger than the original number
- (a + b)2 = a2 + b2 for any values of a and b
2. The cube of any odd number is odd, because 3 3 = 27, 53 = 125, 73 = 343.
What conclusion can we draw from this?
Topic: Estimating results of calculations

The ability to obtain a rough estimate to a calculation is quite useful. It gives you an
approximate value to the answer which is a useful check before doing the full
calculation.
Eg.
31.6 x 6.9
this is roughly 30 x 7 = 210
(Actual answer = 218.04)
19.8
7.6
roughly 20 = 2.5
8
(Actual answer = 2.6052631)
If you had used a calculator to answer 31.6 x 6.9 and got the answer 2180.4,
your estimate (210) would show you something was wrong – possibly an omitted
decimal point.

This can be used in another way too.
Suppose we know 32 x 74 = 2368. what is 3.2 x 74?
Using this information, we can work related calculations:
3.2 is roughly 3; 74 is roughly 75, giving an approximated result 3 x 75 = 225.
So 3.2 x 74 = 236.8
What is 0.32 x 740?
This is roughly 1/3 x 750 = 250.
So 0.32 x 740 = 236.8
What is 32 x 0.74?
This is roughly 32 x ¾ = 24
So 32 x 0.74 = 23.68
What is 32 x 75?
Roughly 30 x 80 = 2400
32 x 75 = 32 x (74 + 1)
= 32 x 74 + 32 x 1
= 2368 + 32
= 2400
Exercise.
1.
Give approximate answers:
5.1 x 19.4 =
38 x 11.3 =
2.
Given 83 x 15 = 1245, underline the correct answer:
8.3 x 15 =
12.45
124.5
8.3 x 150 =
12.45 =
1.5
85 x 15 =
12.45
0.83
1247
1245
124.5
1245
8.3
83
1275
1285
Topic: Factorising algebraic expressions
See also: Brackets sheet

You can think of this as just the ‘opposite process’ to multiplying out brackets.
2a + 2b has a common factor of 2, so 2a + 2b = 2(a + b)

Perimeter of a rectangle of dimensions a cm. by b cm. is a + b + a + b = 2a + 2b cm.
Both terms contain a ’2’, so it can be regarded as a common factor and written
2(a + b).
a
We can interpret this as:
b
the perimeter is (2a + 2b), twice the length plus twice the width
or
the perimeter is 2(a + b), length plus width all doubled.

Similarly, 4xy + 4xw = 4(xy + xw), a 4 is a common factor.
But x is also a common factor, so the full factorisation is 4x(y + w)

xy + xz = x(y + z), as x is a common factor
x3 + x2z = xxx + xxz = x2(x + z), as x2 is a common factor
3x + 6 = 3(x + 2), as 3 is a common factor
4x + 6xy = 2x(2 + 3y), as both 2 and x are common factors
2x + 4y – 6z = 2(x + 2y - 3z)
(You can check that each factorisation is correct by multiplying out the brackets:
eg.2(x + 2y – 3z) = 2x + 4y - 6z)
Exercises
Factorise:
6x + 3y =
8 – 2x =
xy + 10x2 =
6abc – 8bcd =
x3 + x 2 =
6s3 – s =
3xy + 6y2 –9y =
Topic: Factors
Vocab: Factor, multiple, prime, composite number, proper factor

A factor is a whole number that divides exactly into another.
So we can say that 6 is a factor of 18; or 18 is a multiple of 6.

Factors usually come in pairs eg. as 6 is a factor of 18, so is 3, as 18/6 = 3.
So factors of 18 are:
1 and 18; 2 and 9; 3 and 6
ie. {1,2,3,6,9,18}
Factors of 30 are
1 & 30, 2 & 15, 3 & 10, 5 & 6
ie. {1,2,3,5,6,10,15,30}
Note1: 1, 2, 3, 6 are common factors of both 18 and 30.
Note2: a number is a factor of itself: 6 is a factor of 6
Note3: the proper factors of a number do not include the number itself: proper factors of 6 are 1, 2, 3

A prime number has exactly two factors eg. 7 is prime as its only factors are 1 & 7.
Composite numbers are non primes:
18 is composite; 19 is prime.

All numbers can be written as an unique product of prime numbers
18 = 2x3x3 ; 30 = 2x3x5 ; 72 = 2x2x2x3x3 = 23 x 32
Exercises
1. Put the following numbers into one of two sets – prime or composite numbers:
14, 27, 7, 2, 11, 49, 37, 12
2. Find the factors of the following numbers: 21, 31, 15.
Which are prime?
3. Write the following as products of primes:
20 =
32 =
42 =
3. The number 6 is called a perfect number, as it is equal to the sum of its proper
factors, ( 1 + 2 + 3 = 6). Show 28 is also perfect.
Topic: Forming algebraic expressions

m + m = 2m; 2m means ‘twice m’ (or ‘m multiplied by 2’)
n + 2n = 3n; 3n means ‘n multiplied by 3’

y2 is read as ‘y squared’, meaning ‘y multiplied by itself’.

