GCSE Maths flashcards

About this deck

206 free AQA GCSE Maths flashcards across 6 topics. Every card is a key term and its definition, taken word for word from a published Revise Right lesson, and links back to that lesson so you can read the full notes behind it.

Number

29 cards from 18 lessons. Each term links to the lesson it comes from.

Order of Operations (BIDMAS/BODMAS)

• BIDMAS / BODMAS: The conventional priority of operations: Brackets first, then Indices/Orders (powers and roots), then Division and Multiplication (left to right), then Addition and Subtraction (left to right). Every multi-operation expression must be evaluated in this sequence.
• Reciprocal: The reciprocal of a number n is 1/n. For a fraction a/b, the reciprocal is b/a. Multiplying a number by its reciprocal always gives 1. In calculations, dividing by a/b is the same as multiplying by b/a.

Inverse Operations and Simplifying Calculations

• Inverse operation: An operation that reverses the effect of another operation, returning the original value. For example, multiplying by 4 and then dividing by 4 returns you to where you started.

Common Factors and Multiples

• Prime Number: A positive integer greater than 1 that has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23.
• Highest Common Factor (HCF): The largest positive integer that divides exactly into two or more given numbers without remainder.
• Lowest Common Multiple (LCM): The smallest positive integer that is a multiple of two or more given numbers.

Prime Factorisation and Product Notation

• Factor (Divisor): A whole number that divides exactly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Unique Factorisation Theorem (Fundamental Theorem of Arithmetic): Every integer greater than 1 can be written as a product of prime factors in exactly one way, regardless of the order in which the primes are listed. This means prime factorisation is a reliable, unambiguous tool.

Systematic Listing Strategies

• Systematic listing: Recording all possible outcomes in a fixed, logical order — typically alphabetical, numerical, or by fixing one element and varying the others — so that every outcome appears exactly once.

Powers and Roots

• Index notation: A shorthand way of writing repeated multiplication. The base is the number being multiplied; the index (power) is how many times it appears as a factor. So a^n = a × a × ×s × a (n times).

Laws of Indices and Fractional Powers

• Fractional index a^1/n: For any positive integer n and positive real number a: a^1/n = n√(a). The denominator of the index is the root to be taken.
• Fractional index a^m/n: For positive integers m and n and positive real number a: a^m/n = (n√(a))^m = n√(a^m). The denominator gives the root; the numerator gives the power.

Working with Indices and Roots

• Square Root (√(a)): The square root of a is the non-negative value that, when squared, gives a. Written √(a). For example, √(49) = 7 because 7 × 7 = 49. Note: (-7)^2 = 49 too, so technically there are two square roots of 49, but √(49) refers to the positive (principal) root.

Calculating Exactly with Fractions

• Converting a mixed number to an improper fraction: For a mixed number na/b: multiply n × b, then add a, giving nb + a/b. Example: 23/5 = (2 × 5) + 3/5 = 13/5.

Understanding and Calculating with Standard Form

• Standard form: A number written as A × 10^n, where 1 ≤ A < 10 and n is an integer (positive, negative, or zero).

Converting Between Terminating Decimals and Fractions

• Terminating decimal: A decimal with a finite number of digits after the decimal point. Every terminating decimal can be written as an exact fraction. Examples: 0.5 = 1/2, 0.25 = 1/4, 0.375 = 3/8, 3.5 = 7/2.

Fractions in Ratio Problems

• Fraction Ratio: A comparison between quantities expressed as fractions, showing how many parts of one quantity relate to parts of another quantity
• Common Denominator Method: A technique for simplifying fraction ratios by converting all fractions to have the same denominator before comparing numerators

Understanding Fractions and Percentages as Operators

• Fraction as an operator: A fraction a/b applied to a quantity Q produces the result a/b × Q, which equals a × Q/b. The denominator b divides the quantity; the numerator a scales the result.
• Decimal multiplier: The decimal equivalent of a percentage, obtained by dividing the percentage value by 100. Multiplying a quantity by the decimal multiplier gives the percentage of that quantity in a single step.

Standard Units and Compound Measures

• Unit conversion: The process of expressing a measurement in a different unit of the same quantity by multiplying or dividing by the appropriate conversion factor. For example, converting 3.5 km to metres: 3.5 × 1000 = 3500 m.
• Compound measure: A measure formed by combining two or more standard measures using multiplication or division. The units of a compound measure reflect this combination — for example, speed has units of distance ÷ time, written as m/s or km/h.

Estimating and Checking Calculations

• Rounding to 1 significant figure: Reducing a number to its most important digit, making calculations much simpler
• Order of magnitude check: Verifying that an answer is roughly the right size by checking it's in the correct range of powers of 10

Rounding Numbers and Measures

• Decimal places (d.p.): The count of digits after the decimal point. To round to n decimal places, look at the (n+1)th digit after the point: if it is 5 or more, round up the nth digit; if it is 4 or less, leave the nth digit unchanged. This ensures accuracy consistent with decimal representation of values.
• Significant figures (s.f.): Digits counted from the first non-zero digit of a number. Zeros used only to position the decimal point (leading zeros in decimals, trailing zeros in whole numbers before the point) are not significant. To round to n s.f., apply the same 5-or-more rule to the (n+1)th significant digit.

