Download LINEAR HIGHER Maths GCSE Key Facts

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Transcript
Maths GCSE Linear HIGHER – Things to Remember
Remember that the information below is always provided on
the formula sheet at the start of your exam paper
In addition to these formulae, you also need to learn other facts
and formulae
When you first open your exam paper, you should write down all the
key formulae and information that you have revised for the exam
before you start answering questions
The following pages list the key information you need to
remember……
Converting between fractions, decimals and percentages:
F → D → P
÷
x 100
e.g.
2 = 2 ÷ 5 = 0.4, and 0.4 x 100 = 40%
5
5 of 24 = 24 ÷ 8 x 5 = 3 x 5 = 15
8

30% of 64 = 3 lots of 10% of 60 = 3 x 6.4 = 19.2

8 as a percentage of 20 = 8_ = _40 = 40%
20
100
Reverse Percentages
After an increase of 20%, the new amount is 120% of the
original amount
After a decrease of 10%, the new amount is 90% of the
original amount
Ratio
To split £160 in the ratio 3 : 5, you need to split it into 3 +
5 = 8 parts……..
£160 ÷ 8 = £20
And then find what 3 parts and 5 parts are………
3 x £20 = £60
5 x £20 = £100
Standard Form Numbers
3 x 107 = 30 000 000
460 000 = 4.6 x 105
2.4 x 10-5 = 0.000024
0.000000081 = 8.1 x 10-8
nth term of a sequence
the sequence 3, 11, 19, 27…….
has a common difference of 8, so the nth term must
include 8n
but the sequence for 8n is 8, 16, 24, 32…..
So the sequence 3, 11, 19, 27……. must be 8n – 5
Expressions,
Identities
Equations
Formulae
and
An Expression has letters and numbers, but no equal sign
An Equation has an equal sign and usually just one letter
A Formula has an equal sign and will have more than one
letter
An Identity will have the identity sign (≡) instead of an equal
sign
Solving Equations and Re-arranging
Formulae
 Remember to always do the same to all terms on
both sides of the equal sign
 Try to eliminate negatives, brackets and fractions
first
The equation of a straight line
graph
y = mx + c
m is the gradient
of the graph
c is the y-intercept
(where the line
crosses the y axis)
Expanding and Factorising
To expand a bracket, multiply all the terms inside the bracket
e.g. 3(x + 7) = 3x + 21
x(x – 4) = x2 – 4x
Factorising is the opposite of expanding
e.g. 6x – 15 = 3(2x – 5)
x2 + 8x = x(x + 8)
To expand a double bracket, use the FOIL method
e.g. (x +3)(x + 7) = x2 + 7x + 3x + 21
= x2 + 10x + 21
To factorise a quadratic expression such as
x2 + 11x + 24,
you need two numbers that multiply to give 24 and add to
give 11, i.e. 3 and 8, so:
x2 + 11x + 24 = (x +3)(x + 8)
Solving Quadratic Equations
If a quadratic equation factorises, it can be solved like this:
x2 + 3x - 28 = 0
(x + 7) (x - 4) = 0
x = -7 or x = 4
If a quadratic equation of the form ax2 + bx + c = 0 cannot be
factorised, it can be solved by using the quadratic formula:
A quadratic equation can also be solved by using the method
of completing the square, when the quadratic equation is rewritten in the form
(x + p)2 + q = 0
[from this form of the equation we know that the minimum
point on the quadratic graph is (-p, q)]
Laws of Indices
ax x a y = a x + y
e.g. 35 x 37 = 312
ax ÷ a y = a x – y
e.g. x8 ÷ x3 = x5
a-x = 1
ax
e.g. 3-2 = 1
32
a½ = √a
e.g. 6½ = √6
a⅔ = (3√a)2
e.g. 8⅔ = (3√8)2 = 22 = 4
