## Number Types

• Integers are whole numbers (positive, negative, or zero). Examples: -3, 0, 5.
• Decimals represent parts of a whole, using a decimal point. Example: 3.14.
• Fractions represent parts of a whole, written as a numerator over a denominator. Example: 1/2.
• A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11. (Note: 1 is not a prime number).
• Factors are numbers that divide exactly into another number. Factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiples are numbers in another number's times table. Multiples of 3 are 3, 6, 9, 12...
• The Highest Common Factor (HCF) is the largest factor shared by two or more numbers.
• The Lowest Common Multiple (LCM) is the smallest multiple shared by two or more numbers.
• Square numbers are the result of multiplying an integer by itself (e.g., 1x1=1, 2x2=4, 3x3=9).
• Cube numbers are the result of multiplying an integer by itself three times (e.g., 1x1x1=1, 2x2x2=8, 3x3x3=27).

## Operations and Order

• Remember BIDMAS (or BODMAS) for the order of operations:
• Brackets
• Indices (powers and roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)
• Negative numbers:
• Adding a negative is the same as subtracting: 5 + (-3) = 2.
• Subtracting a negative is the same as adding: 5 - (-3) = 8.
• When multiplying or dividing: Same signs give a positive result (+ x + = +, - x - = +). Different signs give a negative result (+ x - = -, - x + = -).

## Fractions, Decimals, Percentages (FDP)

• Conversions:
• Fraction to Decimal: Divide the numerator by the denominator.
• Decimal to Fraction: Write as a fraction over 10, 100, 1000 etc., then simplify.
• Decimal to Percentage: Multiply by 100.
• Percentage to Decimal: Divide by 100.
• Calculations:
• To add/subtract fractions, you must find a common denominator.
• To multiply fractions, multiply the numerators and multiply the denominators.
• To divide fractions, 'flip' the second fraction (reciprocal) and then multiply.
• To find a percentage of an amount, convert the percentage to a decimal and multiply.

## Rounding and Estimation

• Rounding to decimal places: Look at the digit immediately after the required decimal place. If it's 5 or more, round the previous digit up; if it's less than 5, keep the previous digit the same.
• Rounding to significant figures: The first significant figure is the first non-zero digit. Apply the same rounding rule as for decimal places.
• Estimation: Round numbers to 1 significant figure before calculating to get a rough answer. This helps check if your exact answer is reasonable.

## Ratio

• A ratio compares parts of a whole. Ratios can be simplified by dividing all parts by their HCF.
• To share an amount in a given ratio, add the parts of the ratio to find the total parts. Divide the total amount by the total parts to find the value of one part. Then multiply this value by each part of the ratio.

## Fractions, Decimals and Percentages (FDP)

Fractions, decimals, and percentages are different ways of representing parts of a whole. Understanding how to convert between them and perform calculations is crucial.

## Conversions

• Fraction to Decimal: Divide the numerator by the denominator. E.g., 3/4 = 3 ÷ 4 = 0.75.
• Decimal to Percentage: Multiply the decimal by 100. E.g., 0.75 × 100 = 75%.
• Percentage to Decimal: Divide the percentage by 100. E.g., 75% ÷ 100 = 0.75.
• Decimal to Fraction: Write the decimal as a fraction over a power of 10 (e.g., 0.75 = 75/100) and simplify. E.g., 75/100 = 3/4.
• Percentage to Fraction: Write the percentage over 100 and simplify. E.g., 75% = 75/100 = 3/4.

## Comparing and Ordering FDP

To compare or order fractions, decimals, and percentages, convert them all to the same form (usually decimals or fractions with a common denominator) and then compare.

## Working with Fractions

• Equivalent Fractions: Multiply or divide the numerator and denominator by the same number. E.g., 1/2 = 2/4.
• Adding/Subtracting Fractions: Find a common denominator, convert the fractions, then add/subtract the numerators. Keep the denominator the same. E.g., 1/3 + 1/2 = 2/6 + 3/6 = 5/6.
• Multiplying Fractions: Multiply the numerators together and multiply the denominators together. Simplify if possible. E.g., 1/2 × 3/4 = (1×3)/(2×4) = 3/8.
• Dividing Fractions: Use the KFC method: Keep the first fraction, Flip the second fraction (reciprocal), Change the operation to multiplication. E.g., 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3.
• Fraction of an Amount: Divide the amount by the denominator and multiply by the numerator. E.g., 2/3 of 60 = (60 ÷ 3) × 2 = 20 × 2 = 40.

