Maths revison:)

Revision notes for Edexcel Maths Unit 3 Higher tier

HideShow resource information

Maths Revison Notes :)


1 of 44


Rational numbers

  • A whole number, which is either positive or negative.
  • A fraction, where denominator and numeratior are whole numbers, e.g 1/2, -1/4
  • A terminating or reccuring decimal, e.g 0.754, 0.333333333333.

Irrational numbers

  • They are never-ending and non-reating decimal, e.g π (pi).
  • A good sourse of irrational numbers is square roots and cube root (surds).
2 of 44

Surds (Continued..)

A surd is a square root which cannot be reduced to a whole number. For example, square root of 4 = 2 ( is not a surd, as the answer is a whole number. But square root of 5 ( is not a whole number. You could use a calculator to find that square root of 5 = 2.236067977 ( but instead of this we often leave our answers in the square root form, as a surd.

3 of 44

Manipulating surds

The Main Rules

  • ( x( = (
  • ( / ( = (
  • ( ( ² = b
  • You can't do ( +(
  • (a +(² = (a +( (a +( = a² + 2a( +( = a²+ 2a( +b
  • (a + ( b)(a - ( = a² + a( - a( - ((² = a² - b
  • When rationalising the denominator e.g 2/(, you multiply the numerator and denominator by ( so it becomes 2( = 2(
4 of 44

Upper and Lower bounds

Finding the upper and lower bounds

The rule is the real value as much as half the rounded unit added or subtracted the rounded off value.


If a length is given as 2.4m to the nearest o.1m.

The upper bound is half 0.1 added to 2.4, which is 2.4 +0.05 = 2.45

The lower bound is half 0.1 subtracted from 2.4, which is 2.4 - 0.05 = 0.35

5 of 44

Upper and Lower bounds

Maximum and minimum values of a calculation

Maximum answer

  • Upper bound ÷ Lower bound
  • Upper bound - Lower bound
  • Upper bound x Upper bound
  • Upper bound + Upper bound

Minimum answer

  • Lower bound ÷ Upper bound
  • Lower bound - Upper bound
  • Lower Bound x Lower bound
  • Lower bound + Lower bound
6 of 44


Four main things you need to know about Reciprocals are:

  • The Reciprocal of a number is one over the number, e.g Reciprocals of 7 is 1/7
  • You can find the Reciprocals of a fraction by turning it upside down,e.g Reciprocals of 1/4 is 4/1
  • A number multiplied by its Reciprocals gives 1. e.g 7 x 1/7 = 1 and 1/4 x4/1 is 1
  • 0 has no Reciprocals

7 of 44


A ratio is a way of comparing the relative magnitude of different quantities. An example of a ratio is 20 : 30 : 50.

Simplfing ratio: Division by common factors reduces the numbers used in a ratio. The ratio 20 : 30 : 50 becomes 2 : 3 : 5 . More examples,8 : 12 becomes 2 : 3, 20 : 55 becomes 4 : 11 by dividing by 5, 39 : 12 becomes 13 : 4 by dividing by 3, 56 : 24 becomes 7: 3 by dividing by 8.

Proportional division

  • Add up the parts, e.g if its 2:3 than you add 2 and 3.
  • Find amount of one part, e.g say the total amount was 500, once you added 2 and 3 which equals 5, you than divide 500 by five, which will give you one part, which is 100.
  • than to work out 2:3, so 2 x 100 = 200 and 3 x 100= 300
  • to check if correct make sure when you add them they equal total amount (500)
8 of 44



Multiply top and bottom seperatly, e.g 3/5 x 4/7 = 3 x 4/5 x 7 = 12/35


Turn the 2nd fraction upside down and then multiply, e.g 3/4 ÷ 1/3 = 3/4 x 3/1 = 3x3/4x1 = 9/4

Adding and subtracting

You add or subtract numerator as long as the denominator is the same, e.g 2/6 +1/6 =3/6 and 5/7 - 3/7 =2/7

To making denominator the same you divide or multiply the fraction until both fractions have the same denominator

9 of 44

Fractions (continued...)

Finding a fraction of something

Multiply by the numerator and than divide by the denominator, e.g 9/20 of 360 = (9 x 360)÷ 20 = 162

or you can divide first than multiply, e.g 9/20 of 360 = (360÷ 20) x 9

Terminating and Recurring decimals

Recurring decimals which have a patter of numbers which repeat forever, e.g 1/3 = 0.333333333...

