- Created by: I.T.P
- Created on: 22-02-18 21:12
- Angles on a straight line add up to 180 degrees.
- Angles about a point add up to 360 degrees.
- There are 360 degrees in a complete turn. So in half a turn there are 180 degrees, and in a quarter of a turn 90 degrees.
- We sometimes call a quarter of a turn a right angle, and mark it with a square.
- Any angle that is 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 that is between 180 and 360 degrees is called an reflex angle.
- The angles in a triangle add up to 180 degrees, and that the angles in a quadrilateral add up to 360 degrees.
- Parallel lines will never meet, no matter how far they're extended.
- Small arrows are used to show that 2 lines are parallel to each other.
- Vertically opposite angles are equal.
- Corresponding angles are equal.
- Alternate angles are equal.
- Co-interior angles add up to 180 degrees.
- In exam questions look out for parallel lines (squares, rectangles, parallelograms and marked parallel lines) and see which of the above rules apply.
- Look out for lines of equal length. Rememeber that an isosceles triangle has two sides of equal length and that the angles which are directly opposite, these sides are also equal.
- Diagrams are unlikely to be drawn to scale, so do not be tempted to measure!
- The way we describe direction from a point (apart from North, East, South, West) is to use 3-figure bearings. So for less than 100 degrees put an appropriate number of 0s in front. E.g. 020 degrees, 037 degrees, 002 degrees, 007 degrees.
- A compass always points North. Bearings are measured from the north line, always in a clockwise direction.
- Scale drawing allows us to draw large objects on a smaller scale while keeping them accurate. Maps are a great example.
- AU scale drawings must have a scale written on them. Scales are usually expressed as ratios. E.g. 1cm = 100m
There are three types of questions you can be asked when studying scale drawings:
- The first type involves calculating the scale factor of the drawing, in other words what scale label should be present on the diagram.
- The second type of question you could be asked is to work out how large an object from a plan is in real life using measurements and a given scale.
- Finally, you could be asked to produce a scale drawing from a given scale factor and measurements.
Angles in Polygons
- The formula for calulating the sum of the interior angles of a regular polygon is: (n - 2) x 180 degrees where n is the number of sides of the polygon.
- This formula comes from dividing the polygon up into triangles using full diagonals.
- We already know that the interior angles of a triangle add up to 180°. For any polygon, count up how many triangles it can be split into. Then multiply the number of triangles by 180.
- The exterior angle of a polygon and its corresponding interior angle always add up to 180° (because they make a straight line).
- For any polygon, the sum of its exterior angles is 360°.
- Interior angle of a regular polygon = sum of interior angles ÷ number of sides
- We know that the exterior angles of a regular polygon always add up to 360°, so the exterior angle of a regular hexagon is
- The interior angle and its corresponding exterior angle always add up to 180° (for a hexagon, 120° + 60° = 180°).
Primes, Factors and Multiples
- A prime number is a number which is only divisible by 1 and itself.
- The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
- The only even prime number is 2. All other even numbers are divisible by 2.
- The factors of a number are the numbers that divide into it exactly.
- The multiples of a number are the numbers that belong to that number's times table.
- Factors are often given as pairs of numbers, which multiply together to give the original number. These are called factor pairs.
- It is often useful to write a number as the product of its prime factors. This can be done by listing the factor pairs as successive branches in a factor tree. The branches continue to expand until all the factors are prime numbers. The final answer is the product of the prime numbers displayed at the end of these branches.
Prime Factors, HCF and LCM
- Every number can be written as a product of prime numbers.
- Remember: 'product' means 'times' or 'multiply'.
- We can use prime factors to find the highest common factor (HCF) and lowest common multiple (LCM).
- To find the lowest common multiple, we need to think about which list has the most of each factor.
- B rackets
- I ndices
- D ivision
- M ultiplication
- A ddition
- S ubtraction
Rounding Decimal Places
- Decimal place means the position after a decimal point.
- 0 - 4 rounds down
- 5 - 10 rounds up
Rounding Significant Figures
- Any digit after the first significant figure is considered to be a significant figure.
Algebra - Simplifying
- a + 3b + 2a = 3a + 3b
- -5c + 8d + 6c + 3d = c + 11d
- e squared - 2e + 5e squared + 7e = 6e squared + 5e
- 9g + 4h squared - 15g - 2h = 4h squared - 2h - 6g
- 5a x 2 = 10a
- 3b x 4b = 12b squared
- 6c x 6d = 36cd
- 8e squared x 2e = 16e cubed
Algebra - Substitution
a = 2
b = 3
c = 5
- 2a + 3b - c = 2 x a + 3 x b - c
2 x 2 + 3 x 3 - 5
4 + 9 - 5 = 8
- a squared - 5c = 2 squared - 5 x 5
4 - 25 = -21
- abc = a x b x c
2 x 3 x 5 = 30
Expanding Single Brackets
2 ( x + 1 )
3 ( 2 x + 4 )
Multiplying everything in the brackets by the number on the left.
Expanding Double Brackets
( x + 2 ) ( x + 3 )
( x - 4 ) ( x - 7 )
( x + 5 ) ( x - 6 )
Factorising Single Brackets
- " Putting back into the brackets"
2x + 8
= ( x + 4 )
10y - 15
= 5 ( 2y - 3 )
6xy + 15x
= 3x ( 2y + 5 )
Factorising Quadric Expressions
- When factorising a quadratic expression, we want ...
a x squared + b x + c
- Two numbers that ADD together to make B.
