## Number Basics

• Integers are whole numbers (positive, negative, or zero). Examples: -3, 0, 5.
• A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11... (1 is not a prime number).
• A multiple is a number in another number's times table. Examples: Multiples of 4 are 4, 8, 12, 16...
• A factor is a number that divides exactly into another number. Examples: Factors of 10 are 1, 2, 5, 10.
• Square numbers are the result of multiplying a number by itself (e.g., 3 x 3 = 9). Examples: 1, 4, 9, 16, 25...
• Cube numbers are the result of multiplying a number by itself three times (e.g., 2 x 2 x 2 = 8). Examples: 1, 8, 27, 64...

## Operations and Order

• Remember BIDMAS/BODMAS for the order of operations: Brackets, Indices (Powers/Roots), Division/Multiplication (left to right), Addition/Subtraction (left to right).
• Negative numbers: Adding a negative is the same as subtracting (e.g., 7 + (-4) = 3). Subtracting a negative is the same as adding (e.g., 7 - (-4) = 11). When multiplying or dividing, two negative signs make a positive result.

## Fractions, Decimals, and Percentages

• Fractions:
• To simplify, divide the numerator and denominator by their highest common factor (HCF).
• To add or subtract, find a common denominator.
• To multiply, multiply the numerators and multiply the denominators.
• To divide, 'flip' the second fraction (find its reciprocal) and then multiply.
• To find a fraction of an amount, divide by the denominator and multiply by the numerator.
• Decimals: When adding or subtracting, align the decimal points. For multiplication, ignore the decimal points, multiply, then place the decimal point back in the answer based on the total number of decimal places in the original numbers.
• Percentages: 'Per cent' means 'out of 100'.
• To find a percentage of an amount, convert the percentage to a decimal (divide by 100) or fraction, then multiply.
• To increase an amount by a percentage, find the percentage amount and add it, or use a multiplier (e.g., for a 15% increase, multiply by 1.15).
• Conversions:
• Fraction to Decimal: Divide the numerator by the denominator.
• Decimal to Percentage: Multiply by 100.
• Percentage to Decimal: Divide by 100.
• Decimal to Fraction: Write the decimal over 10, 100, 1000 etc., then simplify.

## Ratio and Proportion

• A ratio compares parts of a whole or different quantities. Example: 2:3.
• To simplify a ratio, divide all parts by their HCF.
• To share an amount in a given ratio, add the parts of the ratio, divide the total amount by this sum, then multiply each part of the ratio by the result.

## HCF, LCM, Rounding & Estimation

• Highest Common Factor (HCF): The largest factor shared by two or more numbers.
• Lowest Common Multiple (LCM): The smallest multiple shared by two or more numbers.
• Rounding: To a decimal place, look at the digit after the required place; if it's 5 or more, round up. To significant figures, count from the first non-zero digit.
• Estimation: Round numbers to 1 significant figure before calculating to get a rough answer.

## Fractions, Decimals and Percentages

Fractions, decimals, and percentages (FDP) are different ways of representing parts of a whole. Understanding how to work with and convert between them is a fundamental skill in mathematics.

## Fractions

A fraction represents a part of a whole. It has a numerator (top number) and a denominator (bottom number). The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have.

• Equivalent Fractions: These are fractions that look different but have the same value (e.g., 1/2 and 2/4). You create them by multiplying or dividing both the numerator and denominator by the same number.
• Simplifying Fractions: To simplify a fraction, divide both the numerator and denominator by their highest common factor until they can't be divided any further (e.g., 6/9 simplifies to 2/3 by dividing by 3).
• Adding and Subtracting Fractions: You must have a common denominator before you can add or subtract. Find the lowest common multiple (LCM) of the denominators, convert the fractions, then add or subtract the numerators.
• Multiplying Fractions: Multiply the numerators together and multiply the denominators together (e.g., (1/2) x (3/4) = 3/8).
• Dividing Fractions: Use the 'Keep, Change, Flip' method. Keep the first fraction, change the division to multiplication, and flip (find the reciprocal of) the second fraction (e.g., (1/2) ÷ (3/4) = (1/2) x (4/3) = 4/6 = 2/3).
• Fractions of an Amount: Divide the amount by the denominator and multiply by the numerator (e.g., 2/3 of 30: (30 ÷ 3) x 2 = 10 x 2 = 20).

## Decimals

Decimals are another way to represent parts of a whole, using place value.

• Adding and Subtracting Decimals: Line up the decimal points and then add or subtract as usual.
• Multiplying Decimals: Multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the original numbers and put that many decimal places in your answer (e.g., 0.2 x 0.3 = 0.06).
• Dividing Decimals: If the divisor is a decimal, multiply both the divisor and the dividend by a power of 10 to make the divisor a whole number, then divide.
• Rounding Decimals: Look at the digit to the right of the place you are rounding to. If it's 5 or more, round up; if it's less than 5, keep the digit the same.

## Percentages

Percentages mean 'out of 100'. The symbol '%' means 'per hundred'.