Algebraic expressions can be created one step at a time:
- a number n is doubled and then four is added:
n is doubled
then add 4
2n
2n + 4
-
a number m is cubed, then the result is doubled and five subtracted; then this
result is halved:
m cubed
m3
then double
2m3
subtract 5
2m3 – 5
halved
½(2m3 – 5)
-
to convert degrees Fahrenheit to Centigrade, subtract 32; then multiply by 5
and divide by 9:
degrees Fahrenheit
F
subtract 32
F – 32
multiply by 5
5(F – 32)
divide by 9
5(F – 32)/9 = C
Exercise.
1.
A boy is x years old. His father is five times older.
So father is _____ years old.
2. A boy is m years old. In two years’ time, his father will be six times as old
as the boy will be.
So the father is _________ years old now.
3. N is a whole, positive, even number. What is the next even number?
4. Write down an expression for the mean (average) of numbers p and q.
Topic: Fractions

Equivalent fractions
If you multiply the top (numerator) and bottom (denominator) of a fraction by the
same number, the resulting fraction is equal to the original – they are equivalent to
one another;
eg.
1= 1x3 = 3;
so 1 = 3
2
2x3
6
2
6
2 = 2X4 =8;
5
5 x 4 20

so 2 = 8
5
20
Adding fractions
Fractions can only be added if their denominators are the same
Eg. 1/5 + 2/5 = 3/5
(one fifth and two fifths give three fifths)
So the first stage is to make sure that both fractions have the same denominator –
we need to create equivalent fractions.
Eg. 1/5 + 2/3 ; we’ll make the denominators 15
So
1 = 1x3 = 3 ;
2 = 2 x 5 = 10
5
1x3
15
3 3x5
15
Hence, 1 + 2 = 3 + 10 = 13
5 3
15 15
15

Subtraction – use the same approach as for addition

Multiplication
To multiply two fractions together:
Either – multiply the numerators, then the denominators, then cancel common factors
Or – cancel common factors top and bottom, then multiply numerators, then denominators
Example 1: 8 x 3 = 24 = 2
15
4
60 5
or:
(Divide by common factor 12)
2
8 x 31 = 2
41
5
5 15
Example 2: mixed fractions have to be converted to top heavy ones first
21/10 x 12/3 = 721 x 51 = 7 = 31/2
31
2
210
Exercise.
1.
Equivalent fractions: write 2/9 in three different ways.
4
2. Cancel down to lowest terms:
3. Evaluate:
3
/10 + 5/8 ;
/10
,
28
/42
,
111
/141
22/5 x 11/9
Addition of fractions
We need the same denominators (bottom part) before fractions can be added.
1
Eg1.
/3 + 1/5
look at the lowest common multiple of 3 and 5
(ie. smallest number both 3 and 5 divide into)
LCM (3,5) is 15.
Convert both fractions into fifteenths:
So 1/3 + 1/5 = 5/15 + 3/15 =
Eg2.
3
/4 + 1/6
Converting
8
1 x 5 = 5 and 1 x 3 = 3
3 5 15
5 3 15
/15
LCM (4,6) = 12; (could use 24, but 12 is lowest )
3x3 = 9
4 3
12
1x2 = 2
6 2
12
So 3/4 + 1/6 = 9/12 + 2/12 = 11/12
[If we’d used 24 as denominator, result would be the same, but a common factor
would have to be cancelled out at the end:
Converting:
3 x 6 = 18
4 6
24
So 3/4 + 1/6 = 18/24 + 4/24 =
Eg3.
5
/12 + 1/6
1x4 = 4
6 4 24
22
/24 =
Here LCM is 12, etc.
Multiplication of fractions
11
/12
]
Here common factors between numerator and denominator (top and bottom) can be
cancelled out – the direction of the cancelling lines indicating which pairs of numbers
have been affected:
2
6 x 51 = 2
5
25 93 15
So factor of 3 cancelled from 6 & 9
factor of 5 cancelled from 5 & 25
Topic: Gradient
Vocab: Gradient, slope, coordinates, axis
See also: Straight Lines sheet
The gradient or slope of a straight line is used to determine how steep the line is. It is
possible to have a positive, a negative or a zero gradient.
Definition:
The gradient of a line indicates how fast a quantity is changing.
A working definition is:
Gradient = increase vertically (parallel to y-axis)
increase horizontally (parallel to x-axis)
Process:
Pick two points on the line.
Complete a right angled triangle with
short sides parallel to the two axes
and the hypotenuse as part of the line.
Work out the coordinates of both points.
Find vertical and horizontal differences.
Calculate gradient.
Example1:
Let A(2,3) and B(6,5)
As you move A
B,
both coordinates increase.
y
B
A
2
4
Gradient = 2/4 = ½
x
Example2:
Let A(2,4) and B(6,3).
As A
B, one coordinate
increases and one decreases,
so a negative slope.
y
A
-1
B
4
Gradient = -1/4
x
Example3:
A horizontal line has no vertical increase, so gradient = 0.
Example4:
Plotting Distance vertically and Time horizontally would give
Gradient = increase in distance = speed.
increase in time
So the greater the slope, the faster the speed.
A
Exercises
1.
y
B
Work out the gradient of these lines:
(2,2)
1
x
A
y
2. Which of these lines has: a negative slope
a zero gradient ?
B
x
C
3. On the distance-time graph shown, how can
you interpret the gradients indicated by
line OA, line AB?
d
A
B
t
O
Topic: Grouped data
See also: Averages sheet
 Data is often grouped into classes to give a frequency distribution, as it is easier for
the reader to understand when there is a large amount of data.

