why

Types of Numbers Square Numbers – any number times by itself.

Examples: 4 (2x2),9 (3x3),16 (4x4),etc

Cube Numbers – any numbers times by itself 3 times! (Like volume of a cube!)

Examples: 8 (2x2x2), 27 (3x3x3), 64 (4x4x4),etc

Powers – how many times a number multiplies itself by.

Example: powers of 2:

2² = 2x2 = 4

2³ = 2x2x2= 8

2 = 2x2x2x2= 16

Prime Numbers – they only divide by themselves and one. If the number ends in 1,3,7 or 9, it means that there is a

chance of it being a prime number – but it doesn’t mean it is definitely a prime number! (e.g. 21,27 are NOT prime numbers though they end in 1/7 as they both divide by 3!)

why

Multiples, Factors and Prime Factors

Multiples – the times table of that particular number. E.g. the multiples of 3 = 3,6,9,12,15,18……..

Factors of a number are pairs of numbers that you multiply to get another number.

e.g. factors of 24 = 1x24,2x12,3x8,4x6 (multiply any of those pairs and you’ll get 24!) – just start with 1 and keep going!

Multiple = multiply = just keep multiplying!

why

Examples1x64

2x32

3x –

4x16

5x –

6x –

7x –

8x 8

1x24

2x12

3x8

4x6

5x –

6x4

Factor of 64 – 1,64,2,32,4,16,8

Factors of 24 – 1,24,2,12,3,8,4,6

Repeated numbers – so

STOP!

Repeated numbers – so STOP!

Don’t write a number twice!

Always start with ‘1’ times the number!

3,5,6,7 can’t be divided into 64. So leave it!

why

Factor Tree ‘Factor trees’ are used to find prime factors (prime factors:

a number broken down into a string of prime numbers all multiplied together)

A factor tree looks like this:

420

42 10

256

7

2 3

Factor tree of 420 – it will show you the

prime factors of 420!Each ‘level’ of numbers multiplies to get the number above that level

‘Level’ 1

‘Level’ 2

‘Level’ 3

The numbers at the end (the ones in circles) are the

prime factors!42x10=42

0

2x5=10

6x7=42

2x3=6

Finally! The prime factors of 420 =2,3,7,2 and 5. You would get 420 by multiplying those numbers.So it’s usually written with times signs = 2x3x7x2x5However – examiners like to see numbers in the simplest form, so we should write it like = 2²x3x7x5 (2²= 2x2)

why

LCM LCM- Lowest Common Multiple – The smallest number that

will divide by all of the numbers in the question.

Once you know how to do the prime factors – LCM and HCF are fiddlesticks!

LCM Example: Find the LCM of 42 and 36.

1) Find the prime factors of 42 and 36

2) 42=7x2x3 36=2x2x3x3

3) Circle the common prime factors

4) 42=7x2x3 36=2x2x3x3

5) Write down the ones you have circles ONCE =2x3

6) Then put the ones you haven’t circled in the list= 2x3x2x3x7=252This means if you wrote out all of the multiples

in both numbers, 252 will be the first number that is common to both numbers.

why

HCF

HCF- Highest Common Factor – the biggest number that will divide into all the numbers in the question.

Example of HCF:

Let’s use the same numbers: 42 and 36

We know that the prime factors of -

42=7x2x3 36=2x2x3x3

Now, we’ve just got to circle the same things in both numbers!

HCF of 42 and 36 = 2x3=6This means the highest factor

of both 42 and 36 is 6.

why

1) When multiplying – you add the powers

e.g. 3²x3²=3 (2+2=4)

2) When dividing – take away the powers

e.g. 3/3²=3² (4-2=2)

3) When raising power to another, you multiply them.

e.g. (5²)=5(8)

4) Anything to the power of itself is just itself.

e.g. 2=2 3=3 4=4

5) Anything to the power 0 is 1.

e.g. 3˚=1 4˚=1 5˚=1

6) 1 to the power of anything is still 1.

e.g. 1=1 1=1 1=1

7) Apply powers to top and bottom of fractions.

e.g. (¾) = 9/16

Powers and Roots

8 is the index number, I can’t get index numbers higher than 5 on my laptop..

why

Powers and Roots Continued..

1) Negative powers – turn it upside down and make the power positive.

2) e.g. 7ˉ= 1/7=1/49 (3/5) ˉ = (5/3)

3) Fractional powers- ½= square root, 1/3= cube root, ¼ = fourth root,etc.

e.g. 25½= square root of 25=5

4) Two-Stage Fractional Powers = 64⅜…..the top number of the fraction means the power, bottom number is the root.. So it means the eighth(8) root of 64, then the answer needs to be cubed(3).

