## Types of Numbers

• Integers: Whole numbers, including positive numbers, negative numbers, and zero (e.g., -3, 0, 5).
• Prime Numbers: Numbers with exactly two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11). Note: 1 is not a prime number.
• Factors: Numbers that divide exactly into another number without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiples: Numbers in the times table of another number. For example, multiples of 5 are 5, 10, 15, 20...
• HCF (Highest Common Factor): The largest factor shared by two or more numbers.
• LCM (Lowest Common Multiple): The smallest multiple shared by two or more numbers.

## Operations and Order

• BIDMAS/BODMAS: This acronym dictates the order of operations in calculations:
• Brackets
• Indices (or Orders/Powers/Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)
• Negative Numbers: Remember rules for operations: e.g., subtracting a negative is adding (5 - (-3) = 8); multiplying/dividing two negatives gives a positive.

## Fractions, Decimals, and Percentages

• Fractions: Represent parts of a whole.
• Equivalent Fractions: Have the same value (e.g., 1/2 = 2/4).
• Simplifying: Divide the numerator and denominator by their HCF.
• Adding/Subtracting: Find a common denominator.
• Multiplying: Multiply numerators, multiply denominators.
• Dividing: "Keep, Change, Flip" (keep the first fraction, change division to multiplication, flip the second fraction).
• Decimals: Numbers based on powers of 10. Convert fractions to decimals by dividing the numerator by the denominator.
• Percentages: Mean "per hundred".
• Conversions: To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.
• Percentage of an Amount: Convert the percentage to a decimal or fraction, then multiply.
• Percentage Change: ((New Value - Original Value) / Original Value) × 100.
• Reverse Percentages: If an amount has increased by 20% to £120, the original amount is £120 / 1.20.

## Ratio and Proportion

• Ratio: Compares quantities. Can be simplified like fractions (e.g., 10:15 simplifies to 2:3).
• Dividing in a Ratio: Add the parts of the ratio, divide the total quantity by this sum, then multiply by each part of the ratio.
• Direct Proportion: Two quantities increase or decrease at the same rate (e.g., cost of apples and number of apples).
• Inverse Proportion: As one quantity increases, the other decreases (e.g., speed and time taken for a journey).

## Powers, Roots, and Standard Form

• Powers (Indices): Indicate repeated multiplication (e.g., 3^4 = 3 × 3 × 3 × 3).
• Laws of Indices: Key rules include x^a × x^b = x^(a+b), x^a ÷ x^b = x^(a-b), (x^a)^b = x^(ab), and x^0 = 1.
• Negative Powers: x^-a = 1/x^a.
• Roots: The inverse of powers (e.g., square root, cube root).
• Standard Form: A way to write very large or very small numbers concisely. It's written as A × 10^n, where 1 ≤ A < 10 and n is an integer.

## Estimation and Rounding

• Rounding: To a specified number of decimal places, significant figures, or to the nearest whole number/10/100.
• Estimation: Round numbers to 1 significant figure before performing calculations to get an approximate answer.

## Introduction to Algebra

Algebra uses letters (variables) to represent unknown numbers. It allows us to write general rules and solve problems where values are unknown. An expression is a combination of numbers, variables, and operations (e.g., 3x + 5). An equation has an equals sign, stating two expressions are equal (e.g., 3x + 5 = 11). An inequality uses symbols like <, >, ≤, or ≥ (e.g., x + 2 > 7).

## Simplifying Expressions

To simplify an expression, collect like terms. Like terms have the exact same variable(s) raised to the exact same power (e.g., 3x and 5x are like terms; 3x and 3x² are not).

• Example: 5a + 2b - 3a + 4b = (5a - 3a) + (2b + 4b) = 2a + 6b.

When multiplying or dividing terms, multiply/divide the numbers and the variables separately.

• Example: 3x * 4y = 12xy.

## Expanding Brackets

To expand a single bracket, multiply the term outside the bracket by every term inside the bracket.

