Subjects

analysis	construction	estimation	misconceptions	proof
application	counter-examples	fluency	modelling	reasoning
calculation	deduction	generalisation	patterns	relationships
conjecturing	efficiency	interpretation	predicting	representation
connections	equivalence	limitation	problem-solving	strategy

 

Autumn term 1

Place Value.

This unit enables students to see how the general structure of the place-value system as based on powers of ten naturally extends to decimals and will progress beyond recalling place-value column headings when answering questions.

• Students will know that the general structure of the place-value system with column headings based on powers of 10 naturally extends to decimals.
• Students will know how to use place value to order decimals.

Properties of number: Factors, Multiples, Squares and Cubes.

Students will examine the structure of the numbers involved and explore ways of representing them, for example, by using factor trees and Venn diagrams. Work on exponents and roots provides the foundation for future learning.

• Students will know the definitions of multiple and factor, the square and cube numbers and roots, and correct notation for positive integer exponents, including identifying the correct keys on a calculator.
• Students will know how to write a positive integer uniquely as a product of its prime factors.

 

Autumn term 2

Arithmetic procedures with integers and decimals.

The focus is on developing a strong understanding of the mathematical structures that underpin these standard procedures, and appreciate that these structures are the same for integers and decimals. Developing fluency with a range of calculation approaches and techniques involving combinations of numbers and operations will also take place.

• Students will know the laws of arithmetic and the orders of operations.
• Students will know how to set out the written strategies for all four main operations including with decimals, and will be able to apply this in various situations.

Expressions and equations.

This will involve students using algebraic notation to examine and analyse number structure, and to deepen their understanding. They will be presented with situations where the structure of numbers can be generalised, and will be introduced to conventions concerning the writing of algebraic symbols and techniques for symbolic manipulation.

• Students will know the conventions of algebraic notation and be able to identify like terms in an expression, generalising an understanding of unitising.
• Students will know how to apply their understanding of the distributive law to a range of problem-solving situations and contexts (including collecting like terms, multiplying an expression by a single term and factorising).

 

Spring term 1

Plotting Coordinates.

A key focus will be thinking about x- and y- coordinates as the input and output respectively of a function or rule, and appreciating that the set of coordinates generated and the line joining them can be thought of as a graphical representation of that function.

• Students will know that coordinates can be described and plotted using coordinate notation, and that a graphical representation shows all of the points (within a range) that satisfy a relationship.
• Students will know how to solve a range of problems involving coordinates.

Perimeter and area.

Students will develop a secure and deep understanding of perimeter and area before extending these ideas. When calculating perimeters, students will use the properties of parallelograms, isosceles triangles and trapezia, as well as non-standard shapes, and reason mathematically to deduce missing information.

• Students will know the formulae for a range of shapes including a trapezium.
• Students will know how to derive these formulae, will understand that the areas of composite shapes can be found in different ways, and will use the properties of a range of polygons to deduce their perimeters.

 

Spring term 2

Arithmetic procedures including fractions.

This unit develops students’ understanding of the different ways that numbers can be expressed, develops their proficiency in converting from one form to another, and develops their awareness that different representations of the same number can be used to compare and order numbers.

• Students will know notation relating to fractions and inequalities.
• Students will know how to convert between improper fractions and mixed numbers, simplify fractions through dividing both numerator and denominator by common factors, compare and order fractions and negative numbers, and fluently use addition and subtraction strategies to calculate with fractions and mixed numbers

 

Summer term 1

Understanding multiplicative relationships: Fractions and ratio.

Percentages, fractions, proportionality and ratio, will be considered as contexts in which multiplicative relationships are used and explored. Students should recognise that it is possible to go from any number to any other number by multiplying.

• Students will know that any two numbers can be connected via a multiplicative relationship, expressed as both a ratio and as a fraction, and will know notation and vocabulary relating to fractions and ratio.
• Students will know how to use a ratio table to represent a multiplicative relationship, and solve problems involving fractions and ratios, including dividing quantities and finding an original amount.

 

Summer term 2

Transformation.

In all four transformations, students will recognise that every element of the object undergoes the same transformation. They will consider what is the same and what is different about an object and its image as they work on different transformations.

• Students will know the nature of rotations, enlargements, translations and reflections, and appreciate what changes and what is invariant
• Students will know how to rotate objects using information about centre, size and direction of rotation, how to reflect objects using a range of lines of reflection (including non-vertical and non-horizontal), how to enlarge objects using information about the centre of enlargement and scale factor, and how to translate objects from information given in a variety of forms.