5) Square roots can be positive or negative. e.g. 2² can be 2x2 = 4 but also (-2)² = -2x-2=4

why

Standard Index Form A number in standard index form must always be written in

this way: A x 10ⁿ

This number needs to be between 1 and 10.

‘n’ is how many places the decimal moves.

Examples:1) “Express 3560 in standard

form.”-Move the decimal point until

3560 becomes 3.56 (A must be between 1 and 10).

-The decimal point has moved 3 places, so ‘n’ equals 3. Giving 10³

-3560 is a big number, so n is +3, not -3.

-So, the answer is 3.56x10³

2) “Express 0.0000623 in standard form.”

The decimal point must move 5 places to give 6.23, so the power of 10 is 5.

- Since 0.0000623 is a small number, is must be 10ˉ, not 10

why

Recurring or Terminating There’s one way which you can work out if the fraction is

recurring (like 1/3=0.333333..) or terminating (like 1/20=0.2).

It’s very simple, as long as you know how to work out the prime factors of a number.

If the denominator of the fraction has a prime factor other than 2 or 5, then it’ll be a recurring decimal!

e.g. 1/20 – denominator is 20 – prime factors of 20 are – 2x2x5,so 1/20 is a terminating decimal because the denominators’ prime factors are either 2 or 5.

e.g. 2 – 1/3 – denominator is 3, prime factor – 1x3…so..1/3 is a recurring decimal!

why

Convert recurring decimals to fractions “convert 0.234234234… to fractions in its simplest form.”

First – look at how many numbers are recurring, in this case – 3, because it’s ‘234’ that is just being repeated.

Then, if one digit is recurring (like 0.3333333) then times it by 10;if two digits are recurring (like .23232323) then times it by 100; if three digits are recurring then times it by 1000 and so on.

So, in this case it would by 0.234234234 – three digits are recurring, so times it by 1000.

Then you get: 234.234234234234

Subtract the original number from it!

234.234234234234234

- 0.234234234234234

234

Then, because you times it by a 1000, and took it away from its original (which is times 1), you get 1000-1=999.

So it becomes 234

999

When it is cancelled down, it becomes 26/11.

why

Fractions Multiplying – just multiply across!

e.g. ¾ x ⅜ = 9/32

Dividing – turn the 2nd fraction upside down, then multiply!

e.g. ¾ ÷ ⅜ = ¾ x 8/3 =24/12=2

Adding/Subtracting- if the denominators are the same, just add/subtract the numerator.

e.g. 3/8-1/8=2/8

However, if the denominators are NOT the same, then you have to MAKE THEM the same – it’s called equalising.

e.g. 4/7 + 5/3 = find the LCM of the denominators (7 and 3 in this case), but if you can’t be bothered, then just times them. After that, times the numerators by the opposing denominator too!

=12/21+35/21= 47/21=2 and 5/21

why

Continued… The example – solution!

4

7

5

3+ =

4

7

4x3=12

7x3=21=

5

3=

5x7 =35

3x7=21

Change their denominators first by

multiplying across.

12 35

21 21+

47

21= DON’T add the

denominator!

18

24

e.g. “cancel down

18/24”

6

8

3

4= =

Divide by 3

Divide by 2

Cancelling down – dividing top and bottom number by the same number

until they won’t go any further.

Find a fraction of ‘something’ – multiply the ‘something’ by the TOP of the fraction, then divide by the bottom

e.g. find 9/20 of £360

= (9x360) ÷ 20 = £162

why

Algebra Simplifying – collecting like terms

e.g. “Simplify 2x-4+5x+6”

2x-4+5x+6 = put ‘x’s together = 2x+5x=7x,

then put numbers together = -4+6=+2 answer=7x + 2

Expanding brackets – the item outside the bracket multiplies each separate term inside the bracket.

e.g. 1) 5(x+3) = 5x+15 2) 3x(x+4)=3x+12

Double bracket = (2p-4)(3p+1)=

Follow the arrows = 2px3p,2px1,-4x3p,-4x+1

Then tidy up = 6p +2p-12p-4

= 6p -10p -4

Multiply !

why

D.O.T.S –very important!!

Difference of two squares.

a -b = (a+b)(a-b) This is very important for

factorising! You can easily pick up some marks for very hard algebraic fractions, by knowing this!

You might get a question involving factorising ‘4x -36’

Then, you will be surprised….but if you learn this – you will know the answer is – (2x+6)(2x-6)

Trust me – it is very helpful.

why

Factorising

Putting brackets in!

