Find a skill and its lesson.

Indices, quadratics and simultaneous equations

1. Applies the index laws to simplify algebraic expressions with positive-integer and zero indicesBuilds on: Uses index notation and applies the index laws with numerical bases (Years 7–8); Simplifies algebraic expressions by collecting like terms (Years 7–8)
2. Expands binomial productsBuilds on: Expands and simplifies algebraic expressions using the distributive law
3. Factorises quadratic expressionsBuilds on: Expands binomial products; Factorises algebraic expressions by taking out a common algebraic factor
4. Graphs parabolas of the form y = kx² and y = kx² + c and describes their key featuresBuilds on: Connects algebraic and graphical representations of quadratic and exponential relationships
5. Solves quadratic equations by factorisingBuilds on: Factorises quadratic expressions; Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols
6. Solves simultaneous linear equations graphically and algebraicallyBuilds on: Interprets and applies the gradient/slope–intercept form y = mx + c; Solves linear equations of more than 3 steps and equations with two or more fractions

Financial mathematics (Stage 5)

1. Calculates earnings from wages, salaries, overtime, penalty rates, commission, piecework and leave loadingBuilds on: Calculates a percentage of a quantity, including discounts (Years 7–8); Adds, subtracts and multiplies decimals (Years 5–6); Uses rates to compare quantities, including best buys and speed (Years 7–8)
2. Applies the simple interest formula I = Prn to investment problemsBuilds on: Calculates a percentage of a quantity, including discounts (Years 7–8); Substitutes values into algebraic expressions and formulas (Years 7–8)
3. Calculates taxable income, PAYG tax and net payBuilds on: Calculates earnings from wages, salaries, overtime, penalty rates, commission, piecework and leave loading; Calculates a percentage of a quantity, including discounts (Years 7–8)
4. Compares the true cost of buying on terms, buy now pay later and short-term loansBuilds on: Applies the simple interest formula I = Prn to investment problems; Calculates percentage increase and decrease, including GST (Years 7–8)
5. Calculates compound interest using repeated multiplication and the formula FV = PV(1 + r)^nBuilds on: Applies the simple interest formula I = Prn to investment problems; Applies the index laws to simplify algebraic expressions with positive-integer and zero indices; Calculates percentage increase and decrease, including GST (Years 7–8)
6. Calculates the depreciation of an asset using S = V₀(1 − r)^nBuilds on: Calculates compound interest using repeated multiplication and the formula FV = PV(1 + r)^n

Algebraic techniques (Stage 5)

1. Simplifies algebraic fractions with numerical denominators using the four operationsBuilds on: Simplifies algebraic expressions by collecting like terms (Years 7–8); Adds and subtracts fractions with unlike denominators (Years 5–6)
2. Expands and simplifies algebraic expressions using the distributive lawBuilds on: Expands expressions with brackets using the distributive law (Years 7–8); Multiplies and divides integers, including with the order of operations (Years 7–8)
3. Factorises algebraic expressions by taking out a common algebraic factorBuilds on: Factorises algebraic expressions using the highest common factor (Years 7–8); Applies the index laws to simplify algebraic expressions with positive-integer and zero indices
4. Proves and applies the special products (a − b)(a + b), (a + b)² and (a − b)²Builds on: Expands binomial products
5. Simplifies and operates with algebraic fractions involving indices and pronumerals in the denominatorBuilds on: Simplifies algebraic fractions with numerical denominators using the four operations; Applies the index laws to simplify algebraic expressions with positive-integer and zero indices; Applies the index laws to algebraic expressions involving negative-integer indices
6. Selects factorisation strategies, including grouping in pairs and non-monic quadratic trinomialsBuilds on: Factorises quadratic expressions; Proves and applies the special products (a − b)(a + b), (a + b)² and (a − b)²; Factorises algebraic expressions by taking out a common algebraic factor
7. Operates with algebraic fractions with binomial numerators and simplifies fractions by factorisingBuilds on: Simplifies and operates with algebraic fractions involving indices and pronumerals in the denominator; Selects factorisation strategies, including grouping in pairs and non-monic quadratic trinomials

Indices and surds (Stage 5)