If we had scores: 3,4,9,5,4,3,1,7,8,11,19,4,12,14,15,6,9,9,11,4,3,14,18,16,18 , then,
with little effort, we could work out the actual mean, median, and mode (9.1, 9, 4 resp).

If we had over 100 items though, it might be quite a long task. So we group the data
into classes and indicate how many items fall into each class (the frequency).
Class
0 - <5
5 - <10
10 - <15
15 - <20
f
8
7
5
5
∑f = 25
mid value(x) fx
2.5
20
7.5
52.5
12.5
62.5
17.5
87.5
∑fx = 222.5
So, although we can see from the table that there are eight scores between 0 and
under 5, we cannot actually say what these scores are unless the original data is still
available. If it isn’t, then we take the mid-point of the class as a representative
value for the whole class – so we have 8 scores of 2.5, giving 20 in total, when the
actual total is 26.
Obviously, this can lead to a level of inaccuracy, but it’s the best we can do.
Mean = Total (approx) sum ∑fx = 222.5 = 8.9
Number of scores ∑f
25
Mode = we can only say that the ‘modal class’ is 0 - <5.
Median – we need to create an ogive (cumulative frequency graph), so we need
two more columns in our table.
Class
f
upper bound
cum f
0 - <5
5 - <10
10 - <15
15 - <20
8
7
5
5
<5
<10
<15
<20
8
15
20
25
We can now draw the graph:
Using the graph, we can work out
the median – at the 50% mark reading off horizontal axis.
We can also work out the 25% and 75%
values; the difference between these
gives the interquartile range.
= 32%
= 60%
= 80%
= 100%
%cf
75
25
LQ M UQ
Upper Bd
Topic: Inequalities

We use the symbols ‘=’ (equals, is equal to) and ‘ = ’ (not equal to) quite often, but
there are four others that you need to understand too:
> greater than, < less than, ≥ greater than or equals,
≤ less than or equals

x > 3 means the value of x is greater than 3
(so x could be 3.1, 4, 11, 15.6, . . . )
º
3
x ≥ 3 means that x could also equal 3 itself
•
3

Sometimes we want to limit x to a small range of values – say between 2 and 5 –
so x > 2 and x < 5; this can be written as 2 < x < 5
2
5
Examples
1.
2 < x ≤ 5 means 2<x and x ≤ 5 , x is between 2 (not included) and 5 (included)
2. If x is an integer (whole positive or negative number) and -3 ≤ x < 2,
Then x could be -3, -2, -1, 0, 1.
3. Suppose we know that x is in the set {-5, -4, -3, . . . ., 2, 3} and x < -3 or x > 1.
Then x could be –5, -4, 2, 3.
If we asked for values such that x < -3 and x > 1, that’s impossible, as both
inequalities cannot be true at the same time.
4. The ‘opposite’ (or complement) of x > 3 is not x < 3.
The numbers, which are not in the set of numbers greater than 3, must include all
those less than 3 and 3 itself. So, if x > 3, then the complement is x ≤ 3.
Topic: Interpretation of a graph
See also: Coordinates Sheet

speed
Given - a graph of an object’s speed at certain times.
At time t = 0, object is stationary (ie. speed = 0)
A
B
OA: speed increases/ object accelerates for 2 mins.
AB: speed constant for 2 mins.
BC: object slows down (decelerates) until it stops
O
2
4
after 7 mins.
(NB. the graph does not tell us the object returns to its start point.)
C
7
time
Distance