Autumn term 1

Estimation.

It is essential that students are aware of the general structure of the place-value system as being based on powers of ten and begin to see how this naturally extends to decimals. This learning will support students’ work on significant figures and standard form, as students who can express numbers in these different ways are more likely to have a feel for the size of such numbers and where they fit in the number system. It is also important to emphasise the use of measures in real-life contexts.

• Students will know that the place value of digits in integers and decimal numbers can be used to round to a given degree of accuracy, including up to 3 decimal places and significant figures.
• Students will know how to use a suitable rounding method to estimate and check calculations.

Sequences.

This work extends students’ knowledge of sequences through the exploration of the mathematical structure. Algebraic notation is used to express the structure. It is important that students have time to develop a full understanding of the connection between the notation and the sequence and come to see the nth term as a way of expressing the structure of every term in the sequence.

• Students will know that linear sequences are fundamentally “shifted” times tables.
• Students will know how to identify the common difference in a linear sequence and relate this to its algebraic representation.

Autumn term 2

Graphical Representations of Linear Relationships.

This theme will provide students with opportunities to explore linear relationships and their representation as straight line graphs. Students will appreciate that all linear relationships have certain key characteristics, i.e., the gradient, the y-intercept and the rate of change.

• Students will know that linear relationships can be represented via straight line graphs.
• Students will know how to plot and identify linear graphs. Students will also know how to calculate the y-intercept and gradient of a linear graph and use this information to determine the equation of the straight line.

Solving Linear Equations.

Students will explore how linear equations are effectively the formulation of a series of operations on unknown numbers, and how the solving of such equations is concerned with undoing these operations to find the value of the unknown.

• Students will know that equations represent two mathematical statements that are equal and that this equivalence can be maintained if the same mathematical operation is performed on both sides of the equation.
• Students will know how to solve equations of varying complexity by undoing operations to find the value of the unknown.

Spring term 1

Understand Multiplicative Relationships: Percentages and Proportionality.

Students will focus on percentages and proportionality, and make connections with earlier work on fractions and ratios.

• Students will know that percentages and direct proportion are linked by multiplicative relationships.
• Students will know how to use these multiplicative relationships to solve problems involving percentages and direct proportion.

Spring term 2

• Statistical Representations, Measures and Analysis.

This theme will develop their knowledge of calculating measures of central tendency to include the mode and median, work with grouped data, and be introduced to a measure of spread in statistics: range. This will enable students to engage in more sophisticated data analysis and enable them to construct scatter graphs, bar charts, pie charts and pictograms.

• Students will know that data can be represented in various ways, such as bar charts, pie charts, scatter graphs, and that measures of central tendency and spread can be used to analyse data.
• Students will know how to select an appropriate method to represent dat and how to collect the mean, mode, mean and range for a set of data.

Summer term 1

Perimeter, Area and Volume.

Students will work to develop a secure and deep understanding of perimeter, area and volume. This will build on earlier work in KS3 to learn about the perimeter (circumference) of circles and that the ratio between circumference and diameter is the same for all circles.

• Students will know that pi is the ratio of the diameter of a circle to its circumference. Students will also know that the cross-sectional area of a prism can be used to calculate its volume.
• Students will know how to calculate the perimeter and area of circles and parts of circles. Students will also know how to calculate the surface area and volume of prisms.

Geometrical Properties: Polygons.

The focus is on the reasoning and constructing of proofs for why angle facts and properties relating to polygons hold.

• Students will know that the angles rules learnt earlier in KS3 can be used to prove angle relationships in pairs of parallel lines. Students will also know that the exterior angles of any polygon add to 360o and that the sum of the interior angles in any polygon is linked to the sum of the interior angles of triangles.
• Students will know how find missing angles in pairs of parallel lines and how to calculate exterior and interior angles in any polygon.

Summer term 2

Constructions.

Students will now learn the ruler and compass constructions of: triangles of given lengths, a perpendicular bisector of a line segment, a perpendicular to a given line through a given point, an angle bisector. An important awareness is that these constructions are based on the geometrical properties of a few key shapes.

• Students will know that triangles, rhombuses and kites can be constructed using circles.
• Students will know how to use the properties of rhombuses and kites to construct triangles, perpendicular bisectors, a perpendicular from a point and angle bisectors.