1. Distinguishes rational from irrational numbers within the real numbersBuilds on: Converts between fractions, decimals and percentages (Years 5–6); Calculates squares, cubes, square roots and cube roots (Years 7–8)
2. Establishes and evaluates negative-integer indices for numerical basesBuilds on: Applies the index laws to simplify algebraic expressions with positive-integer and zero indices
3. Describes surds as exact values and the conditions under which they are definedBuilds on: Distinguishes rational from irrational numbers within the real numbers
4. Applies the index laws to algebraic expressions involving negative-integer indicesBuilds on: Establishes and evaluates negative-integer indices for numerical bases; Applies the index laws to simplify algebraic expressions with positive-integer and zero indices
5. Applies the surd results and the four operations to simplify surdsBuilds on: Describes surds as exact values and the conditions under which they are defined; Expresses whole numbers as products of their prime factors (Years 7–8)
6. Uses x⁻¹ as a reciprocal and simplifies expressions of the form (a/b)⁻ⁿBuilds on: Applies the index laws to algebraic expressions involving negative-integer indices
7. Expands expressions involving surds and rationalises denominatorsBuilds on: Applies the surd results and the four operations to simplify surds; Proves and applies the special products (a − b)(a + b), (a + b)² and (a − b)²
8. Converts between surd and index form and evaluates fractional indicesBuilds on: Applies the surd results and the four operations to simplify surds; Applies the index laws to algebraic expressions involving negative-integer indices

Equations and inequalities (Stage 5)

1. Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbolsBuilds on: Solves two-step linear equations, including with brackets (Years 7–8); Expands and simplifies algebraic expressions using the distributive law
2. Solves linear equations involving one algebraic fractionBuilds on: Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols; Simplifies algebraic fractions with numerical denominators using the four operations
3. Solves linear equations arising from formulas and word problemsBuilds on: Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols; Substitutes values into algebraic expressions and formulas (Years 7–8)
4. Solves linear inequalities and graphs their solutions on a number lineBuilds on: Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols; Orders, adds and subtracts positive and negative integers (Years 7–8)
5. Solves cubic equations of the form ax³ = kBuilds on: Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols; Converts between surd and index form and evaluates fractional indices; Calculates squares, cubes, square roots and cube roots (Years 7–8)
6. Solves quadratic equations by completing the squareBuilds on: Proves and applies the special products (a − b)(a + b), (a + b)² and (a − b)²; Solves quadratic equations by factorising; Applies the surd results and the four operations to simplify surds
7. Solves linear equations of more than 3 steps and equations with two or more fractionsBuilds on: Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols; Solves linear equations involving one algebraic fraction; Operates with algebraic fractions with binomial numerators and simplifies fractions by factorising
8. Applies the quadratic formula and uses the discriminant to describe the solutionsBuilds on: Solves quadratic equations by completing the square; Applies the surd results and the four operations to simplify surds; Substitutes values into algebraic expressions and formulas (Years 7–8)
9. Rearranges literal equations by changing the subject of a formulaBuilds on: Solves linear equations of more than 3 steps and equations with two or more fractions; Substitutes values into algebraic expressions and formulas (Years 7–8)
10. Selects and verifies the most appropriate method for solving a quadratic equationBuilds on: Solves quadratic equations by factorising; Solves quadratic equations by completing the square; Applies the quadratic formula and uses the discriminant to describe the solutions
11. Models and solves problems using quadratic equations, including substitution into formulasBuilds on: Selects and verifies the most appropriate method for solving a quadratic equation; Rearranges literal equations by changing the subject of a formula
12. Relates identities and contradictions to coincident and parallel linesBuilds on: Solves simultaneous linear equations graphically and algebraically; Graphs and explains horizontal, vertical and parallel lines

Linear relationships (Stage 5)