Understanding Accuracy and Measurement

• Degree of Accuracy: The precision to which a value is stated. Common examples: nearest whole number, nearest 10, nearest 0.1 (1 decimal place), nearest 0.01 (2 decimal places), or to a given number of significant figures.
• Error Interval: Written as lower bound ≤ x < upper bound, this inequality captures every possible true value of a rounded measurement x. The lower bound is included; the upper bound is excluded.

Limits of Accuracy and Upper and Lower Bounds

• Finding bounds from a rounded value: If a value x is rounded to a degree of accuracy d, then: Lower bound = x - d/2 and Upper bound = x + d/2. The true value satisfies: lower bound ≤ true value < upper bound.

Algebra

61 cards from 37 lessons. Each term links to the lesson it comes from.

Substituting Values into Formulae and Expressions

• Substitution: Replacing each variable in an expression or formula with its given numerical value, then evaluating the result using the correct order of operations (BIDMAS).

Understanding Algebraic Language

• Term: A single number, variable, or product of numbers and variables. Terms are separated from each other by addition or subtraction signs. For example, in 3x^2 + 5x - 7, the three terms are 3x^2, 5x, and -7.
• Factor: A number or expression that divides exactly into another. For example, the factors of 6x are 1, 2, 3, 6, x, 2x, 3x, and 6x. In algebra, factorising means writing an expression as a product of its factors.
• Expression: A mathematical phrase made up of one or more terms connected by operations (+, -, ×, ÷). It contains no equals sign and cannot be solved — only simplified or evaluated. Example: 5x^2 - 3x + 1.
• Equation: A mathematical statement that two expressions are equal, written with an = sign. It can be solved to find the value(s) of the unknown. Example: 5x - 3 = 12 has the solution x = 3.
• Formula: A rule expressed in algebraic symbols that shows how one quantity depends on others. Unlike a one-off equation, a formula works for any valid input values. Example: the area of a rectangle, A = lw, works for any length l and width w.
• Inequality: A mathematical statement comparing two expressions using <, >, ≤, or ≥. The solution is a set of values, not a single number. Example: x + 3 > 7 gives x > 4, meaning any value greater than 4 satisfies it.

Expanding and Factorising Expressions

• Laws of Indices: The laws of indices govern how we manipulate powers. Key laws include: 1) a^m × a^n = a^m+n; 2) (a^m)^n = a^mn; 3) a^m ÷ a^n = a^m-n; 4) a^0 = 1 for any non-zero a. Understanding these laws is crucial for simplifying algebraic expressions. <br> <strong>Example:</strong> Simplify 2x^3 × 3x^2 = 6x^3+2 = 6x^5.
• Factorising: Writing an expression as a product of simpler expressions (factors). Essential for solving equations by setting each factor equal to zero.

Collecting Like Terms and Basic Algebraic Operations

• Like terms: Terms are 'like' when they have identical variable parts — the same letter(s) raised to the same power(s). For example, 4x and -7x are like terms; 4x and 4x^2 are not.
• Highest Common Factor (HCF) in factorising: The largest number and/or variable expression that divides exactly into every term of the expression. Placing the HCF outside the bracket gives the most fully factorised form.

Advanced Algebraic Manipulation

• Collecting like terms: Combining terms that have identical variable parts. For example, 3x^2 + 5x - x^2 + 2x simplifies to 2x^2 + 7x because 3x^2 - x^2 = 2x^2 and 5x + 2x = 7x.
• The ac method for factorising ax^2 + bx + c: Multiply a and c to get the product ac. Find two numbers that multiply to ac and add to b. Rewrite the middle term using those two numbers, then factorise by grouping.

Substituting into and Interpreting Formulae

• Subject of a formula: The variable that stands alone on one side of the equals sign, written without a coefficient. For example, in v = u + at, the subject is v.

Changing the Subject of a Formula

• Changing the subject: To make a chosen variable the subject of a formula, apply inverse operations to both sides in reverse order of operations, until that variable stands alone.

Equations vs Identities and Algebraic Arguments

• Identity: A statement of equality that holds for every value of the variable. Written using the identity symbol (three horizontal bars). Example: 3(x + 2) 3x + 6 is true for all values of x.

Algebraic Proofs

• Algebraic proof: A demonstration, using algebra, that a mathematical statement is true for all relevant values — not just a selection of examples. A proof must start from known facts and reach the stated conclusion through valid algebraic steps.
• Identity (): An equation true for all values of the variable. Written with the symbol rather than =. For example, (x+1)^2 - 1 x^2 + 2x is an identity because it holds for every value of x.

Interpreting Expressions as Functions

• Function (as an expression): A function is a rule, written as an expression, that assigns exactly one output to every input. For example, the expression 3x + 2 is a function: for each value of x you put in, there is one definite result.

Working with Coordinates in All Four Quadrants

• The (x, y) convention: Coordinates are always written as (x, y): the horizontal value first, then the vertical value. A useful memory prompt: you walk along the corridor (x) before you climb the stairs (y).

Plotting Straight-Line Graphs

• The equation y = mx + c: The standard form for a straight-line graph. m is the gradient (steepness and direction) and c is the y-intercept (where the line crosses the y-axis). Every straight-line equation can be written in this form.

