Squares and Rectangles
The formula to work out the area of a rectangle or square is width x length.
The formula to work out the area of a triangle is base x height / 2.
The area of a parallelogram is base x height.
The area of a trapezium is height(a + b)/2.
The area of a circle is , pi is 3.14.
The perimeter is the measured distance around the edge of a shape.
Below is a shape drawn onto 1cm squared paper. The perimeter is 18cm.
The perimeter of a circle is called the circumference. It is calculated with this formula:
Mean, Median, Mode and Range
The mean is calculated by adding up all the data and dividing it by the total peices of data.
The average for the numbers 1 2 5 3 6 8 3 6 would be 4.25.
The median is the middle number (when the numbers are in order).
The mode is the most common (mo = most,mode) number in a set of data.
The range is the difference between the largest and smallest numbers in a set of data. largest - smallest = range
Fractions - Basic
Fractions are shown as:
Fractions can be converted into percentages and decimals too.
To convert into a decimal simply dived the denominator by the numerator.
E.G. 1/2 as a decimal is: 1/2 = 0.5
To convert a fraction to a percentage simply work out the decimal form and times by 10.
E.G. 1/2 as a percentage is: 1/2 = 0.5 0.5 x 10 = 50 50%
Fractions can be improper or proper. An improper fraction has a numerator larger than or equal to the denominator. Whereas a proper fraction has a numerator smaller than the denominator.
E.G. 4/3 as a proper fraction would be: 4/3 = 1.333 1 1/3
Fractions - Addition, Multiplication & Division
To add fractions with common denominators just add the numerators and use the common denominator.
To add fractions that don't have common denominators. First multiply the denominators to find the common factor. Then multiply the numerator/s to create to fractions with common denominators. Add them as you would a normal fraction addition. Simplify the answer if you can.
To multiply a fraction simply multiply the numerators, and then the denominators. Then simplify the answer if necessary.
To divide a fraction simply flip the fraction you want to divide by and change the divide to a multiply.
The numbers are different when they are negative. For example -10 is less than -5.
Adding a negative number is the same as subtracting two positive numbers. For example 5 + (-2) is the same as 5 - 2. Leaving the answer as 3.
Subtracting negative numbers is the same as adding two positive numbers. For example (-5) - (-3) is the same as (-5) + 2. Leaving the answer -3.
Multiplication can be done by following some rules.
- positive x positive = positive
- negative x negative = positive
- positive x negative = negative
- negative x positive = negative
Division follows the exact same rules as multiplication.
A net is a two dimensional representation of a three dimensional shape. There is an example of a net below.
When a net is folded correctly it forms a 3D shape. The net above has 6 equal faces meaning it mus fold to create a cube.
Plans and Elevations are what architects draw to show a 3D shape from side views. They show the shape from above, left and right. An elevation is simply an elevated part on the shape.
There are 360 degrees in a full circle. 180 degrees in a half turn (and a triangle) and 90 degrees in a right angle.
- Any angle less than 90 degrees is called an acute angle.
- Any angle which is between 90 and 180 degrees is called an obtuse angle.
- Any angle which is greater than 180 degrees is called a reflex angle.
All the angles in a triangle add up to a total of 180 degrees. And all angles in any quadrilateral add up to a total of 360 degrees.
Symmetry - Lines and Rotational
A shape is symmetrical if both sides of it are the same when a mirror line is drawn.
Number of lines of symmetry in 2D shapes.
- Triangle: 1 for equilateral triangles
- Square: 4
- Rectangle: 2
- Hexagon: 6
- Decagon: 10
A plane of symmetry would be a line of symmetry through a 3D shape.
Rotational symmetry is is known as a shapes order of rotation. The order of rotation is the number of times a shape can be rotated 90 degrees about the center whilst keeping its original pattern.
Volume is the area of a 3D shape.
To work out the volume of a cube or cuboid do W x L x H
To work out the volume of a Pyramid do Area of the base * Height * 1/3
To work out the volume of a cone do 1/3πr2h
To work out the volume of a sphere simply do 4/3πr3
Product of Prime Factors
To work out 60 as a product of prime factors. You divide the number by the lowest possible prime factor. If you cant move onto the second lowest...
60 --> (2) --> 30 --> (2) --> 15 --> (3) --> 5
First we do 60 / 2 = 30. Then 30 / 2 = 15. Then 15 / 3 = 5.
Meaning that the number 60 as a product of prime factors would be shown as...
2 x 2 x 3 x 5 = 60 or 2² x 3 x 5 = 60
Pythagoras' Theorem states that:
a² + b² = c²
This means that on a triangle, the hypotenuse² (the longest side on a right-angled triangle) is equal to the sum of the two other sides squared.
