Subject Content - GCSE Maths

OCR 1 Number Operations and Integers

This chapter secures core number fluency: integer calculations, whole-number theory, order of operations, and inverse operations. These skills sit underneath later work in algebra, ratio, indices, estimation, and problem solving.

1.01 Calculations with integers

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1.01a	Four rules	Use non-calculator methods to add, subtract, multiply, and divide positive and negative whole numbers.			N2

1.02 Whole number theory

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1.02a	Definitions and terms	Understand and use key number vocabulary, including odd, even, prime, factor, divisor, multiple, common factor, common multiple, square, cube, and root. Use place value accurately.		N2, N4, N6
1.02b	Prime numbers	Identify prime numbers below 20. Express a whole number as a product of prime factors, for example $24 = 2 \times 2 \times 2 \times 3$ . Know that each whole number has one unique prime factorisation.	Identify prime numbers. Use powers when writing prime factorisations, for example $600 = 2^3 \times 3 \times 5^2$ .	N4, N6
1.02c	Highest Common Factor (HCF) and Lowest Common Multiple (LCM)	Find the HCF and LCM of two whole numbers by listing suitable factors and multiples.	Find the HCF and LCM of two whole numbers from their prime factorisations.	N4

1.03 Combining arithmetic operations

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1.03a	Priority of operations	Use the conventional order for calculations involving brackets, the four operations, powers, roots, and reciprocals.			N3

1.04 Inverse operations

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1.04a	Inverse operations	Recognise that addition and subtraction, multiplication and division, and powers and roots are inverse operations. Use inverse operations to simplify and check calculations, including number puzzles and missing-digit problems. e.g. $223 - 98 = 223 + 2 - 100 = 125$ $25 \times 12 = 50 \times 6 = 100 \times 3 = 300$			N3, N6

OCR 2 Fractions, Decimals and Percentages

This chapter develops fluency in equivalent forms of number. Learners move between fractions, decimals, and percentages, calculate with them, compare them, and use them in proportional contexts.

2.01 Fractions

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2.01a	Equivalent fractions	Recognise and use equivalent simple fractions and mixed numbers. e.g. $\frac{2}{6} = \frac{1}{3}$ $2\frac{1}{2} = \frac{5}{2}$			N3
2.01b	Calculations with fractions	Add, subtract, multiply, and divide simple fractions, including proper fractions, improper fractions, mixed numbers, and negative fractions. e.g. $1\frac{1}{2} + \frac{3}{4}$ $\frac{5}{6} \times \frac{3}{10}$ $-3 \times \frac{4}{5}$	Carry out more complex fraction calculations, including calculations with improper fractions. e.g. $\frac{2}{5} + \frac{5}{6}$ $\frac{2}{3} + \frac{1}{2} \times \frac{3}{5}$	See also algebraic fractions, 6.01g.	N2, N8
2.01c	Fractions of a quantity	Calculate a fraction of a quantity. e.g. $\frac{2}{5}$ of £3.50. Express one quantity as a fraction of another. See also ratios and fractions, 5.01c.	Calculate with fractions greater than 1.		N12, R3, R6

2.02 Decimal fractions

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2.02a	Decimals and fractions	Write a simple fraction as a terminating decimal, or write a terminating decimal as a fraction, with or without a calculator. e.g. $0.4 = \frac{2}{5}$ Understand place value in decimals.	Use division to convert a simple fraction to a decimal. e.g. $\frac{1}{6} = 0.1666...$	Convert a recurring decimal to an exact fraction, and convert an exact fraction to a recurring decimal. e.g. $0.\overline{41} = \frac{41}{99}$	N10, N2
2.02b	Addition, subtraction and multiplication of decimals	Add, subtract, and multiply decimals, including negative decimals, without a calculator.			N2
2.02c	Division of decimals	Divide a decimal by a whole number, including negative decimals, without a calculator. e.g. $0.24 \div 6$	Without a calculator, divide a decimal by a decimal. e.g. $0.3 \div 0.6$		N2

2.03 Percentages

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2.03a	Percentage conversions	Convert between fractions, decimals, and percentages. e.g. $\frac{1}{4} = 0.25 = 25\%$ $1\frac{1}{2} = 150\%$		R9
2.03b	Percentage calculations	Understand that percentage means 'number of parts per hundred'. Calculate a percentage of a quantity, and express one quantity as a percentage of another, with or without a calculator.		R9, N12
2.03c	Percentage change	Increase or decrease a quantity by a simple percentage, including with decimal or fractional multipliers. Apply this to original value problems and simple interest. e.g. Add 10% to £2.50 by finding 10% and adding, or by multiplying by $1.1$ or $\frac{110}{100}$ . Calculate the original price of an item costing £10 after a 50% discount.	Express percentage change as a decimal or fractional multiplier. Apply this to percentage change problems, including original value problems. See also growth and decay, 5.03a.	R9, N12

2.04 Ordering fractions, decimals and percentages

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2.04a	Ordinality	Order integers, fractions, decimals, and percentages. e.g. $\frac{4}{5}, \frac{3}{4}, 0.72, 0.9$			N1, N2, R9
2.04b	Symbols	Use $<, >, \le, \ge, =, \ne$ correctly.			N1

OCR 3 Indices and Surds

This chapter introduces index notation, powers, roots, standard form, and exact calculation. Higher-tier learners extend this into fractional indices, surds, and rationalising denominators.

3.01 Powers and roots

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3.01a	Index notation	Use positive integer indices to write repeated multiplication. e.g. $2 \times 2 \times 2 \times 2 = 2^4$	Use negative integer indices to represent reciprocals.	Use fractional indices to represent roots and combinations of powers and roots.	N6, N7
3.01b	Calculation and estimation of powers and roots	Calculate positive integer powers and exact roots. e.g. $2^4 = 16$ , $\sqrt{9} = 3$ , $\sqrt[3]{8} = 2$ . Recognise simple powers of 2, 3, 4, and 5. e.g. $27 = 3^3$	Calculate with integer powers. e.g. $2^{-3} = \frac{1}{8}$ Calculate with roots.	Calculate fractional powers. e.g. $16^{-\frac{3}{4}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{8}$ Estimate powers and roots, for example $\sqrt{51}$ to the nearest whole number.	N6, N7
3.01c	Laws of indices	See also simplifying products and quotients, 6.01c.	Know and apply the laws of indices, including $a^m \times a^n = a^{m+n}$ , $a^m \div a^n = a^{m-n}$ , and $(a^m)^n = a^{mn}$ . See also standard form calculations, 3.02b, and simplifying products and quotients, 6.01c.		N7, A4

3.02 Standard form

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3.02a	Standard form	Interpret and order numbers written in standard form. Convert numbers to and from standard form. e.g. $1320 = 1.32 \times 10^3$ and $0.00943 = 9.43 \times 10^{-3}$			N9
3.02b	Calculations with numbers in standard form	Use a calculator to perform calculations with numbers in standard form.	Add, subtract, multiply, and divide numbers in standard form without a calculator. See also laws of indices, 3.01c.		N9

3.03 Exact calculations

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3.03a	Exact calculations	Use fractions in exact calculations without a calculator.	Use multiples of $\pi$ in exact calculations without a calculator.	Use surds in exact calculations without a calculator.	N2, N8
3.03b	Manipulating surds			Simplify expressions involving surds, including rationalising denominators. e.g. $\sqrt{12} = 2\sqrt{3}$ , $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ , and $\frac{1}{\sqrt{3} + 1} = \frac{\sqrt{3} - 1}{2}$	N8

OCR 4 Approximation and Estimation

This chapter focuses on sensible rounding, estimation, checking calculations, and interpreting limits of accuracy. Learners use these tools to judge whether answers are reasonable.

4.01 Approximation and estimation

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4.01a	Rounding	Round numbers to the nearest whole number, ten, hundred, and so on, or to a given number of significant figures or decimal places.	Round answers to an appropriate level of accuracy.		N15
4.01b	Estimation	Estimate or check a calculation without a calculator by using suitable approximations. e.g. Estimate, to one significant figure, the cost of 2.8 kg of potatoes at 68p per kg.	Estimate or check more complex calculations, including roots, without a calculator. Use the symbol $\approx$ appropriately. e.g. $\sqrt{\frac{2.9}{0.051 \times 0.62}} \approx 10$		N14
4.01c	Upper and lower bounds		Use inequality notation to write an error interval for a number or measurement rounded or truncated to a given accuracy. e.g. If $x = 2.1$ rounded to 1 dp, then $2.05 \le x < 2.15$ . If $x = 2.1$ truncated to 1 dp, then $2.1 \le x < 2.2$ . Apply and interpret limits of accuracy.	Calculate upper and lower bounds for calculations using numbers rounded to a known degree of accuracy. e.g. Calculate the area of a rectangle with length and width each given to 2 sf. Understand the difference between bounds for discrete and continuous quantities. e.g. If there are 200 cars to the nearest hundred, the number of cars $n$ satisfies $150 \le n < 250$ and $150 \le n \le 249$ .	N15, N16

OCR 5 Ratio, Proportion and Rates of Change

This chapter develops ratio, proportional reasoning, direct and inverse proportion, and growth or decay. Learners connect numerical methods with algebraic models.

5.01 Calculations with ratio

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5.01a	Equivalent ratios	Find the ratio of quantities in the form $a:b$ and simplify. Find the ratio of quantities in the form $1:n$ . e.g. $50\text{ cm}:1.5\text{ m} = 50:150 = 1:3$		R4, R5
5.01b	Division in a given ratio	Split a quantity into two parts using the ratio of the parts. e.g. £2.50 in the ratio $2:3$ . Express a division of a quantity into two parts as a ratio. Calculate one quantity from another when the ratio of the two quantities is known.	Split a quantity into three or more parts using the ratio of the parts.	R5, R6
5.01c	Ratios and fractions	Interpret the ratio of two parts as a fraction of a whole. e.g. £9 split in the ratio $2:1$ gives parts $\frac{2}{3} \times £9$ and $\frac{1}{3} \times £9$ . See also fractions of a quantity, 2.01c.		N11, R5, R6, R8
5.01d	Solve ratio and proportion problems	Solve simple ratio and proportion problems. e.g. Adapt a recipe for 6 people to serve 4 people. Understand the relationship between ratio and linear functions.		R5, R8

5.02 Direct and inverse proportion

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5.02a	Direct proportion	Solve simple problems involving quantities in direct proportion, including algebraic proportions. e.g. using equality of ratios: if $y \propto x$ , then $\frac{y_1}{y_2} = \frac{x_1}{x_2}$ or $\frac{y_1}{x_1} = \frac{y_2}{x_2}$ . Apply this to currency conversion problems. See also similar shapes, 9.04c.	Solve more formal direct proportion problems, for example where $y \propto x$ . Recognise that if $y = kx$ , where $k$ is constant, then $y$ is proportional to $x$ .	Formulate equations and solve problems where one quantity is directly proportional to a power or root of another quantity.	R7, R10, R13
5.02b	Inverse proportion	Solve simple word problems involving quantities in inverse proportion or simple algebraic proportions. e.g. speed-time contexts where, if speed is doubled, time is halved.	Solve more formal inverse proportion problems, for example where $y \propto \frac{1}{x}$ . Recognise that if $y = \frac{k}{x}$ , where $k$ is constant, then $y$ is inversely proportional to $x$ .	Formulate equations and solve problems where one quantity is inversely proportional to a power or root of another quantity.	R10, R13

5.03 Discrete growth and decay

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5.03a	Growth and decay	Calculate simple interest, including in financial contexts.	Solve step-by-step problems using multipliers over a given interval, such as compound interest and depreciation. e.g. A car worth £15,000 depreciates by 30%, 20%, and 15% respectively over three years. See also percentage change, 2.03c.	Express exponential growth or decay as a formula. e.g. Amount £A subject to compound interest of 10% p.a. on £100 as $A = 100 \times 1.1^n$ . Solve and interpret answers in growth and decay problems. See also exponential functions, 7.01d, and formulate algebraic expressions, 6.02a.	R9, R16

OCR 6 Algebra

This chapter builds algebraic language, manipulation, formulae, substitution, and proof. Learners progress from basic expressions to higher-tier algebraic fractions and completing the square.

6.01 Algebraic expressions

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6.01a	Algebraic terminology and proofs	Understand and use the concepts and vocabulary of expressions, equations, formulae, inequalities, terms, and factors.	Recognise the difference between an equation and an identity, and show that two algebraic expressions are equivalent. e.g. show that $(x + 1)^2 + 2 = x^2 + 2x + 3$ . Use algebra to construct arguments.	Use algebra to construct proofs and arguments. e.g. prove that the sum of three consecutive integers is a multiple of 3.	A3, A6
6.01b	Collecting like terms in sums and differences of terms	Simplify algebraic expressions by collecting like terms. e.g. $2a + 3a = 5a$			A1, A3, A4
6.01c	Simplifying products and quotients	Simplify algebraic products and quotients. e.g. $a \times a \times a = a^3$ $2a \times 3b = 6ab$ $a^2 \times a^3 = a^5$ $3a^3 \div a = 3a^2$ See also laws of indices, 3.01c.		Simplify algebraic products and quotients using the laws of indices. e.g. $a^{\frac{1}{2}} \times 2a^{-3} = 2a^{-\frac{5}{2}}$ $2a^2b^3 \div 4a^{-3}b = \frac{1}{2}a^5b^2$	N3, A1, A4
6.01d	Multiplying out brackets	Simplify algebraic expressions by multiplying a single term over a bracket. e.g. $2(a + 3b) = 2a + 6b$ $2(a + 3b) + 3(a - 2b) = 5a$	Expand products of two binomials. e.g. $(x - 1)(x - 2) = x^2 - 3x + 2$ $(a + 2b)(a - b) = a^2 + ab - 2b^2$	Expand products of more than two binomials. e.g. $(x + 1)(x - 1)(2x + 1) = 2x^3 + x^2 - 2x - 1$	A1, A3, A4
6.01e	Factorising	Take out common factors. e.g. $3a - 9b = 3(a - 3b)$ $2x + 3x^2 = x(2 + 3x)$	Factorise quadratic expressions of the form $x^2 + bx + c$ . e.g. $x^2 - x - 6 = (x - 3)(x + 2)$ Use the difference of two squares, for example $x^2 - 16 = (x - 4)(x + 4)$ and $x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3})$ .	Factorise quadratic expressions of the form $ax^2 + bx + c$ , where $a \ne 0$ or 1. e.g. $2x^2 + 3x - 2 = (2x - 1)(x + 2)$	A1, A3, A4
6.01f	Completing the square			Complete the square for a quadratic expression. e.g. $x^2 + 4x - 6 = (x + 2)^2 - 10$ $2x^2 + 5x + 1 = 2(x + \frac{5}{4})^2 - \frac{17}{8}$	A11, A18
6.01g	Algebraic fractions			Simplify and manipulate algebraic fractions. e.g. write $\frac{1}{n - 1} + \frac{n}{n + 1}$ as a single fraction. Simplify $\frac{n^2 + 2n}{n^2 + n - 2}$ .	A1, A4

6.02 Algebraic formulae

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6.02a	Formulate algebraic expressions		Formulate simple formulae and expressions from real-world contexts. e.g. cost of car hire at £50 per day plus 10p per mile. The perimeter of a rectangle when the length is 2 cm more than the width.	See, for example, direct proportion 5.02a, inverse proportion 5.02b, and growth and decay 5.03a.	A3, A5, A21, R10
6.02b	Substitute numerical values into formulae and expressions	Substitute positive numbers into simple expressions and formulae to find the value of the subject. e.g. given $v = u + at$ , find $v$ when $t = 1$ , $u = 2$ , and $a = 7$ .	Substitute positive or negative numbers into more complex formulae, including powers, roots, and algebraic fractions. e.g. $v = \sqrt{u^2 + 2as}$ with $u = 2.1$ , $s = 0.18$ , and $a = 9.8$ .		A2, A5
6.02c	Change the subject of a formula	Rearrange a formula to change the subject where the subject appears once. e.g. Make $d$ the subject of $c = \pi d$ . Make $x$ the subject of $y = 3x - 2$ .	Rearrange formulae where the subject appears twice or where a power or reciprocal of the subject appears. e.g. make $t$ the subject of $s = \frac{1}{2}at^2$ , $v = \frac{x}{t}$ , and $2ty = t + 1$ .	Examples may include manipulation of algebraic fractions, 6.01g.	A4, A5
6.02d	Recall and use standard formulae	Recall and use circle formulae: circumference $2\pi r = \pi d$ and area $\pi r^2$ .	Recall and use Pythagoras' theorem $a^2 + b^2 = c^2$ . Recall and use trigonometry formulae: $\sin \theta = \frac{o}{h}$ , $\cos \theta = \frac{a}{h}$ , and $\tan \theta = \frac{o}{a}$ .	Recall and use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Recall and use the sine rule $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ . Recall and use the cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$ . Recall and use the triangle area formula $\frac{1}{2}ab\sin C$ .	A2, A3, A5
6.02e	Use kinematics formulae	Use $v = u + at$ , $s = ut + \frac{1}{2}at^2$ , and $v^2 = u^2 + 2as$ . Here $a$ is constant acceleration, $u$ is initial velocity, $v$ is final velocity, $s$ is displacement from position at time $t = 0$ , and $t$ is the time taken. Knowledge of each letter definition will not be required.			A2, A3, A5

6.03 Algebraic equations

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6.03a	Linear equations in one unknown	Solve linear equations in one unknown algebraically. e.g. solve $3x - 1 = 5$ .	Set up and solve linear equations in mathematical and non-mathematical contexts, including equations with the unknown on both sides. e.g. solve $5(x - 1) = 4 - x$ . Interpret solutions in context.	Examples may include manipulation of algebraic fractions, 6.01g.	A3, A17, A21
6.03b	Quadratic equations		Solve quadratic equations where the coefficient of $x^2$ is 1 by factorising. e.g. solve $x^2 - 5x + 6 = 0$ . Find $x$ for an $x$ cm by $(x + 3)$ cm rectangle of area $40\text{ cm}^2$ .	Know the quadratic formula. Rearrange and solve quadratic equations by factorising, completing the square, or using the quadratic formula. e.g. solve $2x^2 = 3x + 5$ and $\frac{2}{x} - \frac{2}{x + 1} = 1$ .	A18
6.03c	Simultaneous equations		Set up and solve two linear simultaneous equations in two variables algebraically. e.g. solve $2x + 3y = 18$ and $y = 3x - 5$ .	Set up and solve two simultaneous equations in two variables algebraically, including cases where one equation is quadratic or gives a quadratic result. e.g. solve $x^2 + y^2 = 50$ and $2y = x + 5$ .	A19, A21
6.03d	Approximate solutions using a graph	Use a graph to find the approximate solution of a linear equation.	Use graphs to find approximate roots of quadratic equations and the approximate solution of two linear simultaneous equations.	Know that intersection points of a curve and a straight line give the solutions to simultaneous equations where one equation is for the line and one is for the curve.	A11, A17, A18, A19
6.03e	Approximate solutions by iteration			Find approximate solutions to equations using systematic sign-change methods, for example decimal search or interval bisection, where no simple analytical method is available. Specific methods will not be requested in the assessment.	A20, R16

6.04 Algebraic inequalities

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6.04a	Inequalities in one variable	Understand and use the symbols $<, \le, >$ and $\ge$ .	Solve linear inequalities in one variable and express solutions on a number line using conventional notation. e.g. solve $2x + 1 > 7$ and $1 < 3x - 5 \le 10$ .	Solve quadratic inequalities in one variable. e.g. $x^2 - 2x < 3$ . Express solutions in set notation, for example $\{x:x > 3\}$ or $\{x:2 < x \le 5\}$ . See also polynomial and reciprocal functions, 7.01c.	N1, A3, A22
6.04b	Inequalities in two variables			Solve several linear inequalities in two variables and represent the solution set on a graph. See also straight line graphs, 7.02a.	A22

6.05 Language of functions

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6.05a	Functions	Interpret simple expressions as functions with inputs and outputs where appropriate. e.g. $y = 2x + 3$ as $x \mapsto 2x \mapsto 2x + 3 \mapsto y$ .		Interpret the reverse process as the inverse function. Interpret the succession of two functions as a composite function. Knowledge of function notation will not be required. See also translations and reflections, 7.03a.	A7

6.06 Sequences

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6.06a	Generate terms of a sequence	Generate a sequence by spotting a pattern or using a term-to-term rule given algebraically or in words. e.g. continue $1, 4, 7, 10, ...$ and $1, 4, 9, 16, ...$ . Find a position-to-term rule for simple arithmetic sequences, algebraically or in words. e.g. $2, 4, 6, ... , 2n$ and $3, 4, 5, ... , n + 2$ .	Generate a sequence from a formula for the nth term. e.g. nth term $n^2 + 2n$ gives $3, 8, 15, ...$ . Find a formula for the nth term of an arithmetic sequence. e.g. $40, 37, 34, 31, ... , 43 - 3n$ .	Use subscript notation for position-to-term and term-to-term rules. e.g. $x_n = n + 2$ and $x_{n+1} = 2x_n - 3$ . Find a formula for the nth term of a quadratic sequence. e.g. $0, 3, 10, 21, ...$ gives $u_n = 2n^2 - 3n + 1$ .	A23, A25
6.06b	Special sequences	Recognise sequences of triangular, square, and cube numbers, and simple arithmetic progressions.	Recognise Fibonacci and quadratic sequences, and simple geometric progressions $r^n$ , where $n$ is an integer and $r$ is a rational number greater than 0.	Generate and find nth terms of other sequences, for example $1, \sqrt{2}, 2, 2\sqrt{2}, ...$ and $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, ...$ .	A24

OCR 7 Graphs of Equations and Functions

This chapter develops coordinate work, graph plotting, straight-line graphs, curves, transformations of graphs, and interpretation of real-world graphs.

7.01 Graphs of equations and functions

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7.01a	x- and y-coordinates	Work with x- and y-coordinates in all four quadrants.			A8
7.01b	Graphs of equations and functions	Use a table of values to plot graphs of linear and quadratic functions. e.g. $y = 2x + 3$ and $y = 2x^2 + 1$ .	Use a table of values to plot other polynomial graphs and reciprocal graphs. e.g. $y = x^3 - 2x$ , $y = x + \frac{1}{x}$ , and $2x + 3y = 6$ .	Use a table of values to plot exponential graphs. e.g. $y = 3 \times 1.1^x$ .	A9, A14
7.01c	Polynomial and reciprocal functions	Recognise and sketch graphs of simple linear and quadratic functions. e.g. $y = 2$ , $x = 1$ , $y = 2x$ , and $y = x^2$ .	Recognise and sketch graphs of $y = x^3$ and $y = \frac{1}{x}$ . Identify intercepts and, using symmetry, the turning point of graphs of quadratic functions. Find roots of a quadratic equation algebraically.	Sketch graphs of quadratic functions, identifying the turning point by completing the square.	A11, A12
7.01d	Exponential functions			Recognise and sketch graphs of exponential functions in the form $y = k^x$ for positive $k$ .	A12
7.01e	Trigonometric functions			Recognise and sketch the graphs of $y = \sin x$ , $y = \cos x$ , and $y = \tan x$ .	A12
7.01f	Equations of circles			Recognise and use the equation of a circle with centre at the origin.	A16

7.02 Straight line graphs

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7.02a	Straight line graphs	Find and interpret the gradient and intercept of straight lines graphically and using $y = mx + c$ .	Use $y = mx + c$ to find and sketch equations of straight lines. Find the equation of a line through two given points, or through one point with a given gradient.	Identify solution sets of linear inequalities in two variables using dashed and solid line conventions.	A9, A10, A22
7.02b	Parallel and perpendicular lines		Identify and find equations of parallel lines.	Identify and find equations of perpendicular lines. Calculate the equation of a tangent to a circle at a given point. See also equations of circles, 7.01f.	A9, A16

7.03 Transformations of curves and their equations

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7.03a	Translations and reflections			Identify and sketch translations and reflections of a given graph, or the graph of a given equation. Knowledge of function notation will not be required. See also functions, 6.05a. e.g. sketch $y = \sin x + 2$ , $y = (x + 2)^2 - 1$ , and $y = -x^2$ .	A13

7.04 Interpreting graphs

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7.04a	Graphs of real-world contexts	Construct and interpret graphs in real-world contexts. e.g. distance-time graphs, money conversion graphs, and temperature conversion graphs. See also direct proportion, 5.02a, and inverse proportion, 5.02b.	Recognise and interpret graphs that illustrate direct and inverse proportion.		A14, R10, R14
7.04b	Gradients	Understand the relationship between gradient and ratio.	Interpret straight-line gradients as rates of change. e.g. the gradient of a distance-time graph as velocity.	Calculate or estimate gradients of graphs, and interpret them in contexts such as distance-time graphs, velocity-time graphs, and financial graphs. Apply average and instantaneous rates of change, including gradients of chords and tangents, in numerical, algebraic, and graphical contexts.	A14, A15, R8, R14, R15
7.04c	Areas			Calculate or estimate areas under graphs, and interpret these in contexts such as distance-time graphs, velocity-time graphs, and financial graphs.	A15

OCR 8 Basic Geometry

This chapter covers geometric language, ruler and compass construction, angle facts, polygon properties, circle theorems, and three-dimensional shapes.

8.01 Conventions, notation and terms

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8.01a	2D and 3D shapes	Use the terms points, lines, line segments, vertices, edges, planes, parallel lines, and perpendicular lines.	G1
8.01b	Angles	Know the terms acute, obtuse, right, and reflex angles. Use standard conventions for labelling and referring to sides and angles of triangles, for example AB, angle ABC, or side opposite angle A.	G1
8.01c	Polygons	Know the terms regular polygon; scalene, isosceles, and equilateral triangle; quadrilateral, square, rectangle, kite, rhombus, parallelogram, trapezium; pentagon, hexagon, and octagon.	G1
8.01d	Polyhedra and other solids	Recognise the terms face, surface, edge, and vertex, and recognise cube, cuboid, prism, cylinder, pyramid, cone, and sphere.	G12
8.01e	Diagrams	Draw diagrams from written descriptions as required by questions.	G1
8.01f	Geometrical instruments	Use a ruler to construct and measure straight lines. Use a protractor to construct and measure angles. Use compasses to construct circles.	G2, G15
8.01g	x- and y-coordinates	Use x- and y-coordinates in plane geometry problems, including transformations of simple shapes.	G7, G11

Learners are expected to be familiar with these geometrical skills, conventions, notation, and terms, which may be assessed in questions at both tiers.

8.02 Ruler and compass constructions

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8.02a	Perpendicular bisector	Construct the perpendicular bisector and midpoint of a line segment.	G2
8.02b	Angle bisector	Construct the bisector of an angle formed from two lines.	G2
8.02c	Perpendicular from a point to a line	Construct the perpendicular from a point to a line. Construct the perpendicular to a line at a point. Know that the perpendicular distance from a point to a line is the shortest distance to the line.	G2
8.02d	Loci	Apply ruler and compass constructions to construct figures and identify the loci of points, including real-world problems. Understand the term equidistant.	G2

8.03 Angles

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8.03a	Angles at a point	Know and use the sum of angles at a point as 360 degrees.	Apply angle facts to find angles in rectilinear figures and to justify results in simple proofs. e.g. the interior angles of a triangle sum to 180 degrees.	Apply angle properties in more formal proofs of geometrical results.	G3, G6
8.03b	Angles on a line	Know that the sum of angles at a point on a line is 180 degrees.	Apply angle facts to find angles in rectilinear figures and to justify results in simple proofs.	Apply angle properties in more formal proofs of geometrical results.	G3, G6
8.03c	Angles between intersecting and parallel lines	Know and use that vertically opposite angles are equal, alternate angles on parallel lines are equal, and corresponding angles on parallel lines are equal.	Apply angle facts to find angles in rectilinear figures and to justify results in simple proofs.	Apply angle properties in more formal proofs of geometrical results.	G3, G6
8.03d	Angles in polygons	Derive and use the sum of the interior angles of a triangle as 180 degrees. Derive and use the sum of the exterior angles of a polygon as 360 degrees. Find the sum of the interior angles of a polygon. Find the interior angle of a regular polygon.	Apply angle facts to find angles in rectilinear figures and to justify results in simple proofs.	Apply angle properties in more formal proofs of geometrical results.	G3, G6

8.04 Properties of polygons

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8.04a	Properties of a triangle	Know the basic properties of isosceles, equilateral, and right-angled triangles. Give geometrical reasons to justify these properties.	Use these facts to find lengths and angles in rectilinear figures and in simple proofs.	Use these facts in more formal proofs of geometrical results, for example circle theorems.	G4, G6
8.04b	Properties of quadrilaterals	Know the basic properties of square, rectangle, parallelogram, trapezium, kite, and rhombus. Give geometrical reasons to justify these properties.	Use these facts to find lengths and angles in rectilinear figures and in simple proofs.	Use these facts in more formal proofs of geometrical results, for example circle theorems.	G4, G6
8.04c	Symmetry	Identify reflection and rotation symmetries of triangles, quadrilaterals, and other polygons.			G1, G4

8.05 Circles

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8.05a	Circle nomenclature	Understand and use the terms centre, radius, chord, diameter, and circumference.	Understand and use the terms tangent, arc, sector, and segment.		G9
8.05b	Angles subtended at centre and circumference			Apply and prove that the angle subtended by an arc at the centre is twice the angle at the circumference.	G10
8.05c	Angle in a semicircle			Apply and prove that the angle on the circumference subtended by a diameter is a right angle.	G10
8.05d	Angles in the same segment			Apply and prove that two angles in the same segment are equal.	G10
8.05e	Angle between radius and chord			Apply and prove that a radius or diameter bisects a chord if and only if it is perpendicular to the chord.	G10
8.05f	Angle between radius and tangent			Apply and prove that, for a point P on the circumference, the radius or diameter through P is perpendicular to the tangent at P.	G10
8.05g	The alternate segment theorem			Apply and prove that, for a point P on the circumference, the angle between the tangent and a chord through P equals the angle subtended by the chord in the opposite segment.	G10
8.05h	Cyclic quadrilaterals			Apply and prove that opposite angles of a cyclic quadrilateral are supplementary.	G10

8.06 Three-dimensional shapes

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8.06a	3-dimensional solids	Recognise and know the properties of cube, cuboid, prism, cylinder, pyramid, cone, and sphere.			G12
8.06b	Plans and elevations	Interpret plans and elevations of simple 3D solids.	Construct plans and elevations of simple 3D solids, and representations from plans and elevations, for example using isometric paper.		G1, G13

OCR 9 Congruence and Similarity

This chapter covers transformations, congruent triangles, vectors, enlargement, and similarity, including the connection between scale factors, areas, and volumes.