Gradients and Intercepts of Linear Functions

• Gradient (algebraic definition): For a straight line passing through two points (x1, y1) and (x2, y2), the gradient is m = y2 - y1/x2 - x1. This ratio is constant for any two points chosen on the same straight line.

Turning Points and Symmetry of Quadratics

• Axis of symmetry: For a quadratic written as a(x+p)^2 + q, the axis of symmetry is the vertical line x = -p. It passes through the turning point and divides the parabola into two congruent halves.

Quadratic Graphs - Roots and Intercepts

• Roots (x-intercepts): The roots are the x-values where the graph crosses or touches the x-axis. At these points, y = 0. A quadratic can have two distinct roots, one repeated root (where the curve just touches the axis), or no real roots (where the curve does not reach the x-axis at all).
• y-intercept: The y-intercept is where the graph crosses the y-axis. It occurs at x = 0, so its coordinates are always (0, c) for the quadratic y = ax^2 + bx + c. On a graph, read the y-value where the curve meets the vertical axis.

Recognising and Sketching Linear and Quadratic Graphs

• Gradient (m): The rate of change of y with respect to x. Calculated as m = change in ychange in x. A positive value means the line rises from left to right; a negative value means it falls.
• y-intercept (c): The value of y where the line crosses the y-axis, i.e. the point (0, c). Found by substituting x = 0 into the equation.

Straight-Line Graph Skills and Gradients

• Gradient: The gradient of a straight line measures how much y changes for each unit increase in x. It is calculated as m = change in ychange in x = y2 - y1/x2 - x1 between any two points on the line.

Transformations of Linear and Quadratic Graphs

• Vertical translation: f(x) + a: Adding a <em>outside</em> the brackets shifts the graph <strong>up</strong> by a units when a > 0, and <strong>down</strong> by |a| units when a < 0. Every y-value increases by a; x-intercepts change but the y-intercept shifts by a.
• Horizontal translation: f(x + a): Replacing x with (x + a) <em>inside</em> the brackets shifts the graph <strong>left</strong> by a units when a > 0, and <strong>right</strong> by |a| units when a < 0. The direction is the opposite of what the sign suggests.
• Reflection in the x-axis: -f(x): Placing a negative sign <em>outside</em> the brackets negates every y-value. Points above the x-axis move below it, and vice versa. The x-intercepts stay fixed because y = 0 is unchanged by negation.
• Reflection in the y-axis: f(-x): Placing a negative sign <em>inside</em> the brackets negates every x-value. Points to the right of the y-axis move to the left, and vice versa. The y-intercept stays fixed because x = 0 is unchanged by negation.

Interpreting Gradients and Areas Under Graphs

• Gradient of a straight line: The constant rate of change between two points, calculated using the formula: gradient = (change in y) ÷ (change in x)
• Tangent to a curve: A straight line that touches a curve at exactly one point and has the same gradient as the curve at that point

Circle Equations and Properties

• Equation of a circle with centre at the origin: x^2 + y^2 = r^2 where r > 0 is the radius. Every point (x, y) satisfying this equation lies exactly r units from the origin.

Solving Linear Equations Algebraically

• Linear equation: An equation in which the unknown appears to the power of 1 only (no x^2, x^3, etc.). It can always be written in the form ax + b = c for constants a, b, and c, where a eq 0.

Finding Approximate Solutions Using Graphs

• Graphical solution: A value of x read from a graph at the point where two curves (or a curve and a line) intersect. It satisfies the equation formed by setting the two expressions equal, and is described as approximate because it is read by eye from a scale.

Completing the Square and the Quadratic Formula

• Completing the square (standard form): For a monic quadratic (where the coefficient of x^2 is 1): x^2 + bx + c = (x + b/2)^2 - (b/2)^2 + c. The term (b/2)^2 is subtracted to keep the expression equivalent to the original.
• The Discriminant: The discriminant of ax^2 + bx + c = 0 is Δ = b^2 - 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated root. If Δ < 0, there are no real roots.

Solving Quadratic Equations by Factorising

• Zero product property: If A × B = 0, then A = 0 or B = 0 (or both). This is the engine that drives the factorisation method for solving quadratics.

Solving simultaneous equations algebraically

• Elimination method: Add or subtract the two equations (or suitable multiples of them) to eliminate one variable. Solve the resulting single-variable equation, then substitute back to find the second variable.

Graphical solutions to simultaneous equations

• Linear-quadratic simultaneous equations: A system where one equation is linear (giving a straight line) and one is quadratic (giving a parabola). The system can have 2 solutions (line crosses the curve twice), 1 solution (line is tangent to the curve), or 0 real solutions (line misses the curve entirely).

Solving Quadratics, Simultaneous Equations and Rearranging Formulae

• Quadratic equation: An equation of the form ax² + bx + c = 0 where a ≠ 0. The highest power of the variable is 2.
• Discriminant: The value b² - 4ac that determines the nature of quadratic roots. Positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Numerical Solutions Using Iteration

• Suffix notation: A way of labelling successive approximations using subscripts. x1 is the first approximation, x2 is the second, and so on. The general recursive formula is written xn+1 = f(xn), meaning: substitute the current value xn into the right-hand side to produce the next value xn+1.