Autumn term 1

Geometrical Reasoning: Similarity.

In this unit students are required to go beyond intuitively recognising when shapes are similar or congruent, and to think about what can change and what has to stay the same for these properties to hold.

• Students will know that a common scale factor links the corresponding sides of similar shapes but for congruent shapes the sides as well as the angles will be exact.
• Students will know how to find the scale factor to calculate an unknown length of a similar shape and apply the conditions of congruency to identify congruent shapes.

Geometrical Reasoning: Pythagoras’ Theorem.

Learning about this important theorem in mathematics provides an opportunity for students to go beyond knowing that it is true to knowing why, and to think about relationships and structures, to reason with them and to prove results.

• Students will know that Pythagoras’ Theorem can be used to calculate an unknown length of a right angled triangle.
• Students will know how to substitute into the formula and rearrange when necessary to find an unknown length and to solve a variety of problems.

Autumn term 2

Probability.

The introduction of probability will offer students a way to quantify, explore and explain likelihood and coincidence, and to reason about uncertainty. Students will engage in experiments and develop a feel for likely, unlikely, even, certain and impossible chances, before starting to quantify probabilities and the likelihood of different outcomes.

• Students will know that probability can be quantified and applied to a large range of experiments and problems.
• Students will know how to work systematically to calculate and use theoretical probabilities for a range of single and combined events.

Non-linear Relationships.

This work extends students’ knowledge of sequences through exploration of the mathematical structure, not just by spotting the patterns that the structure creates. This learning has connections to other areas of algebra, particularly solving equations (when checking if a number is a term in a sequence) and graphs.

• Students will know that sequences do not just follow a linear pattern.
• Students will know how to identify different types of non-linear sequences, calculate the next term in a sequence and the nth term of sequences.

Spring term 1

Expressions and Formulae.

At the heart of algebra and algebraic thinking is the expression of generality. Algebraic notation and techniques for its manipulation, including conventions governing its use, should naturally arise from exploring the structure of the number system and operations on number.

• Students will know that they can use the distributive law to find the product of a pair of binomials and that formulae can be rearranged to change the subject.
• Students will know how to find products of binomials and how to rearrange formulae to change the subject.

Spring term 2

Trigonometry.

At Key Stage 2, students solved problems involving similar shapes, where the scale factor was known or could be found; earlier in Key Stage 3, students will have extended this work to explore conditions for similarity. This work on similarity and scale factors is now linked to the trigonometric functions and the fundamental ratios of sin θ = opp/hyp, cos θ = adj/hyp and tan θ = opp/adj.

• Students will know that they can apply their knowledge of similar triangles to find unknown lengths in a right angled triangle when given a length and an angle.
• Students will know how to apply their knowledge of trigonometric functions to find unknown lengths.

Summer term 1

Standard Form.

Students will work to further develop their understanding of the different ways that numbers can be expressed and will become more proficient in changing from one form to another. When thinking about very large and very small numbers, working with standard form notation will enable students to further develop their understanding of multiplication and division by powers of ten.

• Students will know that small and large numbers can be written in different ways.
• Students will know how to apply their knowledge of powers of ten to write numbers in standard form.

Summer term 2

Graphical Representations.

The elements here build on the work done in Year 8 and now include interpreting graphical representations. Significant attention is given in this core concept to exploring linear relationships and their representation as straight line graphs. Students should appreciate that all linear relationships have certain key characteristics both in the written algebraic form of the relationship and in its graphical representation.

• Students will know that a point of intersection of two linear graphs satisfies both relationships and hence represents the solution to both those equations.
• Students will know how to model and represent a range of algebraic situations graphically.

The KS4 curriculum provides students with opportunities to build on their learning experiences gained during Key Stage 3.

The aims and objectives of the KS4 curriculum are to enable students to develop fluent knowledge, skills and understanding of mathematical methods and concepts; acquire, select and apply mathematical techniques to solve problems; reason mathematically, make deductions and inferences, and draw conclusions; and comprehend, interpret and communicate mathematical information in a variety of forms appropriate to the information and context.

Transferable skills enable young people to face the demands of further and higher education, as well as the demands of the workplace, and are important in the teaching and learning of our curriculum. Our curriculum aims to develop students’ cognitive skills (non-routine problem solving, expert thinking, metacognition, creativity), systems thinking (decision making and reasoning) and their critical thinking (cognitive skills such as analysing, synthesising and reasoning skills). At same time we seek to improve students’ interpersonal skills (active listening, oral communication, written communication, assertive communication and non-verbal communication) and their capacity for collaborative problem solving (teamwork, establishing and maintaining shared understanding, taking appropriate action, establishing and maintaining team organisation).

