Mathematics: Key Stage 4

Consolidation 1 – Calculations with fractions

• Understand fraction notation, recognising when fractions are equivalent
• Find fractions of a quantity
• Convert between improper fractions and mixed numbers
• Apply the four rules of number to fraction

Y7 U13-15

Y8 U2

Consolidation 2 – Working with FDP

• Appreciate the equivalence of fractions, decimals and percentage and convert between them
• Increase and decrease by a given percentage
• Express one number as a percentage of another, including percentage changes

Y7 U13-15, 19 Y8 U7

Consolidation 3 – Indices and roots

• Recognise and calculate with square numbers and cube numbers, knowing square and cube roots as appropriate
• Understand and use index notation for powers greater than three
• Multiply and divide numbers by any power of 10
• Convert numbers to and from standard form

Y8 U1

Y9 U4

Consolidation 4 – Ratio and proportion 1

• Recognise and use ratio notation
• Simplify ratios
• Compare ratios to fractions, decimals and percentages
• Find missing values in the context of ratio problems

Y8 U8 Y9 U3

Consolidation 5 – Transformations

• Reflect a shape in a given line, including on a coordinate grid using e.g.
• Rotate a shape by a 90⁰, 180⁰ and 270⁰ about a given centre
• Translate a shape by a given vector
• Understand that these transformations do not affect the area of the shape
• and that the distance between pairs of points is the same in the image as in the original shape
• Describe a single transformation using correct mathematical language

Y9 U16

Consolidation 6 – Pythagoras’ Theorem

• Find missing sides in right-angled triangles given the other two sides
• Model practical situations with right-angled triangles and so find missing lengths
• Identify whether a triangle is right-angled by considering the lengths of its side

Y9 U15

Consolidation 7 – Fundamentals of algebra

• Understand and use algebraic notation
• Simplify expressions
• Substitute into formulae, including those from kinematics
• Form and solve linear equations in one variable
• Form and solve inequalities in one variable, including representing solutions on a number line
• Plot straight line graphs
• Understand the equation of a straight line, interpreting the gradient and intercept
• Rearrange formulae where the subject appears only once
• Follow simple algebraic arguments

Y7 U17

Y8 U4

Y9 U5-7, 12

Consolidation 8 – Angles and Shapes

• Know the names of different types of angle
• Know and use:
  • the sum of angles at a point and angles on a straight line
  • vertically opposite angles are equal
  • alternate angles are equal
  • corresponding angles are equal
• Derive and use the sum of the interior angles of a triangle
• Find the sum of the interior angles of a polygon
• Find the interior angle of a regular polygon
• Derive and use the sum of the exterior angles of a polygon
• Identify and know the properties of special quadrilaterals

Y7 U9-11

Y8 U5

Y9 U10-11

Consolidation 9 – Units and decimals

• Understand decimal place value
• Convert between metric units
• Use the four rules of number with decimals
• Round numbers to a given number of decimal places

Y7 U1, 4, 6

Y8 U6, 9

Consolidation 10 – Area and perimeter

• Calculate the area of:
  • Triangles
  • Parallelograms
  • Trapezia
  • Circles
  • Composite 2D shapes made of the above
• Find the surface area of cuboids
• Find the circumference of a circle
• Find the perimeter of composite 2D shapes

Y7 U5-6

Y8 U6, 10, 11

Consolidation 11 – Properties of shapes, constructions and congruence

• Use rulers, protractors and pairs of compasses accurately
• Use the standard ruler and compass constructions for:
  • perpendicular bisector of a line segment
  • constructing a perpendicular to a given line from/at a given point
  • bisecting a given angle
• Understand and use the perpendicular distance from a point to a line as the shortest distance to the line
• Construct triangles and quadrilaterals from given information
• Recognise congruent triangles
• Be aware of the conditions for congruency

Y7 U9-11

Y8 U5

Y9 U8-9

Consolidation 12 – Fundamentals of probability

• Use the language associated with probability
• Find the probability of a single event when there are equally likely items
• Convert between different forms of probability – fractions, decimals and percentages
• Know that P(Not A) = 1 – P(A)
• Solve equations from probability problems

Y9 U18

Consolidation 13 – Real-life graphs

• Construct and interpret graphs of real-life contexts such as:
  • Currency conversion
  • Temperature conversion
  • Distance-time graphs

Y8 U8

Y9 U3

Consolidation 14 – Ratio and proportion 2

• Share a quantity in a given ratio
• Solve simple ratio and proportion problems e.g. recipes

Y8 U8

Y9 U3

Consolidation 15 – Direct and inverse proportion

• Solve problems using unitary method
• Recognise the link between gradient and proportion
• Solve problems involving direct proportion in numerical and algebraic contexts
• Solve problems involving inverse proportion in contexts such as speed, distance and time

Y8 U8

Y9 U3

Consolidation 16 – Fundamentals of statistics

• Construct simple charts for ungrouped data
• Calculate the mean, median and mode and range of ungrouped data
• Find the modal class of grouped data
• Find estimates of the mean, median and range of grouped data
• Make comparisons between sets of data using summary statistics
• Discuss the advantages and disadvantages of different measures used in statistics
• Understand what is meant by an outlier