Solving Linear Equations with Brackets and Unknowns on Both Sides

• Balance method: Solving an equation by applying the same inverse operation to both sides at each step, keeping the scales level until the unknown stands alone.

Creating Algebraic Expressions from Situations

• Algebraic expression: A combination of letters, numbers, and operation signs that represents a quantity. Unlike an equation, it has no equals sign. For example, 5x + 3 is an expression.

Solving Equations from Real Problems

• Deriving an equation: Choosing a letter to represent an unknown quantity, then translating every piece of given information into algebraic expressions until you can write an equation (or two simultaneous equations) that captures the whole problem.
• Simultaneous equations from context: Two equations in two unknowns, each derived from a separate condition stated in the problem. Both equations must be satisfied by the same pair of values.

Representing Solution Sets

• Open circle: A hollow (unfilled) circle on a number line. It marks a boundary value that is NOT included in the solution set. Used for strict inequalities: < and >.
• Closed circle: A filled (solid) circle on a number line. It marks a boundary value that IS included in the solution set. Used for non-strict inequalities: ≤ and ≥.

Advanced Inequalities

• Inequality sign reversal rule: When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign must be reversed. For example, if -2x > 6, dividing both sides by -2 gives x < -3 (sign flips from > to <).

Solving Linear Inequalities

• Inequality symbols: < means strictly less than. > means strictly greater than. ≤ means less than or equal to. ≥ means greater than or equal to. The difference between < and ≤ determines whether the boundary value itself is included in the solution set.
• Inequality reversal rule: When you multiply or divide both sides of an inequality by a negative number, the inequality sign must reverse direction. For example, if -2x > 6, dividing both sides by -2 gives x < -3 (the > becomes <). This rule does not apply when multiplying or dividing by a positive number.

Generating Sequence Terms

• Term-to-term rule: A rule that specifies how each term in a sequence is obtained from the previous term. To use it, you must know at least one term (usually the first) and then apply the rule repeatedly.
• Position-to-term rule (nth term): An algebraic expression in n that gives the value of the term at position n directly. Substituting n = 1 gives the first term, n = 2 gives the second term, and so on, without needing to know any previous term.

Triangular, Square and Cube Numbers and Arithmetic Progressions

• Square Number: The result of multiplying a positive integer by itself. The nth square number is n^2. The sequence begins: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,...
• Cube Number: The result of multiplying a positive integer by itself three times. The nth cube number is n^3. The sequence begins: 1, 8, 27, 64, 125, 216,...
• Arithmetic Progression (AP): A sequence in which each term is obtained by adding a fixed constant (the common difference, d) to the previous term. The sequence has the form a, a+d, a+2d, a+3d, where a is the first term.

Finding the nth Term of Linear Sequences

• Linear sequence: A sequence where the difference between consecutive terms is constant. Also called an arithmetic sequence.
• Common difference: The constant value added to each term to get the next term in a linear sequence, denoted by d.

Ratio, proportion and rates of change

35 cards from 20 lessons. Each term links to the lesson it comes from.

Converting Between Standard and Compound Units

• Conversion Factor: A fraction equal to 1 that changes units without changing the actual quantity. For example, 1000m/1km or 60s/1min.
• Rate: A compound unit showing how one quantity changes with respect to another. Always written as 'something per something else', like miles per hour or pounds per kilogram.

Scale Factors, Scale Diagrams and Maps

• Scale Factor: The number by which all lengths are multiplied when creating a scale diagram. If the scale factor is 0.5, all lengths are halved. If it's 3, all lengths are tripled.
• Map Ratio: A scale written as 1:n, where 1 unit on the map equals n units in reality. The larger the second number, the more the real world has been shrunk down.

Working with Compound Units

• Compound unit: A unit made by combining two base units with division (or occasionally multiplication). The slash symbol '/' means 'per', so m/s means metres per second and £/kg means pounds per kilogram.
• Unit price: The cost of one unit of a product, found by dividing the total price by the total quantity. Expressed as £/kg, pence/litre, £/item, etc. Used to compare value between different pack sizes or products.

Comparing and Scaling in 2D and 3D

• Linear scale factor (LSF): The ratio of a length in the enlarged shape to the corresponding length in the original shape. Written as k where k = new lengthoriginal length.
• Area scale factor (ASF): The square of the linear scale factor. If the linear scale factor is k, then the area scale factor is k^2. Used for any two-dimensional measurement: surface area, cross-section, or base area.

Expressing One Quantity as a Fraction of Another

• Expressing one quantity as a fraction of another: Writing the first quantity as the numerator and the second quantity as the denominator to show their relationship
• Improper fraction: A fraction where the numerator is greater than the denominator, giving a value greater than 1

Simplifying and Interpreting Ratios

• Ratio: A comparison of two or more quantities showing how many times one value contains another, written in the form a:b
• Simplest Form: A ratio where the numbers have no common factors other than 1, achieved by dividing all parts by their highest common factor
• Equivalent Ratios: Different ratios that represent the same proportional relationship, created by multiplying or dividing all parts by the same number

Dividing Quantities Using Ratios

• Ratio Total Method: A systematic approach where you add all ratio parts to find the total number of shares, then calculate each part as a fraction of the whole amount
• Part:Whole Ratio: A ratio comparing one specific part to the complete total, expressed as the part compared to the sum of all parts

Ratio in Real-World Contexts

• Dividing in a ratio: To split a total quantity into shares that are proportional to a given ratio a:b, find the value of one part by dividing the total by (a + b), then multiply each ratio number by that one-part value.