Surds
√a x √b = √(ab)
e.g. √3 x √5 = √15
√a ÷ √b = √(a/b)
e.g. √18 ÷ √3 = √6
When finding MEAN FROM A TABLE, remember to
MULTIPLY the data by the frequency (then sum of data ÷
sum of frequencies)
*for GROUPED DATA, find the MID-POINTS first
Remember to use the Handling
Data cycle when describing how
to test a hypothesis:
1. state what data to collect and
how to collect it
2. state what TYPE of graph or
average to use
3. state that you will compare
your results to the hypothesis
PROBABILITY
OR means ADD the probabilities (OR → +)
AND means MULTIPLY the probabilities (AND → x)
CUMULATIVE FREQUENCY (Curves and Box Plots)
Median = ½ (read off from halfway up the frequency axis)
Lower Quartile (LQ) = ¼ (read off from ¼ of the way
up the frequency axis)
Upper Quartile (UQ) = ¾ (read off from ¾ of the way
up the frequency axis)
Interquartile Range (IQR) = UQ – LQ
When asked to COMPARE data or graphs, try to
compare an AVERAGE (MEAN OR MEDIAN) and
the INTERQUARTILE RANGE
HISTOGRAMS – remember that the AREA of the
bar represents the frequency, so:
HEIGHT OF BAR = FREQUENCY ÷ CLASS WIDTH
STRATIFIED SAMPLING
Take a sample in proportion to the different group
sizes
Group size
Total population size
x sample size
Perimeter and Area
Area of a Triangle = (base x height) ÷ 2
Area of a Circle = π x radius x radius
or
A = π r²
Circumference of a Circle = π x diameter
or
C = πD
Parts of
a Circle
Arc Length (a fraction of the circumference)
Length of an Arc = angle of arc x πD
3600
Sector Area (a fraction of the area of a
circle)
Area of a Sector = angle of sector x πr2
3600
Angles
Don’t forget:
Angles on a straight line add to 1800
Angles at a point add to 3600
Angles in a triangle add to 1800
Angles in a quadrilateral add to 3600
Angles on parallel lines:
Angles
in
Polygons
Bearings – remember: North is 00
Measure angle clockwise
You need to learn these imperial to metric
conversions:
5 miles
= 8 km
1 inch
= 2.5 cm (so, 4 inches = 10 cm
and 12 inches = 30 cm – a large ruler
and 40 inches = 100 cm, or 1 m)
2.2 pounds = 1 kg
(or 1 mile = 1.6 km)
(or 2 lb ≈ 1 kg)
1 gallon
= 4.5 litres
(1 gallon = 8 pints)
1.75 pints = 1 litre ("a litre of water is a pint and three
quarters")
Speed, Distance
and Time
Density, Mass
and Volume
Transforming Shapes (TERRy)
Translate – describe how far across and how far up or down(use a
vector)
Enlarge – describe scale factor and centre of enlargement
Rotate – describe angle and direction of rotation and
rotation
Reflect – describe the equation of the mirror line
Y
centre of
Circle Theorems
In a Right Angled Triangle:
Pythagoras’ Theorem (to find a missing
side)
c2 = a 2 + b 2
(c is the hypotenuse,
the longest side)
a2 = c2 – b2
b2 = c2 – a2
(a and b are the shorter
sides)
SOH CAH TOA (to find a side or an angle)
Sin θ =
opposite
hypotenuse
Cos θ =
adjacent
hypotenuse
Tan θ =
opposite
adjacent
In NON Right Angled Triangles:
C
Use the Sine Rule or Cosine Rule to find missing sides or
angles in NON Right Angled Triangles
A
The Sine rule
a
b
c


sin A sin B sin C
is on the exam paper formula sheet
(this can be used to find missing sides or angles)
The Cosine rule to find a side, a2 = b2 + c2 – 2bc cos A is on the
formula sheet, but you need to learn the
find an angle:
Cosine Rule to
cos A = b2 + c2 – a2
2bc
b
a
c
B