OCR GCSE MATHS

The topics listed below are for OCR GCSE Maths, with exam codes: – Mathematics (Foundation tier): J560F – Mathematics (Higher tier): J560H The list provides everything you need for your OCR GCSE Maths exam, with topics broken in to the headings given by the exam board. More information is available here

[https://www.ocr.org.uk/qualifications/gcse/mathematics-j560-from-2015/specification-at-a-glance/] For samples questions and papers, please click this link:

[https://www.ocr.org.uk/qualifications/gcse/mathematics-j560-from-2015/assessment/]

Everything you need to know about your GCSE (9-1) Maths specifications can be found here.

Number Operations and Integers

Calculations with integers

Subject Content

Four rules

Initial learning for this qualification will enable learners to:

Use non-calculator methods to calculate the sum, difference, product and quotient of positive and negative whole numbers.

Whole number theory

Subject Content

Definitions and terms

Prime numbers

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Initial learning for this qualification will enable learners to:

Understand and use the terms odd, even, prime, factor (divisor), multiple, common factor (divisor), common multiple, square, cube, root. Understand and use place value.

Identify prime numbers less than 20. Express a whole number as a product of its prime factors.
e.g. 24 = 2x2x2x3
Understand that each number can be expressed as a product of prime factors in only one way.

Find the HCF and LCM of two whole numbers by listing.

Foundation tier learners should also be able to:

Identify prime numbers. Use power notation in expressing a whole number as a product of its prime factors.
e.g. 600 = 2³ x 3 x 5²
Find the HCF and LCM of two whole numbers from their prime factorisations.

Combining arithmetic operations

Subject Content

Priority of operations

Initial learning for this qualification will enable learners to:

Know the conventional order for performing calculations involving brackets, four rules and powers, roots and reciprocals.

Inverse operations

Subject Content

Inverse operations

Initial learning for this qualification will enable learners to:

Know that addition and subtraction, multiplication and division, and powers and roots, are inverse operations and use this to simplify and check calculations, for example, in reversing arithmetic in “I’m thinking of a number” or “missing digit” problems.

e.g. 223 – 98 = 223 + 2 – 100 = 125 25 × 12 = 50 × 6 = 100 × 3 = 300

[see also Calculation and estimation of powers and roots, 3.01b]

Fractions, Decimals and Percentages

Fractions

Subject Content

Equivalent fractions
Calculations with fractions
Fractions of a quantity

Initial learning for this qualification will enable learners to:

– Recognise and use equivalence between simple fractions and mixed numbers.

e.g. 6/2 = 1/3

2 1/2 = 5/2

– Add, subtract, multiply and divide simple fractions (proper and improper), including mixed numbers and negative fractions. e.g.

1 1/2 + 3/4

5/6 x 3/10

-3 x 4/5

– Calculate a fraction of a quantity. e.g. 5/2 of £3.50 Express one quantity as a fraction of another. [see also Ratios and fractions, 5.01c]

Foundation tier learners should also be able to:

– Carry out more complex calculations, including the use of improper fractions.
e.g.

2/5 + 5/6
2/3 +1/2 x 3/5

– Calculate with fractions greater than 1.

Decimal fractions

Subject Content

Decimals and fractions

Addition, subtraction and multiplication of decimals

Division of decimals

Initial learning for this qualification will enable learners to:

Express a simple fraction as a terminating decimal or vice versa, without a calculator.

e.g. 0.4 = 2/5

Understand and use place value in decimals.

Add, subtract and multiply decimals including negative decimals, without a calculator.

Divide a decimal by a whole number, including negative decimals, without a calculator.

e.g. 0.24 ÷ 6

Foundation tier learners should also be able to:

Use division to convert a simple fraction to a decimal. e.g. 1/6 = 0.16666…

Without a calculator, divide a decimal by a decimal.
e.g. 0.3 ÷ 0.6

Higher tier learners should additionally be able to:

Convert a recurring decimal to an exact fraction or vice versa.
e.g. 0.4 = 41/99

Percentages

Subject Content

Percentage conversions

Percentage calculations

Percentage change

Initial learning for this qualification will enable learners to:

Convert between fractions, decimals and percentages. e.g.

1/4 = 0.25 = 25%

1 1/2 = 150%

Understand percentage is ‘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.

Increase or decrease a quantity by a simple percentage, including simple decimal or fractional multipliers. Apply this to simple original value problems and simple interest. e.g. Add 10% to £2.50 by either finding 10% and adding, or by multiplying by 1.1 or Calculate original price of an item costing £10 after a 50% discount.

Foundation tier learners should also be able to:

Express percentage change as a decimal or fractional multiplier. Apply this to percentage change problems (including original value problems).