Expressing Multiplicative Relationships as Ratios and Fractions

• Multiplicative relationship: A relationship between two quantities where one quantity is a constant multiple of another, expressed as y = kx where k is the constant multiplier

Understanding Proportion as Equality of Ratios

• Proportion: A mathematical relationship where two ratios are equal, written as a:b = c:d or a/b = c/d
• Cross multiplication: A method for solving proportions by multiplying diagonally across the equals sign: if a/b = c/d, then ad = bc

Connecting Ratios, Fractions and Linear Functions

• Ratio to Fraction Conversion: To convert ratio a:b to fractions, the first quantity becomes a/(a+b) and the second becomes b/(a+b) of the total
• Gradient as Ratio: The gradient m in y = mx + c represents the ratio of change in y to change in x, written as Δy:Δx

Percentage Basics and Percentage of Amounts

• Percentage: A way of expressing a number as a fraction of 100, literally meaning 'per hundred' or 'out of 100 parts'

Comparing and Working with Percentages

• Expressing one quantity as a percentage of another: Divide the quantity you want to express by the reference quantity, then multiply by 100. Written as: Percentage = partreference × 100

Percentage Change and Financial Applications

• Trap: Applying the percentage to the wrong value: Many students see '30% reduction, price is £63' and calculate 30% of £63 = £18.90, then add it back to get £81.90. This is wrong because £63 is already the reduced price — taking 30% of it gives 30% of the wrong number. The correct method divides £63 by the multiplier 0.70 to get £90.

Modelling Direct and Inverse Proportion

• Direct Proportion: y is directly proportional to x, written y x, if and only if y = kx for some non-zero constant k. Equivalently, the ratio y/x = k for all corresponding values of x and y.
• Inverse Proportion: y is inversely proportional to x, written y 1/x, if and only if y = k/x for some non-zero constant k. Equivalently, the product xy = k for all corresponding values of x and y.

Working with Proportion Equations

• Direct proportion equation: An equation of the form y = kx^n, where k is a non-zero constant and n is a positive integer (or sometimes a fraction). It means y is directly proportional to x^n, written y x^n. The graph of y against x^n is always a straight line through the origin.
• Inverse proportion equation: An equation of the form y = k/x^n, where k > 0 and n is a positive integer. It means y is inversely proportional to x^n, written y 1/x^n. The graph of y against x is a curve that falls steeply and never touches either axis (a hyperbola for n = 1).

Gradient as a Rate of Change

• Gradient as a rate of change: On any straight-line graph, the gradient equals the change in the y-axis quantity divided by the change in the x-axis quantity. Its units are always (units of y) per (units of x). This ratio is the rate at which y changes as x increases by 1.

Average and Instantaneous Rates of Change

• Average Rate of Change: For a function y = f(x), the average rate of change between x = a and x = b is the gradient of the chord joining the points (a, f(a)) and (b, f(b)). It is calculated as f(b) - f(a)/b - a.
• Instantaneous Rate of Change: The rate of change of y = f(x) at a single point x = a. It equals the gradient of the tangent to the curve at the point (a, f(a)). On a graph, it is estimated by drawing the tangent carefully and calculating its gradient using two points on that tangent line.

Growth and Decay with Multipliers

• Multiplier: A decimal factor applied to a quantity to produce a percentage change in one step. For an increase of r%: multiplier = 1 + r/100. For a decrease of r%: multiplier = 1 - r/100.
• Compound Interest: A method of calculating interest where each period's interest is added to the principal before the next period's interest is calculated. The total amount grows exponentially because interest is earned on previously accumulated interest.

Compound Growth, Decay and Iteration

• Percentage multiplier: For an increase of r%, the multiplier is 1 + r/100. For a decrease of r%, the multiplier is 1 - r/100. Applying the multiplier n times gives the result after n periods.
• Iterative formula: A recurrence relation of the form xn+1 = f(xn), where each term is calculated by applying a fixed rule to the previous term. A starting value x0 (or x1) must be given to begin the sequence.

Geometry and measures

49 cards from 35 lessons. Each term links to the lesson it comes from.

Geometric Language and Notation

• Line segment: A straight path between two points, with a definite start and end. Written as AB (or AB), where A and B are the endpoints. A full line extends infinitely in both directions.
• Vertex: A point where two or more edges of a polygon meet. Labelled with a capital letter. The plural is vertices. For example, a triangle ABC has vertices A, B, and C.

Triangle Labelling and Drawing Diagrams

• The opposite-side rule: In a triangle with vertices A, B, C: side a is opposite vertex A (i.e. it runs between B and C); side b is opposite vertex B (runs between A and C); side c is opposite vertex C (runs between A and B).

Basic Ruler and Compass Constructions

• Perpendicular bisector: A line that passes through the midpoint of a given line segment at 90°. Every point on the perpendicular bisector is exactly the same distance from each endpoint of the original segment.
• Angle bisector: A ray from the vertex of an angle that divides it into two equal angles. Constructed by drawing an arc from the vertex to mark points on both arms, then drawing crossing arcs from those two points.