Termination decimals are finite, e.g 1/20 = 0.05

The denominator of a fraction tells you if it will be a recuring or terminating decmal when converted. if denominator has a prime number of 2 or 5 its termination, if not then its when converted it will be a Recurring decimals

10 of 44

Recurring Decimals into Fractions

The Method

  • Find the length of the repeating sequence and multiply by whatever will move it all up past the decimal point by a full repeated sequence (10,100,1000), e.g 0.234234234234..., 234 is repeated so that is one full repeated sequence so you multiply by a 1000. 0.234234234234.. x 1000 = 234.234234234...
  • Subtract the original number (r) from the new one, e.g 1000r - r = 234.2342342... - 0.234234234.. which gives 999r=234
  • Than divide to leave r: r=234/999 and cancel if possible: r = 26/11

The easy way out

The fraction always has a repeating unit on the numerator and the same number of 9's as the denominator, e.g 0.44444.. = 4/9, 0.124124124... = 124/999 and so on. If you can cancel down, than you make sure you cancel down.

11 of 44


A percentage is basically over a 100

To find out x% of y, you divide y by a 100 and multiply by x, e.g Find 15% of £46, you divide 46 by 100 and multiply by 15 = £6.90

To Express x as a percentage of y, you put it in a fraction x/y and multiply that fraction by 100, e.g give 40p as a percentage of £3.34, 40/334 than multiply by hundred = 12%

To identify the original value

Example: A house increases in a value of 20% to £72,ooo,Find the original price. Method: 72,000 is 120%, so you divide 72,ooo by 120 to work out 1% which is £600 and then multiply by a 100 to find out 100% which is £60,000 which is the original price. The reaon why 72,000 is 120 persent is beacuse it has increased by 20% on top of the 100% = 120%

12 of 44

Regular Polygons

A polygon is a shape with many sides. A regular polygon is a shape where all sides and angles are the same.

Exterior Angles:

They are the angles the shape has after the sides are extended. Polygons exterior angles always add up to 360 degrees so, 360 ÷ the number of sides = the exterior angle. On a triangle the exterior angle is the sum of the interior angles of the other two vertices.

Interior Angles:

If you know the exterior angle of the polygon then the interior angle is 180 - the exterior angle because exterior angle & the interior angle are on a straight line. To find the sum of the angles inside a polygon we cut the polygon into triangles. The cutting lines must not cross e.g. if we cut the pentagon into 3 triangles. The angle sum in each triangle is 180 degrees. Therefore the angle sum of the pentagon is 3 x 180 = 540 degrees

13 of 44

Circles Equations

  • Area = π x r²
  • Circumference = 2πr or π x d d = diameter, r =radius
  • Area of sector= (Angle/360) ÷ Area of full circle
  • Length of Arc = (Angle/360) x circumferance of whole circle
  • Area of segment: you have to find out the area of the sector as above than you find area of triangle and than subtract the area of triangle from area of sector
14 of 44

Circle geometry

There are six main rules that your ment to know...

· An angle in a semi circle = 90°

· Angles on the same segment are equal

· Angle on circumference is twice angle in centre

· Opposite angles in a quadrilateral (where all sides are touching circumference) add up to 180°

· Angles in opposite segments are equal

· A chord bisector is a diameter

15 of 44


· Volume of sphere = 4/3π r³

· Volume of a prism = cross section area x length

· Volume of a pyramid = 1/3 x Base area x height

· Volume of cone = 1/3 x π

· Volume of Frustum = Volume of original cone - Volume of the removed cone

16 of 44

Surface area

Surface area of 3D is basically the total area of all the outer surfaces add together. for shapes like; prisms,cubes,cuboid and pyramids you can work out surface area with the help of the net of the shape. 3D shapes like spheres, cones and cylinders nets are difficult to draw, so you need to learn these formulae:

  • Cylinder (Curved) surface area: 2πrh
  • Cone (Curved) surface area: πrl (l is slant height)
  • Sphere surface area: 4πr 2

If you want to know surface area for prisms,cubes,cuboid and pyramids, here they are:

Surface Area of a Cube = 6 a 2 (a is the length of the side). Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac (a,b,c are lengths). Surface Area of a prism = (perimeter of shape cross section) x L+ 2 x (Area of shape cross section). Surface Area of a pyramid = Add the area of the base to the sum of the areas of all of the triangular faces

17 of 44


image: showing the elevations of a shape (

18 of 44

Identifying Formulas

Just looking at a formulae you should be able to find out if its for length, area or volume.