1. Finds the gradient of an interval on the Cartesian plane using rise over runBuilds on: Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Orders, adds and subtracts positive and negative integers (Years 7–8)
2. Finds the midpoint of an interval on the Cartesian planeBuilds on: Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Calculates the mean, median, mode and range of a data set (Years 5–6)
3. Graphs linear relationships from tables of values and identifies their interceptsBuilds on: Plots straight-line graphs from a table of values (Years 7–8); Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Substitutes values into algebraic expressions and formulas (Years 7–8)
4. Finds the distance between two points on the Cartesian plane using Pythagoras’ theoremBuilds on: Applies Pythagoras' theorem to find unknown sides in right-angled triangles (Years 7–8); Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Finds the gradient of an interval on the Cartesian plane using rise over run
5. Graphs and explains horizontal, vertical and parallel linesBuilds on: Graphs linear relationships from tables of values and identifies their intercepts; Finds the gradient of an interval on the Cartesian plane using rise over run
6. Interprets and applies the gradient/slope–intercept form y = mx + cBuilds on: Finds the gradient and y-intercept and writes the equation of a line (Years 7–8); Graphs linear relationships from tables of values and identifies their intercepts; Finds the gradient of an interval on the Cartesian plane using rise over run
7. Applies the midpoint and gradient formulas to an interval on the Cartesian planeBuilds on: Finds the midpoint of an interval on the Cartesian plane; Finds the gradient of an interval on the Cartesian plane using rise over run
8. Finds the equations of parallel and perpendicular linesBuilds on: Interprets and applies the gradient/slope–intercept form y = mx + c; Graphs and explains horizontal, vertical and parallel lines
9. Establishes and applies the distance formulaBuilds on: Finds the distance between two points on the Cartesian plane using Pythagoras’ theorem; Applies the surd results and the four operations to simplify surds
10. Identifies line and rotational symmetry in plane shapes and in graphsBuilds on: Classifies triangles and quadrilaterals using their side, angle and symmetry properties (Years 7–8); Graphs parabolas of the form y = kx² and y = kx² + c and describes their key features
11. Describes translations, reflections and rotations on the Cartesian plane using coordinatesBuilds on: Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Identifies line and rotational symmetry in plane shapes and in graphs
12. Converts between gradient–intercept and general form and graphs a line given in any formBuilds on: Interprets and applies the gradient/slope–intercept form y = mx + c; Solves linear equations of more than 3 steps and equations with two or more fractions
13. Uses the point–gradient form to find the equation of a line through a point or through two pointsBuilds on: Converts between gradient–intercept and general form and graphs a line given in any form; Applies the midpoint and gradient formulas to an interval on the Cartesian plane
14. Finds and justifies the equations of parallel and perpendicular lines given in any formBuilds on: Uses the point–gradient form to find the equation of a line through a point or through two points; Finds the equations of parallel and perpendicular lines
15. Solves problems involving geometrical figures by applying coordinate geometry formulasBuilds on: Establishes and applies the distance formula; Applies the midpoint and gradient formulas to an interval on the Cartesian plane; Finds and justifies the equations of parallel and perpendicular lines given in any form

Non-linear relationships (Stage 5)

1. Connects algebraic and graphical representations of quadratic and exponential relationshipsBuilds on: Graphs linear relationships from tables of values and identifies their intercepts; Applies the index laws to simplify algebraic expressions with positive-integer and zero indices; Substitutes values into algebraic expressions and formulas (Years 7–8)
2. Graphs exponential curves of the form y = a^x and describes their featuresBuilds on: Connects algebraic and graphical representations of quadratic and exponential relationships; Establishes and evaluates negative-integer indices for numerical bases
3. Distinguishes between linear, quadratic and exponential relationships from their graphs and equationsBuilds on: Graphs parabolas of the form y = kx² and y = kx² + c and describes their key features; Graphs exponential curves of the form y = a^x and describes their features; Interprets and applies the gradient/slope–intercept form y = mx + c
4. Graphs exponential relationships of the form y = ka^x + c and describes their transformationsBuilds on: Graphs exponential curves of the form y = a^x and describes their features; Applies the index laws to algebraic expressions involving negative-integer indices
5. Derives and graphs the equation of a circle centred at the originBuilds on: Establishes and applies the distance formula; Applies the surd results and the four operations to simplify surds
6. Graphs parabolas in turning-point form and describes their transformationsBuilds on: Graphs parabolas of the form y = kx² and y = kx² + c and describes their key features; Describes translations, reflections and rotations on the Cartesian plane using coordinates
7. Graphs circles with centre (a, b) and finds the centre and radius by completing the squareBuilds on: Derives and graphs the equation of a circle centred at the origin; Solves quadratic equations by completing the square
8. Finds the intercepts, axis of symmetry and vertex of y = ax² + bx + c and sketches it by handBuilds on: Graphs parabolas in turning-point form and describes their transformations; Solves quadratic equations by factorising; Applies the quadratic formula and uses the discriminant to describe the solutions
9. Graphs hyperbolas and describes their features and transformationsBuilds on: Identifies, describes and represents inverse variation; Uses x⁻¹ as a reciprocal and simplifies expressions of the form (a/b)⁻ⁿ; Graphs parabolas in turning-point form and describes their transformations
10. Graphs and compares curves of the form y = kxⁿ and describes their transformationsBuilds on: Graphs parabolas in turning-point form and describes their transformations; Identifies line and rotational symmetry in plane shapes and in graphs
11. Identifies and compares graph types from their equations and finds intersections with a lineBuilds on: Finds the intercepts, axis of symmetry and vertex of y = ax² + bx + c and sketches it by hand; Graphs hyperbolas and describes their features and transformations; Graphs circles with centre (a, b) and finds the centre and radius by completing the square; Solves simultaneous linear equations graphically and algebraically

