Download MR ALI GCSE REVISION NOTES (HIGHER)

MR ALI GCSE REVISION NOTES (HIGHER) - 2016
It’s about…
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Arithmetic with whole
numbers
Decimals
Approximation
Negative numbers
Multiples and Factors
Fractions
Percentages
Can you…?
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Do long multiplication and division e.g. 26 × 5634 or 26 5364
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Do above but with decimals e.g. 2.6 × 56.34 or 1.44 ÷ 1.2
Round using decimal places e.g. 3.14159 = 3.142 (2 d.p.)
Round using significant figures e.g. 3.14159 = 3.1 (2 s.f.)
Use rounding to do approximate calculations e.g. 37.01 ÷ 7.99 ≈ 40 ÷ 8
Choose a suitable degree of accuracy
Calculate with negative numbers e.g. 7 × -2 = -14 or -4 – -7 = +3
Understand the words: multiple, factor, prime, square, cube, root
Write a number in prime factor form e.g. 36 = 22 × 32 , 54 = 2 × 33
Find the hcf (highest common factor) of 2 numbers e.g. 36, 54, hcf = 18
Find the lcm (lowest common multiple) of same e.g. 36, 54, lcm = 108
Work out a fraction of a number e.g. ¾ of 72 = 54
+ and – fractions and mixed numbers e.g. ¾ + ⅔ = 15/12 2½ - 1¼ = 1¼
× and ÷ the above e.g. ¾ × ⅔ = ½ 2½ ÷ 1¼ = 5/2 ÷ 5/4 = 5/2 × 4/5 = 2
Convert between fracs, % and decs e.g. 0.015 = 1.5% = 15  3
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Work out a % of a number e.g. 18% of £42 = 0.18 × £42 = £7.56
and calculate % change e.g. decrease £25 by 10%, £25-2.50=£22.50
work out one number as a % of another e.g.
buy for £12, sell for £14, profit is £2 so
percentage profit is 2÷12 × 100 = 16.7%
work out the original value after a % change e.g.
After a 25% cut, price is £405
75% of original price = £405
1% of original price = £405 ÷ 75=£5.40
Original price = 100 × £5.40 = £540
Do compound interest e.g. £500 for 3 years at 4.5%
After year 1 £500 + 4.5% = £522.50
After year 2 £522.50 + 4.5% = £546.01
After year 3 £546.01 + 4.5% = £570.58
Compound Interest the quick way. £500 for 3 years at 4.5%.
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Ratio
Proportion
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1000
200
Amount of interest is £570.58 - £500 = £70.58
Scale according to a ratio e.g.
An alloy is mixed in the ratio 5:7
If I have 12kg of metal A
How much metal B do I need?
12 ÷ 5 × 7 = 16.8kg
Share in a given ratio e.g.
share £400 in the ratio 3:2
£400 ÷ 5 = £80,
3 × £80 = £240 2 × £80 = 160
Apply ratio to ‘best value’ e.g. 300ml @ 50p or 400ml @ 65p
300 ÷ 50 = 6ml per penny BEST VALUE
400 ÷ 65 = 6.15ml per penny
Apply to speed and density problems (TRICKIER)
. Know how to rearrange into
and
What is the weight of a piece of rock which has a
volume of
and a density of
(which
means 2.25 grams per cubic cm).
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
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Powers with numbers
Powers with algebra
Fractional Powers
Standard Form
Surds
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Work with fractional indices e.g. 81 / 3  2
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Add and subtract with standard form e.g.
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Substitution
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Expand brackets
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Factorise into one
Linear Equations
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Equations from
problems
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Do equations with letters on both sides e.g.
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Solve equations with brackets e.g.
Use trial and
Solve simultaneous
3/ 2
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27
8
Substitute numbers into formulae e.g. x = 5 and y = -3
3x2 – y
= 3 × 52 - -3 = 75 + 3 = 78
Expand and simplify brackets e.g.
3(2x + 1) - 2(x – 4)
= 6x + 3 - 2x + 8
= 4x + 11
Factorise into a single bracket e.g.
t2 – 5t = t(t – 5)
Solve simple linear equations e.g
3x + 7 = 25
3x
= 25 – 7
3x
= 18
x
= 6
Solve equations with negative coefficients e.g. 10 – 2x = -4
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improvement
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9
 