The rule is:

  • Areas always have lengths square (L²)
  • Volumes always have Lengths cubed (L3)
  • Lengths always are on their own (L)


4πr 2 + 6d² ( Area) Lwh + 6r²L (Volume) 4πr + 15 L (Length)

19 of 44

Converting Area and Volume measurements

1 m2 = 100cm x 100cm = 10,000 cm2

To change area measurements fom m2 to cm2 we multiply area in m2 by 10,000 and to change cm2 to m2 we divide area in cm2 by 10,000.

1 m³ = 100cm x 100cm x 100cm = 1,000,000 cm³

To change area measurements from m³ to cm³ we multiply area in m³by 1,000,000 and to change cm³ to m³ we divide area in cm2 by 1,000,000.

20 of 44

Distance-time graphs

You need to know three main points for Distance-time graphs:

  • At any point, Gradeint = Speed, But watch out for units!
  • The steeper the graph, the faster its going.
  • Flat sections is where it is stopped
21 of 44

Example of a Distance-time graphs


22 of 44

Example of Distance-time graphs


23 of 44

Velocity-time graph

You need to know four main points for velocity-time graphs:

· At any point Gradient = Acceleration (m/s²)

· Negative slopes (going down) means deceleration

· Flat sections mean steady speed

· Area under graph = distance travelled

24 of 44

Example of a Velocity-time graph


25 of 44

Loci and Construction

Equidistant line from a fixed point(a circle)

All the points P lie on a locus of points the same distance from O

loci #3 (**-loci-3.jpg)

26 of 44

Loci and Construction

The Locus of points which are a fixed distance from a given line, would be an oval type shape.


27 of 44

Loci and Construction

The Locus of points that are equidistant from two given points A and B below, would be a perpendicular straight line.

( the points along the straight line are equidistant from A and B.

Let each point be the centre of a circle, the distance from A to B the radii. Draw circles note the intersections. Draw a line connecting the intersections, this line is equidistant.

Note: Draw a line connecting A to B, this line is perpendicular to the line connecting the intersections.

28 of 44

Loci and Construction

The locus of points that are equidistant from two given lines X and Y, would be a straight line bisecting the angle of the lines.

( Using a fixed radius start where the lines meet draw an arc that intersects both lines. from these new points 2nd and 3rd, draw arcs that intersect with each other.

Connect this new point to the 1st point for the equidistant line. The angle between the two lines is now halved.

29 of 44

Loci and Construction

Constructing Accurate Angles: accurate angles can be achieved with just a compass and a ruler. The radius of all the circles have to be the same.

Draw the first circle at the end of the line.

Draw the second circle with the centre at the point where the first circle intersects the straight line.

Where both circles intersect with each other a line drawn from the first point will be exactly 60(

30 of 44

Congruent and similar

  • Congruent - same size and same shape.
  • Similar - same shape but diffrent size.

For a triangle to be congruent, at least one of these rules must apply:

  • *** - All 3 sides are same,
  • AAS - 2 Angles and a side are the same,
  • SAS - 2 sides are equal and so is the angle between them,
  • PHS - A right angle, hypotenuse and another side are the same.

For a shape that is similar its shape is diffrent but all the angles stay the same.

In Transformation,

  • Translation, Rotation and Reflection are all congruent
  • Enlargements are similar
31 of 44


o if the scale factor is larger than 1, shape gets bigger

o If the scale factor is less then 1, the shape gets smaller

o If the scale factor is negative, shape is rotated 180

o The scale factor tells you the distance between the original point and the new point form centre of enlargement e.g. point Original point a is 3 cm away from centre of enlargement and it is enlarged by a scale factor of 3 so you do 3 x 3 = 9, so the new point is 9 cm away from centre of enlargement.

Area and volume of enlagement, For scale factor n:

  • Sides - n times bigger
  • Area - n² times bigger
  • volume - n³ times bigger
32 of 44

Pythagoras' Theorem

a² + b² = c²

Only works with right-angle triangle

See full size image ( sure you get the sides right.