Y7 U20

Y8 U13

Y9 U19-20

Core 1 – Indices and standard form

• Understand the meaning of higher powers and know how to find these, and corresponding roots, using a calculator when necessary
• Understand the difference between decimal approx and exact values of roots
• Derive, understand and use the rules of indices with integer values
• Be able to perform calculations involving standard form, with and without a calculator as appropriate

K1, K3, K9

Core 2 – Geometric Change

• Solve problems involving simple and compound interest
• Solve other problems involving repeated change, such as depreciation
• Solve problems involving growth and decay

K1, K2

Core 3 - Sequences

• Recognise and describe arithmetic and geometric sequences
• Generate terms of a sequence given a rule
• Find a formula for the n^th term of a linear and geometric sequences
• Explain whether a number is a member of a given sequence

K7

Core 4 – Enlargement & similarity

• Enlarge shapes from a given centre (positive integer & fractional scale factors)
• Find the centre of enlargement given a shape and its image
• Understand the meaning of similarity
• Find missing sides in pairs of similar shapes, including similar triangles

K4, K5

Core 5 – Bearings and trigonometry

• Solve problems involving bearings
• Understand and use the trigonometric ratios sin, cos and tan
• Understand the link between similar triangles and trigonometry
• Derive and use the exact values of sin θ and cos θ for θ = 0⁰, 30⁰, 45⁰, 60⁰ and 90⁰; know the exact value of tan θ for θ = 0⁰, 30⁰, 45⁰, 60⁰

K4, K6, K7, K8

Core 6 – Algebraic reasoning

• Formulate simple formulae from real-world situations
• Simplify more complex algebraic expressions
• Use reasoning to show whether two expressions are equivalent
• Solve problems involving co-ordinates and mid-points of line segments
• Find the equation of a straight line given sufficient information 
• Prove whether two straight lines are parallel

K7, K14

Core 7 – Geometric reasoning

• Use angle facts to find missing angles in increasingly complex situations
• Use angle facts to justify results in simple proofs
• Use the known properties of triangles and quadrilaterals to follow and to derive simple proofs in rectilinear figures, including key angle and area facts

K8

Core 8 - Vectors

• Represent a two-dimensional vector as a column vector
• Add and subtract vectors
• Multiply a vector by a scalar

Core 9 – Working with 3D shapes

• Recognise the vocabulary associated with 3D solids
• Construct and interpret plans and elevations of 3D solids

Core 10 – Calculations with 3D shapes

• Calculate the surface area and volume of spheres, cones, pyramids
• Compare the surface area and volume of solid shapes
• Form and solve equations related to 3D shapes
• Give answers in terms of p if appropriate
• Understand the limits of accuracy, using inequality notation
• Convert between area and volume units

K9, K10

Core 11 – Geometric reasoning

• Apply ruler and compass constructions to construct figures
• Identify the loci of points and use these to solve real-world problems
• Prove pairs of triangles are congruent using SSS, ASA, AAS and RHS

K11

Core 12 – Sampling

• Know the difference between a sample and a population
• Understand different types of sampling
• Find the sizes of groups in a stratified sample
• Discuss the reliability of different types of sample

K1, K12

Core 13 – Theoretical and Experimental probability

• Understand what is meant by relative frequency
• Understand why relative frequency is sometimes used as an estimate for probability
• Compare theoretical probability with result obtained by experiments
• Use Venn diagrams and two-way tables to solve probability problems
• Construct Venn diagrams and two-way tables to solve probability problems
• Use the addition law for probability, understanding when events are mutually exclusive

K1, K12

Core 14 – Combined events

• Systematically list sample spaces
• Use the product rule for counting*
• Understand the multiplication rule for independent and dependent events
• Use tree diagrams to solve probability problems
• Construct tree diagrams to solve probability problems

K1, K12

Core 15 - Quadratics

• Expand products of two binomials
• Factorise quadratic expressions of the form
• Solve quadratic equations of the form  by factorising
• Develop proofs using quadratic expressions

K7

Core 16 – Non-linear graphs

• Draw and recognise quadratic graphs
• Use quadratic graphs to find the approximate solution to quadratic equations
• Identify intercepts and the turning points of graphs of quadratic functions
• Use tables of values to plot polynomial graphs
• Use tables of values to plot reciprocal graphs
• Recognise and sketch graphs of ,   and

K7, K13

Core 17 – Simultaneous equations

• Set up and solve (algebraically) two linear simultaneous equations in two variables, interpreting the solution in context
• Find the approximate solution to two simultaneous equations using a graph

K7

Core 18 - Circle

• Identify and use the terms centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
• Calculate the length of an arc of a circle
• Calculate the area of a sector of a circle

K10

Core 19 - Variation

• Solve formal problems involving direct proportion where
• Solve formal problems involving inverse proportion where

K4, K14, K15

Core 20 - Proof

• Follow and form logical arguments in both algebra and geometry

K7

Core 21 – Representing data

• Choose the most appropriate pictorial representation for a particular set of data
• Recognise when graphs and charts can be misleading
• Choose the most appropriate measures for representation of a set of data
• Interpret and construct line graphs for time series data
• Identify outliers