4
Note: turn into regular numbers with adding and subtracting
Multiply and divide with standard form e.g. 6.3  105  2.1 103  3  102
Note: calculate in standard form when multiplying and dividing
Use a calculator for standard form i.e. use the EXP button
Work with surds e.g 3√2 +5√8= 3√2 + 10√2 = 13√2
bracket
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3 / 2
Can you…?
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4
 
9
543000  5.4  105  2.3  104  540000  23000  563000  5.63  105
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It’s about…
54  53
1
 5 2 
9
5
25
4 2 3
12a b c
Tidy up powers with algebra e.g.
 3a 2 c
4a 2 b 2 c 2
Tidy up powers of numbers e.g.
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10
10+4
14
14÷ - 2
-7
5x + 4
2x + 4
2x
x
= -4+2x
= -2x
= -2x
= x
= x
= 3x +12
= 12
= 8
=4
3(4 – x) = 2(x + 1)
12 – 3x = 2x + 2
12
= 5x + 2
10
= 5x
2
=x
Form an equation from a problem e.g. perimeter is 26cm, find x
equations
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(x-1)
Rearrange Formulae
(2x+2)
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x-1+x-1+2x+2+2x+2 = 26
6x +2 = 26
6x
= 24
x
=4
solve x3+2x=20
x
x3
2x
x3+2x
(trial and
2
8
4
12
low
improvement)
3
27
6
33
high
keep going until answer is accurate enough
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
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4x + 3y = 37
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2x + y = 17
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×2 4x + 2y = 34
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-
y =3
Substitute y = 3 into  4x + 9 = 37
4x
= 28
x
=7
Rearrange a simple formula e.g.
y = 3x – 7 make x the subject
y + 7 = 3x
Do simultaneous equations e.g.
y7
x
x
and that’s the answer!
Rearrange formulae with squares e.g e = ½mv2
2e = mv2
2e
 v2
m
2e
v
m
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Quadratics
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and that’s the answer!
Expand 2 brackets e.g. (5x + 4)(3x – 2) = 15x2 + 2x – 8
Factorise into 2 brackets e.g. x2 + 5x – 24 = (x + 8)(x – 3)
Solve quadratic equations by factorising e.g.
x2 + 5x – 24 = 0 so (x + 8)(x – 3)=0 so x = -8 or x = 3
Solve quadratic equations using the formula e.g.
2x2 + 3x – 5 = 0 so a = 2, b = 3 and c = -5 a
2
and use  b  b  4ac   2  4  4  2  (5)   2  44  4.63 or -8.63
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Linear graphs
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Further graphs
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Algebraic Fractions
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Sequences
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Harder rearranging
2a
4
4
Draw any straight line graph e.g. draw y = 3x – 2 from x=-2 to x=4
find the gradient of any straight line i.e. count units and do rise
run
Find the equation of a graph (Note: remember y = mx+ c)
e.g. if gradient = 2 and y-intercept = 3, y = 2x + 3
Solve simultaneous equations using graphs
i.e. draw both graphs and see where they cross
Draw graphs of quadratic equations
e.g. draw x2 + x – 6 = 0 between x = -3 and x = +4 (fill in a table)
Use quadratic graphs to solve equations
e.g. solve x2 + x – 6 =0.5 using the above graph (see where y=0.5)
3
1 3x  ( x  4)
2x  4
Work with algebraic fractions e.g.
 

( x  4)
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Find the nth term of a sequence e.g.
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Finding rules from diagrams eg.
Make a table of values
x
x( x  4)
x( x  4)
20, 17, 14, 11….
Going down in 3s so try -3n
1st term comes out at -3×1=-3
We need 20, so have to add 23
Formula is tn = -3n + 23
Tables 1
2
3
4
5
Seats
4
7
10 13 16
Seats going up by 3 so try 3t.
1st term is 3 x 1 = 3. 2nd term is 3 x 2 = 6.
Need to add + 1 so
Rule is: s = 3t + 1
To find how many seats for 50 tables
s = 3(50) + 3 = 153 seats.
To find how many tables are needed for 100 guests 100 = 3t + 3
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
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Quadratic sequences
e.g. 5, 11, 21, 35, 53
6 10 14 18 (1st diff)
4 4
4
(2nd diff)
nd
Halve the 2 diff (4) so the rule has ‘2x2’ in it.
1st term: 2x 12=2 (need 5 so plus 3). Eg. 2x2 + 3.
2nd term: 2 x 22 = 8 (add 3 is 11 so we are right)
Rule is tn = 2x2 + 3
Factorise to rearrange formulae e.g a – b = ax
(A-grade stuff)
a
= ax + b
a – ax = b
a(1-x) = b
b
and that’s the answer!
a
1 x
Rearrange with fractions and √s e.g
a – b = ax
Check that the dimensions of a formula are consistent
e.g. do these expressions represent length, area or volume?:
πr2 + ab
a + 2b
a + bc
1) Replace each letter with the letter m (π and numbers don’t count – cross off).
2) Simplify as much as possible.
3) Decide if the formula is a length, area or volume.
Dimensions
a + 2b = m + m = 2m
πr2 + ab = m2 + m2 = 2m2
=
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=m3
LENGTH
AREA
VOLUME
a + bc = m + m2 (can’t be simplified)
NONE OF THESE
Convert a α b to a = kb and a α 1/b to a = k/b and a α 1/b2 to a α k/b2
Solve problems of direct proportion e.g. A is directly proportional to t
A is 45 when t is 5
Find A when t is 8
Aαt
 A  kt
45  5k
k 9
 A  9t
Direct and inverse
proportion
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If t = 8 A = 9×8 = 72
Solve problems of inverse prop e.g. C is inversely proportional to f2
C is 20 when f is 3
Find C when f is 5
1
f2
k
C  2
f
k
k
 20  2 
9
3
180  k
C
C 
Inequalities
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So when f = 5
Solve simple inequalities e.g
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Solve double inequalities e.g.
180
f2
C = 180/25 = 7.2
3x+4<5
3x < 1
x < ⅓or
–8 < 5x+2 < 22
-10<5x
or 5x < 20
-2 < x
or x < 4
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
So
-2 < x < 4
-2 < x < 4 x is an integer
so x= -1, 0, 1, 2, 3
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Do above for integer values e.g.
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Illustrate above on a number line
remember solid dots for ≤ or ≥, hollow dots for < or >
moves
the graph by +a in
the y direction
(i.e. moves the
graph up by a).
moves
the graph by -a in
the x direction
(i.e. moves the
graph by a to the
left).
moves the
graph by –a in
the y direction
(i.e. moves the
graph down by
a).
moves the
graph by +a in
the x direction
(i.e. moves the
graph by a to
the right).
Transforming graphs
1. Area
2. Volume of Prisms
3. Volume of Pyramids
4. Volume and surface area
of spheres
stretches
the graph by a
factor of a in the y
direction.
stretches the
graph by a
factor of 1/a in
the x direction
(squashes by a
factor of a).
is the
graph of
reflected in the y
axis.
is the
graph of
reflected in the
x axis.
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Calculate area of a rectangle: b × h (LEARN)
Calculate area of a triangle: ½ × b × h (LEARN)
Calculate area of a trapezium: ½ ×( l1×l2) × h
(LEARN)
Calculate circumference and area of a circles: C = πd = 2πr (LEARN)
A = πr2
(LEARN)
Calculate area and perimeter of sectors of circles e.g.
4cm
P = 4 + 4 + (60/360 x 2 x π x 4) = 12.2cm
60° A = 60/360 x π x 42 = 8.37cm2
Use the above to calculate the areas of compound shapes
1. Chop into shapes you know
2. Work out areas separately
3. Add the separate areas
Use cross-sectional area × length to calculate volume of prism
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
Area = A
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Use ⅓ × base area × height to calculate volume of pyramids (LEARN)
Calculate volume and surface area of cone: V = 1/3 πr2h (ON
Curved surface area = πr3l
