MR ALI GCSE REVISION NOTES (HIGHER) - 2016
It’s about… / Can you…?· Arithmetic with whole numbers
· Decimals
· Approximation
· Negative numbers
· Multiples and Factors / · Do long multiplication and division e.g. 26 × 5634 or
· 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
· Fractions
· Percentages / · 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% =
· 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%.
Amount of interest is £570.58 - £500 = £70.58
· Ratio
· Proportion / · 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).
· Powers with numbers
· Powers with algebra
· Fractional Powers
· Standard Form
· Surds / · Tidy up powers of numbers e.g.
· Tidy up powers with algebra e.g.
· Work with fractional indices e.g.
· Add and subtract with standard form e.g.
Note: turn into regular numbers with adding and subtracting
· Multiply and divide with standard form e.g.
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
It’s about… / Can you…?
· Substitution
· Expand brackets
· Factorise into one bracket
· Linear Equations
· Equations from problems
· Use trial and improvement
· Solve simultaneous equations
· Rearrange Formulae / · 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
10 = -4+2x
10+4 = -2x
14 = -2x
14÷ - 2 = x
-7 = x
· Do equations with letters on both sides e.g. 5x + 4 = 3x +12
2x + 4 = 12
2x = 8
x = 4
· Solve equations with brackets e.g. 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
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
· Do simultaneous equations e.g. 4x + 3y = 37
2x + y = 17
×2 4x + 2y = 34
- 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
and that’s the answer!
· Rearrange formulae with squares e.g e = ½mv2
2e = mv2
and that’s the answer!
Quadratics / · 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
and use 4.63 or -8.63
Linear graphs / · 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
· 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
Further graphs / · 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)
· Algebraic Fractions
· Sequences
· Harder rearranging / · Work with algebraic fractions e.g.
· Find the nth term of a sequence e.g. 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
· Finding rules from diagrams eg.
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
· Quadratic sequences e.g. 5, 11, 21, 35, 53
6 10 14 18 (1st diff)
4 4 4 (2nd diff)
Halve the 2nd 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
and that’s the answer!
· Rearrange with fractions and √s e.g a – b = ax
Dimensions / · Check that the dimensions of a formula are consistent
e.g. do these expressions represent length, area or volume?:
a + 2b πr2 + ab 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.
a + 2b = m + m = 2m LENGTH
πr2 + ab = m2 + m2 = 2m2 AREA
= =m3 VOLUME
a + bc = m + m2 (can’t be simplified) NONE OF THESE
Direct and inverse proportion / · 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
A α t
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
So when f = 5 C = 180/25 = 7.2
Inequalities / · Solve simple inequalities e.g 3x+4<5
3x < 1
x < ⅓or
· Solve double inequalities e.g. –8 < 5x+2 < 22
-10<5x or 5x < 20
-2 < x or x < 4
So -2 < x < 4
· Do above for integer values e.g. -2 < x < 4 x is an integer
so x= -1, 0, 1, 2, 3
· Illustrate above on a number line
remember solid dots for ≤ or ≥, hollow dots for < or >
Transforming graphs / moves the graph by +a in the y direction (i.e. moves the graph up by a). / / 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 left). / / moves the graph by +a in the x direction (i.e. moves the graph by a to the right). /
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. /
1. Area
2. Volume of Prisms
3. Volume of Pyramids
4. Volume and surface area of spheres / · 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
Area = A l Volume = A x l (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
· Use ⅓ × base area × height to calculate volume of pyramids (LEARN)
· Calculate volume and surface area of cone: V = 1/3 πr2 h (ON
Curved surface area = πr3l SHEET)
· Calculate volume and surface area of a sphere: A = 4 πr2 (ON
V = 4/3 πr3 SHEET)
· Calculate volumes of compound shapes (chop into shapes you know)
1. Pythagoras
2. Trigonometry (right angled triangles SOHCAHTOA:
/ · Use Pythagoras to find missing sides a2 + b2 = c2 (c is the long side)
62 + 82 = x2 52 + y2 = 132
6cm x 36+64= x2 5cm 13cm 25+ y2 = 169
100 = x2 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