Ordering fractions, decimals and percentages

Subject Content

Ordinality

Symbols

Initial learning for this qualification will enable learners to:

Order integers, fractions, decimals and percentages.

e.g. 4/5 , 3/4 , 0.72, –0.9

Use <, >, ≤, ≥, =, ≠

Indices and Surds

Powers and roots

Subject Content

Index notation

Calculation and estimation of powers and roots

Laws of indices

Initial learning for this qualification will enable learners to:

Use positive integer indices to write, for example,
2x2x2x 2 = 2⁴

Calculate positive integer powers and exact roots.

e.g.

2⁴= 16

√9 =3

³√8 = 2

Recognise simple powers of 2, 3, 4 and 5. e.g. 27 = 3³ [see also Inverse operations, 1.04a]

Foundation tier learners should also be able to:

Use negative integer indices to represent reciprocals.

Calculate with integer powers.

e.g. 2¯³ = 1/8

Calculate with roots.

Know and apply:

a^mx aⁿ= a^(m+n)

a^m ÷ aⁿ= a^(m-n)

(a^m)ⁿ= a^(mn)

Higher tier learners should additionally be able to:

Use fractional indices to represent roots and combinations of powers and roots.

Calculate fractional powers.

e.g.

6^-3/4= 1/∜(16)³ =1/8

Estimate powers and roots. e.g. √51 to the nearest whole number

Standard form

Subject Content

Standard form

Calculations with numbers in standard form

Initial learning for this qualification will enable learners to:

Interpret and order numbers expressed in standard form. Convert numbers to and from standard form.

e.g. 1320 = 1.32 x 10³

0.00943 = 9.43 x 10¯³

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]

Exact calculations

Subject Content

Exact calculations

Manipulating surds

Initial learning for this qualification will enable learners to:

Use fractions in exact calculations without a calculator.

Foundation tier learners should also be able to:

Use multiples of π in exact calculations without a calculator.

Higher tier learners should additionally be able to:

Use surds in exact calculations without a calculator.

Simplify expressions with surds, including rationalising denominators.

e-g √12 = 2√3

1/ √3 = √3 / 3

1/ √3 +1 = (√3 -1 ) / 2

Approximation and Estimation

Approximation and estimation

Subject Content

Rounding

Estimation

Upper and lower bounds

Initial learning for this qualification will enable learners to:

Round numbers to the nearest whole number, ten, hundred, etc. or to a given number of significant figures (sf) or decimal places (dp).

Estimate or check, without a calculator, the result of a calculation by using suitable approximations. e.g. Estimate, to one significant figure, the cost of 2.8 kg of potatoes at 68p per kg.

Use inequality notation to write down an error interval for a number or measurement rounded or truncated to a given degree of accuracy. e.g. If x = 2.1 rounded to 1 dp, then 2.05 ≤ x < 2.15 If x = 2.1 truncated to 1 dp, then 2.1 ≤ x < 2.2 . Apply and interpret limits of accuracy.

Foundation tier learners should also be able to:

Round answers to an appropriate level of accuracy.

Estimate or check, without a calculator, the result of more complex calculations including roots.
Use the symbol ≈ appropriately.

e-g

√(2.9/(0.051×0.62) ≈ 10

Calculate the upper and lower bounds of a calculation using numbers rounded to a known degree of accuracy. e.g. Calculate the area of a rectangle with length and width given to 2 sf. Understand the difference between bounds of discrete and continuous quantities. e.g. If you have 200 cars to the nearest hundred then the number of cars n satisfies:

150 ≤ n < 250 and

150 ≤ n < 249

Ratio, Proportion and Rates Of Change

Calculations with ratio

Subject Content

Equivalent ratios

Division in a given ratio

Ratios and fractions

Solve ratio and proportion problems

Initial learning for this qualification will enable learners to:

Split a quantity into two parts given the ratio of the parts. e.g. £2.50 in the ratio 2 : 3 Express the division of a quantity into two parts as a ratio. Calculate one quantity from another, given the ratio of the two quantities.

Interpret a ratio of two parts as a fraction of a whole. e.g. £9 split in the ratio 2 : 1 gives parts 2/3 x £ 9 and 1/3 x £ 9. [see also Fractions of a quantity, 2.01c]

Solve simple ratio and proportion problems. e.g. Adapt a recipe for 6 for 4 people. Understand the relationship between ratio and linear functions.

Foundation tier learners should also be able to:

Split a quantity into three or more parts given the ratio of the parts.

