# 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