Loci Problems and Geometric Constructions

• Construction: An accurate drawing produced using only a ruler (for straight lines) and compasses (for arcs). All construction arcs must be left visible — they are part of the answer and earn method marks.

Angles on Parallel Lines

• Alternate angles (Z-angles): Formed on opposite sides of the transversal, between the two parallel lines. Alternate angles are equal. Spot them by tracing a Z (or N) shape across the diagram.
• Corresponding angles (F-angles): Formed on the same side of the transversal, one at each parallel line, in matching positions. Corresponding angles are equal. Spot them by tracing an F shape along the diagram.

Angle Sums in Polygons

• Interior Angle Sum Formula: For any polygon with n sides: Sum of interior angles = (n - 2) × 180°. This is derived by splitting the polygon into (n-2) triangles from a single vertex.

Angles at Points and on Lines

• Angles at a point: When any number of rays meet at a single point, the angles between them sum to exactly 360°. This is because they together form one complete rotation.
• Angles on a straight line: Angles formed on one side of a straight line at a common point sum to 180°. This is sometimes called the straight-angle property or supplementary angles on a line.

Properties of Special Quadrilaterals

• Square: A quadrilateral with four equal sides, four right angles (90°), diagonals that are equal, bisect each other at right angles, and four lines of symmetry (including both diagonals).
• Rhombus: A parallelogram with all four sides equal. Its diagonals bisect each other at right angles and also bisect the vertex angles. It has two lines of symmetry (along the diagonals) and rotational symmetry of order 2.
• Kite: A quadrilateral with two pairs of adjacent (next-to-each-other) sides that are equal. One diagonal is a line of symmetry and bisects the other diagonal at right angles. One pair of opposite angles (between the unequal sides) are equal.

Properties of Triangles and Other Plane Figures

• Interior angle sum of a triangle: For any triangle, the three interior angles sum to 180°. Written formally: if the angles are a, b, and c, then a + b + c = 180°.
• Order of rotational symmetry: The number of times a shape looks identical to its original position during one full 360° rotation. A shape with no rotational symmetry has order 1 (it only matches itself at the start).

Triangle Congruence Criteria

• Congruent triangles: Two triangles are congruent if they are identical in shape and size. All three pairs of corresponding sides are equal and all three pairs of corresponding angles are equal. The symbol for congruence is (or in some texts, but Exam questions use the word 'congruent').
• Included angle: The angle formed between two given sides of a triangle. In SAS, the equal angle must be the one directly between the two equal sides — not an angle at the far end of one side.

Angle Facts and Properties

• Vertically Opposite Angles: When two straight lines intersect, the angles formed opposite each other at the intersection are equal. They are called vertically opposite because they share a vertex and face each other across it.
• Exterior Angle of a Triangle: When one side of a triangle is extended, the angle formed outside the triangle at that vertex is the exterior angle. It equals the sum of the two interior angles that are not adjacent to it (the two 'remote' interior angles).

Congruence, Similarity and Geometric Proofs

• SSS (Side-Side-Side): All three pairs of corresponding sides are equal. If you know every side length matches, the triangles must be congruent.

Transformations and Similar Shapes

• Congruent shapes: Two shapes are congruent if they are exactly the same size and shape. Every corresponding side length and every corresponding angle is equal. Rotations, reflections, and translations all produce congruent images.
• Similar shapes: Two shapes are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. One shape is an enlargement of the other. The ratio of corresponding sides is called the scale factor.

Basic Transformations

• Column Vector: A column vector (a, b) describes a translation: a is the horizontal movement (positive = right, negative = left) and b is the vertical movement (positive = up, negative = down).

Combined Transformations and Invariance

• Translation (column vector form): A translation moves every point of a shape by the same displacement. Written as (a, b), where a is the movement in the x-direction and b is the movement in the y-direction. Positive a means right; negative a means left. Positive b means up; negative b means down.

Circle Definitions and Properties

• Centre: The fixed interior point that is equidistant from every point on the circle. It lies inside the circle and is not itself on the circumference.

Circle Theorems and Geometric Proof

• Angle at the centre theorem: The angle subtended by an arc at the centre of a circle is equal to twice the angle subtended by the same arc at any point on the remaining circumference. Written formally: AOB = 2 ACB, where O is the centre and C is any point on the major arc.
• Cyclic quadrilateral: A quadrilateral with all four vertices on the circumference of a circle. Because each pair of opposite angles subtends arcs that together make the full circle (360°), and each arc produces a central angle of twice the inscribed angle, the two opposite inscribed angles must sum to 180°.

Using Circle Theorems to Find Angles

• Subtended angle: An angle is subtended by an arc (or chord) at a point when the two lines forming the angle both meet the endpoints of that arc. The arc 'opens up' the angle at that point.

Solving Problems on Coordinate Axes

• Perpendicular gradient rule: If a line has gradient m, any line perpendicular to it has gradient -1m. This is called the negative reciprocal. For example, a line with gradient 3 is perpendicular to a line with gradient -13.

Properties of 3D Shapes

• Face: A flat surface on a 3D shape. A cube has 6 faces, all of which are squares.
• Polyhedron: A 3D shape with flat polygonal faces, straight edges, and vertices. Cubes, cuboids, prisms, and pyramids are all polyhedra. Cylinders, cones, and spheres are not polyhedra because they have curved surfaces.

