How to learn Stage 7: Mathematics

Cambridge Primary, Lower Secondary and IGCSE: Mathematics, Stage 7

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1. Statistics & Probability: Probability

The Probability Scale: Understand probability as a measure on a scale from 0 (impossible) to 1 (certain); use the language of probability including likely, unlikely, certain, and impossible
Experimental probability: Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability
The 0-to-1 Probability Scale: Understand probability as a measure expressed as a number between 0 (impossible) and 1 (certain); place events on the probability scale; express probabilities as fractions, decimals, and percentages
Likelihood Language: Use probability language to describe and compare the likelihood of everyday events using words such as certain, likely, even chance, unlikely, impossible

2. Statistics & Probability: Statistics

Statistical Analysis Vocabulary: Read, write, and use the vocabulary of statistical analysis — mean, median, mode, range, frequency, data, sample, average, chart, table, graph, pie chart, scatter graph, correlation — with understanding of what each term describes
Comparing measurements: Describe, interpret, and compare distributions of a single variable using appropriate measures of central tendency (mean, median, mode) and spread (range), including the effect of outliers

3. Geometry & Measure: Position and transformations

Coordinate Transformations: Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original
Transformations on a grid: Represent and carry out geometric transformations on squared paper or a coordinate grid: reflections (in horizontal, vertical, and diagonal mirror lines, including the axes), translations (described as a vector or as left/right/up/down moves), and rotations (90° or 180° about a stated centre point); describe each transformation precisely using the correct language; identify which transformation maps one shape onto its image by comparing position, orientation, and size
Transformations on a Grid: Identify, describe, and represent the position of a shape following a reflection or translation using appropriate language; know that the shape has not changed

4. Geometry & Measure: Geometrical reasoning, shapes and measurements

Classifying shapes by properties: Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties
Types of angles (age 11+): Use conventional geometric terms and notation to describe, sketch, and draw points, lines, parallel and perpendicular lines, right angles, regular polygons, and reflectively/rotationally symmetric polygons
Properties of triangles and quadrilaterals: Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths
Angles with parallel lines: Understand and use the relationship between parallel lines cut by a transversal: corresponding angles, alternate interior angles, and co-interior (same-side interior) angles; use these to find unknown angles
Lines, Rays & Angles: Draw and identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines; identify these in two-dimensional figures
Angle Sum Rules: Know that angles at a point sum to 360° (one whole turn), angles on a straight line sum to 180°, and vertically opposite angles are equal; use these facts to find missing angles
Angles in triangles (age 10+): Find unknown angles in triangles, quadrilaterals, and regular polygons using angle sum properties
Angle sums in triangles and polygons: Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons
Parts of a circle: Illustrate and name parts of circles, including radius, diameter, and circumference; know that the diameter is twice the radius
Circles: Circumference & Area: Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts
3-D shapes (age 11+): Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections
Counting Unit Cubes: Measure volumes by counting unit cubes using cubic cm, cubic in, cubic ft, and other units
Using Simple Formulae: Use simple formulae expressed in words or symbols to calculate values (e.g. perimeter = 2 × (length + width))
Area of Triangles & Parallelograms: Calculate the area of parallelograms and triangles using formulae (A = b × h for parallelograms, A = ½ × b × h for triangles)
Unit Conversions: Convert freely between related standard units of measurement (time, length, area, volume/capacity, mass) using decimal notation to up to three decimal places where appropriate
What Is an Angle?: Understand that an angle is a geometric shape formed by two rays sharing a common endpoint (vertex); recognise angles in real-life contexts and 2-D shapes

5. Algebra: Sequences, functions and graphs

Algebraic Transformations: Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs; move between word problems, algebraic representations, tables, and graphical representations
Linear Function Graphs: Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-intercept in context
Plotting Linear Graphs: Plot linear graphs by generating a table of values, reduce a two-variable linear equation to the form y = mx + c, and calculate gradients from two points on a line
Generating Sequences: Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither
Nth-Term Rules: Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence

6. Algebra: Expressions, equations and formulae

Expressions & Equations Vocabulary: Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors; distinguish between an expression (no equals sign), an equation (equals sign), and an inequality (inequality sign)
Collecting Like Terms: Simplify algebraic expressions by collecting like terms — combine terms with the same variable and power (e.g., 3a + 2b + 5a = 8a + 2b) while maintaining equivalence
Substituting into Formulae: Substitute numerical values into formulae and expressions including scientific formulae; evaluate expressions by replacing variables with given values and computing the result using correct order of operations

7. Number: Fractions, decimals, percentages, ratio and proportion

Percentages as Fractions: Define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal; express one quantity as a percentage of another; compare quantities using percentages; work with percentages greater than 100%
Proportional Reasoning Vocabulary: Know and use advanced vocabulary of multiplicative reasoning — direct proportion, inverse proportion, ratio, rate, unit rate, compound unit, scale factor — accurately in problem-solving contexts
Ratio Notation: Use ratio notation to describe the relationship between two or more quantities, simplify ratios to their simplest form, and convert between ratio and fraction representations
Percentages (age 9+): Know and use the vocabulary of ratio and proportion — ratio, proportion, percentage, scale, equivalent, unequal, relative size, part-to-part, part-to-whole, and out of — and understand the difference between ratio (comparing parts to parts) and proportion (comparing a part to the whole)
Decimal & Percent Notation: Read, write, and use decimal and percentage notation correctly — decimal, decimal point, tenths, hundredths, thousandths, percentage, per cent, % symbol, convert, terminating decimal — and understand the relationships between fractions, decimals, and percentages as three ways of expressing the same value

8. Number: Place value, ordering and rounding

Decimal place value: Round decimals to any place using place-value understanding; round any whole number to a required degree of accuracy
Decimal place value (age 11+): Round numbers and measures to an appropriate degree of accuracy including to a specified number of decimal places or significant figures

9. Number: Integers, powers and roots

Ratio (age 11+): Use conventional notation for the priority of operations including brackets, powers, roots, and reciprocals; apply BIDMAS/BODMAS consistently to evaluate complex numerical expressions
Square and cube numbers: Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations
Factors, multiples, and primes: Find all factor pairs for a whole number in the range 1–100; identify common factors and common multiples of two numbers; use these concepts to solve problems
Positive and Negative Numbers: Understand positive and negative numbers as describing quantities with opposite directions or values; use them in context such as temperature, floors in a building, and bank balances