How to do this-

1) Take out the biggest number that goes into all numbers

2) Take each letter in turn and take out the highest power that will go into EVERY term

3) Open the brackets and fill in all the bits needed to reproduce each term.

Example: Factorise 15xy+20xyz-35xyz

Answer: 5xy(3x+4yz-7xz)Biggest

number that will go into

15,20 and 35

Highest power that x and y

would go into all three terms

Z is not in all three terms, so

can’t come out as a common factor.

why

Algebraic Fractions Multiplying – it’s the same, just multiply across.

Dividing – turn the second fraction upside down, then multiply across…it’s the same rule…again..

Adding/Subtracting – get the same denominator by multiplying across. As long as you can do fractions, this is a piece of cake!

st10w

35stw

6x =

35stw

60w =

7st

12w

st10w

6

35stw÷ =

35stw

60w =

7st

12wst

10w x

why

Midpoint of a line To find the midpoint of a line – just divide!

“Find the midpoint of the line AB.

A has the co-ordinates of (4,5), B has the co-ordinates of (6,3)”

Add the Xs and the Ys together separately, then divide by two.

4+6/2=5

5+3/2=4

Then put the Xs and Ys back. (5,4)

The first number is x,

second number is y,

always.

why

y=mx+c m- the gradient of the line c- the y-intercept (where the graph hits the y

axis) The gradient of a line = y divided by x.

m is always the number with x, c is always the number on its own, so don’t get it mixed up when they give you ‘y=2+4x’ – put it back – ‘y=4x+2’

why

Distance-time graph

Gradient=speed

The steeper the line, the faster the speed.

Straight lines means it has stopped.

why

Parallel and perpendicular lines

Parallel lines have the same gradient (m), but different y intercept.

So the lines y=2x+3,y=2x-1,y=2x+6 are all parallel.

The gradient of two perpendicular lines multiply to give -1.

If the gradient of the first line is m, the gradient of the other line will be -1/m, because mx-1/m=-1.

why

The gradients of these lines are 2 and -1/2.The product of the gradients is 2 x -1/2= -1.

why

GeometryAngles in a

triangle adds up to 180.

Angles on a straight line adds up to 180.

Angles in quadrilateral (4-

sided shapes) adds up to 360.

Angles around a

point adds up to 360.

Isosceles triangles – 2

sides the same, two angles the

same.

Angle d = a+b

why

Continued…

a

a b

a a

a

a and a are the same a and b add

up to 180a and a are the same

These are called

alternate angles

These are called

corresponding angles

These are called

supplementary angles

why

Polygons

Exterior Angle

Interior angle

Exterior angle=

360/number of sides.

Interior angle = 180-exterior

angle

Sum of interior angles = (number of sides - 2) x 180

why

Symmetry Line Symmetry- draw one or more mirror lines

across a picture and both sides will fold together exactly.

Rotational Symmetry- rotate the shape into different positions and it will look exactly the same.

why

Circles

A tangent is a straight line that just touches the outside of the circle.

A chord is a line drawn across the inside of a circle.

An arc is just part of the circumference of the circle.

Chord

Tangent

The angle between a tangent and the radius is always 90 degrees!!

Same length tangents

why

Areas

why

Net, Volume of prism Surface area of the solid = area of net

why

Volume of PrismA prism is a solid object which is the same shape all the way through (such as a triangular prism and a cube).

Volume of prism = cross section x length

3cm

4cm

10cm

=area of cross-section (triangle) x length =(4x3)/2x10

=60cm cubed

why

Plan and Elevation You can look at an object from different points of view –

plan (from the top), front (from the front!) and side (obviously, from the side XD).

In your exam, you would then have to draw an image of the different points of views. So – get practising~

why

Converting Units

Metric Units Imperial Units

1 (kg) 2.2 pounds (lb)

1 litre 1¾ pints

4.5 litres 1 gallon

8 km 5 miles

30 cm 1 foot (ft)

These are the ones you need you know for your exam.

Jayden drives at an average speed of 60mph. How long will it take him to drive 120 km?

8km=5 miles

60/5=12

8x12=96

60mph=96km/h

Time = Speed/Distance

=120/96=1¼ hours

why

Speed, Distance and Time

This is used to find out the above…but you would’ve been using it for a long time, so I will not ramble~

You can think of this equation as SDT – SoDiT- SOD IT!

“A car travels 90 miles at 36 miles per hour. How long does it take?”

T=D/S = 90/36=2.5 hours.

But- Remember to get the units right!

“A boy walks 800m in 10 minutes. Find his speed in km/h”

800m=0.8km 10minutes =0.1667 hours Then divide 0.8km by 0.1667hours to get

4.8km/h