Reading Plans and Elevations of 3D Shapes

• Plan (view from above): The plan is what you see when you look straight down at the top of the shape. Imagine a bird hovering directly overhead — the outline it sees is the plan. It shows the length and width of the object, but gives no direct information about height.

Standard Units of Measure

• Length: A one-dimensional measure of distance between two points. Standard SI unit: metre (m). Common GCSE units: millimetre (mm), centimetre (cm), metre (m), kilometre (km).

Measuring in Geometric Figures and Scale Drawings

• Bearing: A three-figure angle, measured in degrees, clockwise from north. Bearings range from 000° (north) to 359°. Used to specify direction on maps and in navigation problems.

Volumes of 3D Shapes

• Right prism: A 3D solid whose cross-section is the same shape and size all the way along its length, with the cross-section perpendicular (at 90°) to the length. Volume = Area of cross-section × length.

Circumference from Radius and Diameter

• Exact form: An answer left in terms of π (for example 12π cm) rather than converted to a decimal. Examiners often ask for exact answers — leaving π in gives a perfectly precise result.

Area of Circles and Composite Figures

• Radius vs Diameter: The radius r is half the diameter d. If a question gives you the diameter, halve it before substituting into A = π r^2. This is the single most common source of errors on circle area questions.

Scale Factors in Similar Figures

• Linear scale factor (k): The ratio of any corresponding length in the larger (or smaller) figure to the matching length in the original figure. Written as k = new lengthoriginal length.

2D and 3D Trigonometric Problem Solving

• The Sine Rule: In any triangle ABC with sides a, b, c opposite to angles A, B, C respectively: asin A = bsin B = csin C. Use this rule when you know two angles and any side (AAS or ASA), or two sides and a non-included angle (SSA).
• Angle between a line and a plane: Drop a perpendicular from one end of the line to the plane. The angle between the line and the plane is the angle formed at the foot of that perpendicular, inside the right-angled triangle created. This is always the smallest angle between the line and any line in the plane.

Pythagoras and Trigonometry in Right-Angled Triangles

• Pythagoras' Theorem: In any right-angled triangle with hypotenuse c and shorter sides a and b: a^2 + b^2 = c^2 The hypotenuse is always the side opposite the right angle — it is always the longest side.
• Labelling Sides for Trigonometry: Relative to angle θ in a right-angled triangle: the <strong>hypotenuse</strong> is opposite the right angle (longest side); the <strong>opposite</strong> side is directly across from θ; the <strong>adjacent</strong> side is next to θ (but not the hypotenuse).

Sine, Cosine and Tangent Ratios

• Hypotenuse (H): The side directly opposite the right angle. It is always the longest side. This label never changes regardless of which angle you choose.

Solving Triangles with the Cosine Rule

• The Cosine Rule: For a triangle with sides a, b, c and angle A opposite side a: a^2 = b^2 + c^2 - 2bccos A Use this rule when you know two sides and the included angle (to find the third side), or when you know all three sides (to find an angle).

Solving Triangles with the Sine Rule

• Matched pair: A side and the angle directly opposite it in the same triangle. You must have at least one complete matched pair (both the side length and its opposite angle) before you can apply the sine rule.

Describing Translations as 2D Vectors

• 2D Column Vector (for a translation): Written as (a, b), where a is the number of units moved horizontally and b is the number of units moved vertically. The top number always describes left–right movement; the bottom number always describes up–down movement.

Using Vectors in Geometric Proofs

• Vector addition along a path: To find AC, write it as a chain: AC = AO + OC. Reversing a vector changes its sign: AO = -OA.

Vector Notation and Operations

• Vector: A quantity described by both a magnitude (size) and a direction. Drawn as an arrow where the length represents magnitude and the arrowhead shows direction. Written in bold as a or with an arrow as AB.

Probability

16 cards from 11 lessons. Each term links to the lesson it comes from.

Recording and Analysing Probability Experiments

• Frequency: The number of times a particular outcome occurs in an experiment. Recorded as a whole number count for each category.
• Frequency Tree: A branching diagram used to record and organise frequency data across two or more categorical attributes. Each branch shows a subgroup count, and the counts at each level must sum to the total at the previous branch.

Expected Outcomes in Probability Experiments

• Expected number of outcomes: Expected number of outcomes = probability of the event × number of trials. This gives the predicted frequency of an event over repeated experiments, assuming the probability stays constant each trial.

Relative Frequency and Theoretical Probability

• The 0 to 1 Probability Scale: Every probability is a number between 0 and 1 inclusive. A probability of 0 means the event is impossible. A probability of 1 means the event is certain. Values in between describe how likely an event is: closer to 0 means unlikely, closer to 1 means likely, and 0.5 means equally likely to happen or not happen.
• Relative Frequency: Relative frequency is calculated by dividing the number of times an event occurs by the total number of trials in an experiment. It gives an experimental estimate of probability.