As a school we follow the Edexcel GCSE 9-1 specification which is aligned with the DfE KS4 Mathematics Programme of Study. Themes covered are:

Number 1.

In this unit, students will develop a secure understanding of estimation, approximation and bounds. They will apply and interpret limits of accuracy when rounding or truncating, including upper and lower bounds.

• Students will know that numbers can be estimated, rounded and truncated, and that where numbers have been rounded there are limits of accuracy which can be represented using error interval notation.
• Students will know how to round numbers to decimal places and significant figures and when it is appropriate to do so; how to estimate and check calculations using approximation and estimation; and how to apply and interpret limits of accuracy, including performing calculations using upper and lower bounds.

Algebra 1.

In this unit, students will consolidate their knowledge and skills relating to simplifying and manipulating expressions involving sums, products and powers. Students will argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs.

• Students will know the definitions of expression, equation, identity, variables and unknowns. They will know that the laws of arithmetic can be extended to algebraic expressions.
• Students will know how to use the laws of arithmetic to manipulate algebraic expressions, including forming algebraic proofs, and will know the processes for solving different types of equations.

Number 2.

In this unit, students will work with fractions, decimals and surds, extending to algebraic expressions. Students will calculate with indices and standard form.

• Students will know the conventional notation for indices, standard form, surds and priority of operations. They will know the laws of indices, including negative and fractional powers, and they will know common fraction, decimal and percentage equivalences.
• Students will know how to convert between fractions, decimals and percentages, including recurring decimals, and how to calculate with indices, surds, and numbers written in standard form.

Geometry and Measures 1.

In this unit, students will consolidate their knowledge of area and perimeter, and extend to calculating surface area and volumes of shapes including spheres, pyramids, cones and composite solids, including finding arc lengths, angles and areas of sectors of circles. Students will apply Pythagoras’ Theorem, trigonometric ratios and the sine and cosine rules, to find angles, lengths and areas of triangles, and extend to 3D shapes.

• Students will know formulae relating to area, volume, surface area, Pythagoras’ theorem and trigonometry. They will know exact trigonometric values.
• Students will know how to use the formulae to solve problems involving angles, lengths, areas and volumes of 2D and 3D shapes, including compound shapes.

Ratio, Proportion and Rates of Change 1.

In this unit, students will develop their understanding of ratio and proportion, and extend to properties of similar shapes. Students will recognise and interpret graphs and equations describing direct and inverse proportion. Students will set up, solve and interpret answers in growth and decay problems, including compound interest and general iterative processes.

• Students will know the notation and vocabulary relating to ratio of amounts and equations of proportionality. They will know that any two quantities can be connected via a multiplicative relationship, expressed as both a ratio and as a fraction, and will extend this to quantities represented as expressions.
• Students will know how to solve problems involving ratio and proportion, and how to construct and interpret equations of proportionality, including those represented algebraically and graphically.

Algebra 2.

In this unit, students will deduce expressions to calculate the nth term of linear and quadratic sequences, translate simple stations or procedures into algebraic expressions or formulae, and where appropriate interpret simple expressions as functions with inputs and outputs, extending to inverse and composite functions.

• Students will know notation and vocabulary relating to sequences and functions, including linear, non-linear and iterative sequences, and composite and inverse functions.
• Students will know how to find nth terms of a range of sequences, solve problems involving formulae and functions, and approximate solutions to equations using iteration.

Geometry and Measures 2.

In this unit, students will consolidate their knowledge of the four transformations, extending to fractions and negative scale factors for enlargements. Students will describe the changes and invariance achieved by combinations of transformations. Students will apply the angle rules, including parallel lines and polygons, and apply and prove the standard circle theorems.

• Students will know angle rules including those relating to parallel lines, polygons and circles. They will know the nature of rotations, enlargements, translations and reflections, and appreciate what changes and what is invariant
• Students will know how to solve problems involving missing angles in a range of 2D shapes, including circles, and how to describe transformations, including single and combined transformations.

Statistics 1

In this unit, students will infer properties of populations or distributions from a sample, and will apply statistics to describe a population.

• Students will interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency and spread.
• Students will know how to use a variety of sampling methods and understand the limitations of each. Students will know how to calculate a range of measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers).

