Numbers and the number system

• use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor and lowest common multiple
• use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5
• recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions
• Write a number as a product of its prime factors
• Use prime factorisations to find the HCF and LCM of two numbers
• Solve problems using highest common factors or lowest common multiples

Visualizing and constructing

• use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries
• use the standard conventions for labelling and referring to the sides and angles of triangles
• Use the centre and scale factor to carry out an enlargement with a positive integer scale factor
• Find the centre of enlargement
• Find the scale factor of an enlargement
• Use scale diagrams, including maps
• Use the concept of scaling in diagrams

Investigating properties of shapes

• identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres
• derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus; and triangles and other plane figures using appropriate language

Autumn 2

Calculating

• understand and use place value (e.g. when working with very large or very small numbers, and when calculating with decimals)
• apply the four operations, including formal written methods, to integers and decimals
• use conventional notation for priority of operations, including brackets recognise and use relationships between operations, including inverse operations (e.g. cancellation to simplify calculations and expressions)

Counting and comparing

• order positive and negative integers, decimals and fractions
• use the symbols =, ≠, <, >, ≤, ≥

Measuring space

• use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money, etc.)
• use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate
• change freely between related standard units (e.g. time, length, area, volume/capacity, mass) in numerical contexts

• measure line segments and angles in geometric figures

Investigating angles

• Measure accurately
• Convert between measures
• Solve problems involving measurement

Exploring fractions, decimals and percentages

• express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1
• define percentage as ‘number of parts per hundred’ express one quantity as a percentage of another

Proportional reasoning

• use ratio notation, including reduction to simplest form
• divide a given quantity into two parts in a given part:part or part:whole ratio

Algebraic proficiency: tinkering

• understand and use the concepts and vocabulary of expressions, equations, formulae and terms
• use and interpret algebraic notation, including: ab in place of a × b, 3y in place of y + y + y and 3 × y, a² in place of a × a, a³ in place of a × a × a, a/b in place of a ÷ b, brackets
• simplify and manipulate algebraic expressions by collecting like terms and multiplying a single term over a bracket
• where appropriate, interpret simple expressions as functions with inputs and outputs
• substitute numerical values into formulae and expressions
• use conventional notation for priority of operations, including brackets

Pattern sniffing

• Recognise simple arithmetic progressions
• Use a term-to-term rule to generate a linear sequence
• Use a term-to-term rule to generate a non-linear sequence

Calculating fractions, decimals and percentages

• apply the four operations, including formal written methods, to simple fractions (proper and improper), and mixed numbers
• interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively
• compare two quantities using percentages
• solve problems involving percentage change, including percentage increase/decrease

Summer 1

Checking, approximating and estimating

• round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures)
• estimate answers; check calculations using approximation and estimation, including answers obtained using technology
• recognise and use relationships between operations, including inverse operations (e.g. cancellation to simplify calculations and expressions)

Solving equations and inequalities

• recognise and use relationships between operations, including inverse operations (e.g. cancellation to simplify calculations and expressions)
• solve linear equations in one unknown algebraically

Calculating space

• use standard units of measure and related concepts (length, area, volume/capacity)
• calculate perimeters of 2D shapes
• know and apply formulae to calculate area of triangles, parallelograms, trapezia
• calculate surface area of cuboids
• know and apply formulae to calculate volume of cuboids
• understand and use standard mathematical formulae

Mathematical movement

• work with coordinates in all four quadrants
• understand and use lines parallel to the axes, y = x and y = -x
• solve geometrical problems on coordinate axes
• identify, describe and construct congruent shapes including on coordinate axes, by considering rotation, reflection and translation
• describe translations as 2D vectors

Presentation of data

• interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data and know their appropriate use

Measuring data

• Find the mode of set of data
• Find the median of a set of data including when there are an even number of numbers in the data set
• Calculate the mean from a frequency table
• Find the mode from a frequency table
• Find the median from a frequency table
• Calculate and understand the range as a measure of spread (or consistency)
• Analyse and compare sets of data, appreciating the limitations of different statistics (mean, median, mode, range)

Mathematical Movement

• Solve geometrical problems on coordinate axes
• Write the equation of a line parallel to the x-axis or the y-axis
• Identify and draw the lines y = x and y = -x
• Construct and describe reflections in horizontal, vertical and diagonal mirror lines (45° from horizontal)
• Describe a translation as a 2D vector
• Construct and describe rotations using a given angle, direction and centre of rotation
• Solve problems involving rotations, reflections and translations