Given – a graph of distance (from Wrexham) against time.
OA: car moves off from Wrexham for 1 min.
AB: distance from Wrexham doesn’t change
Stationary
Or moving in a circle, centre Wrexham!
BC: returns to Wrexham in 30 secs.
A
C
O
(BC is steeper than OA, so return speed is greater than outward speed.)
Exercise
1
3
4
time
h
Graph represents man climbing ladder,
plotting height (m.) versus time (secs).
What could be happening between
- A and B ?
- C and D ?
B
C
A
B
O
D
t
Topic: Nth terms
If a sequence of numbers changes in a regular way, it is often possible to create a
general (or nth ) term for the whole sequence. If we know the form of this general term,
we can use it to predict new terms of the sequence.

Suppose the nth term is (4n + 3).
By substituting values of n into this formula, we can generate the sequence itself:
When n = 1, 4n + 3 = 4x1 +3 = 7
When n = 2, 4n + 3 = 4x2 + 3 = 11
When n = 3, 4n + 3 = 4x3 + 3 = 15 etc
So the sequence begins 7, 11, 15, . . . .

Suppose the sequence begins:
2
Write term number (n) underneath:
1
So the term value is double the term number.
General (nth) term = 2n

Sequence:
3
5
7
9
11
Term number (n):
1
2
3
4
5
The difference between successive terms is 2.
So the general term is (2n + k); we need to work out the value of k.
When n = 1, term value = 3 , so 2n + k = 2x1 + k = 3, giving k = 1.
So general term is (2n + 1).

Sequence:
2
5
8
11
14
17
Term number:
1
2
3
4
5
6
Difference is 3 each time.
So general term is (3n + k).
When n = 1, term value is 2, so 3n +k = 3x1 + k = 2, giving k = -1.
So general term is (3n – 1).

If we have a diagram to work with, rather than just a sequence, we can use a
different approach.
4
2
6
3
8
4
10
5
Term No.
1
•
• •
2
•
•
• • •
3
.....
•
•
•
• • • •
n
Term value
3
5
7
?
Vertical dots
Horizl dots
2
1
3
2
4
3
Exercise
1. Find the nth terms:
3, 6, 9, 12, . . . . .
6, 10, 14, 18, 22, . . . .
2½, 3¼, 4, 4¾, . . . .
2. Find number of matches in the n
th
picture:
n+1
n
General term = 2n + 1
Topic: Order of arithmetic operations

In mathematics, there are certain operational conventions that need to be followed
in order to avoid confusion. For example, does 3 + 4 x 5 give 35 or 23?

The order in which operations should be performed is easily remembered with the
mnemonic BODMAS Brackets
Of
Divide, Multiply
(With two operations on the same level,
Add, Subtract
work from left to right through expression)

Examples
3 + 4 x 5 = 3 + 20 = 23
(3 + 4) x 5 = 7 x 5 = 35
12 ÷ 3 x 4 = 4 x 4 = 16
1
/5 of 10 + 3 = 2 + 3 = 5
Exercise
Evaluate:
14 + 6 ÷3 =
3 x (4 – 2) =
7+4x3=
(4.6 + 7.1) x 2 – 1.3 =
5 + 3 –2 + 1 = 8 –2 + 1 = 6 + 1 = 7
10 ÷ 2 + 3 x 2 = 5 + 6 = 11
10 ÷ (2 + 3) x 2 = 10 ÷ 5 x 2 = 2 x 2 = 4
Topic: Order of size


To compare numbers, it is best to convert them to decimals:
1
/5 = 0.2 ; 3/4 = 0.75 ; 1/8 = 0.125;
so 0.125 < 0.2 < 0.75 or
1
/8 < 1/5 < 3/4
To convert a fraction into a decimal, divide numerator by denominator
Eg.
0. 75
3
/4 = 4 )3.00
Exercise.
1. Put in numerical order:
2. Put in numerical order:
3
/8
0.3
1
5
/8
61
2
/100
/3
/3
2
/5
65%
4
/9
0.62
Topic: Rational/Irrational numbers

Rational numbers can be written as the ratio of two whole numbers (ie fractions). So
any number that can be written in this way is rational:
2 = 2/1 = 4/2 ; -5 = -5/1

Irrational numbers cannot be written as fractions eg. π, √3, 2 + √5
The decimal expansion does not repeat, but does ‘go on for ever’:
π = 3.1415926535897932384626433832795. . . .
√3= 1.73205080756887729352744634150587. . .
(NB1. Whilst the square roots of most numbers are irrational, some are not
eg √9 = 3 exactly, so is rational.
NB2. Recurring decimals are rational: 1/3 = 0.33333333333. . . . . .
2
/7 = 0.2857142857142857. . . . .
)
Topic: Probability

Probability is an indication of the chance that something will happen. It can be
worked out accurately in some cases, but, in others, it can only be approximated by
experiment.

