Programme of Study: Key Stage 3, Year 9
Skills available for England Key Stage 3, Year 9 maths objectives
Dimension
Print skill plan
Working mathematically
Develop fluency
Consolidate their numerical and mathematical capability from key stage 2 and extend their understanding of the number system and place value to include decimals, fractions, powers and roots.
Classify numbers
Integers
Rational numbers
Indices
Square roots and cube roots
Standard form
Consolidate their numerical and mathematical capability from key stage 2 and extend their understanding of the number system and place value to include decimals, fractions, powers and roots.
Classify numbers
Integers
Rational numbers
Indices
Square roots and cube roots
Standard form
Select and use appropriate calculation strategies to solve increasingly complex problems.
Select and use appropriate calculation strategies to solve increasingly complex problems.
Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships.
Direct proportions
Inverse variation
Variable expressions
One-variable equations
Linear functions
Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships.
Direct proportions
Inverse variation
Variable expressions
One-variable equations
Linear functions
Substitute values in expressions, rearrange and simplify expressions, and solve equations.
Order of operations
Indices
Squares and square roots
Variable expressions
One-variable equations
Two-variable equations
Substitute values in expressions, rearrange and simplify expressions, and solve equations.
Order of operations
Indices
Squares and square roots
Variable expressions
One-variable equations
Two-variable equations
Move freely between different numerical, algebraic, graphical and diagrammatic representations [for example, equivalent fractions, fractions and decimals, and equations and graphs].
Standard form
Equivalent ratios
Direct proportions
Inverse variation
Percents, fractions and decimals
One-variable equations
Linear functions
Move freely between different numerical, algebraic, graphical and diagrammatic representations [for example, equivalent fractions, fractions and decimals, and equations and graphs].
Standard form
Equivalent ratios
Direct proportions
Inverse variation
Percents, fractions and decimals
One-variable equations
Linear functions
Develop algebraic and graphical fluency, including understanding linear and simple quadratic functions.
Functions
Linear functions
Quadratic functions
Develop algebraic and graphical fluency, including understanding linear and simple quadratic functions.
Functions
Linear functions
Quadratic functions
Use language and properties precisely to analyse numbers, algebraic expressions, 2-D and 3-D shapes, probability and statistics.
Integer operations
Properties of operations and equality
Angles
Circles
Three-dimensional figures
Statistics
Probability
Use language and properties precisely to analyse numbers, algebraic expressions, 2-D and 3-D shapes, probability and statistics.
Integer operations
Properties of operations and equality
Angles
Circles
Three-dimensional figures
Statistics
Probability
Reason mathematically
Extend their understanding of the number system; make connections between number relationships, and their algebraic and graphical representations.
Classify numbers
Integers
Rational numbers
Inequalities
Extend their understanding of the number system; make connections between number relationships, and their algebraic and graphical representations.
Classify numbers
Integers
Rational numbers
Inequalities
Extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically.
Ratios and rates
Scale drawings
Proportions
Direct proportions
Metric units
Imperial units
Rate of change
Dilations
Similarity
Extend and formalise their knowledge of ratio and proportion in working with measures and geometry, and in formulating proportional relations algebraically.
Ratios and rates
Scale drawings
Proportions
Direct proportions
Metric units
Imperial units
Rate of change
Dilations
Similarity
Identify variables and express relations between variables algebraically and graphically.
Direct proportions
Inverse variation
Sequences
Variable expressions
Functions
Linear functions
Identify variables and express relations between variables algebraically and graphically.
Direct proportions
Inverse variation
Sequences
Variable expressions
Functions
Linear functions
Make and test conjectures about patterns and relationships; look for proofs or counter-examples.
Make and test conjectures about patterns and relationships; look for proofs or counter-examples.
Begin to reason deductively in geometry, number and algebra, including using geometrical constructions.
Begin to reason deductively in geometry, number and algebra, including using geometrical constructions.
Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning.
Mixed operations
Ratios and rates
Interpret when the structure of a numerical problem requires additive, multiplicative or proportional reasoning.
Mixed operations
Ratios and rates
Explore what can and cannot be inferred in statistical and probabilistic settings, and begin to express their arguments formally.