Variation and rates of change (Path)

1. Identifies, describes and represents direct variationBuilds on: Simplifies ratios and divides a quantity in a given ratio (Years 7–8); Uses rates to compare quantities, including best buys and speed (Years 7–8); Interprets and applies the gradient/slope–intercept form y = mx + c
2. Identifies, describes and represents inverse variationBuilds on: Identifies, describes and represents direct variation
3. Recognises graphs of direct and inverse variation and solves variation problemsBuilds on: Identifies, describes and represents direct variation; Identifies, describes and represents inverse variation; Graphs linear relationships from tables of values and identifies their intercepts
4. Interprets graphs that increase or decrease at a constant rateBuilds on: Recognises graphs of direct and inverse variation and solves variation problems; Interprets and applies the gradient/slope–intercept form y = mx + c; Interprets distance–time graphs (Years 7–8)
5. Interprets and constructs graphs with a variable rate of changeBuilds on: Interprets graphs that increase or decrease at a constant rate

Polynomials (Path)

1. Defines polynomials and their terminology and uses P(x) notationBuilds on: Applies the index laws to simplify algebraic expressions with positive-integer and zero indices; Substitutes values into algebraic expressions and formulas (Years 7–8); Applies the index laws to algebraic expressions involving negative-integer indices
2. Adds, subtracts and multiplies polynomialsBuilds on: Defines polynomials and their terminology and uses P(x) notation; Expands binomial products
3. Divides a polynomial by a linear polynomial and expresses the result as P(x) = D(x)Q(x) + R(x)Builds on: Adds, subtracts and multiplies polynomials; Divides two- and three-digit numbers by a one-digit number, with remainders (Years 3–4)
4. Applies the remainder and factor theorems to factorise polynomials and find their zeroesBuilds on: Divides a polynomial by a linear polynomial and expresses the result as P(x) = D(x)Q(x) + R(x); Selects factorisation strategies, including grouping in pairs and non-monic quadratic trinomials
5. Graphs polynomials from their zeroes using multiplicity and the leading termBuilds on: Applies the remainder and factor theorems to factorise polynomials and find their zeroes; Graphs and compares curves of the form y = kxⁿ and describes their transformations
6. Compares transformations of polynomial graphsBuilds on: Graphs polynomials from their zeroes using multiplicity and the leading term; Describes translations, reflections and rotations on the Cartesian plane using coordinates

Logarithms (Path)

1. Defines logarithms and converts between index and logarithmic formBuilds on: Applies the index laws to simplify algebraic expressions with positive-integer and zero indices; Graphs exponential curves of the form y = a^x and describes their features; Converts between surd and index form and evaluates fractional indices
2. Compares the graphs of exponential and logarithmic functionsBuilds on: Defines logarithms and converts between index and logarithmic form; Graphs exponential relationships of the form y = ka^x + c and describes their transformations
3. Deduces and applies the laws of logarithmsBuilds on: Defines logarithms and converts between index and logarithmic form; Applies the index laws to simplify algebraic expressions with positive-integer and zero indices
4. Solves simple equations involving exponents or logarithmsBuilds on: Deduces and applies the laws of logarithms; Solves linear equations of up to 3 steps, including pronumerals on both sides and grouping symbols
5. Examines logarithmic scales and explains their use in contextBuilds on: Deduces and applies the laws of logarithms; Compares the graphs of exponential and logarithmic functions

Functions and other graphs (Path)