Always check if answers are sensible

33 of 44


To find or plot a bearing you must rememeber the 3 key words:

  • FROM - Find the word from in the question, and put your pencil on the diagram at the point your going from.
  • NORTH LINE - A the point you are going from, draw a north line.
  • CLOCKWISE - Draw an angle clock wise from the northline to the pont or line. this is the angle your meant to measure - bearing

All bearing are meant to have 3 figures, e.g 018, 360, 034

34 of 44



  • Label three sides of the triangle's sides O (opposite), A (adjacent) and H (hypotenuse)
  • You need to memorise SOH CAH TOA and write it down, the way i memorise it is : Silly Old Homework Can Always Help Trignometry Out Alot
  • Descide which two sides are involved,O,H A,H or O,A
  • And then using the prevoius step decide which formulae to use: Sinθ = O/H , Cosθ= A/H and Tanθ =O/A
  • Translate into number and work it out and make sure answers are sensible.

Important Points

  • H is the longest side. Oppisite is side oppisite the angle being used (θ ) and adjasent is the other side that is next to the angle being used (θ ).
  • When working out the angle - use reverse sin,cos,tan.(make sure your calculator is on DEG mode)
  • you only use trigonometry in right- angled triangles
35 of 44


A vector quantity has both direction and magnitude (size).

(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)

image: a grid with a diagonal line marked A and B at each respective end. There's an arror in the centre pointing in the upward direction labelled a. (

For example this arrow represents a vector. The direction is given by the arrow, while the length of the line represents the magnitude.This vector can be written as: AB (arrow above) ( , a, or 3 over 4 ( .In print, a is written in bold type. In handwriting, the vector is indicated by putting a squiggle underneath the letter: the letter with a squiggle underneath it indicating a vector (

36 of 44

Vectors (continued...)

If two vectors have the same magnitude and direction, then they are equal

Adding vectors: Look at the graph below to see the movements between PQ, QR and PR.(a over b) + (c over d) = (a + c over b + d) ( Vector PQ (arrow above) ( followed by vector QR (arrow above) ( represents a movement from P to R. PQ (arrow above) + QR (arrow above) = PR (arrow above) ( Written out the vector addition looks like this

(2 over 5) + (4 over -3) = (6 over 2) (

image: a grid with the points P, Q and R marked. From P to Q the direction of the line is upward, from Q to R the direction is downward (

37 of 44

Subtracting Vectors

Subtracting a vector is the same as adding a negative version of the vector. (Remember that making a vector negative means reversing its direction.)

(a over b) - (c over b) = (a - c over b - d) (

image: a grid with points X, Y and Z joined. The line between X and Y has an arrow indicating an upward direction, the line from Y to Z also had an arrow indicating an upward direction. ( at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors XY (arrow above) ( and ( You could say it is vector XY (arrow above) ( followed by a backwards movement along ( So we can write the path from X to Z as

XY (arrow above) - ZY (arrow above) = XZ (arrow above) ( Written out in numbers it looks like this:

(4 over 2) - (1 over 2) = (3 over 0) (

38 of 44

3D Pythagoras and Trigonometry

  • On the diagram, make a right-angled triangle using the line, a line in the plane and a line betwwen them 2.
  • Draw the right-angled triangle again so its clear to see. Label the sides. you might have to use Pythagoras to work out a legth.
  • If question askes for angle, use Trigonometry to calculate angles.
39 of 44

Drawing Straght Line Graphs

  • Get the equation in to the form y = mx + c
  • Put a dot on the y axis at the value of c
  • Than go along one unit and up or down by the value of m (step) and make another dot
  • Repeat the same 'step' in both directions
  • check gradient is correct

Finding the equation of a straight line.

  • From the axes identify 2 variables
  • Find the gradient and y-intercept from the graph
  • Use this information to write the equation
40 of 44

Graphs to Learn

y= ax²+bx+c (n shape when negative, u when positive)


y = ax³+bx²+cx+d


41 of 44

Graphs to Learn (continuatation...)

y = A ÷ x or xy = A


y = kx (where k is a positive number)


42 of 44

Graphs to Learn (continuation...)

x² + y² = r²


43 of 44

Quadratic Graphs


Quadratic graphs are always symmetrical u shapes or n shapes if negative.

44 of 44


Callum Pullen


hello! and welcome to revising is annoying! today we will show you that the flashcards dont work!!!!!




♬ Shivani ♬


This is good, but do you have anything for unit 1 & 2 revision because i'm doing my exams on those 2 topics in 2 weeks time :(



Great notes thanks:)



Realli gud notes

thanxx =D


Tom Lovick


Amazing!! thanks, need good notes for final revision tomorrow!!



great stuff



Thankyou, these are really helpful, just one bit, on congruent and simliar should *** not be SSS because all 3 sides are the same?

sinem ozer


how do you knoiw which units you are doing?