SHEET)
Calculate volume and surface area of a sphere: A = 4 πr 2
(ON
V = 4/3 πr3 SHEET)
Calculate volumes of compound shapes (chop into shapes you know)
Use Pythagoras to find missing sides a2 + b2 = c2 (c is the long side)
62 + 82 = x2
52 + y2 = 132
2
6cm
x 36+64= x
5cm
13cm 25+ y2 = 169
2
100 = x
y2 = 169-25
8cm
x = 10cm
y
y2 = 144
y = 12cm
Use Pythagoras to solve problems
e.g. a 4m ladder is leaning against a wall, the base is 1m from the
wall. How far up the wall does the ladder reach?
DRAW A DIAGRAM
a 2 + b 2 = c2
h
4m 12 + h2 = 42
1 + h2 = 16
h2 = 16 – 1 = 15
1m
h = √15 = 3.87m
Use Trig to find missing sides and angles: (always labels sides first)
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(o) ?
12cm (h)
42°
opposite
sin x 
hypotenuse
opposite
sin 42 
12
Opposite = 12sin42°
= 8.01cm
adjacent
hypotenuse
opposite
tan x 
adjacent
cos x 
Angles
(LEARN)
Calculate surface areas of prisms/pyramids (draw nets to count faces)
e.g.
note it has 5 faces –
work out area of each one
and add
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Volume = A x l
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1. Pythagoras
2. Trigonometry (right
angled triangles
SOHCAHTOA:
opposite
sin x 
hypotenuse
l
(a) 2m x°
tan x 
tan x 
h
(o) 3m
opposite
adjacent
3
 1.5
2
tan-1(1.5) = 56.3°
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Use Trig to solve problems: (draw a diagram and labels sides first)
e.g. a 4m ladder is leaning against a wall, the base is 1m from the
wall. What angle does the ladder make with the ground?
DRAW A DIAGRAM
cosx = adjacent
AND LABEL SIDES
4m
hypotentuse
cos x = ¼ = 0.25
x°
x = cos-1(0.25)
a 1m
= 75.5°
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Use angle sum of a triangle = angles on a straight line = 180° (1×180°)
Use angle sum of a quadrilateral = angle sum around a point = 360°
Use angle sum of a pentagon = 540 (3×180°)
Use angle sum of an n-sided polygon add up to (n-2) × 180°
Remember that where two lines cross, opposite angles are equal
In parallel lines, alternate (Z) and corresponding (F) angles are equal
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
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Recall that in parallel lines, interior or allied (C) add to 180o.
Alternate ( Z ) angles are equal therefore a = 74o
Angles on a straight line add to 180o therefore
b = 180o – a
b = 180o – 74o
b = 106o
Allied (C) angles add to 180o therefore
c = 180o – a
c = 180o – 74o
c = 106o
(Also note that b and c are corresponding (F) angles which means
they are equal)
Corresponding angles are equal therefore d = 65o
Opposite angles are equal therefore e = 65o
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Circle Theorems (all 7)
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draw and label the radius, diameter and circumference of a circle
draw and label a chord, sector, segment or tangent to a circle
Radius : the distance from centre to circumference
(Diameter = 2 x radius)
Tangent: a straight
line which touches
the circumference
of the circle (once
only)
Chord: a straight line
which connects any
two points on the
circumference
Segment: a chord divides a circle into two segments (major and minor)
Circumference: distance around the edge of a circle
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Use angles subtended by the same chord are equal
a=b=c
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Use angle at centre = 2 × angle at circumference
7
MR ALI GCSE REVISION NOTES (HIGHER) - 2016
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Use opposite angles in a cyclic quadrilateral add up to 180°
Use angle between a tangent and a radius is 90°
Use tangents to a circle from an external point to the points of contact are equal in
length.
Use angle subtended (standing on) a diameter is 90°
Use the alternate segment theorem
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Reflect a shape in a given line
Reflect shape A in the line x = -1
x=-1
A
NOTE: you need to be able to draw
x=…-3, -2, -1, 0 1, 2, 3, 4 etc
y= …-3, -2, -1, 0, 1, 2, 3, erc
y=x
and y=-x
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Rotate a shape around a given
Rotate shape A 90° clockwise
About the point (0,0)
point
A
NOTE: use tracing paper to do this
Transformations