Direct and inverse proportion

Subject Content

Direct proportion

Inverse proportion

Initial learning for this qualification will enable learners to:

Solve simple problems involving quantities in direct proportion including algebraic proportions. e.g. Using equality of ratios, if y ∝ x , then y1/y2 = x1/x2

or y1/x1 = y2/x2 . Currency conversion problems.

Solve simple word problems involving quantities in inverse proportion or simple algebraic proportions.
e.g. speed–time contexts (if speed is doubled, time is halved).

Foundation tier learners should also be able to:

Solve more formal problems involving quantities in direct proportion (i.e. where y ∝ x ).
Recognise that if y = kx, where k is a constant, then y is proportional to x.

Solve more formal problems involving quantities in inverse proportion (i.e. where x ∝ 1/y ). Recognise that if y xk = , where k is a constant, then y is inversely proportional to x.

Higher tier learners should additionally be able to:

Formulate equations and solve problems involving a quantity in direct proportion to a power or root of another quantity.

Formulate equations and solve problems involving a quantity in inverse proportion to a power or root of another quantity.

Discrete growth and decay

Subject Content

Growth and decay

Initial learning for this qualification will enable learners to:

Calculate simple interest including in financial contexts.

Foundation tier learners should also be able to:

Solve problems step-by step involving multipliers over a given interval, for example, compound interest, depreciation, etc.
e.g. A car worth £15 000 new depreciating by 30%, 20% and 15% respectively in three years.

Higher tier learners should additionally be able to:

Express exponential growth or decay as a formula.
e.g. Amount £A subject to compound interest of 10% p.a. on £100 as
100 = 1.1 ⁿ
Solve and interpret answers in growth and decay problems.
[see also Exponential functions, , Formulate algebraic expressions ]

Algebra

Algebraic expressions

Subject Content

Algebraic terminology and proofs

Collecting like terms in sums and differences of terms

Simplifying products and quotients

Multiplying out brackets

Factorising

Completing the square

Algebraic fractions

Initial learning for this qualification will enable learners to:

Understand and use the concepts and vocabulary of expressions, equations, formulae, inequalities, terms and factors.

Simplify algebraic expressions by collecting like terms.
e.g. 2a + 3a = 5a

Simplify algebraic products and quotients.

e.g. a x a x a = a³

2a x 3b = 6ab

a² x a³ = a⁵

3a³ ÷ a = 3a²

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

Take out common factors.

3a – 9b = 3(a-3b)

2x + 3x² = x(2 + 3x)

Foundation tier learners should also be able to:

Expand products of two binomials.

(x-1)(x-2) = x² – 3x + 2

(a+2b) (a-b) = a² + ab – 2b²

Factorise quadratic expressions of the form

e.g. x² -x -6 = (x-3) (x+2)

x² – 16 = (x-4) (x+4)

x² – 3 = (x – √3)(x +√3)

Higher tier learners should additionally be able to:

Simplify algebraic products and quotients using the laws of indices.

e.g. a½ x 2a¯³ = 2a^-5/2

2a² b³ ÷ 4a¯³ b = 1/2 a⁵ b²

Expand products of more than two binomials.

e.g. (x+1)(x-1)(2x+1) = 2x³ + x² – 2x -1

Factorise quadratic expressions of the form ax² + bx + c (where a ≠ 0 or 1)

e.g.
2x² + 3x -2 = (2x -1 ) (x+2)

Complete the square on a
quadratic expression.

x² + 4x – 6 = (x+2)² – 10

2x² +5x+1 = 2 (x + 5/2)² – 17/8

Simplify and manipulate algebraic fractions.

e.g. Write (1/n-1 ) + (n/n+1) as a single fraction.

Simplify (n² + 2n) / (n² + n -2)

Algebraic formulae

Subject Content

Formulate algebraic expressions

Substitute numerical values into formulae and expressions

Change the subject of a formula

Recall and use standard formulae

Use kinematics formulae

Initial learning for this qualification will enable learners to:

Rearrange formulae to change the subject, where the subject appears once only. e.g. Make d the subject of the formula c = πd . Make x the subject of the formula y = 3x – 2.

Recall and use: Circumference of a circle

2πr = πd

Area of a circle πr²

Use:
v = u + at

s = ut +1/2 at ²

v ² = u ² +2as

where a is constant
acceleration, u is initial
velocity, v is final velocity,
s is displacement from position
when t = 0 and t is time taken.

Foundation tier learners should also be able to:

Formulate simple formulae and expressions from realworld 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.