Theoretically
Probability
of event
=
number of ways in which event can happen
total number of possible ways things can happen
Example: What is the probability of getting an even number with an ordinary six sided
die?
The die can fall in six ways altogether
Number of ways to get even number = 3 (2 or 4 or 6)
So Prob (even) = 3 = ½
6

Experimentally,
Relative frequency =
number of times we get required result
total number of trials
In many cases, we cannot work out the probability of an event occurring
theoretically, so we have to conduct trials to calculate the relative frequency.
After a great number of trials, the relative frequency should ‘settle down’ to a
particular value – which is then taken to be the probability that the required
event will occur.
Example: What is the chance a drawing pin lands tip up?
not
This cannot be determined theoretically, so we will have to drop the pin
many times and count the number of times it lands tip up.
No. of throws
10
No. tip up
5
Rel. freq.
½ = 0.5
25
100
1000
2500
9
44
421
1051
0.36
0.44
0.421
0.4204
So relative frequency seems to be settling down to about 0.42, which is
taken as the probability that the pin ends tip up.
With only a small number of trials, it is unlikely that the RF will be
anywhere near the final value, but, after a large number of trials, the RF
should be fairly steady.
Topic: Pythagoras’ Theorem
Hypotenuse
Side1

This is an important result which only holds true
when we have a right angled triangle.
Side2
If we know the lengths of the three sides, then:
(Hypotenuse)2 = (Side1)2 + (Side2)2
2
2
5
2
eg. 5 = 25 ; 3 + 4 = 9 + 16
So a 3, 4, 5 triangle is right angled as 52 = 32 + 42
3
4

Example 1: Calculate the length of the diagonal of a rectangle with sides 2 by 5 cm.
Let diagonal length be y cm.
Pythagoras’ theorem:
y2 = 22 + 52
= 4 + 25
=29
2
5
So y = √29 = 5.385 to 3 dec pl.

Example 2: A 10m. long ladder is leaning against a wall with
its top touching the wall 9m. above the ground.
How far is the base of the ladder from the wall?
9
Let base be d metres from the wall.
Pythagoras’ theorem: 102 = 92 + d2
100 = 81 + d2
10
d2 = 100 – 81 = 19
d = √19 = 4.36 to 3 sig figs
d
Exercise.
y
3
Find m.
7
3
m
(Hint: find y first)
Qu. Why can’t we just say m2 = 32 + 72 ?
Topic: Ratio

This is a way of comparing two quantities, provided they are given in the same units:
Eg.
Two trees are 6m and 10m tall.
Ratio of their heights is 6 : 10 (or 3 : 5); first is 3/5 the height of the second
In a class, there are 20 boys and 15 girls.
So ratio of boys to girls is 20 : 15 (or 4 : 3)

Ratios are usually expressed in the form a : b and cancelled down to lowest terms,
using common factors. The numbers are usually left as whole numbers; thus we
usually write 2 : 3, not 1 : 1.5

Maps use scale factors – say 1cm on map is 1km on the ground. Then the ratio is given
as 1: 100000. (NB. same units)
Example 1:
A model of a classic car is built in the ration 1 : 20.
If the model is 25cm long, how long is the actual car?
Every 1cm on the model is equivalent to 20cm on actual car
So 25cm on the model is equivalent to 20 x 25 cm = 500cm = 5m. on car.
Example 2:
The height of the actual car above is 1.8m, what is the height of the model?
Ratio is 1 : 20, so model is 1/20 the size of the car.
So height of model = 1/20 of 1.8m = 9 cm.
(NB. ratio only affects linear dimensions – number of wheels does not
increase in ratio 1 : 20, nor angles between spokes!)
Example 3:
The ratio of boys to girls in a group is 3 : 5. If there are 56 children, how many
of them are boys?
3:5 means 3 boys for every 5 girls, so 8 children, 3/8 are boys.
So out of 56 children, there are 3/8 of 56 = 21 boys.
Example 4:
120 gm of a mixture contains substances A, B, and C in the ratio 2:3:1. How much
of each is there in the mixture?
2:3:1 means 2 parts A, 3 parts B and 1 part C – a total of 6 parts.
So each part is 120/6 gm = 20 gm.
Thus, 40gm of A, 60gm of B, 20gm of C.
Exercise
1.
Express in lowest terms:
4:6
10 : 25
45 : 27
2.
A model of a wedge is made in the ratio 1 : 12.
If the length and the slope of the actual wedge are 108cm and 18º respectively,
what are the corresponding values on the model?
Topic: Rounding

If asked your age, you are unlikely to say 19 years 3 months 2 weeks and 4 days. You
usually ‘round’ (approximate) to the nearest whole year (ie. you’re 19).
Of course, if you are 19 years 10 months, you would say ‘nearly 20’.