Understanding risk I

• relate relative expected frequencies to theoretical probability, using appropriate language and the 0 - 1 probability scale
• record describe and analyse the frequency of outcomes of probability experiments using tables
• construct theoretical possibility spaces for single experiments with equally likely outcomes and use these to calculate theoretical probabilities
• apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one

Visualising and constructing

• Use the centre and scale factor to cary out an enlargement with a positive integer scale factor
• Find the centre and scale factor of an enlargement
• Use scale diagrams, including maps and the concept of scaling in diagrams
• Interpret plans and elevations
• Understand and use bearings
• Construct scale diagrams involving bearings
• Solve geometrical problems using bearings

Numbers and the number system

• use the concepts and vocabulary of prime numbers, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem
• round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures)
• interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer

Calculating

• apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative
• use conventional notation for priority of operations, including brackets, powers, roots and reciprocals

Algebraic proficiency: tinkering

• use and interpret algebraic notation, including: a²b in place of a × a × b, coefficients written as fractions rather than as decimals
• understand and use the concepts and vocabulary of factors
• simplify and manipulate algebraic expressions by taking out common factors and simplifying expressions involving sums, products and powers, including the laws of indices
• substitute numerical values into scientific formulae
• rearrange formulae to change the subject

Exploring fractions, decimals and percentages

• work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 or 3/8)

Proportional reasoning

• express the division of a quantity into two parts as a ratio; apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)
• identify and work with fractions in ratio problems
• understand and use proportion as equality of ratios
• express a multiplicative relationship between two quantities as a ratio or a fraction
• use compound units (such as speed, rates of pay, unit pricing)
• change freely between compound units (e.g. speed, rates of pay, prices) in numerical contexts
• relate ratios to fractions and to linear functions

Pattern sniffing

• generate terms of a sequence from either a term-to-term or a position-to-term rule
• deduce expressions to calculate the nth term of linear sequences

Investigating angles

• understand and use alternate and corresponding angles on parallel lines
• derive and use the sum of angles in a triangle (e.g. to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)

Calculating fractions, decimals and percentages

• interpret fractions and percentages as operators
• work with percentages greater than 100%
• solve problems involving percentage change, including original value problems, and simple interest including in financial mathematics
• calculate exactly with fractions

Solving equations and inequalities

• solve linear equations with the unknown on both sides of the equation
• find approximate solutions to linear equations using a graph

Calculating space

• compare lengths, areas and volumes using ratio notation
• calculate perimeters of 2D shapes, including circles
• identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference
• know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr²
• calculate areas of circles and composite shapes
• know and apply formulae to calculate volume of right prisms (including cylinders)

Algebraic proficiency: visualizing

• plot graphs of equations that correspond to straight-line graphs in the coordinate plane
• identify and interpret gradients and intercepts of linear functions graphically
• recognise, sketch and interpret graphs of linear functions and simple quadratic functions
• plot and interpret graphs and graphs of non-standard (piece-wise linear) functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance and speed

Understanding Risk II

• apply systematic listing strategies
• record describe and analyse the frequency of outcomes of probability experiments using frequency trees
• enumerate sets and combinations of sets systematically, using tables, grids and Venn diagrams
• construct theoretical possibility spaces for combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
• apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments

Presentation of data

• interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate graphical representation involving discrete, continuous and grouped data
• use and interpret scatter graphs of bivariate data
• recognise correlation

Measuring data

• interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers)
• apply statistics to describe a population

Calculating fractions, decimals and percentages

• interpret fractions and percentages as operators
• work with percentages greater than 100%
• solve problems involving percentage change, including original value problems, and simple interest including in financial mathematics
• calculate exactly with fractions

Measuring data

• interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers)
• apply statistics to describe a population

Calculating

• calculate with roots, and with integer indices
• calculate with standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer
• use inequality notation to specify simple error intervals due to truncation or rounding
• apply and interpret limits of accuracy

Visualizing and constructing

• 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)
• use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line
• construct plans and elevations of 3D shapes

Algebraic proficiency: tinkering

• understand and use the concepts and vocabulary of identities
• know the difference between an equation and an identity
• simplify and manipulate algebraic expressions by expanding products of two binomials and factorising quadratic expressions of the form x² + bx + c
• argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments
• translate simple situations or procedures into algebraic expressions or formulae