Substitute positive or negative numbers into more complex formulae, including powers, roots and algebraic fractions.
e.g. v = √(u² + 2as) with

u = 2.1,s = 0.18,a = -9.8

Rearrange formulae to change the subject, including cases where the subject appears twice, or where a power or reciprocal of the subject appears.
e.g. Make t the subject of the formulae
(i) s = 1/2 at²
(ii) v = x/t
(iii) 2ty = t+1

Recall and use:
Pythagoras’ theorem
a² + b² = c²
Trigonometry formulae
sinΘ = o/h, cos Θ = a/h, tanΘ = o/a

Higher tier learners should additionally be able to:

[Examples may include manipulation of algebraic fractions, 6.01g]

Recall and use: The quadratic formula

x = -b ± √(b² – 4ac)

Sine rule = a/SinA = b/SinB = c/Sinc

Cosine rule

a² = b² + c² – 2bcCosA

Area of a triangle

1/2ab SinC

Algebraic Equations

Subject Content

Linear equations in one unknown

Quadratic equations

Simultaneous equations

Approximate solutions using a graph

Approximate solutions by iteration

Initial learning for this qualification will enable learners to:

Solve linear equations in one unknown algebraically.
e.g. Solve 3x – 1 = 5

Use a graph to find the approximate solution of a linear equation.

Foundation tier learners should also be able to:

Set up and solve linear equations in mathematical and non-mathematical contexts, including those with the unknown on both sides of the equation.
e.g.

Solve 5 (x – 1 ) = 4 – x
Interpret solutions in context.

Solve quadratic equations with coefficient of x equal to 1 by factorising. e.g. Solve

x² – 5x +6 = 0

Find x for an x cm by (x + 3) cm rectangle of area 40cm² .

Set up and solve two linear simultaneous equations in two variables algebraically.
e.g. Solve simultaneously
2x + 3y = 18 and
y = 3x – 5

Use graphs to find approximate roots of quadratic equations and the approximate solution of two linear simultaneous equations.

Higher tier learners should additionally be able to

[Examples may include manipulation of algebraic fractions, 6.01g]

Know the quadratic formula. Rearrange and solve quadratic equations by factorising, completing the square or using the quadratic formula.

e.g. 2x² = 3x + 5

2/x – 2/(x+1) = 1

Set up and solve two simultaneous equations in two variables algebraically (including where one is a quadratic or that result in a quadratic). e.g. Solve simultaneously

x² + y² = 50 and

2y = x+5

Know that the coordinates of the points of intersection of a curve and a straight line are the solutions to the simultaneous equations for the line and curve.

Find approximate solutions to equations using systematic sign-change methods (for example, decimal search or interval bisection) when there is no simple analytical method of solving them. Specific methods will not be requested in the assessment.

Algebraic Inequalities

Subject Content

Inequalities in one variable

Inequalities in two variables

Initial learning for this qualification will enable learners to:

Understand and use the symbols <, >, ≤ and ≥

Foundation tier learners should also be able to:

Solve linear inequalities in one variable, expressing solutions on a number line using the conventional notation.

e.g 2x + 1 ≥ 7

1 < 3x – 5 ≤ 10

Higher tier learners should additionally be able to:

Solve quadratic inequalities in one variable.

e.g x² – 2x < 3

Express solutions in set
notation.

e.g { x : x ≥ 3}

{ x : 2 < x ≤ 5}

[see also Polynomial and reciprocal functions, 7.01c]

Solve (several) linear inequalities in two variables, representing the solution set on a graph.
[see also Straight line graphs, 7.02a]

Language of Functions

Subject Content

Functions

Initial learning for this qualification will enable learners to:

Interpret, where appropriate, simple expressions as functions with inputs and outputs. e.g. y = 2x + 3 as

x → x2 → +3 → y

Higher tier learners should additionally be able to:

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]

Sequences

Subject Content

Generate terms of a sequence

Special sequences

Initial learning for this qualification will enable learners to:

Generate a sequence by spotting a pattern or using a term-to-term rule given algebraically or in words. e.g. Continue the sequences 1, 4, 7, 10, …

1, 4, 9, 16, …

Find a position-to-term rule for simple arithmetic sequences, algebraically or in words.

e.g. 2, 4, 6, … 2n

3, 4, 5, … n + 2

Recognise sequences of triangular, square and cube numbers, and simple
arithmetic progressions.

Foundation tier learners should also be able to:

Generate a sequence from a formula for the nth term.
e.g. nth term = n² + 2n gives
3, 8, 15, …
Find a formula for the nth term of an arithmetic sequence.
e.g. 40, 37, 34, 31, … 43 – 3n

Recognise Fibonacci and quadratic sequences, and
simple geometric progressions (rⁿ where n is an integer and r is a rational number > 0).

Higher tier learners should additionally be able to:

Use subscript notation for position-to-term and term-to term rules.

e.g Xn = n +2

Xn+1 = 2Xn -3

Find a formula for the nth term of a quadratic sequence.
e.g. 0, 3, 10, 21, …
Un = 2 n² – 3n + 1

Generate and find nth terms of other sequences.

