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Mathematics: Key Stage 4

Consolidation

Autumn 1: Numbers

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

Autumn 2: Geometry

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

 

Spring 1: 2D Geometry

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

 

Spring 2 : Geometry & Number

 

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

Summer 1: Probability

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

Summer 2:  Algebra

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

Autumn 1: Algebra

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

Autumn 2: Statistics

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 and Higher

Autumn 1:  Numbers

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 nth term of a linear and geometric sequences
  • Explain whether a number is a member of a given sequence

K7

Autumn 2: Geometry

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

 

Spring 1: 2D Geometry

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

 

Spring 2:  Geometry & Numbers

 

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

 

Summer 1: Handling Data

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

Summer 2: Algebra

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

Autumn 1: Algebra & Geometry

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

Autumn 2: Number and Stats

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

 

Autumn 1: Numbers

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

C3

Autumn 2: Geometry

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

 

Spring 1: 2D Geometry

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

 

Spring 2: Geometry & Numbers

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

Summer 1: Handling Data

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

 

Summer 2:  Algebra

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

Autumn 1: Algebra and Geometry

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

Autumn 2: Number and Stats

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

Spring 1: Functions

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

Spring 2: Further graphs

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