We also do this with other types of data:
Whole numbers
246
246
is 250 to the nearest 10
is 200 to the nearest 100
Decimal numbers
6.236 is 6.2 to 1 decimal place
6.236 is 6.24 to 2 dec pl
If the digit in the place just beyond what is required is 5 or more, we round up
by 1:
3.2783
is 3.278 to 3 dec pl
3.2783
is 3.28 to 2 dec pl
3.2783
is 3.3 to 1 dec pl
3.885
3.885
14.697
14.697

is 3.89 to 2dec pl
is 3.9 to 1 dec pl
is 14.70 to 2 dec pl (NB. Zero needed here)
is 14.7 to 1 dec pl
Another system uses significant figures – this will include all digits, not just those after the
decimal point:
Eg.
3.146
3.155
30.08
= 3.15 to 3 sig figs
= 3.2 to 2 sig figs
= 30.1 to 3 sig figs
NB. If you have leading zeros, these are included, but not counted:
0.01278 = 0.0128 to 3 sig figs
0.00219 = 0.0022 to 2 sig figs
Exercise
1. Write the following correct to 3 decimal places: 2. 4567
43.23589
0.00245
1.2945
2.
Write the following correct to 3 significant figures:
2.3456
3.00763
0.012875
0.0024961
Topic: Straight lines
See also: Gradient sheet
In general, equations of the form 2x + 3y = 6 and y = 3x – 1 give graphs which are straight lines.
They can always be written in the form y = mx + c.
(The equation 2x + 3y = 6 can be re-written 3y = -2x +6, then y = -2/3 x + 2 )
When in this form, m gives the slope or gradient of the line and c gives the intercept on the yaxis.
y
B
A
A: y = 2x –1
B: y = x + 2
D
C: y = 1 – x (y = -x + 1)
D: y = x
E: 2x + 3y +6 (y = -2/3 x + 2)
E
x
C
A is the steepest line, as it has a slope of 2.
B has slope of 1; C has slope of –1.



Lines intersect the y-axis (the intercepts on the y-axis):
A: at –1
B: at 2
C: at 1
D: at 0
E: at 2
Lines with the same slope (m value) are parallel – look at B and D above.
Example: Find the equation of the line passing
through (0, 1) and (3, 7).
6 =2
3
Intercept on y-axis, c = 1
So equation is y = 2x + 1
y
(3, 7)
Slope, m =
6
1
3
x

To plot a straight line, we usually choose three values of x and work out the corresponding y
coordinates by substituting in the equation.
Eg. When y = 2x + 1, if
x = 0, y = 2x0 + 1 = 1
x = 1, y = 2x1 + 1 = 3
x = 2, y = 2x2 + 1 = 5
Now plot the points (0,1), (1,3), (2,5) on graph paper.
Exercises
1. Write in the form y = mx + c :
Which line is the steepest?
y – 2x = 1
2y + 4 = 3x
1 + 3x = 3y
2.
Where do the lines in Qu.1 cross the y-axis?
3.
What is the equation of the line that has slope 2, passing through the point (0, 2)?
3.
What is the equation of the line passing through the points (0, -1) and (2, 5)?
Topic: Surface area of 3D shapes

We need to think of the net of the shape. The ‘faces’ of 3D shapes are either flat
[or curved]. Flat surfaces form polygons – triangles, rectangles, circles . . . .
[Curved surfaces could be on spheres, cylinders, cones, . . . . . ]

Area of triangle
= ½x base x height
Area of rectangle
= length x height
Area of circle
= πr2
r

Curved surface of cylinder: opens out to give a rectangle (and two circles)
r
h
= 2πrh + 2πr2
2πr
r
h
r
]
Example1:
Cuboid 6 x 4 x 3 cm.
Surface area = 2A + 2B + 2C
A: 3 x 4 = 12 cm2
B: 3 x 6 = 18 cm2
4
3
6
C
B
A
C: 4 x 6 = 24 cm2
Example2:
So SA = 108cm2.
Wedge 3 x 4 x 5cm., 4 cm wide
5
3
Net would have 3 rectangles
and two right angles triangles
So SA = 3x4 + 4x5 + 4x4 + 2x(½x3x4)
= 12 + 20 + 16 + 12
= 60 cm2