## Working with Percentages

• Percentage of an Amount (Non-Calculator): Find 10% (divide by 10), 1% (divide by 100), or 50% (divide by 2) and build up the required percentage. E.g., 30% of 80: 10% of 80 = 8, so 30% = 3 × 8 = 24.
• Percentage of an Amount (Calculator): Convert the percentage to a decimal and multiply by the amount. E.g., 35% of 120 = 0.35 × 120 = 42.
• Expressing One Number as a Percentage of Another: (Part / Whole) × 100. E.g., 15 out of 50 = (15/50) × 100 = 30%.
• Percentage Increase/Decrease: (Change / Original Amount) × 100. E.g., price from £40 to £50: Change = £10. (10/40) × 100 = 25% increase.
• Simple Interest: Interest = (Principal × Rate × Time) / 100. Rate is usually annual percentage.

## Recurring Decimals

Some fractions, like 1/3, convert to decimals that repeat indefinitely, written with a dot above the repeating digit(s). E.g., 1/3 = 0.3̇.

## Ratio

Ratio is used to compare quantities of the same type. For example, if there are 3 red apples and 2 green apples, the ratio of red to green is 3:2.

• Simplifying Ratios: Divide all parts of the ratio by their highest common factor (HCF). For example, 10:15 simplifies to 2:3 by dividing both by 5.
• Sharing in a Ratio: To share a total amount in a given ratio (e.g., 2:3):

1. Add the parts of the ratio to find the total number of parts (2 + 3 = 5 parts).

2. Divide the total amount by the total number of parts to find the value of one part.

3. Multiply the value of one part by each number in the ratio to find the individual shares.

• Ratios in the form 1:n or n:1: To express a ratio like 10:25 in the form 1:n, divide both sides by the first number (10 ÷ 10 : 25 ÷ 10 = 1:2.5). To express it as n:1, divide both sides by the second number.

## Proportion

Proportion describes how quantities relate to each other.

• Direct Proportion: Two quantities are in direct proportion if as one increases, the other increases by the same factor, or as one decreases, the other decreases by the same factor. For example, if 3 pencils cost £1.50, then 6 pencils (double the amount) would cost £3.00 (double the cost).
• Finding a Unit Value: Often, you'll find the value of one item (the unit rate) first. If 4 oranges cost £1.20, one orange costs £1.20 ÷ 4 = £0.30.
• Best Buys: To find the best value, calculate the unit price (cost per item, or cost per unit of weight/volume) for each option and compare them.

## Rates of Change

Rates of change measure how one quantity changes in relation to another. They always involve units for both quantities (e.g., km/h, £/kg).

• Speed, Distance, Time:
• Speed = Distance ÷ Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

*Remember to use consistent units (e.g., km and hours, or metres and seconds).*

• Density, Mass, Volume:
• Density = Mass ÷ Volume
• Mass = Density × Volume
• Volume = Mass ÷ Density
• Other Rates: This can include flow rates (litres per minute), pay rates (£ per hour), or currency exchange rates (£ per $). Always pay attention to the units given in the problem.

## Algebra Basics and Equations

Algebra uses letters (called variables) to represent unknown numbers, helping us solve problems and describe relationships.

## Algebraic Notation

• Multiplication: `3 x a` becomes `3a`. `a x b` becomes `ab`.
• Division: `a ÷ b` becomes `a/b`.
• Powers: `a x a` is `a²` (a squared).

## Simplifying Expressions

To simplify means to make an expression easier.

• Collecting Like Terms: Combine terms that are "alike". `3a + 2b + a - b` simplifies to `4a + b`.
• Multiplying Terms: Multiply numbers, then letters. `2a x 3b = 6ab`. `4x x 2x = 8x²`.

## Expanding and Factorising

• Expanding Brackets: Multiply the term outside by *every* term inside.
• `2(x + 3) = 2x + 6`
• Factorising: The reverse of expanding. Find the highest common factor (HCF) and put it outside a bracket.
• `3x + 9 = 3(x + 3)`

## Substitution

Replace letters with their given numerical values.

• If `a = 5` and `b = 2`, then `2a + b = 2(5) + 2 = 10 + 2 = 12`.

## Solving Linear Equations

An equation has an equals sign (`=`). To solve means finding the variable's value.

• Keep the equation balanced: whatever you do to one side, do to the other.
• Steps:

1. Deal with any brackets (expand).

2. Collect like terms on each side.

3. Move all terms with the variable to one side, and numbers to the other (using inverse operations).

4. Divide to find the variable's value.

• Example: `2x - 3 = 7`
• Add 3 to both sides: `2x = 10`
• Divide both sides by 2: `x = 5`
• Example with brackets: `4(x + 1) = 16` -> `4x + 4 = 16` -> `4x = 12` -> `x = 3`.

