## Number Basics

• Integers are whole numbers (positive, negative, or zero).
• A multiple is the result of multiplying a number by an integer (e.g., multiples of 3 are 3, 6, 9...).
• A factor is a number that divides exactly into another number (e.g., factors of 12 are 1, 2, 3, 4, 6, 12).
• A prime number has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11...). 2 is the only even prime number.
• Square numbers are the result of multiplying an integer by itself (e.g., 1, 4, 9, 16...). The square root is the inverse operation.
• Cube numbers are the result of multiplying an integer by itself three times (e.g., 1, 8, 27...). The cube root is the inverse operation.

## Operations and Order

• Use BIDMAS/BODMAS for the order of operations: Brackets, Indices (or Orders/Powers), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• Negative numbers: Adding a negative is subtraction (e.g., 5 + (-3) = 2). Subtracting a negative is adding (e.g., 5 - (-3) = 8).
• Multiplying or dividing two numbers with the same sign gives a positive result. Different signs give a negative result.

## Fractions, Decimals, and Percentages (FDP)

• Fractions: To add/subtract, find a common denominator. To multiply, multiply numerators and denominators. To divide, "flip" the second fraction and multiply.
• Conversions:
• Fraction to Decimal: Divide numerator by denominator.
• Decimal to Percentage: Multiply by 100.
• Percentage to Decimal: Divide by 100.
• Percentage to Fraction: Write over 100 and simplify.
• Percentages of an amount: Convert percentage to decimal or fraction and multiply.
• Increase/Decrease by a percentage: Find the percentage amount, then add/subtract from the original. Or, use a multiplier (e.g., 10% increase = multiply by 1.10).

## Multiples, Factors, and Primes

• 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.
• Prime Factor Decomposition: Expressing a number as a product of its prime factors (e.g., 12 = 2 x 2 x 3 or 2² x 3). This is useful for finding HCF and LCM.

## Rounding and Estimation

• Rounding: To a given number of decimal places (d.p.), look at the digit after the last required digit. If 5 or more, round up; otherwise, stay the same.
• To a given number of significant figures (s.f.), count from the first non-zero digit. Apply the same rounding rule.
• Estimation: Round numbers in a calculation to 1 significant figure before performing the operation. This gives a rough answer.

## 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 operations is crucial for GCSE Maths Foundation.

## Converting Between FDP

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

## Ordering FDP

To order fractions, decimals, and percentages, it's usually easiest to convert them all to the same format, most commonly decimals. Then compare their values.

## Fractions

• Equivalent Fractions: Multiply or divide both the numerator and denominator by the same number. E.g., 1/2 = 2/4.
• Improper Fractions and Mixed Numbers: An improper fraction has a numerator larger than or equal to its denominator (e.g., 7/3). A mixed number combines a whole number and a fraction (e.g., 2 1/3). You must be able to convert between them.
• Adding and Subtracting Fractions: Find a common denominator (a common multiple of the denominators). Convert both fractions to equivalent fractions with this common denominator, then add or subtract the numerators. Keep the denominator the same.
• Multiplying Fractions: Multiply the numerators together, and multiply the denominators together. Simplify the result. E.g., (1/2) × (3/4) = (1×3)/(2×4) = 3/8.
• Dividing Fractions: "Keep, Change, Flip" (KCF). Keep the first fraction, change the division to multiplication, and flip (find the reciprocal of) the second fraction. Then multiply. E.g., (1/2) ÷ (3/4) = (1/2) × (4/3) = 4/6 = 2/3.
• Fraction of a Quantity: Multiply the fraction by the quantity. E.g., 1/3 of 60 = (1/3) × 60 = 20.
• Expressing one number as a fraction of another: Write (first number / second number) and simplify.

## Percentages

• Percentage of a Quantity: Convert the percentage to a decimal (divide by 100) or a fraction, then multiply by the quantity. E.g., 20% of 50 = 0.20 × 50 = 10.
• Percentage Increase/Decrease:
• Method 1: Calculate the percentage amount, then add or subtract it from the original amount.
• Method 2 (Using Multipliers): For an increase of X%, multiply the original amount by (1 + X/100). For a decrease of X%, multiply by (1 - X/100). E.g., Increase 50 by 10%: 50 × (1 + 0.10) = 50 × 1.1 = 55.
• Expressing one number as a percentage of another: (part / whole) × 100%.