K16

Core 22 – Correlation and best fit

• Plot and use scatter graphs to identify correlation
• Understand that correlation does not imply causality
• Draw (by eye) lines of best fit
• Interpolate and extrapolate to make estimates, knowing the limitations of this
• Spot outliers on a scatter graph

K16

Core 23 – Compound and other measures

• Use and apply compound units such as density and pressure
• Know and apply Density = Mass ÷ Volume
• Use and apply compound units in algebraic contexts

Higher 1 – Further Number

• Estimate powers and roots of any given positive number
• Calculate with fractional indices
• Understand and use surds
• Rationalise denominators
• Change recurring decimals into their corresponding fractions and vice versa

C1

Higher 2 – Recurrence relations

• Understand the meaning of iteration and use iterative processes
• Use recurrence formulae

C3

Higher 3 – Further sequences

• Find the formula for the n^th term of a quadratic sequence
• Find missing terms in, and find the formula for the n^th term of, geometric sequences with ratios that are surds

C3

Higher 4 – Further transformations

• Enlarge shapes from a given centre, inc negative integer & fractional scale factors
• Consider the effects of combining reflections, rotations and translation

C4

Higher 5 – Triangles from 3D

• Apply Pythagoras’ theorem to problems in three dimensions, including repeated use of the theorem e.g. in finding the length of the diagonal of a cuboid
• Identify right-angled triangles in three-dimensional shapes and use trigonometry to find missing sides and angles

C5

Higher 6 – Trigonometric Graphs

• Evaluate the sine, cosine and tangent of angles greater than 90⁰
• Recognise and sketch the graphs of and 
• Solve simple trig equations using graphs

C5, C6

Higher 7 - Proof

• Use algebra to construct proofs of arguments
• Verify whether two straight lines are perpendicular
• Use vectors to prove geometric arguments

C6

Higher 8 – Further inequalities

• Express solutions to inequalities using set notation
• Solve several inequalities in two variables, representing  the solution on a graph

C6, C8

Higher 9 – Upper and lower bounds

• Find the upper and lower bounds of a calculation using numbers that have been rounded to a given degree of accuracy
• Understand the difference between the bounds of discrete and continuous quantities

C10

Higher 10 –Similar areas and volume

• Understand and use the relationship between lengths, areas and volumes of similar shapes

C4, C10

Higher 11 – Trigonometry in all triangles

• Use the sine rule to find missing sides and angles in any triangle
• Use the cosine rule to find missing sides and angles in any triangle
• Find the area of a triangle using the formula Area

C5, C11

Higher 12 – Conditional probability

• Understand what is meant by conditional probability
• Calculate conditional probabilities
• Extend their work on Venn diagrams to include those with three regions
• Solve more complex problems involving tree diagrams

C13, C14

Higher 13 – Further quadratics

• Factorise quadratic expressions of the form
• Complete the square of a quadratic expression
• Rearrange and solve quadratic equations by factorisation, completing the square or the use of the quadratic formula
• Set up and solve two simultaneous equations where one is linear and one is quadratic
• Solve quadratic inequalities
• Sketch graphs of quadratic functions, finding the turning point by completing the square

C15, C16, C17

Higher 14 – Exponential graphs

• Plot graphs of the form  from a table of values.
• Recognise and sketch the graphs of exponential functions

C2, C16

Higher 15 – Algebraic fractions

• Simplify algebraic fractions
• Manipulate algebraic fractions

C15

Higher 16 – Circle theorems

• Prove and use:
  • The angled subtended by an arc at the centre of a circle is twice that at the circumference
  • The angle in a semi-circle is a right angle
  • Angles in the same segment are equal
  • Opposite angles of a cyclic quadrilateral sum to 180⁰
  • A radius bisects a chord if and only if it is perpendicular to the chord
  • The angle between a tangent and a radius is a right angle
  • The alternate segment theorem

C18, C20

Higher 17 – Circle geometry

• Recognise and use the equation of a circle, centre the origin
• Find the equation of a tangent to a circle at a given point

C17, C18

Higher 18 – Variation with Powers

• Solve problems involving a quantity directly or inversely proportional to a power or a root of another quantity

C19

Higher 19 – Further statistics

• Construct and interpret histograms with equal and unequal class intervals
• Plot and interpret cumulative frequency diagrams
• Calculate estimates of statistical measures from graphical representations of grouped data
• Draw and interpret box plots
• Use the median and interquartile range to compare distributions

C21

Higher 20 – Functions

• Understand and use function notation
• Find the inverse of a function
• Interpret the succession of two functions as a composite function
• Identify and sketch the graphs of translations and reflections of a given graph

C16

Higher 21 – Further graphs

• Find approximate solutions to equations by using:
  • Trial and improvement/decimal search
  • Sign change methods
• Calculate estimates of gradients of graphs using gradients of chords
• Interpret gradients of real-world graphs
• Calculate estimates areas under graphs
• Interpret areas under real-world graph