Explore what can and cannot be inferred in statistical and probabilistic settings, and begin to express their arguments formally.
Solve problems
Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems.
Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems.
Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics.
Price lists
Unit prices
Consumer maths
Linear functions
Two-dimensional figures
Statistics
Experimental and theoretical probability
Compound events
Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics.
Price lists
Unit prices
Consumer maths
Linear functions
Two-dimensional figures
Statistics
Experimental and theoretical probability
Compound events
Begin to model situations mathematically and express the results using a range of formal mathematical representations.
Direct proportions
Inverse variation
Variable expressions
One-variable equations
Linear functions
Quadratic functions
Polynomials
Data and graphs
Begin to model situations mathematically and express the results using a range of formal mathematical representations.
Direct proportions
Inverse variation
Variable expressions
One-variable equations
Linear functions
Quadratic functions
Polynomials
Data and graphs
Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems.
Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems.
Number
Understand and use place value for decimals, measures and integers of any size.
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 symbols =, ≠, <, >, ≤, ≥.
Integers
Rational numbers
Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥.
Integers
Rational 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.
Prime or composite
Prime factorisation
HCF and LCM
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.
Prime or composite
Prime factorisation
HCF and LCM
Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative.
Add and subtract integers
Multiply and divide integers
Add and subtract rational numbers
Multiply and divide rational numbers
Mixed operations with rational numbers
Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative.
Add and subtract integers
Multiply and divide integers
Add and subtract rational numbers
Multiply and divide rational numbers
Mixed operations with rational numbers
Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals.
Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals.
Recognise and use relationships between operations including inverse operations.
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.
Indices
Square roots
Cube roots
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.
Indices
Square roots
Cube roots
Interpret and compare numbers in standard form A × 10n1 ≤ A < 10, where n is a positive or negative integer or zero.
Interpret and compare numbers in standard form A × 10n1 ≤ A < 10, where n is a positive or negative integer or zero.
Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8).
Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8).
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%.
Percents, fractions and decimals
Percent problems
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%.
Percents, fractions and decimals
Percent problems
Interpret fractions and percentages as operators.
Percents of numbers
Percent of change
Interpret fractions and percentages as operators.
Percents of numbers
Percent of change
Use standard units of mass, length, time, money and other measures, including with decimal quantities.
Money
Metric units
Imperial units
Use standard units of mass, length, time, money and other measures, including with decimal quantities.
Money
Metric units
Imperial units
Round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures].
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 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.
Use a calculator and other technologies to calculate results accurately and then interpret them appropriately.
Appreciate the infinite nature of the sets of integers, real and rational numbers.
Appreciate the infinite nature of the sets of integers, real and rational numbers.
Algebra
Use and interpret algebraic notation, including:
Use and interpret algebraic notation, including:
ab in place of a × b
ab in place of a × b
3y in place of y + y + y and 3 × y
3y in place of y + y + y and 3 × y
a2 in place of a × a, a3 in place of a × a × a; a2b in place of a × a × b
a2 in place of a × a, a3 in place of a × a × a; a2b in place of a × a × b
a/b in place of a ÷ b
a/b in place of a ÷ b
coefficients written as fractions rather than as decimals
coefficients written as fractions rather than as decimals
brackets
brackets
Substitute numerical values into formulae and expressions, including scientific formulae.
Sequences
Expressions
Substitute numerical values into formulae and expressions, including scientific formulae.
Sequences
Expressions
Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors.
Expressions
Inequalities
Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors.
Expressions
Inequalities
Simplify and manipulate algebraic expressions to maintain equivalence by:
Simplify and manipulate algebraic expressions to maintain equivalence by:
collecting like terms
collecting like terms
multiplying a single term over a bracket
Linear expressions
Distributive property
multiplying a single term over a bracket
Linear expressions
Distributive property
taking out common factors
Linear expressions
Factors of linear expressions
taking out common factors
Linear expressions
Factors of linear expressions
expanding products of two or more binomials
Multiply a polynomial by a monomial
Multiply polynomials
expanding products of two or more binomials
Multiply a polynomial by a monomial
Multiply polynomials
Understand and use standard mathematical formulae; rearrange formulae to change the subject.
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.