## Ratio

Ratios compare how much of one quantity there is compared to another. They show the relative sizes of two or more values.

• Simplifying Ratios: To simplify a ratio, divide all parts by their highest common factor. For example, 10:15 simplifies to 2:3 (dividing both by 5). Ratios should always be in their simplest form.
• Writing Ratios in the form 1:n or n:1: Divide both sides by the first number to get 1:n, or by the second number to get n:1. E.g., 4:10 becomes 1:2.5 (divide by 4).
• Sharing a Quantity in a Given Ratio: This is a common problem type.

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

2. Divide the total quantity 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 share for each part.

*Example*: Share £70 in the ratio 3:4. Total parts = 3+4=7. One part = £70/7 = £10. Shares are 3 x £10 = £30 and 4 x £10 = £40.

## Proportion

Proportion describes how quantities relate to each other. If two quantities are proportional, a change in one causes a predictable change in the other.

• Direct Proportion: Two quantities are in direct proportion if they increase or decrease at the same rate. If one quantity doubles, the other quantity also doubles.
• Unitary Method: A very useful method for solving direct proportion problems. First, find the value of one unit.

*Example*: If 4 apples cost £2.00, how much do 7 apples cost?

1. Find cost of 1 apple: £2.00 / 4 = £0.50.

2. Find cost of 7 apples: £0.50 x 7 = £3.50.

• Recipes: Scaling recipes up or down is a practical application of proportion. If you need to make twice as much, you must double all ingredients. If you need half, halve them.

## Rates of Change

Rates of change involve how one quantity changes in relation to another quantity, often over time or space.

• Speed, Distance, Time: These are linked by specific formulas:
• Speed = Distance / Time
• Distance = Speed x Time
• Time = Distance / Speed

*Remember to use consistent units (e.g., km/h, m/s). You might use a formula triangle to help remember these.

• Density, Mass, Volume: Similar to speed, these are linked by:
• Density = Mass / Volume
• Mass = Density x Volume
• Volume = Mass / Density
• Best Buys / Unit Cost: To find the best value for money when comparing items, calculate the unit cost (the cost per item, per gram, per litre, etc.) for each option and compare. The lowest unit cost indicates the best buy.
• Currency Conversion: To convert between different currencies, you'll use an exchange rate. You will either multiply or divide by the exchange rate depending on the direction of conversion.

*Example*: If £1 = €1.15, to convert £50 to euros, multiply: 50 x 1.15 = €57.50. To convert €100 to pounds, divide: 100 / 1.15 = £86.96 (to 2 d.p.).

## Algebra Basics and Equations

Algebra uses letters, called variables, to represent unknown numbers. An expression is a combination of numbers, variables, and operations (like `2x + 5`). An equation sets two expressions equal to each other (like `2x + 5 = 11`). A formula is a special type of equation that shows the relationship between different variables (e.g., `Area = length × width`).

Simplifying Expressions

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

• Example: `3a + 4b + 2a - b`
• Combine `3a` and `2a` to get `5a`.
• Combine `4b` and `-b` (which is `-1b`) to get `3b`.
• Simplified expression: `5a + 3b`.

Remember that `x` and `x²` are not like terms.

Substitution

Substitution means replacing variables in an expression or formula with given numerical values and then calculating the result.

• Example: If `a = 3` and `b = 4`, find the value of `2a + b`.
• Substitute: `2(3) + 4`
• Calculate: `6 + 4 = 10`.

Always follow the order of operations (BIDMAS/BODMAS).

Expanding and Factorising Single Brackets

Expanding a bracket means multiplying each term inside the bracket by the term outside.

• Example: `3(x + 5)`
• Multiply `3` by `x` and `3` by `5`.
• Result: `3x + 15`.

Factorising into a single bracket is the reverse of expanding. You find the highest common factor (HCF) of all terms and place it outside the bracket.

• Example: Factorise `4y + 12`.
• The HCF of `4y` and `12` is `4`.
• Result: `4(y + 3)`.

Solving Linear Equations

To solve a linear equation, you need to find the value of the unknown variable. The goal is to isolate the variable on one side of the equation. Do the same operation to both sides to keep the equation balanced.

• Example 1: `x + 7 = 15`
• Subtract `7` from both sides: `x = 8`.
• Example 2: `3x - 5 = 10`
• Add `5` to both sides: `3x = 15`.
• Divide by `3` on both sides: `x = 5`.
• Example 3: `2(x + 3) = 14`
• Expand: `2x + 6 = 14`.
• Subtract `6`: `2x = 8`.
• Divide by `2`: `x = 4`.