Proportional reasoning

• solve problems involving direct and inverse proportion including graphical and algebraic representations
• apply the concepts of congruence and similarity, including the relationships between lengths in similar figures
• change freely between compound units (e.g. density, pressure) in numerical and algebraic contexts
• use compound units such as density and pressure

Pattern sniffing

recognise and use Fibonacci type sequences, quadratic sequences

Solving equations and inequalities

• understand and use the concepts and vocabulary of inequalities
• solve linear inequalities in one variable
• represent the solution set to an inequality on a number line

Calculating space

• identify and apply circle definitions and properties, including:  tangent, arc, sector and segment
• calculate arc lengths, angles and areas of sectors of circles
• calculate surface area of right prisms (including cylinders)
• calculate exactly with multiples of π
• know the formulae for: Pythagoras’ theorem, a² + b² = c², and apply it to find lengths in right-angled triangles in two dimensional figures

Conjecturing

• use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)
• apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

Algebraic proficiency: visualising

• identify and interpret gradients and intercepts of linear functions algebraically
• use the form y = mx + c to identify parallel lines
• find the equation of the line through two given points, or through one point with a given gradient
• interpret the gradient of a straight line graph as a rate of change
• recognise, sketch and interpret graphs of quadratic functions
• recognise, sketch and interpret graphs of simple cubic functions and the reciprocal function  y = 1/x with x ≠ 0
• plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

Solving equations and inequalities II

• solve, in simple cases, two linear simultaneous equations in two variables algebraically
• derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution
• find approximate solutions to simultaneous equations using a graph

Understanding risk II

• calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
• enumerate sets and combinations of sets systematically, using tree diagrams
• understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

Presentation of data

• interpret and construct tables, charts and diagrams, including tables and line graphs for time series data and know their appropriate use
• draw estimated lines of best fit; make predictions
• Understand that correlation does not indicate causation; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing

• Calculations, checking and rounding
• Indices and roots
• Factors, multiples and primes
• Indices and standard form

• Algebra basics
• Equations
• Sequences
• Averages and range

• Representing and interpreting data
• Scatter graphs
• Fractions 

• Fractions
• Percentages
• Ratio and proportion 

• Polygons, angles and parallel lines
• Pythagoras’ theorem and trigonometry

• Graphs and real life graphs
• Linear graphs and coordinate geometry
• Quadratic, cubic and other graphs 

• Statistics and sampling
• The averages
• Perimeter and area

• Perimeter, area and circles
• 3D forms and volumes
• Accuracy and bounds

• 3D forms and volume
• Real life graphs
• Straight line graphs

• Transformations
• Constructions, loci and bearings

• Transformations
• Ratio

• Solving quadratic and simultaneous equations
• Inequalities
• Probability

• Proportion
• Pythagoras’ theorem and trigonometry
• Probability

• Probability
• Multiplicative reasoning
• Similarity and congruence

• Multiplicative reasoning
• Plans and elevations

• Graphs of trigonometric functions
• Further trigonometry

• Constructions, loci and bearings
•  Quadratic equations: expanding and factorising

• Collecting data

• Data
• Population and sampling
• Collecting discrete data

Students learn the key definitions of statistical words, how to sample data.

• Planning and collecting data
• Qualitative and discrete data

• Continuous data
• Tabulation
• Measures of central tendency – mode, median and mean

Students will convert raw data to summary statistics, design, construct and present summary tables.

• Measures of dispersion
• Box plots, skewness, calculating and representing outliers

• Scatter graphs
• Time series
• Introduction to probability

• Complete a mock controlled assessment

Autumn 1

• Quadratic equations: graphs
• Circles, cylinders, cones and spheres
• Fractions and reciprocals
• Similarity and congruence in 2D

• Cumulative frequency, box plots and histograms
• Quadratics, expanding more than two brackets, sketching graphs, graphs of circles, cubes and quadratics
• Circle theorems

• Similarity and congruence in 2D
• Vectors

• Circle geometry
• Changing the subject of formulae (more complex), algebraic fractions, solving equations arising from algebraic fractions, rationalising surds, proof

• Rearranging equations, graphs of cubic and reciprocal functions and simultaneous equations

• Vectors and geometric proof
• Reciprocal and exponential graphs; Gradient and area under graphs
• Direct and inverse proportion

• Revision

• Revision

• Revision

• Revision

• Public exam

• Public exam

• Calculating moving averages, seasonal and cyclic trends

• Simple probability and theoretical probability

• Interpreting index numbers in context and simple calculations
• Binomial distribution 

• Normal distribution and standardised scores
• Quality assurance

• Revision for exam