4
4
NB. Volume of 3D shape:
Cuboid = length x width x height cm3
Wedge = ½ x L x B x H cm3
(Note units)
Topic: Tree diagrams
See also: Fractions Sheet
This is a way to represent the probabilities of combined/successive events. Each route
represents a particular sequence of events. Probabilities on the route are multiplied to
find the overall probability of that sequence. When more than one path satisfy the
conditions of the problem, their overall probabilities are added.
Example 1: Tossing two coins (or one coin twice)
1st toss
2nd toss
½
H
P(HH) = ½x½=¼
½
½
T
P(HT) = ¼
½
½
H
P(TH) = ¼
½
T
P(TT) = ¼
1
H
T
(NB. These should add to 1)
So probability of having one head and one tail = P(HT or TH) = P(HT) + P(TH) = ½
Example 2: A bag contains 2 red and 3 white balls; draw two balls without replacement
R
2
/5
¼
R
P(RR) = 2/20
¾
W
P(RW) = 6/20
(NB. Probability on second draw
is over 4 as we have one
ball in the bag)
less
3
/5
W
2
/4
R
P(WR) = 6/20
2
/4
W
P(WW) = 6/20
20
/20 = 1
(Check!)
So probability of no white = P(RR) = 2/20
Probability of at least one white = 1 – P(no white) = 1 – 2/20 = 18/20
[or = P(RW or WR or WW) = 6/20 + 6/20 +6/20 = 18/20 ]
Example 3:
A
Win
B
Throw die (1 – 6); if 6, A moves one square and wins
if 1,2,3,4,5, B moves one square.
Continue throwing die. Who is more likely to win?
P (B wins) = P (B moves 5 times)
= 5 x 5 x 5 x 5 x 5 = 0.4 (approx)
6 6 6 6 6
P ( A wins) = 1 – P (B wins) = 0.6 (approx)
1/6
5/6
1st throw
6 (A wins)
not 6
(B moves)
1/6
5/6
6 (A wins)
not 6
(B moves)
etc…..
2nd throw
3rd throw
Answers
Averages
1.
2.
3.
4.
Mode - £15; Mean -£15.83; Median - £15
Mode - £15; Mean -£28.33; Median - £15
9.86 years
5.385
Brackets
6 + 4x
x²- x
3xy + 3yz
2w²- 6w
x²+ 7x + 12
x²+ x – 2
p² - 4
2m² + 5m + 2
Circle
1.
48cm
238cm²
2. 100.5m
3. 942.5mm²
Co-ordinates
1. (3, -1)
4a + 8b + 12c
ab + ac - a²
x³ - abx
8x – 4y + 12z
2. (2, 3) – (-2, 3)
(1, 1) – (-1, 1)
(3, 0) – (-3, 0)
(0, 2) – (0, 2)
(-2, -1) – (2, -1)
Disproving a Hypothesis
There are alternative answers but the following are some examples:
1.
3 + 4 + 5 = 8, not odd;
(½)²= ¼, which is less than ½, so it is smaller;
(1 + 1)²= 4; 1 + 1 = 2
2. Only that the cubes of the three numbers shown are odd.
Estimating results of calculations
1.
5 x 20 = 100
Various answers can be given, including:
40 x 10 = 400
40 x 11 = 440
38 x 10 = 380
2. 124.5
1245
8.3
1275
Factorising algebraic expressions
3 (2x + y)
2 (4 – x)
x ( y + 10x)
2bc (3a – 4d)
x²(x + 1)
s (6s² - 1)
3y (x + 2y - 3)
Factors
1.
Prime – 2, 7, 11, 37
Composite – 12, 14, 27, 49
2. 21 = 1 x 21
3x7
31 = 1 x 31
PRIME
15 = 1 x 15
3x5
3. 20 = 2²x 5
32 = 25
42 = 2 x 3 x 7
3. 28 = 1 + 2 + 4 + 7 + 14
4.
Forming algebraic expressions
1.
2.
3.
4.
5x
6( m + 2) –2
N+2
p+q
2
Fractions
1.
2
9
2.
2
5
4
18
2
3
6
27
8 (Answers may vary)
36
37
47
3. 37
40
4. 2 2
3
Gradient
1. m = 1; m = ½
2. C
B
3. OA –object is moving away from origin over a period of time
AB – object is staying at the same distance from the origin, so either stopped or
circling
Interpretation of a Graph
A–B
C–D
Man is stationary on the ladder
Man is coming down ladder very quickly, or falling off.
Nth terms
1.
3n
4n + 2
3/4n + 1.75 or 3/4n + 7/4 = 3n + 7
4
Order of arithmetic operations
16
6
19
22.1
Order of size
3 = 0.375
8
0.3
1 = 0.3 (recurring)
3
2 = 0.4
5
4 = 0.4 (recurring)
9
So order of size is: 0.3
1/3 3/8
2/5
4/9
Pythagoras’ Theorem
M = 8.7