## Geometry, Measures and Area: Foundation Essentials

This topic covers understanding shapes, their properties, how to measure them, and how they move in space. It's a core part of GCSE Mathematics.

## Angles

Understanding angle rules is crucial.

• 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°.
• Vertically opposite angles are always equal.
• For parallel lines cut by a transversal (a line crossing them):
• Corresponding angles are equal (they form an 'F' shape).
• Alternate angles are equal (they form a 'Z' shape).
• Interior (or consecutive) angles add up to 180° (they form a 'C' shape).

## 2D and 3D Shapes

Know the names and basic properties of common shapes.

• 2D Shapes: Recognise and describe properties of triangles, quadrilaterals (square, rectangle, parallelogram, rhombus, trapezium, kite), and regular polygons. Understand line symmetry (mirror lines) and rotational symmetry (how many times a shape looks the same when rotated 360°).
• 3D Shapes: Identify cuboids, prisms, pyramids, cylinders, cones, and spheres. Know their number of faces, edges, and vertices. A net is a 2D pattern that can be folded to form a 3D shape.

## Perimeter, Area and Volume

These are about measuring the size of shapes.

• Perimeter is the total distance around the outside of a 2D shape. You calculate it by adding up the lengths of all its sides.
• Area is the amount of surface a 2D shape covers.
• Rectangle: Area = length × width
• Triangle: Area = ½ × base × height (height must be perpendicular to the base)
• Parallelogram: Area = base × perpendicular height
• Trapezium: Area = ½ × (a + b) × height (where 'a' and 'b' are the parallel sides)
• For compound shapes, split them into simpler shapes (rectangles, triangles) and add their individual areas.
• Volume is the amount of space a 3D object occupies.
• Cuboid: Volume = length × width × height
• For any prism: Volume = Area of cross-section × length

## Measures and Conversions

Be proficient in converting between common metric units:

• Length: 1 kilometre (km) = 1000 metres (m), 1 m = 100 centimetres (cm), 1 cm = 10 millimetres (mm).
• Mass: 1 tonne = 1000 kilograms (kg), 1 kg = 1000 grams (g).
• Capacity: 1 litre (L) = 1000 millilitres (ml).

Remember to multiply when converting to a smaller unit and divide when converting to a larger unit.

## Transformations and Bearings

• Translation: Moving a shape from one position to another without rotating, reflecting, or changing its size. It's described by a vector (e.g., (3, -2)).
• Reflection: Flipping a shape over a mirror line (line of reflection). Each point in the original shape is the same distance from the mirror line as its corresponding point in the reflected shape.
• Rotation: Turning a shape around a fixed point. To describe a rotation, you need the centre of rotation, the angle of rotation, and the direction (clockwise or anticlockwise).
• Bearings: Used to describe direction. They are always measured clockwise from North, and always given as three figures (e.g., 045°, 180°, 270°).

## Probability Basics

Probability measures the likelihood of an event happening. It's always a value between 0 and 1, where 0 means impossible and 1 means certain.

• 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).
• A sample space diagram or tree diagram can help list all possible outcomes for two or more events.
• Experimental probability (or relative frequency) is based on results from an experiment: (Number of times an event happens) / (Total number of trials).

## Types of Data

• Qualitative data describes qualities or characteristics (e.g., favourite colour).
• Quantitative data deals with 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, weight).

## Data Representation

• Frequency tables and tally charts organise raw data.
• Bar charts and pictograms display discrete data.
• Pie charts show proportions of a whole, where the angle for each sector is (frequency / total frequency) * 360°.
• Line graphs are used for continuous data, often showing trends over time.
• Scatter graphs show the relationship between two variables.
• Correlation describes this relationship: positive (as one increases, the other increases), negative (as one increases, the other decreases), or no correlation. A line of best fit can be drawn on a scatter graph to show the trend.

## Averages and Spread

These are ways to summarise data.

• Mean: The sum of all values divided by the number of values. It's affected by outliers.
• Median: The middle value when data is arranged in order. If there's an even number of values, it's the average of the two middle values.
• Mode: The value that appears most frequently. There can be one, more than one, or no mode.
• Range: The difference between the highest and lowest values. It measures the spread of the data.

Comparing datasets often involves looking at an average (mean, median, mode) and the range.