• Example: 2(x + 3) = 2*x + 2*3 = 2x + 6.

To expand double brackets, multiply each term in the first bracket by each term in the second bracket (often remembered as FOIL: First, Outer, Inner, Last).

• Example: (x + 2)(x + 3) = x*x + x*3 + 2*x + 2*3 = x² + 3x + 2x + 6 = x² + 5x + 6.

## Solving Linear Equations

The goal is to isolate the variable. Whatever you do to one side of the equation, you must do to the other side to keep it balanced.

• One-step: x + 5 = 12 => x = 12 - 5 => x = 7.
• Two-step: 2x - 3 = 7 => 2x = 7 + 3 => 2x = 10 => x = 10 / 2 => x = 5.
• Variables on both sides: 5x - 2 = 2x + 10 => 5x - 2x = 10 + 2 => 3x = 12 => x = 4.
• With brackets: 3(x + 1) = 15 => 3x + 3 = 15 => 3x = 12 => x = 4.

## Inequalities

Solving linear inequalities is similar to solving equations, but with one key difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

• Example: x + 3 > 8 => x > 5.
• Example: -2x < 6 => x > -3 (sign reversed because divided by -2).

## Substitution

Substitution means replacing variables with given numerical values and then evaluating the expression or formula.

• Example: If x = 3 and y = 4, find the value of 2x + y.

2(3) + 4 = 6 + 4 = 10.

## Ratio, Proportion & Rates of Change

This topic covers how quantities relate to each other, how they scale, and how one changes with respect to another. It's fundamental for understanding many real-world applications.

## Ratio

A ratio compares two or more quantities. Ratios can be part-to-part (e.g., apples to oranges) or part-to-whole (e.g., apples to total fruit). Ratios are often written with a colon (e.g., 2:3).

• Simplifying Ratios: To simplify a ratio, divide all parts by their highest common factor until they are in their simplest integer form. For example, 10:15 simplifies to 2:3 by dividing by 5.
• Sharing in a Ratio: To share a quantity in a given ratio (e.g., 2:3), first find the total number of parts (2 + 3 = 5). Then, divide the total quantity by the total parts to find the value of one part. Finally, multiply the value of one part by each number in the ratio.
• Equivalent Ratios: Ratios that represent the same proportion, such as 1:2 and 5:10.
• Ratios can be converted to fractions or percentages. For a ratio A:B, the fraction of the whole for A is A/(A+B).

## Proportion

Proportion describes how quantities are related when one changes in response to another.

• Direct Proportion: Two quantities are in direct proportion if they increase or decrease at the same rate. If `y` is directly proportional to `x`, then `y = kx`, where `k` is the constant of proportionality. As `x` doubles, `y` doubles.
• Inverse Proportion: Two quantities are in inverse proportion if one increases as the other decreases, such that their product remains constant. If `y` is inversely proportional to `x`, then `y = k/x`, where `k` is the constant of proportionality. As `x` doubles, `y` halves.
• Solving Proportion Problems: Often involves finding the constant `k` first, or using the unitary method (finding the value for one unit, then scaling up or down).

## Rates of Change

A rate of change describes how one quantity changes in relation to another quantity. These often involve compound measures, which combine two or more units (e.g., distance and time).

• Speed: The rate at which distance is covered over time. Formula: Speed = Distance / Time.
• Density: The rate of mass per unit volume. Formula: Density = Mass / Volume.
• Pressure: The rate of force applied per unit area. Formula: Pressure = Force / Area.
• Flow Rate: The volume of liquid or gas passing a point per unit time (e.g., litres/minute).
• Understanding these formulas allows you to calculate any of the variables if the others are known (e.g., Time = Distance / Speed, Mass = Density × Volume).

## Geometry & Measures: Key Concepts

This topic covers understanding shapes, their properties, measurements, and how they move in space.