Direct proportions
Variable expressions
One-variable equations
Linear functions
Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs.
Direct proportions
Variable expressions
One-variable equations
Linear functions
Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement).
Two-step equations
Multi-step equations
Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement).
Two-step equations
Multi-step equations
Work with coordinates in all four quadrants.
Work with coordinates in all four quadrants.
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.
Direct proportions
Linear functions
Quadratic functions
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.
Direct proportions
Linear functions
Quadratic functions
Interpret mathematical relationships both algebraically and graphically.
Direct proportions
Direct and inverse variation
Variable expressions
One-variable equations
Functions
Linear functions
Interpret mathematical relationships both algebraically and graphically.
Direct proportions
Direct and inverse variation
Variable expressions
One-variable equations
Functions
Linear functions
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.
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.
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.
Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs.
Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs.
Generate terms of a sequence from either a term-to-term or a position-to-term rule.
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.
Recognise arithmetic sequences and find the nth term.
Recognise geometric sequences and appreciate other sequences that arise.
Recognise geometric sequences and appreciate other sequences that arise.
Ratio, proportion and rates of change
Change freely between related standard units [for example time, length, area, volume/capacity, mass].
Metric units
Imperial units
Change freely between related standard units [for example time, length, area, volume/capacity, mass].
Metric units
Imperial units
Use scale factors, scale diagrams and maps.
Use scale factors, scale diagrams and maps.
Express one quantity as a fraction of another, where the fraction is less than one and greater than one.
Express one quantity as a fraction of another, where the fraction is less than one and greater than one.
Use ratio notation, including reduction to simplest form.
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.
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.
Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction.
Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction.
Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions.
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.
Percent problems
Percent of change
Consumer maths
Solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics.
Percent problems
Percent of change
Consumer maths
Solve problems involving direct and inverse proportion, including graphical and algebraic representations.
Ratios and rates
Proportions
Scale drawings
Direct proportions
Solve problems involving direct and inverse proportion, including graphical and algebraic representations.
Ratios and rates
Proportions
Scale drawings
Direct proportions
Use compound units such as speed, unit pricing and density to solve problems.
Use compound units such as speed, unit pricing and density to solve problems.
Geometry and measures
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).
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 2-D shapes (including circles), areas of circles and composite shapes.
Circles
Compound figures
Area between two shapes
Calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes.
Circles
Compound figures
Area between two shapes
Draw and measure line segments and angles in geometric figures, including interpreting scale drawings.
Draw and measure line segments and angles in geometric figures, including interpreting scale drawings.
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.
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.
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.
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.
Triangles and quadrilaterals
Circles
Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies.
Triangles and quadrilaterals
Circles
Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures.
Translations
Reflections
Rotations
Mixed transformations
Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures.
Translations
Reflections
Rotations
Mixed transformations
Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids.
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.
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.
Understand and use the relationship between parallel lines and alternate and corresponding angles.
Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons.
Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons.
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.
Triangles and quadrilaterals
Similar and congruent figures
Pythagoras' theorem
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.
Triangles and quadrilaterals
Similar and congruent figures
Pythagoras' theorem
Use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles.
Pythagoras' theorem
Trigonometric ratios
Use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles.
Pythagoras' theorem
Trigonometric ratios
Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D.
Three-dimensional figures
Volume and surface area
Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D.
Three-dimensional figures
Volume and surface area
Interpret mathematical relationships both algebraically and geometrically.
Polynomials
Angles
Area
Similar and congruent figures
Interpret mathematical relationships both algebraically and geometrically.
Polynomials
Angles
Area
Similar and congruent figures
Probability
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.
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.
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.
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.
Theoretical probability
Compound events
Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities.
Theoretical probability
Compound events
Statistics
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).
Mean, median, mode, and range
Outliers
Distributions in line plots
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).
Mean, median, mode, and range
Outliers
Distributions in line plots
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.
Tables
Bar graphs
Line graphs
Line plots
Stem-and-leaf plots
Frequency charts
Pie charts
Choose the best type of graph
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.
Tables
Bar graphs
Line graphs
Line plots
Stem-and-leaf plots
Frequency charts
Pie charts
Choose the best type of graph
Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs.
Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs.