This document contains a summary of various mathematical concepts taught at different levels, including:
1) Working with fractions, decimals, and percentages by converting between them.
2) Dividing a quantity according to a given ratio.
3) Using proportional reasoning to solve problems involving scaling quantities up or down.
4) Performing calculations with fractions including addition, subtraction, multiplication, and division.
5) Solving linear equations through methods like expanding brackets and isolating variables.
6) Understanding sequences by identifying term-to-term and position-to-term rules to generate additional terms.
7) Plotting linear graphs by substituting values of x into the equation and plotting the
MATHS SYMBOLS - OTHER OPERATIONS (1) - ABSOLUTE VALUE - ROUNDING to INTEGER - PLUS or MINUS - RECIPROCAL - RATIO - PROPORTIONS and FIRST PROPERTIES - BRACKETS - EQUALITY SIGN - APPROXIMATELY EQUAL - NOT EQUAL - LESS - MUCH LESS - LESS THAN or EQUAL TO - GREATER - MUCH GREATER THAN - GREATER THAN or EQUAL TO - PROPORTIONALITY - DEFINITION
1. 6/3 Divide a quantity into a given ratio
Level 6
PROMPT sheet ~ Put headings
~Find how many shares in total
~ Amount ÷ no. shares = value of one share
6/1 Equivalent fractions, decimals &
percentages e.g. Divide £240 between A and B in ratio
of 3:5
Percentage to decimal to fraction A:B
27
27% = 0.27 = 3 : 5 = 8 shares
100
One share = £240 ÷ 8 = £30
7
7% = 0.07 = A = 3 shares = 3 x £30 = £90
100
70 7 B = 5 shares = 5 x £30 = £150
70% = 0.7 = =
100 10
Decimal to percentage to fraction 6/4 Use proportional reasoning
3
0.3 = 30% =
10 Change an amount in proportion
0.03 = 3% =
3 e.g. If 6 books cost £22.50
100 Find the cost of 11. (find cost of 1 first)
0.39 = 39% =
39 Change amounts to compare
100 e.g. A pack of 5 pens cost £6.10
Fraction to decimal to percentage A pack of 8 pens cost £9.20
4 80
= = 80% = 0.8 Which is the best buy? (find cost of 40 of each)
5 100
Change to 100
6/5 Calculate with fractions
3
= 3 ÷ 8 = 0.375 = 37.5% Add & subtract fractions
8
~Make the denominators the same
1 7 4
e.g. + -
6/2 Increase/Decrease by a percentage 5 10 5
2 7 12 10
= + = -
To increase £12 by 5% 10 10 15 15
= 1.05 x £12 (100% + 5% = 105%) =
9
=
2
OR 10 15
= £12 + 5% of £12
Multiply fractions
To decrease £50 by 15%
7
= 0.85 x £50 (100% - 15% = 85%) ~Write 7 as
1
OR ~Multiply numerators & denominators
= £50 – 15% of £50 2 4 2
e.g. 5 x x
3 5 3
5 2 8
= x =
1 3 15
=3
10 1
=
3 3
2. Divide fractions 6/7 Solve linear equations
7
~Write 7 as
1 ~Multiply out brackets first
~Flip numerator & denominator after ÷ ~If there are letters on both sides get rid
~Multiply numerators & denominators of the smaller first
~Do the same to both sides
e.g. 5 ÷
2 4
÷
2 e.g.
3 5 3 To solve 5(x – 3) = 3x + 7 (expand bracket)
5 3 4 3
= x = x 5x – 15 = 3x + 7(-3x from both sides)
1 2 5 2
2x – 15 = + 7 (+15 to each side)
=7 =1 =1
15 1 12 2 1
= = 2x = 22 (÷2 both sides)
2 2 10 10 5
2 2
Calculate fraction of quantity x = 11
4
To find of a quantity ÷5x4
5
6/8 Sequences
4
e.g. of £20 = 20 ÷ 5 x 4 = £16
5 Understand position and term
Position 1 2 3 4
Term 3 7 11 15
6/6 Solve an equation by trial &
improvement method +4
Term to term rule = +4
~ Find 2 consecutive numbers that the Position to term rule is x 4 - 1
solution lies between (because position 1 x 4 – 1 = 3)
~ Then choose the half way number nth term = n x 4 -1 = 4n - 1
~ Keep making improvements until the
required accuracy achieved Generate terms of a sequence
If the nth term is 5n + 1
e.g. To solve x3 – 3x = 6 (correct to 1dp) 1st term (n=1) = 5x1 + 1 = 6
2nd term (n=2) = 5x2 + 1= 11
Try x = x3 – 3x Comment
3rd term (n=3) = 5x3 + 1 = 16
2 23 – 2x2=4 Too small
3 33 – 3x3=28 Too big
6/9 Plot graphs of linear equations
3
2.5 2.5 – 3x2.5=8.125 Too big
~Substitute values of x into the equation
2.3 2.33 – 3x2.3=5.267 Too small
~Plot the points in pencil
2.4 2.43 – 3x2.4=6.624 Too big ~Join the points with a ruler and pencil
~They should be in a straight line
2.35 2.353 – 3x2.35=5.928 Too small
e.g. y = 3x – 1
Solution is nearer 2.4 than 2.3 x -2 -1 0 1 2
So x = 2.4 (correct to 1dp) y -7 -4 -1 2 5
4. 6/16 2D representations of 3D shapes 6/17 Enlarge a shape
3D drawing on isometric paper You need to know:
(notice NO horizontal lines) Centre e.g. ( 5, 4)
Scale factor e.g. 2
3 views of a 3D shape
Plan view
6/18 Translate & Reflect a shape
Translate a shape
You need to know:
Vector from A to B e.g. 3 Right
-4 Down
front elevation
side –vie w
Side view Plan view Front elevation A
B
Nets
Notice:
The new shape stays the same way up
The new shape is the same size
Cube Cuboid Square based
pyramid USE TRACING PAPER TO HELP
5. Reflect a shape Construct triangle given 3 sides
You need to know: (Use a pair of compasses
Angle e.g. 900 Leave the arcs on)
Direction e.g. clockwise
Centre of rotation e.g.(0,0)
3cm 5cm
7cm
Construct triangle given angles
(Use a protractor)
USE TRACING PAPER TO HELP 230 570
7cm
6/19 Constructions
6/20 Use formulae
Perpendicular bisector of a line
Area of circle
Draw a straight line through where
Area of circle = π x r2
the arcs cross above and below.
= π x r2
= π x 52 5cm
= 78.5cm2
Circumference of circle
Area of circle = π x d
=πx8
= 25.1cm 8cm
Bisector of a line Volume of cuboid
Volume = l x w x h
Draw a line from where the arcs =5x3x2
cross to the vertex of the angle = 30cm3 3cm
2cm
5cm
Surface area of cuboid
Front = 5x3 = 15
Back = 5x3 = 15
Top = 5x2 = 10
Bottom = 5x2 = 10 Total Surface Area =62cm2
Side = 3x2 = 6
Side = 3x2 = 6
6. 6/23 Presentation of data 6/24 Find all possible outcomes
Construct a pie chart Outcomes can be presented:
In a list
In a table or sample space
Transport Frequency Angle
Example of a sample space
0
Car 13 x 9 117 To show all possible outcomes from spinning a
spinner and rolling a dice
Bus 4x9 360
4
3
Walk 15 x 9 135
4
1
2
Cycle 8x9 72
Dice
Total frequency = 40
3600 ÷ 40 = 90 per person + 1 2 3 4 5 6
1 2 3 4 5 6 7
Construct a frequency polygon
Spinner
2 3
(points plotted at the midpoint of the bars)
3 4
20
4 5
15
6/25 Sum of mutually exclusive outcomes =1
F req uency
10 If 2 outcomes cannot occur together,
They are mutually exclusive
If 2 outcomes A and B are mutually
5 exclusive
P(A) + p(B) = 1
If 3 outcomes A B and C are mutually
0 exclusive
P(A) + p(B) + p(C) = 1
0 10 20 30 40 50
S cien ce m ark
Construct a scatter graph
e.g. If outcomes A, B and C are mutually
exclusive and
1 50 × p(A) = 0.47
1 30 p(B) = 0.31
p(C) = 1 – (0.47 + 0.31)
×
= 1 – 0.78
110
L en g th o f
= 0.22
sh o e la ce (c m )
90
×
70 × ×
×
50 ×
0 2 4 6 8 10 12 14 16
N u m b e r o f ey es in th e s h o e