Because the triangle is not a right-angled triangle.
Ratio
1. 2:3 2:5 5:3
2. 9cm length
18º slope
Rounding
1.
2.457
43.236
0.002
1.296
2. 2.35
3.01
0.0129
0.00250
Straight lines
1.
Y = 2x + 1
Y = 1.5x – 2
or y = 3/2x - 2
Y = x + 1/3
So the first one is the steepest.
2. a ( 0 , 1 )
3. y = 2x + 2
b ( 0, -2 )
4. y = 3x - 1
c ( 0, 1/3 )
Index
Topic
Algebraic expressions
Approximate answers
Area of circle
Area of rectangle
Area of triangle
Axes / Axis
BODMAS
Brackets (factorising)
Brackets (multiplying)
Circumference
Common factor
Complement
Composite number
Coordinates
Cumulative frequency graph
Decimal place
Decimals (rounding)
Decimals (fraction conversion)
Denominator
Diameter
Sheet
Forming algebraic expressions
Estimating results of calculations
Surface area of 3D shapes /
Circles
Surface area of 3D shapes
Surface area of 3D shapes
Coordinates / Gradient / Straight
lines
Order of arithmetic operations
Factorising algebraic expressions
Brackets
Circles
Factorising algebraic expressions /
Ratio
Inequalities
Factors
Coordinates / Gradient
Grouped data
Rounding
Rounding
Order of size
Fractions
Circles
Digits
Distance / speed graph
Distance / time graph
Equations of straight lines
Equivalent fractions
Estimating answers
Factorisation
Factor
Fractions
Fractions (addition)
Fractions (decimal conversion)
Fractions (multiplication)
Fractions (subtraction)
Frequency Distribution
Gradient
Graph interpretation
Greater than / equals to
Hypotenuse
Hypothesis
Inequalities
Integer
Interception on y – axis
Interquartile range
Irrational numbers
Less than / equals to
Lowest term
Map references
Mean
Median
Mid value
Mode
Multiple
Net
Nth term
Numerator
Operational conventions
Origin
Perfect number
Perimeter
Plotting straight line graph
Estimating results of calculations
Interpretation of a graph
Coordinates/Interpretation of a
graph
Straight lines
Fractions
Estimating results of calculations
Factorising algebraic expressions
Factors
Fractions
Fractions
Order of size
Fractions / Tree diagrams
Fractions
Averages / Grouped data
Gradient / Straight lines
Interpretation of a graph
Inequalities
Pythagoras’ Theorem
Disproving a hypothesis
Inequalities
Inequalities
Straight lines
Grouped data
Rational / Irrational numbers
Inequalities
Fractions / Ratio
Coordinates
Averages / Grouped data
Averages / Grouped data
Grouped data
Averages / Grouped data
Factors
Surface area of 3D shapes
Nth terms
Fractions
Order of arithmetic operations
Coordinates
Factors
Circles
Straight lines
Polygon
Prime number
Probability
Proper factor
Proving a hypothesis
Pythagoras’ theorem
Radius
Ratio
Rational numbers
Recurring decimals (conversion)
Reflection of points
Relative frequency
Right angled triangle
Rounding
Sequence of numbers
Scale factors
Sigma
Significant figure
Straight line graph
Surface area
Surface area of cuboid
Surface area of cylinder
Surface area of wedge
Term number
Tree diagrams
Volume of 3D shape
Y = mx + c
Y – axis
Surface area of 3D shapes
Factors
Probability / Tree diagrams
Factors
Disproving a hypothesis
Pythagoras’ Theorem
Circles
Ratio
Rational / Irrational numbers
Rational / Irrational numbers
Coordinates
Probability
Pythagoras’ Theorem
Rounding
Nth terms
Ratio
Averages / Grouped data
Rounding
Straight lines
Surface area of 3D shapes
Surface area of 3D shapes
Surface area of 3D shapes
Surface area of 3D shapes
Nth terms
Tree diagrams
Surface area of 3D shapes
Straight lines
Straight lines