The equation can be rearranged to work out other sides of a triangle.
b² = c² - a² and a² = c² - b²
Transformations - Translations & Reflections
If we translate an object, we move it up or down or from side to side. But we do not change its shape, size or direction.
When an object is transformed by a reflection the object is always the same distance from the mirror line. A mirror line is written as:
x = 1 or y=1 The 1 is the coordinate and the x/y is the axis.
Transformations - Rotations & Enlargements
A rotation is when a shape is rotated about its center point by 90 degrees at a time.
The scale factor will tell you what to multiply the sides of the shape to get the enlarged version of the shape.
To estimate a tricky looking question like 104 x 9.2 first round of the numbers to make the calculation easier.
Round the 104 to 100 and the 9.2 to 9. This gives us a much more easy calculation of 100 x 9 = 900.
To right the answer we use a simple that looks like this.
This symbol means approximately equal to. We would use this when writing out answers that are approximate.
Algebra - Simplifying Equations & Expanding Bracke
Equations can be expressed in short or long form. For example:
3x + 4x is equal to 7x
2y x 3y is equal to 5y²
4m x 3t is equal to 7mt
Brackets help to organize information in an to be easily solved or read.
3(y + 4)
We would first multiply 3 by y to get 3y. Then we would multiply 3 by 4 to get 7. We then put pack the addition to get the expanded equation. 3y + 7
Algebra - Index Notation
An index notation is a number that tells us how many times to multiply the number it is after by. For example:
5 ² is equal to 5 x 5
34 is equal to 3 x 3 x 3 x 3
The square of a number is the number multiplied by itself. Or m ². So the square root of a number is the number that is squared.
is 3 because 3 x 3 = 9.
Algebra - Simultaneous Equations
Solve these simultaneous equations and find the values of x and y.
- Equation 1: 2x + y = 7
- Equation 2: 3x - y = 8
Add the two equations to eliminate the ys:
- 2x + y = 7
- 3x - y = 8
- 5x = 15
- x = 3
- Now you can put x = 3 in either of the equations.
- Substitute x = 3 into the equation 2x + y = 7:
- 6 + y = 7
- y = 1
So the answers are x = 3 and y = 1
Probability - Basic
The probability of an outcome can be represented as a fraction, percentage, decimal or in word form.
1/5 2/5 3/5 4/5 5/5
20% 40% 60% 80% 100%
0.2 0.4 0.6 0.8 1.0
Impossible Unlikely Even Likely Certain
Equation for calculating probability is...
number of desired outcomes / total outcomes = probability
3 / 10 = 3/10 = 30% = 0.3
Probability - Tree Diagrams & AND/OR Rules
A tree diagram is a diagram that shows the probability of all the outcomes for something. For example the probability of throwing 3 heads in a is...
1/2 x 1/2 x 1/2 = 1/8
So the probability of throwing 2 heads in a row is 1/8.
The AND/OR Rule
Probability(A and B) = Prob(A) x Prob(B)
Probability(A or B) Prob(A) + Prob(B)
E.G When taking coloured counters out of a bag, the probability of the second counter being a certain colour with be different depending out the previous counter that was taken out.
Probability - Expectation & Relative Frequency
The formula for expectation is:
expected successes = probability of success x number of trials
Say we have a circle divided into 4 equal sections. When we spin a spinner 240 times, how many times would we expect it to land on 3?
1/4 x 240 = 60/240
We can estimate the probability of something from the relative frequency.
relative frequency = number of successful outcomes / total attempts
Expanding Double Brackets
To expand double brackets use the mnemonic:
The equation (x + 9) (8 - 3) would be expanded to get:
8x - 3x + 72 - 27
Which we can simplify to get:
5x + 45
If two quantities are in direct proportion, as one increases, the other increases by the same percentage.
If y is directly proportional to x, this can be written as y ∝ x
If a 15 shoe costs £30 then the equation is
15 ∝ 30
We then replace the ∝ with =K to get the full equation of 15 = 30k
Then we modify this equation to get K = 15 / 30
15 / 30 = 0.5 or k = 0.5
Mean from Grouped Data
The table bellow shows grouped data. Each category of heights is not exact, but a group.
To find the mean we use the equation ∑fx / ∑f. This will give us the average height.
Number of People (f) Midpoint (x) fx 101-120 1 110.5 110.5 121-130 3 125.5 376.5 131-140 5 135.5 677.5 141-150 7 145.5 1018.5 151-160 4 155.5 622 161-170 2 165.5 331 171-190 1 180.5