2D & 3D Shapes: Area, Perimeter, Volume, Surface Area

• Perimeter: The total distance around the outside of a 2D shape. Units are linear (e.g., cm, m).
• Area: The amount of surface a 2D shape covers. Units are squared (e.g., cm², m²).
• Rectangle: `Area = length × width`
• Triangle: `Area = ½ × base × height`
• Circle: `Area = πr²`, `Circumference = 2πr` or `πd`
• Trapezium: `Area = ½ × (a + b) × height` (where a and b are parallel sides)
• Volume: The amount of space a 3D object occupies. Units are cubed (e.g., cm³, m³).
• Cuboid: `Volume = length × width × height`
• Prism: `Volume = Area of cross-section × length`
• Cylinder: `Volume = πr²h`
• Surface Area: The total area of all the faces of a 3D object. Calculate the area of each face and sum them.

Angles

• Angles on a straight line sum to 180°.
• Angles around a point sum to 360°.
• Angles in a triangle sum to 180°.
• Angles in a quadrilateral sum to 360°.
• Parallel Lines: Look for alternate angles (Z-angles, equal), corresponding angles (F-angles, equal), and interior angles (C-angles, sum to 180°).
• Polygons: Sum of interior angles = `(n-2) × 180°` (where n is number of sides). Sum of exterior angles is always 360°.

Transformations

• Reflection: Flip a shape over a mirror line (e.g., x=2, y-axis).
• Rotation: Turn a shape around a centre of rotation by a specific angle and direction (clockwise/anti-clockwise).
• Translation: Slide a shape using a vector (e.g., (3, -2) means 3 right, 2 down).
• Enlargement: Change the size of a shape using a scale factor and a centre of enlargement. If scale factor is negative, the image is inverted and on the opposite side of the centre.

Pythagoras Theorem & Basic Trigonometry

• Pythagoras Theorem: For right-angled triangles only: `a² + b² = c²`, where 'c' is the hypotenuse (longest side, opposite the right angle).
• Trigonometry (SOH CAH TOA): For right-angled triangles only.
• SOH: `Sin(θ) = Opposite / Hypotenuse`
• CAH: `Cos(θ) = Adjacent / Hypotenuse`
• TOA: `Tan(θ) = Opposite / Adjacent`

Bearings

• Bearings are three-figure angles, measured clockwise from North (000° to 359°).

Units & Conversions

Be familiar with converting between common metric units for length (mm, cm, m, km), area (cm², m²), volume (cm³, m³), and capacity (ml, cl, L). Remember `1 L = 1000 cm³`.

## Probability Basics

Probability measures the likelihood of an event happening. It's always a value between 0 and 1 (or 0% and 100%).

• 0 means impossible.
• 1 means certain.
• 0.5 (or 50%) means an even chance.

The basic formula for the theoretical probability of an event is:

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

All outcomes must be equally likely.

## Types of Events

• Mutually Exclusive Events: Events that cannot happen at the same time (e.g., rolling a 1 and a 6 on a single die roll). For mutually exclusive events A and B:

P(A or B) = P(A) + P(B)

• Complementary Events: Two events are complementary if one event happening means the other cannot, and together they cover all possibilities (e.g., raining or not raining). If A is an event, A' (not A) is its complement:

P(A') = 1 - P(A)

• Independent Events: The outcome of one event does not affect the outcome of another (e.g., flipping a coin twice). For independent events A and B:

P(A and B) = P(A) × P(B)

• Dependent Events: The outcome of one event *does* affect the outcome of another (e.g., drawing two cards from a deck without replacement). These are often solved using tree diagrams.

## Representing Probabilities

• Sample Space Diagrams: Tables or lists that show all possible outcomes of two or more events (e.g., rolling two dice).
• Tree Diagrams: Used for sequences of events. Branches show the probability of each outcome. Probabilities along branches are multiplied for 'and' events, and probabilities at the end of branches are added for 'or' events.
• Venn Diagrams: Visual representations of sets and their relationships, useful for showing overlaps ('and'), unions ('or'), and complements ('not').