• Percentages of an Amount:
• Non-Calculator: Find 10% (divide by 10), 1% (divide by 100), 50% (divide by 2), 25% (divide by 4), and combine these to find other percentages.
• Calculator: Convert the percentage to a decimal (divide by 100) and multiply by the amount (e.g., 20% of 60 = 0.20 x 60 = 12).
• Percentage Increase/Decrease: Calculate the percentage of the amount, then add it for an increase or subtract it for a decrease.

## Conversions (FDP)

• Fraction to Decimal: Divide the numerator by the denominator.
• Decimal to Fraction: Write the decimal as a fraction over a power of 10 (e.g., 0.7 = 7/10, 0.25 = 25/100) and simplify.
• Decimal to Percentage: Multiply by 100.
• Percentage to Decimal: Divide by 100.
• Fraction to Percentage: Convert to a decimal first, then multiply by 100.
• Percentage to Fraction: Write the percentage over 100 and simplify (e.g., 75% = 75/100 = 3/4).

## Ratio

Ratio is used to compare quantities. It shows how much of one thing there is compared to another. For example, if a recipe uses 1 part sugar to 3 parts flour, the ratio of sugar to flour is 1:3.

Simplifying Ratios

To simplify a ratio, divide all parts of the ratio by their highest common factor (HCF). This makes the numbers in the ratio as small as possible while keeping the same proportion.

• Example: Simplify 10:15. The HCF of 10 and 15 is 5. Divide both by 5: 10÷5 : 15÷5 = 2:3.

Ratios can also be written in the form 1:n or n:1. To do this, divide all parts by the first or second number respectively.

• Example: Write 5:20 in the form 1:n. Divide both by 5: 5÷5 : 20÷5 = 1:4.

Sharing in a Ratio

To share a quantity in a given ratio:

1. Add the parts of the ratio to find the total number of 'shares'.

2. Divide the total quantity by the total number of shares to find the value of one share.

3. Multiply each part of the ratio by the value of one share.

• Example: Share £70 in the ratio 3:4.

1. Total shares: 3 + 4 = 7

2. Value of one share: £70 ÷ 7 = £10

3. Amounts: 3 × £10 = £30 and 4 × £10 = £40. (Check: £30 + £40 = £70).

## Proportion

Direct proportion means that as one quantity increases, the other quantity increases at the same rate, and vice versa. Their ratio remains constant. You often use the unitary method to solve these problems (finding the value of one unit).

• Example: If 4 pens cost £2.40, how much do 7 pens cost?
• Cost of one pen: £2.40 ÷ 4 = £0.60.
• Cost of 7 pens: 7 × £0.60 = £4.20.

## Rates of Change

A rate is a ratio that compares two different types of quantities, often with different units.

Speed, Density, and Pressure

These are common rates of change, and you should know their formulas:

• Speed = Distance ÷ Time (e.g., miles per hour, metres per second)
• Density = Mass ÷ Volume (e.g., grams per cubic centimetre, kilograms per cubic metre)
• Pressure = Force ÷ Area (e.g., Newtons per square metre, Pascals)

Currency Conversion

To convert between currencies, you use an exchange rate. You either multiply or divide by this rate.

• Example: If £1 = €1.10, how many Euros are £60?
• Multiply the amount in pounds by the exchange rate: 60 × 1.10 = €66.
• To convert Euros back to Pounds, you would divide by the exchange rate.

## Introduction to Algebra

Algebra uses letters (called variables) to represent unknown numbers.

• An expression is a combination of numbers, variables, and operations (e.g., `3x + 5`). It doesn't have an equals sign.
• An equation is a statement that two expressions are equal (e.g., `2x + 1 = 9`). It always has an equals sign.
• A formula is a special type of equation that shows the relationship between different variables (e.g., `A = lw` for area).

## Simplifying Expressions

To simplify an expression, you collect like terms. Like terms have the exact same variable part.

• Example: `3a + 2b + a - b` simplifies to `4a + b`. (Remember `a` is `1a`, and `-b` is `-1b`).
• When multiplying terms, multiply the numbers and then the variables: `2a * 3b = 6ab`.
• When dividing, simplify the numbers and variables: `10x / 5 = 2x`.

## Expanding and Factorising

• Expanding involves multiplying out brackets. Multiply the term outside the bracket by *every* term inside.
• Example: `3(x + 4) = 3x + 12`.
• Factorising is the reverse of expanding. You find the highest common factor (HCF) of the terms and put it outside the bracket.
• Example: `4x + 8 = 4(x + 2)`. (The HCF of 4x and 8 is 4).

## Substitution

Substitution means replacing variables with given numerical values and then calculating the result.

• Example: If `x = 5`, find the value of `2x + 3`.
• `2(5) + 3 = 10 + 3 = 13`.

## Solving Linear Equations

The goal is to find the value of the unknown variable. Use inverse operations to isolate the variable. Whatever you do to one side of the equation, you must do to the other.