1, √2, 2, 2√2, ….

1/2, 2/3, 3/4, ….

Graphs of Equations and Functions

Graphs of equations and functions

Subject Content

x- and y-coordinates

Graphs of equations and functions

Polynomial and reciprocal functions

Exponential functions

Trigonometric functions

Equations of circles

Initial learning for this qualification will enable learners to:

Work with x- and y-coordinates in all four quadrants.

Use a table of values to plot graphs of linear and quadratic functions.

e.g y = 2x + 3

y = 2x² + 1

Recognise and sketch the graphs of simple linear and quadratic functions.

e.g. y = 2
x = 1
y = 2x

y = x²

Foundation tier learners should also be able to:

Use a table of values to plot other polynomial graphs and reciprocals.

e.g. y = x³ – 2x

y = x +1/x

2x + 3y = 6

Recognise and sketch graphs of:

y = x³, y = 1/x
Identify intercepts and, using symmetry, the turning point of graphs of quadratic functions.
Find the roots of a quadratic equation algebraically.

Higher tier learners should additionally be able to:

Use a table of values to plot exponential graphs.

e.g. y = 3 x 1.1^x

Sketch graphs of quadratic functions, identifying the turning point by completing the square.

Recognise and sketch graphs of exponential functions in the form y = kx for positive k.

Recognise and sketch the graphs of

y = sinx , y = cosx and y = tanx

Recognise and use the equation of a circle with centre at the origin.

Straight line graphs

Subject Content

Straight line graphs

Parallel and perpendicular lines

Initial learning for this qualification will enable learners to:

Find and interpret the gradient and intercept of straight lines, graphically and using
y = mx + c.

Foundation tier learners should also be able to:

Use the form 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 and find equations of parallel lines.

Higher tier learners should additionally be able to:

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]

Transformations of curves and their equations

Subject Content

Translations and reflections

Higher tier learners should additionally be able to:

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 the graph of

y = sinx + 2

y = (x + 2)² – 1

y = -x²

Interpreting graphs

Subject Content

Graphs of real-world contexts

Gradients

Areas

Initial learning for this qualification will enable learners to:

Construct and interpret graphs in real-world contexts.
e.g. distance-time money conversion temperature conversion [see also Direct proportion, 5.02a, Inverse proportion, 5.02b]

Foundation tier learners should also be able to:

Understand the relationship between gradient and ratio.

Recognise and interpret graphs that illustrate direct and inverse proportion.

Interpret straight line gradients as rates of change.
e.g. Gradient of a distancetime graph as a velocity.

Higher tier learners should additionally be able to:

Calculate or estimate gradients of graphs, and interpret in contexts such as distance-time graphs, velocity-time graphs and financial graphs.
Apply the concepts of average and instantaneous rate of change (gradients of chords or tangents) in numerical, algebraic and graphical contexts

Calculate or estimate areas under graphs, and interpret in contexts such as distance-time graphs, velocity-time graphs and financial graphs.

Basic Geometry

Conventions, notation and terms

Subject Content

2D and 3D shapes

Angles

Polyhedra and other solids

Diagrams

Geometrical instruments

x- and y-coordinates

Initial learning for this qualification

Foundation tier learners

Higher tier learners

Use the terms points, lines, line segments, vertices, edges, planes, parallel lines, perpendicular lines.

Know the terms acute, obtuse, right and reflex angles. Use the standard conventions for labelling and referring to the sides and angles of triangles. e.g. AB, ∠ABC, angle ABC, a is the side opposite angle A

Know the terms:

• regular polygon

• scalene, isosceles and equilateral triangle

• quadrilateral, square, rectangle, kite, rhombus, parallelogram, trapezium

• pentagon, hexagon, octagon.

Recognise the terms face, surface, edge, and vertex, cube, cuboid, prism, cylinder, pyramid, cone and sphere

Draw diagrams from written descriptions as required by questions.

Use a ruler to construct and measure straight lines.
Use a protractor to construct and measure angles.
Use compasses to construct circles.

Use x- and y-coordinates in plane geometry problems, including transformations of simple shapes.

Ruler and compass constructions

Subject Content

Perpendicular bisector

Angle bisector

Perpendicular from a point to a line

Loci

Foundation tier learners should also be able to:

Construct the perpendicular bisector and midpoint of a line segment.

Construct the bisector of an angle formed from two lines.

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.

Apply ruler and compass constructions to construct figures and identify the loci of points, to include real-world problems. Understand the term ‘equidistant’.

