# GCSE Intermediate Mathematics

I have my GCSE Maths Year 9 Practice Exam coming up in about 4 days. I started making these revision cards about a week ago. Here are the topics the cards cover:

- Reading Scales
- Perimeter and Area
- Mode, Mean, Median & Range
- Fractions
- Negative Numbers
- 3D Shapes
- Angles
- Symmetry
- Algebra
- Estimation
- Pie Charts
- Probability
- Scatter Graphs
- Transformations
- Bearings
- Trial and Improvement
- Nth Term Rules
- Plans and Elevation
- Ratio
- Percentages
- Product of Prime Factors
- Pythagoras' Theorem
- Volume
- Compound Interest
- Direct Proportion
- Loci
- Similarity and Proportion
- Trigonometry
- Volume of Prisms
- Mean from grouped data
- Frequency Polygons
- Nth term rules
- Expanding double brackets

- Created by: Tom
- Created on: 16-06-11 15:57

## Area

**Squares and Rectangles**

The formula to work out the area of a rectangle or square is ** width x length**.

**Triangles**

The formula to work out the area of a triangle is **base x height / 2**.

**Parallelogram**

The area of a parallelogram is *base x height.*

**Trapezium**

The area of a trapezium is *height(a + b)/2.*

**Circle**

The area of a circle is , pi is 3.14.

## Perimeter

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

**Mean**

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.***Median**

The median is the middle number (when the numbers are in order).

**Mode**

The mode is the most common (mo = most,mode) number in a set of data.

**Range**

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:

numerator

denominator

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

**Addition**

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.

**Multiplication**

To multiply a fraction simply multiply the numerators, and then the denominators. Then simplify the answer if necessary.

**Division**

To divide a fraction simply flip the fraction you want to divide by and change the divide to a multiply.

## Negative Numbers

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

**is the same as**

*5 + (-2)***. Leaving the answer as**

*5 - 2***.**

*3*** Subtracting** negative numbers is the same as adding two positive numbers. For example

**is the same as**

*(-5) - (-3)***. Leaving the answer**

*(-5) + 2***.**

*-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*

Division follows the exact same rules as multiplication.

## 3D Shapes

**Nets**

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.

## Angles

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

**Line Symmetry**

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**

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

Volume is the area of a 3D shape.

**Cube/Cuboid**

To work out the volume of a cube or cuboid do *W x L x H*

**Pyramid**

To work out the volume of a Pyramid do **Area of the base * Height * 1/3**

**Cone**

To work out the volume of a cone do ^{1}/_{3}πr^{2}h

**Sphere**

To work out the volume of a sphere simply do ^{4}/_{3}πr^{3}

## 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

*Then*

**30 / 2 = 15.**

**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

Pythagoras' Theorem states that:

**a****²** **+ b****²****= c****² **

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.

*b***² = c****² - a*** ² * and

**a**

**² = c**

**² - b**

**²**## Transformations - Translations & Reflections

**Translations**

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.

**Reflections**

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

**Rotations**

A rotation is when a shape is rotated about its center point by 90 degrees at a time.

**Enlargements**

The scale factor will tell you what to multiply the sides of the shape to get the enlarged version of the shape.

## Estimation

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

**Simplifying Equations**

Equations can be expressed in short or long form. For example:

**3x + 4x** is equal to **7x**

**2y x 3y** is equal to **5y****²**

**4m x 3t** is equal to **7mt**

**Expanding Brackets**

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

**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**

**3 ^{4}** 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.

For example:

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 **y**s:

- 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

**Tree Diagrams**

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)*

**Conditional Probability**

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

**Expectation**

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**

**Relative Frequency**

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:

**F**irst

**O**utside

**I**nside

**L**ast

The equation **(x + 9) (8 - 3)** would be expanded to get:

**8x - 3x + 72 - 27**

Which we can simplify to get:

**5x + 45**

## Direct Proportion

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**

For example:

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.

Height (cm)

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

180.5

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