# Mathematics

## Key Stage 3

The national curriculum for Mathematics aims to ensure that all students:

- Become fluent in the fundamentals of Mathematics, including through varied and frequent practice with increasingly complex problems over time, so that students develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
- Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

At KS3 decisions about progression are based on the security of students’ understanding and their readiness to progress to the next stage. Students who grasp concepts rapidly are challenged by further rich and sophisticated problems before any acceleration through new content in preparation for key stage 4. Those who are not sufficiently fluent are encouraged to consolidate their understanding, through additional practice, before moving on.

## Year 7

### Number

In this unit students have the opportunity to:

- Understand and use place value for decimals, measures and integers of any size
- Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers
- Use the concepts and vocabulary of prime numbers, factors (or divisors),multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
- Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
- Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
- Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
- Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2)
- Define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%
- Use standard units of mass, length, time, money and other measures, including with decimal quantities
- Round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]
- Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b
- Use a calculator and other technologies to calculate results accurately and then interpret them appropriately

### Algebra

In this unit students have the opportunity to:

- Use and interpret algebraic notation, including coefficients written as fractions rather than as decimals brackets
- Substitute numerical values into formulae and expressions, including scientific formulae
- Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors
- Simplify and manipulate algebraic expressions to maintain equivalence by collecting like terms, multiplying a single term over a bracket, taking out common factors, expanding products of two or more binomials.
- Understand and use standard mathematical formulae; rearrange formulae to change the subject
- Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)
- Recognise, sketch and produce graphsof linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane
- Interpret mathematical relationships both algebraically and graphically
- Reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically
- Use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations
- Generate terms of a sequence from either a term-to-term or a position-to-term rule
- Recognise arithmetic sequences and find the nth term as well as geometric sequences and appreciate other sequences that arise

### Ratio, Proportion and Rates of Change

In this unit students have the opportunity to:

- Change freely between related standard units [for example time, length, area, volume/capacity, mass]
- Express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1.
- Use ratio notation, including reduction to simplest form
- Divide a given quantity into two parts in a given part : part or part : whole ratio; express the division of a quantity into two parts as a ratio
- Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
- Solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial Mathematics
- Solve problems involving direct and inverse proportion, including graphical and algebraic representations

### Geometry and Measures

In this unit students have the opportunity to:

- Derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)
- Calculate and solve problems involving - perimeters of 2D shapes (including circles), areas of circles and composite shapes
- Describe, sketch and draw using conventional terms and notations ( points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric)
- Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies
- Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
- Understand and use the relationship between parallel lines and alternate and corresponding angles
- Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
- Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D
- Interpret mathematical relationships both algebraically and geometrically

### Probability

In this unit students have the opportunity to:

- Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0–1 probability scale. Students will learn that that the probabilities of all possible outcomes sum to 1.
- Enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
- Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities

### Statistics

In this unit students have the opportunity to:

- Describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)
- Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data

## Year 8

### Number

In this unit students have the opportunity to:

- Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
- Use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
- Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
- Recognise and use relationships between operations including inverse operations
- Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
- Interpret and compare numbers in standard form
- Work interchangeably with terminating decimals and their corresponding fractions
- Define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentagesgreater than 100%
- Appreciate the infinite nature of the sets of integers, real and rational numbers

### Probability

In this unit students have the opportunity to:

- Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0–1 probability scale
- Understand that the probabilities of all possible outcomes sum to 1
- Enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
- Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities

### Algebra

In this unit students have the opportunity to:

- Use and interpret algebraic notation
- Substitute numerical values into formulae and expressions, including scientific formulae
- Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors
- Simplify and manipulate algebraic expressions to maintain equivalence by collecting like terms, multiplying a single term over a bracket, taking out common factors expanding products of two or more binomials
- Understand and use standard mathematical formulae; rearrange formulae to change the subject
- Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs
- Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)
- Recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane
- Reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically
- Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs
- Recognise arithmetic sequences and find the nth term
- Recognise geometric sequences and appreciate other sequences that arise

### Ratio, Proportion and Rates of Change

In this unit students have the opportunity to:

- Use ratio notation, including reduction to simplest form
- Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
- Solve problems involving direct and inverse proportion, including graphical and algebraic representations
- Use compound units such as speed, unit pricing and density to solve problems

### Geometry and Measures

In this unit students have the opportunity to:

- Derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)
- Calculate and solve problems involving: perimeters of 2D shapes (including circles), areas of circles and composite shapes
- Derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line
- Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
- Use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles
- Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies
- Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
- Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
- Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
- Understand and use the relationship between parallel lines and alternate and corresponding angles
- Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D
- Interpret mathematical relationships both algebraically and geometrically

### Statistics

In this unit students have the opportunity to:

- Describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)
- Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data
- Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs

## Year 9

### Algebra

Pupils will develop further their understanding of sequence, algebraic manipulation including use of brackets and factorising. This will develop their skills from year 7 by introducing extended use on indices. They will explore the graphs of linear functions with an increasing degree of sophistication. Pupils will also learn how to create tables of values for quadratic and cubic functions and plot their graphs.

They will further extend their understanding of forming and solving equations. Use and represent inequalities and solve quadratic and cubic equations using a systematic trial and improvement method.

### Data Handling

Pupils will also explore how they can calculate an estimate for the mean, and locate the class interval where the mode and median lie from grouped data. Throughout the unit pupils will be encouraged to think about comparing two sets of data using the techniques outlined. They will develop their understanding of scatter graphs and how pairs of scatter graphs can be used to compare strength of correlation.

They continue to develop the concepts of theoretical and experimental probability

### Shape and Measure

In this unit they will explore and extend methods for finding the area and volume of shapes. Pupils will spend most of their time investigating circles and prisms and applying methods and formulas to problems.

They will continue to develop their knowledge and understanding of the angle and other associated geometric properties of shapes. They focus on transformations on coordinate axes and how shapes and their properties change under single and combined transformations.

### Numbers

Extensive work is carried out on multiplying and dividing decimal numbers by powers of 10. They will continue to learn how to compare and order decimal numbers and negative numbers in context and consolidate efficient written methods for adding and subtracting whole numbers and decimals with up to 2 places. Decimals and percentages are revisited.

## Key Stage 4

What qualification will the course lead to? GCSE Mathematics

Which Examination Board? Edexcel

### Course content

This is a linear qualification, with three terminal exams that must be taken at the end of the course in year 11.

The grading scale will be from nine to one (nine representing the highest grade). There are two tiers of entry available: Foundation (grades one to five) and Higher (grades four to nine).

The assessment for each tier of entry consists of three written exam papers. Each paper has an assessment time of one and a half hours and is worth 33.3% of the final grade. Paper one is non calculator and papers two and three are calculator.

There will be a greater emphasis on mathematical reasoning, communication and problem solving skills.

The content is divided in to six main areas:

1. Number

2. Algebra

3. Ratio, proportion and rates of change

4. Geometry and measures

5. Probability

6. Statistics