1. Graphs regions corresponding to linear inequalities in one and two variablesBuilds on: Solves linear inequalities and graphs their solutions on a number line; Converts between gradient–intercept and general form and graphs a line given in any form
2. Defines relations and functions and applies the vertical line testBuilds on: Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Identifies and compares graph types from their equations and finds intersections with a line
3. Uses function notation f(x) and evaluates f(c)Builds on: Defines relations and functions and applies the vertical line test; Substitutes values into algebraic expressions and formulas (Years 7–8)
4. Determines the domain and range of a functionBuilds on: Uses function notation f(x) and evaluates f(c); Solves linear inequalities and graphs their solutions on a number line
5. Graphs and describes transformations of y = f(x)Builds on: Determines the domain and range of a function; Graphs parabolas in turning-point form and describes their transformations; Compares transformations of polynomial graphs

Pythagoras, trigonometry and similarity

1. Identifies similar figures and uses scale factorsBuilds on: Simplifies ratios and divides a quantity in a given ratio (Years 7–8); Finds unknown angles using angles at a point, on a line and in triangles (Years 7–8); Classifies triangles and quadrilaterals using their side, angle and symmetry properties (Years 7–8)
2. Uses sine, cosine and tangent ratios to find unknown sidesBuilds on: Applies Pythagoras' theorem to find unknown sides in right-angled triangles (Years 7–8); Identifies similar figures and uses scale factors
3. Enlarges, reduces and interprets scale drawingsBuilds on: Identifies similar figures and uses scale factors; Simplifies ratios and divides a quantity in a given ratio (Years 7–8)
4. Uses trigonometry to find unknown anglesBuilds on: Uses sine, cosine and tangent ratios to find unknown sides
5. Solves problems involving angles of elevation and depressionBuilds on: Uses sine, cosine and tangent ratios to find unknown sides; Uses trigonometry to find unknown angles
6. Solves practical problems involving true and compass bearingsBuilds on: Uses sine, cosine and tangent ratios to find unknown sides; Solves problems involving angles of elevation and depression; Finds unknown angles formed when parallel lines are cut by a transversal (Years 7–8)

Measurement, area and volume (Stage 5)

1. Interprets unit prefixes for very small and very large measurementsBuilds on: Measures length in centimetres and metres (Years 3–4); Uses index notation and applies the index laws with numerical bases (Years 7–8)
2. Solves practical problems involving the area of composite shapesBuilds on: Finds the area of composite shapes (Years 7–8); Uses formulas to find the area of parallelograms, trapeziums, rhombuses and kites (Years 7–8); Calculates the circumference and area of circles (Years 7–8)
3. Calculates the surface area of prisms and cylindersBuilds on: Finds the area of composite shapes (Years 7–8); Calculates the volume of prisms and cylinders (Years 7–8); Uses formulas to find the area of parallelograms, trapeziums, rhombuses and kites (Years 7–8)
4. Finds the precision, absolute error and percentage error of a measurementBuilds on: Interprets unit prefixes for very small and very large measurements; Calculates a percentage of a quantity, including discounts (Years 7–8)
5. Solves problems involving the surface area of composite solidsBuilds on: Calculates the surface area of prisms and cylinders; Solves practical problems involving the area of composite shapes
6. Finds the volume of right prisms with composite and circular cross-sectionsBuilds on: Calculates the volume and capacity of right prisms with various cross-sections (Years 7–8); Solves practical problems involving the area of composite shapes
7. Rounds numbers to a given number of significant figures and judges accuracyBuilds on: Adds, subtracts and multiplies decimals (Years 5–6); Finds the precision, absolute error and percentage error of a measurement
8. Calculates the volume and capacity of composite solids made from prisms and cylindersBuilds on: Finds the volume of right prisms with composite and circular cross-sections; Calculates the volume of prisms and cylinders (Years 7–8)
9. Represents and calculates with numbers in scientific notationBuilds on: Applies the index laws to simplify algebraic expressions with positive-integer and zero indices; Rounds numbers to a given number of significant figures and judges accuracy

Further trigonometry (Path)