Angles

Subject Content

Angles at a point

Angles on a line

Angles between intersecting and parallel lines

Angles in polygon

Initial learning for this qualification will enable learners to:

Know and use the sum of the angles at a point is 360º.

Know that the sum of the angles at a point on a line is 180º.

Know and use: vertically opposite angles are equal alternate angles on parallel lines are equal corresponding angles on parallel lines are equal.

Derive and use the sum of the interior angles of a triangle is 180º.
Derive and use the sum of the exterior angles of a polygon is 360º.
Find the sum of the interior angles of a polygon. Find the interior angle of a regular polygon.

Foundation tier learners should also be able to:

Apply these angle facts to find angles in rectilinear figures, and to justify results in simple proofs.
e.g. The sum of the interior angles of a triangle is 180º

Higher tier learners should additionally be able to:

Apply these angle properties in more formal proofs of geometrical results.

Properties of polygons

Subject Content

Properties of a triangle

Properties of quadrilaterals

Symmetry

Initial learning for this qualification will enable learners to:

Know the basic properties of isosceles, equilateral and rightangled triangles. Give geometrical reasons to justify these properties.

Know the basic properties of the square, rectangle, parallelogram, trapezium, kite and rhombus.
Give geometrical reasons to justify these properties.

Identify reflection and rotation symmetries of triangles, quadrilaterals and other polygons

Foundation tier learners should also be able to:

Use these facts to find lengths and angles in rectilinear figures and in simple proofs..

Higher tier learners should additionally be able to:

Use these facts in more formal proofs of geometrical results, for example circle theorems.

Circles

Subject Content

Circle nomenclature

Angles subtended at centre and circumference

Angle in a semicircle

Angles in the same segment

Angle between radius and chord

Angle between radius and tangent

The alternate segment theorem

Cyclic quadrilaterals

Initial learning for this qualification will enable learners to:

Understand and use the terms centre, radius, chord, diameter and circumference.

Foundation tier learners should also be able to:

Understand and use the terms tangent, arc, sector and segment.

Higher tier learners should additionally be able to:

Apply and prove:
the angle subtended by an arc at the centre is twice the angle at the circumference

Apply and prove:
the angle on the circumference subtended by a diameter is a right angle.

Apply and prove:

two angles in the same segment are equal.

Apply and prove:
a radius or diameter bisects a chord if and only if it is perpendicular to the chord.

Apply and prove:
for a point P on the circumference, the radius or diameter through P is perpendicular to the tangent at P.

Apply and prove:
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.

Apply and prove:
the opposite angles of a cyclic quadrilateral are supplementary.

3-dimensional solids

Subject Content

3-dimensional solids

Plans and elevations

Initial learning for this qualification will enable learners to:

Recognise and know the properties of the cube, cuboid, prism, cylinder, pyramid, cone and sphere

Interpret plans and elevations of simple 3D solids.

Foundation tier learners should also be able to:

Construct plans and elevations of simple 3D solids, and representations (e.g. using isometric paper) of solids from plans and elevations.

Congruence and Similarity

Plane isometric transformations

Subject Content

Reflection

Rotation

Translation

Combinations of transformations

Initial learning for this qualification will enable learners to:

Rotate a simple shape clockwise or anti-clockwise through a multiple of 90º about a given centre of rotation.

Use a column vector to describe a translation of a simple shape, and perform a specified translation.

Foundation tier learners should also be able to:

Identify the centre, angle and sense of a rotation from a simple shape and its image under rotation.

Higher tier learners should additionally be able to:

Perform a sequence of isometric transformations (reflections, rotations or translations), on a simple shape. Describe the resulting transformation and the changes and invariance achieved.

Congruence

Subject Content

Congruent triangles

Applying congruent triangles

Initial learning for this qualification will enable learners to:

Identify congruent triangles.

Foundation tier learners should also be able to:

Prove that two triangles are congruent using the cases:
3 sides (SSS)
2 angles, 1 side (ASA)
2 sides, included angle (SAS)
Right angle, hypotenuse, side (RHS).

Apply congruent triangles in calculations and simple proofs. e.g. The base angles of an isosceles triangle are equal.

Plane vector geometry

Subject Content

Vector arithmetic

Column vectors

Foundation tier learners should also be able to:

Understand addition, subtraction and scalar multiplication of vectors.

Represent a 2-dimensional vector as a column vector, and draw column vectors on a square or coordinate grid.

Higher tier learners should additionally be able to:

Use vectors in geometric arguments and proofs.

Similarity

Subject Content

Similar triangles

Enlargement

Similar shapes

Initial learning for this qualification will enable learners to:

Identify similar triangles.

Enlarge a simple shape from a given centre using a whole number scale factor, and identify the scale factor of an enlargement.