Translate a shape by a given vector
Translate shape A by the vector
A
1 
 
  3
 Enlarge a shape with a given centre
and scale factor
Enlarge by scale factor 2 centre (2, 4)
x
A
NOTE: find the distance of each corner
from the centre of enlargement and
multiply by the scale factor
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Carry out
two of the above transformations in succession
e.g. reflect the triangle A in the line x = -1, call the new shape B THEN rotate triangle B
180 about the
8
MR ALI GCSE REVISION NOTES (HIGHER) - 2016
point (-1,0), call this shape C
 Fully describe a transformation
e.g. describe the SINGLE
TRANSFORMATION that maps A to C
answer: reflection in the x-axis
B
A
C
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Constructions and Loci
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 Construct any triangle
Given 3 sides
e.g. To construct a triangle of sides 8 cm, 7cm and 6 cm
1. Draw a line of 8cm long with a ruler
2. Set compass to 7 cm, place at either end of the line and draw an
arc
3. Set compass to 6 cm, place at the other end of the line and
draw an arc to intersect the first one
4. Draw straight lines from the point of intersection to
both ends of the line
Given 1 side and 2 angles
e.g. To construct a triangle of side, 9 cm with angles of 35 o and 65o
1. Draw a straight line 9cm long
2. Use a protractor to draw angles of
and 650 on either end of line
3. Draw straight lines from both
until they intersect to form the
triangle
350
ends
Given 2 sides and with an included angle
e.g. To construct a triangle of sides, 9 cm and 7cm
with an
o
angle of 40
1. Draw a straight line 9cm long
2. Use a protractor to draw an angle of 400 on either end of line
3. Mark off a length of 7cm along that line
4. Join end points to form the triangle
5.
 Construct angle bisectors of a line
1. Place compass at A, and draw an arc
2. Place compass at X and draw an arc
3. Do the same at Y (with the compass set in the same distance)
4. Draw the angle bisector from A through the point of intersection, B

Construct perpendicular bisectors of a line
1. Place compass at one end of the line, set it over halfway and draw 2
arcs above and below the line
9
MR ALI GCSE REVISION NOTES (HIGHER) - 2016
2.
3.

Without changing the compass, do exactly the same at the other end of
the line
Draw a straight line through the points of intersection. This is the
perpendicular bisector
Solving locus problems
The locus of a point that remains a constant distance form a fixed point is a
circle
The locus of a point on a straight line is a pair of parallel lines
The locus of a point on a finite line is a pair of parallel lines joined by 2
semicircles
Real Life Graphs

Interpret distance-time graphs
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
Graph shows Jamie’s
journey form home to his
friend’s house.
Left home at 10:10

Stopped for 30 mins
Arrived home at 11:50
BE CAREFUL WITH UNITS!
 So, speed on the way there =
 Speed on way home =
=
= 40
= 0.5
 Total distance travelled = 40km
 Conversion graphs
 Odd graphs (e.g. containers filling with water)

Find missing lengths
m
Don’t forget to note when the missing value is only
part of the unknown length:
Similarity
m
Advanced Trigonometry

Use scale factors to find areas and volumes:
E.g. If length scale factor = 2
then Area scale factor = 2x2 = 22 = 4
and Volume scale factor = 2x2x2 = 23 = 8

Solve 3D problems
Calculate the length of the diagonal.
Length of diagonal across the base =
= 10.77cm
Length of diagonal =
= 12.33cm

use the sine and cosine rules for triangles without right angles
a
b
c


sin A sin B sin C
or
sin A sin B sin C


a
b
c
11
MR ALI GCSE REVISION NOTES (HIGHER) - 2016
x
30

sin 50 sin 60
x
30cm
60o
x
50o
30  sin 50
sin 60
x  26.54cm
10cm
θ
40o
sin  sin 40

14
10
14  sin 40
sin  
10
sin   0.8999
14cm
  64.15
a 2  b 2  c 2  2bcCosA
5m
or
b2  c2  a 2
CosA 
2bc
x 2  4.52  52  2  4.5  5  Cos115
115o
4.5m
x 2  64.2678
x  8.02m
x
8cm
9cm
Cos 
Cos  0.7454
θ
12cm
area 
1
ab sin C
2
  41.81
area 
1
 13  15  sin 30
2
area  48.75cm 2
13cm
30o
9 2  12 2  82
2  9  12
15cm
Recognise the graphs of sin cos and tan and solve simple trig equations (A*)
y = sinx
y = cosx
12
MR ALI GCSE REVISION NOTES (HIGHER) - 2016
y = tanx
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
Recognise transformed trigonometric graphs
Solve simple trig equations (A*)
Degree of accuracy
Rule of thumb
For problems involving x & ÷ round to lowest number of significant figures of the
numbers given in the question
For problems involving + & - round to lowest number of decimal places of the numbers
given in the question
E.g. One side of a rectangle is measured as 12.53cm, the other side is 17.3cm
Perimeter: 12.53 + 17.3 + 12.53 + 17.3 = 59.66. Round to one decimal place, 59.7 cm
Area: 12.53 x 17.3 = 216.769. Round to 3 significant figures, 217 cm 2
Accuracy
Upper and lower bounds.
Given a measurement e.g. 17.6m,
The lower bound is the lowest possible measurement that would be rounded up to 17.6 i.e.
17.55.
The upper bound is the highest possible measurement that would be rounded down to 17.6
i.e.
.
Solve problems involving upper and lower bounds eg.
A square playground has a side of length 25m to the nearest metre.
(a) What is the upper bound for the perimeter?
(b) What is the lower bound for the area?



Vectors

Use vectors written a, b, c
Add, subtract and multiply vectors in component form
Recognise parallel vectors (one must be a scalar multiple of the other)
and use to determine whether points are on the same straight line
(vectors defined using these points must be parallel and have a point in common).
Translate between the above and AB etc
B (7,3)
A (4,1)
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016

Use vectors to solve geometrical problems.
a+b
b + 3a
a + b – 4a + b = 2b – 3a
Questionnaires
A good question will not be:

Too personal. E.g. how old are you?

Full of technical jargon.

Leading. E.g. don’t you think that ________?

Open. E.g. what do you think of _______?
A good question will:

Have a yes/no, number, or multiple choice answer.
Multiple choices will cover all possible answers and will not have overlapping intervals.

Questionnaires

Sampling

Averages

Frequency polygons &
Histograms

Cumulative frequency
Bad e.g. How much time do you spend watching TV on average per day?
(THESE ARE NO GOOD –THEY OVERLAP)
o 0 – 30 minutes
o 30 – 60 minutes
o 1-2 hours
o 2 hours or more
HERE’S HOW YOU FIX THE PROBLEM
How much time do you spend watching TV on average per day?
1. Less than 30 minutes
2. More than 30 minutes, less than one hour
3. At least one hour but less than two hours
4. More than two hours hours
Sampling
Describe how to take a sample using the following methods
 Random
 Systematic
 Cluster
 Stratified random
Describe the pros and cons of the above sampling methods
 Calculate the required group sizes for a stratified random sample of 100 students
Year group
Boys
Girls
7
152
160
8
145
155
9
130
133
Sample
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
Year group
7
8
9
Boys
17
17
15
Girls
18
18
15
Size of sample group = __group size__ x desired sample size
Population size

Given a list of numbers eg.
Test scores: 9, 10, 12, 15, 15, 15, 17, 18, 18, calculate the…
 Mean (14.22)
 Median (15)
 Mode (15)
 Upper quartile (17.5)
 Lower quartile (11)
 Inter-quartile range (6.5)
 Range (9)
 Draw a box-plot
 Given a frequency table eg.
Number of pets
Frequency
0
8
1
12
2
16
3
4
Calculate the…
 mean (1.4)
 median (1.5)
 mode (2)
 Draw a bar chart

Given a grouped frequency table eg.
Height
Frequency
140 ≤ h < 150
4
150 ≤ h < 160
9
160 ≤ h < 170
11
170 ≤ h < 180
6




Calculate an estimate of the mean? (161.33)
Which class contains the median? 160 ≤ h < 170
Which is the modal class? 160 ≤ h < 170
Draw a histogram and frequency polygon
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016
15
10
5
140


150
160
170
180
Draw a cumulative frequency graph
Estimate the median and quartiles from the cumulative frequency graph and
draw a box plot
30
25
20
15
10
5
140
150
160
170
180
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016

Given a grouped frequency table with uneven class intervals
Height
Frequency
100 ≤ h < 150
4
150 ≤ h < 160
9
160 ≤ h < 170
11
170 ≤ h < 180
6



Calculate an estimate of the mean? (161.33)
Draw a histogram
Complete the table and the graph
Time (t) in
minutes
0 < t ≤ 10
Frequency
20
Frequency
density
10 < t ≤ 15
15 < t ≤ 30
30 < t ≤ 50
62
50 < t ≤ 60
23
0


20
30
40
Time (minutes)
50
60
70
Bar charts



Data Presentation
10
Gaps between the bars
The horizontal axis represents the type of data, it is NOT a continuous number scale
The vertical axis represents frequency
Histograms
No gaps between the bars
The horizontal axis has a continuous scale
The vertical axis represents frequency density, which is
(frequency of the class interval)
(width of class interval)
 Area of each bar represents the class frequency of the bar



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MR ALI GCSE REVISION NOTES (HIGHER) - 2016

Pie charts
Working out
Total frequency: 22+36+8+24 = 90
Midlands: (22/90) x 3600 = 880
London:(36/90)x 3600 = 1440
Southern England (8/90) x 3600 = 320
Northern England: (24/90) x 3600=960
 Scatter Graphs
 Plot points
 Draw line of best fit
 Comment on correlation
 Relative frequency
Rob throws a biased dice 100 times. The table shows his results.
Probability
Score
Frequency
1
8
2
10
3
15
4
12
5
21
6
34
Find an estimate for the probability that he will get a 6.
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MR ALI GCSE REVISION NOTES (HIGHER) - 2016

Probability trees
Loren has two bags.
The first bag contains 3 red counters and 2 blue counters.
The second contains 2 red counters and 5 blue counters. Loren takes one
counter at random from each bag.
Complete the probability tree diagram.
Counter from
first bag
Counter from
second bag
Red
2
7
Red
3
5
......
Blue
Red
......
......
Blue
......
Blue



What is the probability that she …
takes 2 red counters? (6/35)
takes 2 different coloured counters? (19/35)
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