Probability 1

In this unit, students will consolidate their understanding of probability using a range of representations, extending to conditional probability.

• Students will know notation and vocabulary relating to probability, and will know that probabilities can be described using words or values represented as fractions, decimals or percentages.
• Students will know how to calculate probabilities of single and combined events, of independent and dependent events, and calculate probabilities using expected frequencies, and expressions represented algebraically.

Geometry and Measures 3 In this unit, students will convert between related compound units in numerical and algebraic contexts.

• Students will know formulae relating to compound measures including speed, density and pressure, and common conversions relating to measures.
• Students will know how to solve problems involving compound measures, including those involving time, and those which involve conversion between standard units in numerical and algebraic contexts.

Algebra 3.

In this unit, students will consolidate their understanding of solving equations, and extend to solving simultaneous and quadratic equations, using a variety of methods.

• Students will know notation relating to equations and inequalities and formulae relating to quadratic equations.
• Students will know how to form, solve and interpret different types of equations and inequalities, including simultaneous and quadratic equations and inequalities, and how to represent these on a number line, using set notation, and on a graph.

Algebra 4.

In this unit, students will recognise, sketch and interpret graphs of a variety of functions. Students will find the equations of straight lines and circles, including tangents. Students will sketch translations and reflections of given functions. Students will calculate or estimate gradients of graphs, and points on a curve, and areas under graphs, and interpret results.

• Students will know the shapes and properties of different types of graphs, and their general equations.
• Students will know how to calculate or estimate, and interpret gradients, intercepts and areas under graphs, how to find equations of straight line graphs including tangents, and how to sketch translations and reflections of a given function.

Statistics 2

In this unit, students will interpret and construct tables and graphs for data including grouped, discrete and continuous data.

• Students will interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate graphical representation involving discrete, continuous and grouped data.
• Students will know how to construct and interpret a range of graphs and charts, and will be able to use these to make comparisons and predictions.

Geometry and Measures 4.

In this unit, students will apply addition and subtraction of vectors, multiplication of vectors by a scalar and diagrammatic and column representation of vectors. Students will use vectors to construct geometric arguments and proofs.

• Students will know vocabulary and notation relating to vectors, including the representation of vectors and combinations of vectors diagrammatically.
• Students will know how to calculate using column vectors, and how to solve problems involving diagrammatic vectors, including using vectors to construct geometric arguments and proofs.

Mathematics has a language all of its own – not just its own vocabulary but the way sentences and maths questions are structured. Maths is clothed by polysemous vocabulary (words that have multiple meanings) that can foster ambiguity and unhelpful misconceptions, for example, prime, expression, converse, volume, mean, parallel, interest. Not only that, some maths words are homophones (words having the same pronunciation but different meanings, origins, or spellings) with more common words e.g. pi and pie, sine and sign, weight and wait, mode and mowed. Even simple mathematical operations, like subtraction, can be described in lots of ways in typical talk, such as ‘subtract’, ‘minus’, ‘difference’ or ‘take away’. Then you add in words like ‘decrease’, ‘reduce’ and ‘take off’. Teachers are constantly translating the language of maths. Teachers of maths will testify to the challenge of multi-step word problems. We can be confident then that reading in maths really does matter. Precision with mathematical talk will matter. Explicitly teaching on ‘how to read like a mathematician’ will prove beneficial for those students who are struggling with word problems. Reading matters in maths, more than students may assume.

Reading is built into the maths curriculum in several ways.

• Guided reading tasks are incorporated into the start of each unit. Some of these provide background information linked to the upcoming unit of work; others provide information about history of maths, key figures and great moments; some show how maths links to the real world. Sharples guided reading strategies are used for these tasks, with a combination of teacher led and student led reading. The linked questions provide an opportunity for students to demonstrate that they have understood what they have read.
• Students have regular exposure to worded and problem solving questions to develop students’ comprehension skills and support them in developing strategies to decode words and understand what the question is asking them to do.
• Reading lists are shared in each pupil unit booklet – these include books linked to the topic being studied, as well as those that are more recreational.
• Key words are shared within the pupil unit booklets and explicit vocabulary instruction is incorporated into lessons. Each unit includes a glossary of mathematical terms, along with accompanying diagrams and representations that may help.

Click here to view the suggested reads poster.
Please note: the images of the book covers are clickable hyperlinks to the book.