Compare lengths, areas and volumes using ratio notation and scale factors.

Foundation tier learners should also be able to:

Prove that two triangles are similar.

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.

Apply similarity to calculate unknown lengths in similar figures. [see also Direct proportion, 5.02a]

Higher tier learners should additionally be able to:

Perform and recognise enlargements with negative scale factors.

Understand the relationship between lengths, areas and volumes of similar shapes. [see also Direct proportion, 5.02a]

Mensuration

Units and measurement

Subject Content

Units of measurement

Compound units

Maps and scale drawings

Initial learning for this qualification will enable learners to:

Use and convert standard units of measurement for length, area, volume/capacity, mass, time and money.

Use and convert simple compound units (e.g. for speed, rates of pay, unit pricing). Know and apply in simple cases: speed = distance ÷ time

Use the scale of a map, and work with bearings. Construct and interpret scale drawings.

Foundation tier learners should also be able to:

Use and convert standard units in algebraic contexts.

Use and convert other compound units (e.g. density, pressure).
Know and apply:
density = mass ÷ volume
Use and convert compound units in algebraic contexts.

Perimeter calculations

Subject Content

Perimeter of rectilinear shapes

Circumference of a circle

Perimeter of composite shapes

Initial learning for this qualification will enable learners to:

Calculate the perimeter of rectilinear shapes.

Know and apply the formula circumference = 2πr² = πd to calculate the circumference of a circle.

Apply perimeter formulae in calculations involving the perimeter of composite 2D shapes.

Foundation tier learners should also be able to:

Calculate the arc length of a sector of a circle given its angle and radius.

Area calculations

Subject Content

Area of a triangle

Area of a parallelogram

Area of a trapezium

Area of a circle

Area of composite shapes

Initial learning for this qualification will enable learners to:

Know and apply the formula: area = 1/2 x base × height.

Know and apply the formula: area = base × height. [Includes area of a rectangle]

Calculate the area of a trapezium.

Know and apply the formula area = πr² to calculate the area of a circle.

Apply area formulae in calculations involving the area of composite 2D shapes.

Foundation tier learners should also be able to:

Calculate the area of a sector of a circle given its angle and radius.

Higher tier learners should additionally be able to:

Know and apply the formula: area = 1/2 ab sin C .

Volume and surface area calculations

Subject Content

Polyhedra

Cones and spheres

Pyramids

Initial learning for this qualification will enable learners to:

Calculate the surface area and volume of cuboids and other right prisms (including cylinders).

Calculate the surface area and volume of spheres, cones and simple composite solids (formulae will be given).

Calculate the surface area and volume of a pyramid (the formula 1/3 area of base × height will be given).

Triangle mensuration

Subject Content

Pythagoras’ theorem

Trigonometry in right-angled triangles

Exact trigonometric ratios

Sine rule

Cosine rule

Foundation tier learners should also be able to:

Know, derive and apply Pythagoras’ theorem a² + b² = c² to find lengths in right-angled triangles in 2D figures.

Know and apply the trigonometric ratios, sin θ, cos θ and tan θ and apply them to find angles and lengths in right-angled triangles in 2D figures. [see also Similar shapes, 9.04c]

Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90° Know the exact value of tan θ for θ = 0°, 30°, 45° and 60°

Higher tier learners should additionally be able to:

Apply Pythagoras’ theorem in more complex figures, including 3D figures.

Apply the trigonometry of right-angled triangles in more complex figures, including 3D figures.

Know and apply the sine rule, a/sinA = b/sinB = c/sinC, to find lengths and angles.

Know and apply the cosine rule, a² = b² + c² – 2bccosA , to find lengths and angles.

Probability

Basic probability and experiments

Subject Content

The probability scale

Relative frequency

Relative frequency and probability

Equally likely outcomes and probability

Initial learning for this qualification will enable learners to:

Use the 0-1 probability scale as a measure of likelihood of random events, for example, ‘impossible’ with 0, ‘evens’ with 0.5, ‘certain’ with 1.

Record, describe and analyse the relative frequency of outcomes of repeated experiments using tables and frequency trees.

Use relative frequency as an estimate of probability.

Calculate probabilities, expressed as fractions or decimals, in simple experiments with equally likely outcomes, for example flipping coins, rolling dice, etc. Apply ideas of randomness and fairness in simple experiments. Calculate probabilities of simple combined events, for example rolling two dice and looking at the totals. Use probabilities to calculate the number of expected outcomes in repeated experiments.

Foundation tier learners should also be able to:

Understand that relative frequencies approach the theoretical probability as the number of trials increases.