## Experimental Probability and Expected Outcomes

Experimental Probability (also called Relative Frequency) is found by conducting an experiment:

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

As the number of trials increases, experimental probability tends to get closer to theoretical probability.

The expected number of outcomes for an event over a given number of trials is calculated as:

Expected Number = P(Event) × Total number of trials

## Introduction to Statistics

Statistics involves the collection, organisation, analysis, and interpretation of data. Data can be primary (collected directly by you for a specific purpose) or secondary (data collected by someone else, often for a different purpose).

## Types of Data

• Qualitative Data: Describes qualities or characteristics that cannot be measured numerically (e.g., favourite colour, type of car).
• Quantitative Data: Deals with numbers and can be measured or counted.
• Discrete Data: Can only take specific, distinct values, often counted (e.g., number of siblings, shoe size).
• Continuous Data: Can take any value within a given range, often measured (e.g., height, temperature, time).

## Sampling

When it's not possible to collect data from an entire population, a smaller group called a sample is used. A good sample should be representative of the population.

• Random Sampling: Every member of the population has an equal chance of being selected. This helps reduce bias.
• Stratified Sampling: The population is divided into distinct subgroups (strata), and a random sample is taken from each stratum in proportion to its size in the population. This ensures all subgroups are fairly represented.
• Systematic Sampling: Selecting items at regular intervals from an ordered list (e.g., every 10th person).
• Opportunity Sampling: Using people who are conveniently available at the time of the study.

## Presenting Data

Visual representations make data easier to understand.

• Frequency Tables: Organise data by showing how often each value or category occurs.
• Bar Charts: Used for discrete or qualitative data. Bars are separate.
• Pie Charts: Show proportions of a whole. The angle for each sector is calculated as (frequency / total frequency) * 360°.
• Histograms: Used for continuous data with grouped frequency. Bars *must* touch. The area of each bar is proportional to the frequency. If class widths are unequal, you must calculate frequency density (frequency / class width).
• Line Graphs: Show trends over time or continuous change.
• Scatter Graphs: Display the relationship (correlation) between two variables. A line of best fit can be drawn to show the trend and make predictions (interpolation within the data range, extrapolation outside).
• 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.

## Analysing Data: Averages (Measures of Central Tendency)

These describe the 'centre' or typical value of a data set.

• Mode: The value that appears most frequently. For grouped data, it's the modal class (the class with the highest frequency).
• Median: The middle value when the data is arranged in order. For 'n' values, it's the ((n+1)/2)th value. For grouped data, you estimate its position.
• Mean: The sum of all values divided by the number of values. For frequency tables: (Σfx) / (Σf). For grouped frequency tables: estimate using the midpoint of each class: (Σ(midpoint × frequency)) / (Σf).

## Analysing Data: Spread (Measures of Dispersion)

These describe how spread out or varied the data is.

• Range: The highest value minus the lowest value. It's simple but can be heavily affected by outliers.
• Interquartile Range (IQR): The difference between the Upper Quartile (Q3) and the Lower Quartile (Q1). Q1 is the value 1/4 of the way through the ordered data, and Q3 is 3/4 of the way. IQR is less affected by outliers than the range.

## Comparing Data Sets

When comparing two data sets, always refer to both a measure of average (e.g., mean or median) and a measure of spread (e.g., range or IQR) to draw comprehensive conclusions about their central tendency and consistency.

## Introduction to Trigonometry

Trigonometry is the study of the relationship between the sides and angles of triangles. At GCSE level, it primarily focuses on right-angled triangles and extends to non-right-angled triangles using specific rules.

## Right-Angled Trigonometry (SOH CAH TOA)

For a right-angled triangle, we label the sides relative to a chosen non-right angle:

• Hypotenuse: The longest side, always opposite the right angle.
• Opposite: The side directly opposite the chosen angle.
• Adjacent: The side next to the chosen angle, not the hypotenuse.