1. Derives and applies the exact trigonometric ratios for 30, 45 and 60 degreesBuilds on: Uses sine, cosine and tangent ratios to find unknown sides; Applies Pythagoras' theorem to find unknown sides in right-angled triangles (Years 7–8); Uses index notation and applies the index laws with numerical bases (Years 7–8)
2. Applies the sine rule to non-right-angled trianglesBuilds on: Uses sine, cosine and tangent ratios to find unknown sides; Uses trigonometry to find unknown angles
3. Uses the unit circle to define and graph the trigonometric functionsBuilds on: Uses sine, cosine and tangent ratios to find unknown sides; Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Graphs parabolas of the form y = kx² and y = kx² + c and describes their key features
4. Applies the cosine rule to non-right-angled trianglesBuilds on: Applies the sine rule to non-right-angled triangles; Substitutes values into algebraic expressions and formulas (Years 7–8)
5. Relates the gradient of a line to its angle of inclinationBuilds on: Finds the gradient and y-intercept and writes the equation of a line (Years 7–8); Uses trigonometry to find unknown angles; Uses the unit circle to define and graph the trigonometric functions
6. Applies Pythagoras' theorem and trigonometry to solve 3-dimensional problemsBuilds on: Applies Pythagoras' theorem to find unknown sides in right-angled triangles (Years 7–8); Uses sine, cosine and tangent ratios to find unknown sides; Uses trigonometry to find unknown angles; Solves practical problems involving true and compass bearings
7. Uses the area rule and selects the appropriate rule for non-right-angled trianglesBuilds on: Applies the sine rule to non-right-angled triangles; Applies the cosine rule to non-right-angled triangles
8. Solves trigonometric equations, including the ambiguous caseBuilds on: Uses the unit circle to define and graph the trigonometric functions; Derives and applies the exact trigonometric ratios for 30, 45 and 60 degrees; Uses the area rule and selects the appropriate rule for non-right-angled triangles

Further area and volume (Path)

1. Finds the surface area of right pyramids and right conesBuilds on: Calculates the surface area of prisms and cylinders; Applies Pythagoras' theorem to find unknown sides in right-angled triangles (Years 7–8); Solves problems involving the surface area of composite solids
2. Finds the surface area of spheres and composite solidsBuilds on: Finds the surface area of right pyramids and right cones; Solves problems involving the surface area of composite solids
3. Finds the volume of right pyramids and right conesBuilds on: Calculates the volume and capacity of composite solids made from prisms and cylinders; Finds the surface area of right pyramids and right cones
4. Finds the volume of spheres and related composite solidsBuilds on: Finds the volume of right pyramids and right cones; Finds the surface area of spheres and composite solids

Congruence, similarity and geometrical proof (Path)

1. Applies interior and exterior angle results for polygonsBuilds on: Uses the angle sum of a quadrilateral to find unknown angles (Years 7–8); Finds unknown angles formed when parallel lines are cut by a transversal (Years 7–8)
2. Identifies congruent figures and matches their sides and anglesBuilds on: Classifies triangles and quadrilaterals using their side, angle and symmetry properties (Years 7–8); Identifies similar figures and uses scale factors
3. Establishes and uses the congruence tests for trianglesBuilds on: Identifies congruent figures and matches their sides and angles
4. Establishes and uses the similarity tests for trianglesBuilds on: Identifies similar figures and uses scale factors; Establishes and uses the congruence tests for triangles
5. Relates area and volume ratios to the similarity ratio of similar figures and solidsBuilds on: Establishes and uses the similarity tests for triangles; Calculates the volume and capacity of composite solids made from prisms and cylinders; Enlarges, reduces and interprets scale drawings
6. Finds unknown sides and angles in plane shapes, giving geometrical reasonsBuilds on: Applies interior and exterior angle results for polygons; Establishes and uses the congruence tests for triangles; Establishes and uses the similarity tests for triangles
7. Constructs formal proofs of congruent and similar trianglesBuilds on: Establishes and uses the congruence tests for triangles; Establishes and uses the similarity tests for triangles; Finds unknown sides and angles in plane shapes, giving geometrical reasons
8. Proves properties and tests for triangles and quadrilateralsBuilds on: Constructs formal proofs of congruent and similar triangles; Applies interior and exterior angle results for polygons

Circle geometry (Path)

1. Proves and applies chord and angle properties of circlesBuilds on: Calculates the circumference and area of circles (Years 7–8); Constructs formal proofs of congruent and similar triangles; Proves properties and tests for triangles and quadrilaterals
2. Proves and applies tangent and secant properties of circlesBuilds on: Proves and applies chord and angle properties of circles

Introduction to networks (Path)