- Two numbers that MULTIPLY together to make C.
1. Find the biggest number that goes into both numerator and denominator.
2. Divide the numerator and denominator by this number.
Adding and Subtracting Fractions
- When adding or subtracting fractions you must always find a common denominator.
Multiplying and Dividing Fractions
When multiplying you simply multiply the numerator by numerator and the denominator by denominator.
When dividing, follow the "Keep, Flip, Change" rule.
Mixed Numbers and Improper Fractions
Mixed to Improper =
You convert to improper when adding, subtracting, multiplying or dividing.
Impoper to Mixed =
Adding and Subtracting Decimals
- The number of decimal places in the question will be the same in the answer.
- We need to make the number we are dividing by a whole number by multiplying through by a power of 10 (e.g. 10, 100, 1000)
Converting a recurring decimal into a fraction
Fractions into recurring decimals
Coordinates and Linear Graphs
- To generate coordinates, you simply substitute the values for x into the linear function.
Identifying the Gradient and Intercept
- The general equation for a straight line is ...
y = m x + c
- Where 'm' is the gradient and 'c' is where the line intercepts the y axis. The gradient of a line tells us how steep the line is.
- The formula for finding the gradient of a straight line from a graph is ...
gradient = change in y / change in x
gradient = rise / thread
intercept = read of the graph
Midpoints and Finding the Equation of a Line
Finding the Midpoint Line
To calculate the midpoint of a line segment you substitute in the coordinates you are given into the following formula ...
Midpoint = ( )
Finding the Nth Term
n 1 2 3 4 5
2 4 6 8 10
1 3 5 7 9
n 1 2 3 4 5
-3 -6 -9 -12 -15
-2 -5 -8 -11 -14
Generating a Sequence
- Calculate the first 5 terms of the sequence.
3n + 1
1st term is when n = 1:
3(1) + 1 = 4
2nd term is when n = 2:
3(2) + 1 = 7
3rd term is when n = 3:
3(3) + 1 = 10
Finding whether a number is a term in a sequence
Is 101 a term in the sequence 5n + 6?
5n + 6 = 101
5n = 95 = subtract 6
n = 19 = divided by 5
101 is the 19th term of the sequence 5n + 6.
Is 140 a term of the sequence 7n - 5?
7n - 5 = 140
7n = 145 = plus 5
n = 20.71 = divided by 7
No, 140 is not a term in the sequence 7n - 5 because you can't have decimals in sequences.
Finding Quadratic Nth Terms
n 1 2 3 4 5
n 1 4 9 16 25
6 9 14 21 30
- A cumilative frequency polygon shows how the cumalative frequency changes.
Box Plots - Quartiles
- The quartiles split the data into quarters.
- The lower quartile is a 1/4 of the way into the data.
- The upper quartile is 3/4 of the way into the data.
- The median is 2/4 ( ie 1/2 ) of the way into the data.
- Always order your data smallest to biggest.
- 'n' is all of the data.
- The interquartile range ( IQR ) is the upper quartile - lower quartile.
- It tells us how spread out the middle 50% of the data is.
- Smaller IQR means more consistency and vice versa.
- A box plot is a way of illustrating key information about a set of data.
- They are also very useful for comparing the distribution of two sets of data ( e.g boys vs girls ).
Stem Leaf Diagrams
- Pie charts show proportion, not amounts.
- To find the multiplier you do: 360 divided by the total.
- To find each angle you do: Frequency x Multiplier.
Interpreting Pie Charts
To calculate a quantity you...
360 degrees x Total
Frequency Density = Frequency / Class Width
Frequency = Frequency Density x Class Width
Class Width = Frequency / Frequency Density
Histogram = The bars touch ( continuous data )
The area of a bar represents the frequency, not the height of the bar.
Time Series Graphs and Moving Averages
- can show short and long term trends
- time always plotted along the x - axis
- must explain fluctuations in the context of the question
Types of Fluctuations:
- Scatter diagrams are used to show whether there is a relationship between two sets of data.
- The 'line of best fit' is a line that goes roughly through the middle of all the scatter points on a graph.
- We can use a line of best fit to estimate other values that may be missing from the scatter diagram.
Finding one quantity as a percentage of another
Percentage Increase and Decrease
- To multiply by 1 gives you 100% of your value. So, to increase by a percentage, you do...
Increase = Orginal Value x ( 1 + percentage )
Decrease = Original Value x ( 1 - percentage )
1) Find the difference between the 2 values ( second value - first value )
2) Calculate ( difference / orginal value ) x 100
1) Find the total current percentage ( 100 + increase )
2) Current amount / total percentage
3) x 100
After % increase
After % decrease
Interest - Simple Interest
- Simple Interest is where the same amount of money is added on at the end of each year.
Interest - Compound Interest
- Compound interest is where the interest is calculated on the balance at the end of each year.
Real Life Graphs
Distance - Time Graphs
- Distance is represented on the y - axis.
- Time is represented on the x - axis.
The steeper the gradient the faster the object is moving.
The gradient represents speed.
Velocity Time Graphs
- Area of a square / rectangle / rhombus / parallelogram...
Base x Height
- Area of a triangle...
Base x Height / 2
- Area of a trapezium...
a + b
2 x h