We use the acronym SOH CAH TOA to remember the three basic trigonometric ratios:

• SOH: Sin(angle) = Opposite / Hypotenuse
• CAH: Cos(angle) = Adjacent / Hypotenuse
• TOA: Tan(angle) = Opposite / Adjacent

Finding Missing Sides

To find a missing side, you need one angle (other than the right angle) and one side. Choose the appropriate ratio based on the known side and the side you want to find. Rearrange the formula to solve for the unknown side.

Finding Missing Angles

To find a missing angle, you need at least two sides. Use the inverse trigonometric functions: sin⁻¹, cos⁻¹, or tan⁻¹. For example, if sin(x) = 0.5, then x = sin⁻¹(0.5).

## Exact Trigonometric Values

You should memorise the exact values for sine, cosine, and tangent for common angles (0°, 30°, 45°, 60°, 90°) as these can appear in non-calculator questions. For example, sin(30°) = 1/2, cos(60°) = 1/2, tan(45°) = 1.

## Non-Right-Angled Triangles (Higher Tier)

For triangles without a right angle, we use the Sine Rule, Cosine Rule, and a specific area formula.

The Sine Rule

Used when you have:

1. Two angles and one side.

2. Two sides and a non-included angle.

Formula: a/sinA = b/sinB = c/sinC (where a, b, c are side lengths and A, B, C are the angles opposite those sides).

The Cosine Rule

Used when you have:

1. Three sides (to find an angle).

2. Two sides and the included angle (the angle between the two known sides, to find the third side).

Formula to find a side: a² = b² + c² - 2bc cosA

Formula to find an angle: cosA = (b² + c² - a²) / 2bc

Area of a Non-Right-Angled Triangle

Used when you know two sides and the included angle.

Formula: Area = 1/2 ab sinC (where a and b are two sides, and C is the included angle between them).

## Introduction to Vectors

A vector is a quantity that has both magnitude (size or length) and direction. This is different from a scalar quantity, which only has magnitude (e.g., speed, mass, temperature). In GCSE Maths, vectors are often used to describe displacement – the movement from one point to another.

## Representing Vectors

Vectors can be represented in different ways:

• Column Vectors: This is the most common representation at GCSE. A 2D column vector is written as `[x, y]`, where 'x' represents the horizontal movement and 'y' represents the vertical movement. For example, `[3, 2]` means 3 units right and 2 units up.
• Vector Notation: Sometimes vectors are represented by a single bold letter (e.g., a) or an arrow above two points (e.g., `AB` with an arrow over it, meaning the vector from point A to point B).

## Vector Operations

1. Adding and Subtracting Vectors

To add or subtract two vectors, you simply add or subtract their corresponding components.

• If a = `[x1, y1]` and b = `[x2, y2]`:
• a + b = `[x1 + x2, y1 + y2]`
• a - b = `[x1 - x2, y1 - y2]`

Geometrically, adding vectors means placing them 'nose to tail' to find the resultant vector.

2. Multiplying a Vector by a Scalar

A scalar is just a number. When you multiply a vector by a scalar, you multiply each component of the vector by that scalar. This changes the magnitude of the vector but not its direction (unless the scalar is negative, which reverses the direction).

• If a = `[x, y]` and 'k' is a scalar:
• ka = `[kx, ky]`

## Magnitude of a Vector

The magnitude (or length) of a vector `[x, y]` can be found using Pythagoras' theorem, as the x and y components form the two shorter sides of a right-angled triangle.

• Magnitude of a = `|[x, y]|` = `sqrt(x^2 + y^2)`

## Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. This means they point in the same or opposite direction.

• If a = kb (where k is a scalar), then a and b are parallel.

For example, `[4, 6]` is parallel to `[2, 3]` because `[4, 6] = 2 * [2, 3]`.