9.01 Plane isometric transformations

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9.01a	Reflection	Reflect a simple shape in a given mirror line, and identify the mirror line from a shape and its image.	Identify a mirror line $x = a$ , $y = b$ , or $y = \pm x$ from a simple shape and its image under reflection.		G7
9.01b	Rotation	Rotate a simple shape clockwise or anti-clockwise through a multiple of 90 degrees about a given centre of rotation.	Identify the centre, angle, and sense of rotation from a simple shape and its image under rotation.		G7
9.01c	Translation	Use a column vector to describe a translation of a simple shape, and perform a specified translation.			G7, G24
9.01d	Combinations of transformations			Perform a sequence of isometric transformations on a simple shape. Describe the resulting transformation and the changes or invariance achieved.	G8

9.02 Congruence

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9.02a	Congruent triangles	Identify congruent triangles.	Prove that two triangles are congruent using SSS, ASA, SAS, and RHS conditions.		G5, G7
9.02b	Applying congruent triangles		Apply congruent triangles in calculations and simple proofs. e.g. the base angles of an isosceles triangle are equal.		G6, G19

9.03 Plane vector geometry

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9.03a	Vector arithmetic		Understand addition, subtraction, and scalar multiplication of vectors.	Use vectors in geometric arguments and proofs.	G25
9.03b	Column vectors		Represent a 2-dimensional vector as a column vector, and draw column vectors on a square or coordinate grid.		G25

9.04 Similarity

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9.04a	Similar triangles	Identify similar triangles.	Prove that two triangles are similar.		G6, G7
9.04b	Enlargement	Enlarge a simple shape from a given centre using a whole-number scale factor, and identify the scale factor of an enlargement.	Identify the centre and scale factor, including fractional scale factors, of an enlargement of a simple shape, and perform such an enlargement on a simple shape.	Perform and recognise enlargements with negative scale factors.	R2, R12, G7
9.04c	Similar shapes	Compare lengths, areas, and volumes using ratio notation and scale factors.	Apply similarity to calculate unknown lengths in similar figures. See also direct proportion, 5.02a.	Understand the relationship between lengths, areas, and volumes of similar shapes. See also direct proportion, 5.02a.	R12, G19

OCR 10 Mensuration

This chapter covers units, compound measures, scale drawings, perimeter, area, volume, surface area, Pythagoras' theorem, and trigonometry.

10.01 Units and measurement

GCSE (9-1) content Ref.	Subject content	Initial learning enables learners to...	Foundation tier learners should also be able to...	DfE Ref.
10.01a	Units of measurement	Use and convert standard units of measurement for length, area, volume/capacity, mass, time, and money.	Use and convert standard units in algebraic contexts.	N13, R1, G14
10.01b	Compound units	Use and convert simple compound units, for example for speed, rates of pay, and unit pricing. Know and apply, in simple cases, $\text{speed} = \text{distance} \div \text{time}$ .	Use and convert other compound units, for example density and pressure. Know and apply $\text{density} = \text{mass} \div \text{volume}$ . Use and convert compound units in algebraic contexts.	N13, R1, R11, G14
10.01c	Maps and scale drawings	Use the scale of a map and work with bearings. Construct and interpret scale drawings.		R2, G15

10.02 Perimeter calculations

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10.02a	Perimeter of rectilinear shapes	Calculate the perimeter of rectilinear shapes.		G17
10.02b	Circumference of a circle	Know and apply $\text{circumference} = 2\pi r = \pi d$ to calculate the circumference of a circle.	Calculate the arc length of a sector of a circle when given its angle and radius.	G17, G18
10.02c	Perimeter of composite shapes	Apply perimeter formulae in calculations involving the perimeter of composite 2D shapes.		G17, G18

10.03 Area calculations

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10.03a	Area of a triangle	Know and apply $\text{area} = \frac{1}{2} \times \text{base} \times \text{height}$ .		Know and apply $\text{area} = \frac{1}{2}ab\sin C$ .	G16, G23
10.03b	Area of a parallelogram	Know and apply $\text{area} = \text{base} \times \text{height}$ . This includes the area of a rectangle.			G16
10.03c	Area of a trapezium	Calculate the area of a trapezium.			G16
10.03d	Area of a circle	Know and apply $\text{area} = \pi r^2$ to calculate the area of a circle.	Calculate the area of a sector of a circle when given its angle and radius.		G17, G18
10.03e	Area of composite shapes	Apply area formulae in calculations involving composite 2D shapes.			G17, G18

10.04 Volume and surface area calculations

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10.04a	Polyhedra	Calculate the surface area and volume of cuboids and other right prisms, including cylinders.		G16
10.04b	Cones and spheres		Calculate the surface area and volume of spheres, cones, and simple composite solids. Formulae will be given.	N8, G17
10.04c	Pyramids		Calculate the surface area and volume of a pyramid. The formula $\frac{1}{3} \times \text{area of base} \times \text{height}$ will be given.	G17

10.05 Triangle mensuration

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10.05a	Pythagoras' theorem	Know, derive, and apply Pythagoras' theorem $a^2 + b^2 = c^2$ to find lengths in right-angled triangles in 2D figures.	Apply Pythagoras' theorem in more complex figures, including 3D figures.	G6, G20
10.05b	Trigonometry in right-angled triangles	Know and apply the trigonometric ratios $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ to find angles and lengths in right-angled triangles in 2D figures. See also similar shapes, 9.04c.	Apply trigonometry of right-angled triangles in more complex figures, including 3D figures.	R12, G20
10.05c	Exact trigonometric ratios	Know exact values of $\sin \theta$ and $\cos \theta$ for $\theta = 0\degree, 30\degree, 45\degree, 60\degree, 90\degree$ . Know exact values of $\tan \theta$ for $\theta = 0\degree, 30\degree, 45\degree, 60\degree$ .		R12, G21
10.05d	Sine rule		Know and apply the sine rule $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ to find lengths and angles.	G22
10.05e	Cosine rule		Know and apply the cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$ to find lengths and angles.	G22

OCR 11 Probability

This chapter covers probability scales, relative frequency, enumeration, Venn diagrams, tree diagrams, and addition and multiplication laws.

11.01 Basic probability and experiments

GCSE (9-1) content Ref.	Subject content	Initial learning enables learners to...	Foundation tier learners should also be able to...	DfE Ref.
11.01a	The probability scale	Use the 0-1 probability scale as a measure of likelihood for random events, for example impossible with 0, evens with 0.5, and certain with 1.		P3
11.01b	Relative frequency	Record, describe, and analyse relative frequency outcomes from repeated experiments using tables and frequency trees.		P1
11.01c	Relative frequency and probability	Use relative frequency as an estimate of probability.	Understand that relative frequencies approach theoretical probability as the number of trials increases.	P3, P5

11.02 Probability diagrams and rules

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11.02b	Enumeration	Use systematic listing strategies.		Use the product rule for counting outcomes of combined events.	N5
11.02c	Venn diagrams and sets	Use a two-circle Venn diagram to enumerate sets and calculate related probabilities. Use simple set notation to describe simple sets of numbers or objects. e.g. A = {even numbers}, B = {mathematics learners}, C = {isosceles triangles}.	Construct a Venn diagram to classify outcomes and calculate probabilities. Use set notation to describe a set of numbers or objects, for example $D = \{x:1 < x < 3\}$ and $E = \{x:x \text{ is a factor of } 280\}$ . Knowledge of intersection, union, and complement notation will not be required.	Construct tree diagrams, two-way tables, or Venn diagrams to solve more complex probability problems, including conditional probability. Structured diagrams may not be given. Knowledge of intersection, union, and complement notation will not be required.	P6, P9
11.02d	Tree diagrams		Use tree diagrams to enumerate sets and record probabilities of successive events. Tree frames may be given and partly completed in some cases.		P6, P9
11.02e	The addition law of probability	Use the addition law for mutually exclusive events. Use $p(A) + p(\text{not }A) = 1$ .	Derive or informally understand and apply the formula $p(A \text{ or } B) = p(A) + p(B) - p(A \text{ and } B)$ .		P4
11.02f	The multiplication law of probability and conditional probability		Use tree diagrams and other representations to calculate probabilities of independent and dependent combined events.	Understand conditional probability and calculate it from first principles in known contexts. e.g. in a random cut of a pack of 52 cards, calculate the probability of drawing a diamond given that a red card is drawn. Derive or informally understand and apply $p(A \text{ and } B) = p(A \text{ given } B)p(B)$ . Know that events A and B are independent if and only if $p(A \text{ given } B) = p(A)$ .	P8, P9

OCR 12 Statistics

This chapter covers sampling, representing data, summary statistics, grouped data, misleading graphs, bivariate data, and outliers.

12.01 Sampling

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12.01a	Populations and samples		Define the population in a study and understand the difference between population and sample. Infer properties of populations or distributions from a sample. Understand simple random sampling and bias in sampling.		S1

12.02 Interpreting and representing data

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12.02a	Categorical and numerical data	Interpret and construct charts appropriate to the data type, including frequency tables, bar charts, pie charts, pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, and multiple or composite bar charts.	Design tables to classify data. Interpret and construct line graphs for time-series data and identify trends, for example seasonal variation.		S2
12.02b	Grouped data			Interpret and construct diagrams for grouped data as appropriate, including cumulative frequency graphs and histograms with equal or unequal class intervals.	S3, S4

12.03 Analysing data

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12.03a	Summary statistics	Calculate the mean, mode, median, and range for ungrouped data. Find the modal class, and calculate estimates of the range, mean, and median for grouped data, understanding why these are estimates. Describe a population using statistics. Make simple comparisons. Compare data sets using like-for-like summary values. Understand the advantages and disadvantages of summary values.		Calculate estimates of mean, median, mode, range, quartiles, and interquartile range from graphical representations of grouped data. Draw and interpret box plots. Use the median and interquartile range to compare distributions.	S4, S5
12.03b	Misrepresenting data	Recognise graphical misrepresentation through incorrect scales, labels, and similar issues.			S4
12.03c	Bivariate data	Plot and interpret scatter diagrams for bivariate data. Recognise correlation.	Interpret correlation in context, and understand the distinction between correlation and causation. Draw a line of best fit by eye and use it to make predictions. Interpolate and extrapolate from data, noting the limitations of these techniques.		S6
12.03d	Outliers	Identify an outlier in simple cases.	Appreciate that data may contain errors from values that do not fit. Recognise outliers on a scatter graph.		S4