• One-step equations:
• `x + 5 = 12` (subtract 5 from both sides) -> `x = 7`
• `3x = 15` (divide by 3 on both sides) -> `x = 5`
• Two-step equations:
• `2x - 1 = 9`
• Add 1 to both sides: `2x = 10`
• Divide by 2 on both sides: `x = 5`
• Equations with brackets: Expand first, then solve.
• `2(x + 3) = 10`
• `2x + 6 = 10`
• `2x = 4`
• `x = 2`

## Inequalities

An inequality uses signs like `<` (less than), `>` (greater than), `≤` (less than or equal to), `≥` (greater than or equal to). Solving them is similar to solving equations, but if you multiply or divide by a negative number, you must reverse the inequality sign.

• Example: `x + 3 < 7` (subtract 3 from both sides) -> `x < 4`.

## Geometry, Measures and Area

This topic covers understanding shapes, their properties, how to measure them, and calculate their perimeter, area, and volume.

Angles

Angles are fundamental. A right angle is 90°. An acute angle is less than 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180°.

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Angles in a triangle add up to 180°.
• When two straight lines intersect, vertically opposite angles are equal.
• For parallel lines cut by a transversal: alternate angles are equal (Z-shape), corresponding angles are equal (F-shape), and interior (or co-interior) angles add up to 180° (C-shape).

2D Shapes: Perimeter and Area

The perimeter is the total distance around the outside of a 2D shape. The area is the amount of surface a 2D shape covers.

• Rectangle: Perimeter = 2(length + width), Area = length × width.
• Triangle: Area = ½ × base × height.
• Parallelogram: Area = base × perpendicular height.
• Trapezium: Area = ½ × (a + b) × height (where a and b are parallel sides).
• Circle: Circumference (perimeter) = πd or 2πr. Area = πr². Remember, d = 2r (diameter = 2 × radius).

3D Shapes: Volume

Volume is the amount of space a 3D object occupies.

• Cuboid: Volume = length × width × height.
• Prism: Volume = Area of cross-section × length. (e.g., for a triangular prism, calculate the area of the triangular face, then multiply by the length).

Measures and Scale

• Units: Be able to convert between common units of length (mm, cm, m, km), mass (g, kg, tonnes), and capacity (ml, l).
• Bearings: Used for direction, measured clockwise from North, given as 3 figures (e.g., 045°).
• Scale Drawings: Understand and use scales (e.g., 1:100 means 1 cm on the drawing is 100 cm in real life).

## Probability Basics

Probability is a measure of how likely an event is to happen. It's always a value between 0 (impossible) and 1 (certain).

• A probability can be written as a fraction, decimal, or percentage.
• P(event) = (Number of favourable outcomes) / (Total number of possible outcomes).
• The sum of probabilities of all possible outcomes for an event is always 1.
• Mutually exclusive events are events that cannot happen at the same time. For two mutually exclusive events A and B, P(A or B) = P(A) + P(B).

## Experimental Probability and Relative Frequency

• Theoretical probability is what we expect to happen based on calculations (e.g., probability of rolling a 6 on a fair die is 1/6).
• Experimental probability (or relative frequency) is based on the results of an experiment or observation.
• Relative Frequency = (Number of times an event occurs) / (Total number of trials).
• As the number of trials increases, the experimental probability usually gets closer to the theoretical probability.
• The expected number of outcomes for an event in a given number of trials is P(event) × Number of trials.

## Statistics: Collecting and Presenting Data

Statistics involves collecting, organising, analysing, and interpreting data.

• Qualitative data describes qualities (e.g., colours). Quantitative data involves numbers.
• Discrete data can only take specific values (e.g., number of siblings).
• Continuous data can take any value within a range (e.g., height).
• Data can be collected using questionnaires or by observation. Be aware of bias.
• Common ways to present data:
• Frequency tables organise data into categories.
• Bar charts and pictograms show frequencies for discrete data.
• Pie charts show proportions of a whole.
• Line graphs show trends over time.
• Stem and leaf diagrams display data while retaining original values.
• Two-way tables show relationships between two categorical variables.

## Averages and Spread

These help summarise and compare data sets.

• Mean: The sum of all values divided by the number of values. It's affected by outliers.
• Median: The middle value when data is ordered from smallest to largest. If there are two middle values, find their average.
• Mode: The value that appears most frequently. A data set can have no mode, one mode, or multiple modes.
• Range: The difference between the highest and lowest values. It measures the spread of the data.
• For data in a frequency table, estimate the mean by multiplying each value by its frequency, summing these products, and dividing by the total frequency.

## Scatter Graphs and Correlation

• A scatter graph shows the relationship between two variables.
• Correlation describes the relationship:
• Positive correlation: As one variable increases, the other tends to increase.
• Negative correlation: As one variable increases, the other tends to decrease.
• No correlation: No clear relationship.
• An outlier is a data point far from the others.
• A line of best fit can be drawn on a scatter graph to show the general trend, used for making predictions (interpolation within the data range, extrapolation outside).