Combined events and probability diagrams

Subject Content

Sample spaces

Enumeration

Venn diagrams and sets

Tree diagrams

The addition law of probability

The multiplication law of probability and conditional probability

Initial learning for this qualification will enable learners to:

Use tables and grids to list the outcomes of single events and simple combinations of events, and to calculate theoretical probabilities.
e.g. Flipping two coins.
Finding the number of orders in which the letters E, F and G can be written.

Use a two-circle Venn diagram to enumerate sets, and use this to 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}

Use the addition law for mutually exclusive events. Use p(A) + p(not A) = 1

Foundation tier learners should also be able to:

Construct a Venn diagram to classify outcomes and calculate probabilities.
Use set notation to describe a set of numbers or objects.
e.g.D = { x : 1 < x < 3}

E = { x : x is a factor of 280}
[Knowledge of intersection (∩), union ( ) and complement (‘) notation will not be required.]

Use tree diagrams to enumerate sets and to record the probabilities of successive events (tree frames may be given and in some cases will be partly completed).

Derive or informally understand and apply the formula p(A or B) = p(A) + p(B) – p(A and B).

Use tree diagrams and other representations to calculate the probability of independent and dependent combined events.

Higher tier learners should additionally be able to:

Recognise when a sample space is the most appropriate form to use when solving a complex probability problem. Use the most appropriate diagrams to solve unstructured questions where the route to the solution is less obvious.

Use the product rule for counting numbers of outcomes of combined events.

Construct tree diagrams, twoway tables or Venn diagrams to solve more complex probability problems (including conditional probabilities; structure for diagrams may not be given). [Knowledge of intersection (∩), union ( ) and complement (‘) notation will not be required.]

Understand the concept of 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 a red card is drawn. Derive or informally understand and apply the formula p(A and B) = p(A given B)p(B). Know that events A and B are independent if and only if p(A given B) = p(A).

Probability

Sampling

Subject Content

Populations and samples

Foundation tier learners should also be able to:

Define the population in a study, and understand the difference between population and sample. Infer properties of populations or distributions from a sample. Understand what is meant by simple random sampling, and bias in sampling.

Interpreting and representing data

Subject Content

Categorical and numerical data

Grouped data

Initial learning for this qualification will enable learners to:

Interpret and construct charts appropriate to the data type; including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data. Interpret multiple and composite bar charts.

Foundation tier learners should also be able to:

Design tables to classify data. Interpret and construct line graphs for time series data, and identify trends (e.g. seasonal variations).

Higher tier learners should additionally be able to:

Interpret and construct diagrams for grouped data as appropriate, i.e. cumulative frequency graphs and histograms (with either equal or unequal class intervals).

Analysing data

Subject Content

Summary statistics

Misrepresenting data

Bivariate data

Outliers

Initial learning for this qualification will enable learners to:

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, and understand why they 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.

Recognise graphical misrepresentation through incorrect scales, labels, etc.

Plot and interpret scatter diagrams for bivariate data. Recognise correlation.

Identify an outlier in simple cases.

Foundation tier learners should also be able to:

Interpret correlation within the context of the variables, and appreciate 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, and be aware of the limitations of these techniques.

Appreciate there may be errors in data from values (outliers) that do not ‘fit’. Recognise outliers on a scatter graph.

Higher tier learners should additionally be able to:

Calculate estimates of mean, median, mode, range, quartiles and interquartile range from graphical representation of grouped data. Draw and interpret box plots. Use the median and interquartile range to compare distributions.

OCR GCSE MATHS

Number Operations and Integers

Calculations with integers

Whole number theory

Combining arithmetic operations

Inverse operations

Fractions, Decimals and Percentages

Fractions

Decimal fractions

Percentages

Ordering fractions, decimals and percentages

Indices and Surds

Powers and roots

Standard form

Exact calculations

Approximation and Estimation

Approximation and estimation

Ratio, Proportion and Rates Of Change

Calculations with ratio

Direct and inverse proportion

Discrete growth and decay

Algebra

Algebraic expressions

Algebraic formulae

Algebraic Equations

Algebraic Inequalities

Language of Functions

Sequences

Graphs of Equations and Functions

Graphs of equations and functions

Straight line graphs

Transformations of curves and their equations

Interpreting graphs

Basic Geometry

Conventions, notation and terms

Ruler and compass constructions

Angles

Properties of polygons

Circles

3-dimensional solids

Congruence and Similarity

Plane isometric transformations

Congruence

Plane vector geometry

Similarity

Mensuration

Units and measurement

Perimeter calculations

Area calculations

Volume and surface area calculations

Triangle mensuration

Probability

Basic probability and experiments

Combined events and probability diagrams

Probability

Sampling

Interpreting and representing data

Analysing data

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