1. Describes networks using vertices, edges and degreeBuilds on: Classifies triangles and quadrilaterals using their side, angle and symmetry properties (Years 7–8); Interprets line graphs and two-way tables (Years 5–6)
2. Identifies planar graphs and applies Euler's formulaBuilds on: Describes networks using vertices, edges and degree
3. Explains and finds Eulerian trails and circuitsBuilds on: Identifies planar graphs and applies Euler's formula

Comparing data sets

1. Determines the five-number summary and interquartile range of a datasetBuilds on: Calculates the mean, median, mode and range of a data set (Years 5–6); Chooses appropriate measures of centre and spread, including the effect of outliers (Years 7–8); Constructs and interprets dot plots and stem-and-leaf plots (Years 7–8)
2. Constructs and compares box plots using five-number summariesBuilds on: Determines the five-number summary and interquartile range of a dataset; Calculates the mean, median, mode and range of a data set (Years 5–6); Constructs frequency tables and histograms for numerical data (Years 7–8)
3. Uses standard deviation to measure and compare the spread of datasetsBuilds on: Calculates the mean, median, mode and range of a data set (Years 5–6); Determines the five-number summary and interquartile range of a dataset; Calculates squares, cubes, square roots and cube roots (Years 7–8)
4. Compares two or more datasets using parallel box plots, shape and summary statisticsBuilds on: Constructs and compares box plots using five-number summaries; Describes the shape of data distributions, including symmetry, clusters and outliers (Years 7–8); Determines the five-number summary and interquartile range of a dataset

Bivariate data and lines of best fit

1. Identifies bivariate data and distinguishes association from causationBuilds on: Classifies data as categorical or numerical, discrete or continuous (Years 7–8); Interprets line graphs and two-way tables (Years 5–6)
2. Draws scatter plots and describes the relationship between two variablesBuilds on: Identifies bivariate data and distinguishes association from causation; Plots and reads points in all four quadrants of the Cartesian plane (Years 5–6); Interprets line graphs and two-way tables (Years 5–6)
3. Fits a line of best fit by eye and uses it to interpolate and extrapolateBuilds on: Draws scatter plots and describes the relationship between two variables; Finds the gradient and y-intercept and writes the equation of a line (Years 7–8)

Multistage probability and simulation

1. Distinguishes independent and dependent events and lists outcomes for multistage experimentsBuilds on: Uses tables and tree diagrams for two-step experiments (Years 7–8); Calculates theoretical probabilities for single-step experiments (Years 7–8); Uses complementary events to calculate probabilities (Years 7–8)
2. Calculates probabilities for multistage chance experiments, with and without replacementBuilds on: Distinguishes independent and dependent events and lists outcomes for multistage experiments; Multiplies and divides fractions and mixed numerals (Years 7–8)
3. Designs, runs and evaluates probability simulations using digital toolsBuilds on: Runs experiments and compares relative frequency with theoretical probability (Years 7–8); Calculates probabilities for multistage chance experiments, with and without replacement

Statistical inquiry (Path)

1. Plans and conducts a statistical inquiry and reports its findingsBuilds on: Compares two or more datasets using parallel box plots, shape and summary statistics; Fits a line of best fit by eye and uses it to interpolate and extrapolate; Uses standard deviation to measure and compare the spread of datasets
2. Critically evaluates sampling, surveys and statistical claims in media reportsBuilds on: Plans and conducts a statistical inquiry and reports its findings; Identifies bivariate data and distinguishes association from causation

Venn diagrams, sets and conditional probability (Path)

1. Represents data in Venn diagrams and two-way tables and converts between themBuilds on: Uses tables and tree diagrams for two-step experiments (Years 7–8); Calculates theoretical probabilities for single-step experiments (Years 7–8); Interprets line graphs and two-way tables (Years 5–6)
2. Uses set language and notation to describe eventsBuilds on: Represents data in Venn diagrams and two-way tables and converts between them
3. Describes and calculates probabilities of compound events, mutually exclusive or notBuilds on: Uses set language and notation to describe events; Represents data in Venn diagrams and two-way tables and converts between them; Uses complementary events to calculate probabilities (Years 7–8)
4. Calculates conditional probabilities and interprets conditional languageBuilds on: Represents data in Venn diagrams and two-way tables and converts between them; Describes and calculates probabilities of compound events, mutually exclusive or not; Distinguishes independent and dependent events and lists outcomes for multistage experiments