I like the notes, they're really useful. = ] Although on card 15, 'Circle Geometry', the first bullet point says 'An angle in a semi circle is 90 degrees', which I found confusing. Obviously a 90 degree angle can be formed when the radius meets the base, but its not really clear. But otherwise, apart from the few spelling/grammar mistakes (which don't really matter, given its Maths), really good and useful. Thanks :)

Thomas Neal


Very helpful thankyou! sitting my exam 2day wish me luck ;)



well done....brilliant notes...

Ahmed =)


Excellent notes, Arwa, thank you!!



Really helpful..... :)


But.... in the upper and lower bounds... Upperbound+ uppper bound = max answer & UB x UB too....... ;)

Anyways..... they're really amazing

Sitting my exams 2morrow....

wish me luck.. cos i need it lots :S



Great notes, cheers ;D

Abby Mackay


im so greatful these are brill. i just needed something to refresh the different topics in my mind and something breif and you did just that, plain simple, and helpful.

Abby Mackay


im so greatful these are brill. i just needed something to refresh the different topics in my mind and something breif and you did just that, plain simple, and helpful.



great notess! thnxz! x

Alison King


this is brilliant, i now understand circles :)



good notes, but in the bounds, 2.4 - 0.05 isn't 0.35



bare manz



James Stewart wrote: bare manz 



Lindsey wrote:Thankyou, these are really helpful, just one bit, on congruent and simliar should *** not be SSS because all 3 sides are the same?


Danae Josephine


These are wicked helpful! Thank you sooo much(:

Stephanie Clarke


Good explanation and content. 

Harsharanjit Mustota


i've seen better



I hope ill get an A for this :D

This is fantastic <3

Jennifer Wright


amazing!!!! :')



wooo! exam in three hours time and i havent done anything! hopefully these will help:)



this is really useful, thank you. i was really struggling with some of this, and i was especially terrified after my non-calculator exam the other day. i'll feel a lot more confident going into the exam tomorrow now, thanks!!

GEEK twO ;)


Awhhh dish iss SOOO good mann,, fukin swear dwn i dnt  understnd it alll... fanku bruv fanku ;)



looooooooooooolllllllllllllll eazii nw

GEEK twO ;)


Truss me blad!!!!! eloo geeks!



wag 1 blad you kwl an tht do yu understand it



wow gr8 notes they r soooo gr8 

got my mock on monday hope i get a b xoxo 


Amandeep Kaur Dhillon


these are really good but the diagrams on the loci & construstion notes aren't there- do you think you could put them up again and in order? Thanks for the notes :)

Hannah-Louise Healey


I love these notes,really useful,i find it hard to understand text books but these notes really helped.. Great stuff thanks! :))



not what i had in mind but any how



Fab Notes thanks:)



some mistakes, e.g fractions continued "a patter of numbers" and the line "if denominator has a prime number of 2 or 5 its termination, if not then its when converted it will be a Recurring decimals" doesn't make sense, but a very helpful

Max Reeley


Rukhsar wrote:

 you need to revise english.



♬ Shivani ♬ wrote:

Need it for 1 n 2 



Can someone help? where it says 'angles in a semi circle = 90degrees'

what does that mean???????? 

Pearl Alexus


Wow these are really good, sorry is im not doing this at the mo!oh and the angles in a semi circle means: if you take a circle and draw a triangle in half of it (in a semi-circe) the angle which touches the circumference of the circle will be 90 degress/ a right angle

holly flannery


Rukhsar wrote:

Realli gud notes

thanxx =D


Maybe you should check out some English notes... Just saying.

holly flannery


Rukhsar wrote:

Realli gud notes

thanxx =D


Maybe you should check out some English notes... Just saying.



A great set of revision notes for higher level students.



A really good set of notes.

:) Thanks



Hi these cards would be good for my revision but it won't let me print them, can you help me please 



Hi this notes were really helpful and easy to understand........

:) :) :)



Bmt these revision notes are bare good still.

Alice kirby25


I couldn`t print them 



I cant print these cant



PDF won't load.

Needs fixing



Amazing but sadly all the links dont work :(






A bit too much to look at. Try adding some pics or animations to make it more visually appealling.

Similar Mathematics resources:

See all Mathematics resources »See all Algebra resources »