Exhaustive Sets and Mutually Exclusive Events

• Exhaustive set of outcomes: A collection of outcomes is exhaustive if it includes every possible result of the experiment. No outcome outside the set can occur. Consequence: the probabilities of all outcomes in an exhaustive set must sum to 1.
• Mutually exclusive events: Two or more events are mutually exclusive if they cannot occur simultaneously in a single trial. If event A happens, event B cannot also happen in that same trial. There is no overlap between them. This can be visualized using a number line diagram where each event takes up a distinct section.

Experimental Probability, Relative Frequency and Sample Size

• Convergence of Relative Frequency: As the number of trials in an unbiased experiment increases, the relative frequency of an outcome tends towards (gets closer to) the theoretical probability of that outcome. This is sometimes called the Law of Large Numbers at GCSE level.

Probability Scale, Equally Likely Outcomes and Expected Frequency

• Probability scale: A number line from 0 to 1 on which every event can be placed. A probability of 0 means the event cannot happen; a probability of 1 means it will definitely happen. All other probabilities fall strictly between these two values.

Systematic Enumeration Using Tables, Grids and Venn Diagrams

• Sample space diagram (grid): A two-way table listing all possible outcomes when two events are combined. Each cell shows one outcome. The total number of cells equals the total number of equally likely outcomes.
• Complement of a set: All outcomes in the universal set that are NOT in set A. Written A'. On a Venn diagram, A' is everything outside circle A, including the 'neither' region and the part of B that does not overlap A.

Constructing Possibility Spaces and Calculating Theoretical Probabilities

• Theoretical probability: P(event) = number of favourable outcomestotal number of equally likely outcomes This formula only works when every outcome in the possibility space is equally likely.

Calculating Probabilities from Tree Diagrams

• Branch probability: The probability written on a single branch of a tree diagram. It represents the probability of that one outcome at that stage of the experiment. All branches from the same point must sum to 1.

Tree Diagrams for Combined Events

• The Multiplication Rule (AND): For two events A and B: P(A and B) = P(A) × P(B|A). When A and B are independent, this simplifies to P(A) × P(B) because P(B|A) = P(B).
• The Addition Rule (OR, mutually exclusive outcomes): For two mutually exclusive outcomes (they cannot both occur): P(A or B) = P(A) + P(B). On a tree diagram, you add the probabilities of separate branches that all lead to the outcome you want.

Conditional Probability from Tables, Tree Diagrams and Venn Diagrams

• Conditional probability: The probability of event A occurring given that event B has already occurred. Written P(A B). The vertical bar is read as 'given'. The sample space is restricted to outcomes where B has occurred.

Statistics

16 cards from 10 lessons. Each term links to the lesson it comes from.

Populations, Samples and Sampling

• Population: In statistics, a population is the entire group of individuals or items that a study is interested in. It does not have to mean people — it could be all the cars in a car park, all the apples in a supermarket, or all the test scores from a year group.
• Sample: A sample is a subset of the population that is selected and studied. Data collected from the sample is used to draw conclusions about the wider population.

Making Inferences from Samples

• Inference: Using the results from a sample to draw a conclusion about the wider population. For example, if 40% of a sample prefers option A, you infer that approximately 40% of the population also prefers option A.

Comparing Populations with Statistical Measures

• Mean: The mean is calculated by adding all the values together and dividing by how many values there are. It uses every data value, so it is sensitive to extreme values (outliers).
• Range: Range = largest value − smallest value. It measures how spread out the data is. A small range means the data is consistent; a large range means there is more variation in the population.

Choosing and Interpreting Statistical Charts

• Bar chart: A chart for categorical data where each category is represented by a bar of equal width. The height of the bar shows the frequency. Bars are separated by gaps to show the data is categorical, not continuous.
• Pictogram: A chart for categorical data that uses repeated symbols to represent frequency. A key states what each whole symbol represents. Half or quarter symbols can be used for non-integer multiples.

Constructing Statistical Diagrams and Comparing Distributions

• Class interval: A range of values grouped together in a frequency table, such as 10–20 or 20–30. The midpoint of a class interval is found by adding the lower and upper boundaries and dividing by 2.

Histograms and Cumulative Frequency Curves

• Histogram (equal class widths): A bar chart for continuous or grouped data where bars have no gaps and equal widths. The vertical axis shows frequency directly. The area of each bar equals its frequency.
• Cumulative Frequency: The total frequency of all values up to and including a given class boundary. Found by adding each class frequency to the sum of all previous class frequencies. Plotted against the upper class boundary to form a cumulative frequency curve.

Choosing Statistical Diagrams and Summary Measures

• Discrete data: Data that takes only specific, countable values with no meaningful values in between. Examples: number of goals scored (0, 1, 2, 3…), number of pets, shoe size. You cannot score 2.7 goals.
• Grouped data: Data placed into class intervals, such as 10 ≤ h < 20. This is used when the range of values is large or when individual values are not needed. Both discrete and continuous data can be grouped.

Box Plots and Quartiles

• Lower Quartile (Q1): The median of the lower half of the data set (the values below the overall median). It marks the 25th percentile — one quarter of the data lies at or below this value.

Scatter Graphs and Correlation

• Correlation: Correlation describes the relationship between two variables shown on a scatter graph.

Lines of Best Fit, Interpolation and Extrapolation

• Interpolation: Using the line of best fit to estimate a value inside the range of data already plotted.
• Extrapolation: Using the line of best fit to estimate a value outside the range of the data already plotted.