## Inequalities

Similar to equations but use `<` (less than), `>` (greater than), `≤` (less than or equal to), `≥` (greater than or equal to).

• Solve like equations, BUT if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
• Example: `-2x < 6` becomes `x > -3`.

## Forming Expressions and Equations

Translate word problems into algebra.

• "A number `n` increased by 5" is `n + 5`.
• "Double a number `x` and then subtract 3" is `2x - 3`.
• "The cost of `p` pens at £1.50 each is £6" is `1.50p = 6`.

## Angles and Lines

Angles are fundamental to geometry. An acute angle is less than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, and a reflex angle is greater than 180° but less than 360°.

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Vertically opposite angles are equal.

When two parallel lines are intersected by a transversal line:

• Corresponding angles are equal (form an 'F' shape).
• Alternate angles are equal (form a 'Z' shape).
• Interior (or consecutive) angles add up to 180° (form a 'C' or 'U' shape).

## 2D Shapes and Their Properties

• The sum of angles in a triangle is 180°.
• The sum of angles in a quadrilateral is 360°.
• Isosceles triangles have two equal sides and two equal base angles.
• Equilateral triangles have all sides equal and all angles equal to 60°.
• Shapes can have line symmetry (a mirror line) and rotational symmetry (how many times it looks the same in a 360° turn).

## Perimeter, Area and Volume

Perimeter is the total distance around the outside of a 2D shape.

Area is the amount of surface a 2D shape covers.

• Rectangle: 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: Area = πr² (where r is the radius)
• Circumference of a circle: C = πd or C = 2πr (where d is the diameter)

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

• Cuboid: Volume = length × width × height
• Prism: Volume = area of cross-section × length
• Surface area of a cuboid: Sum of the areas of all its faces.

## Units and Conversions

Remember common metric conversions:

• Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm.
• Area: 1 m² = 10000 cm² (100x100), 1 cm² = 100 mm² (10x10).
• Volume: 1 m³ = 1000000 cm³ (100x100x100), 1 litre = 1000 ml = 1000 cm³.

## Bearings

Bearings are used to describe direction. They are always measured:

• From North (000°).
• Clockwise.
• As a three-figure number (e.g., 045°, 180°, 270°).

## Probability Basics

Probability measures how likely an event is, from 0 (impossible) to 1 (certain). It can be a fraction, decimal, or percentage.

The sum of probabilities for all possible outcomes is always 1.

P(not A) = 1 - P(A).

## Calculating Probability

For equally likely outcomes, theoretical probability is:

P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes)

• A sample space lists all possible outcomes.
• Mutually exclusive events cannot happen at the same time. P(A or B) = P(A) + P(B).
• Use two-way tables or simple tree diagrams to list outcomes for combined events. For tree diagrams, multiply probabilities along branches.

## Experimental Probability

Experimental probability (or relative frequency) is based on trials:

P(Event) = (Number of times event occurred) / (Total number of trials)

More trials generally lead to experimental probability being closer to theoretical probability.

## Data Collection and Types

• Qualitative data describes qualities (e.g., colour).
• Quantitative data involves numbers:
• Discrete: counted values (e.g., number of pets).
• Continuous: measured values (e.g., height).
• Data can be primary (collected by you) or secondary (existing data).
• Design clear questionnaires avoiding bias or overlapping response options.

## Representing Data

• Frequency tables organise data.
• Bar charts for discrete/qualitative data (gaps between bars).
• Pictograms use symbols.
• Pie charts show proportions (total 360 degrees).
• Line graphs for continuous data, often over time.
• Stem and leaf diagrams show distribution while keeping original values.
• Scatter graphs show relationships between two variables.

## Analysing Data (Averages & Spread)

• Measures of central tendency (averages):
• Mode: Most frequent value.
• Median: Middle value when ordered.
• Mean: Sum of values / number of values.
• Range: Highest value - Lowest value (measures spread).
• Use averages and range to compare different data sets.

## Scatter Graphs and Correlation

• A scatter graph plots paired data.
• Correlation describes the relationship:
• Positive: As one increases, the other increases.
• Negative: As one increases, the other decreases.
• No correlation: No clear relationship.
• Draw a line of best fit to show the trend. Use it for interpolation (estimates within data range) but be careful with extrapolation (outside data range).