Find a skill and its lesson.

Reading, viewing and listening to texts

1. Rereads and reinterprets to analyse layers of meaningBuilds on: Re-reads closely to uncover main ideas and layers of meaning (Years 7–8)
2. Justifies a personal response by analysing how the text positions the readerBuilds on: Explains a personal response and where it comes from (Years 7–8); Explains how texts position readers to accept a point of view (Years 7–8)
3. Reads complex texts strategicallyBuilds on: Reads strategically: pathways, context cues and reading aloud (Years 7–8)
4. Sustains varied reading and responds at lengthBuilds on: Sustains wide reading across diverse perspectives (Years 7–8)
5. Evaluates own reading strategies and the value of readingBuilds on: Reflects on what reading has taught you (Years 7–8); Sustains varied reading and responds at length
6. Analyses the main ideas and thematic concerns of a textBuilds on: Identifies themes and explains how they develop across a text (Years 7–8); Rereads and reinterprets to analyse layers of meaning

Understanding and responding to texts A

1. Analyses complex characters and their structural rolesBuilds on: Analyses how characters are constructed and what they represent (Years 7–8)
2. Analyses figurative language within larger patterns of meaningBuilds on: Identifies imagery, simile, metaphor and personification and analyses their effect (Years 7–8)
3. Examines focalisation, reliability and authorial voiceBuilds on: Analyses narrative voice, focaliser and person (Years 7–8)
4. Analyses narrative conventions across genres, modes and contextsBuilds on: Analyses narrative conventions and the identities they depict (Years 7–8); Analyses genre conventions and how texts exhibit or challenge them (Years 7–8)
5. Analyses how representations are shaped and how texts use or subvert conventionsNot written yetBuilds on: Understands texts as acts of representation and describes them with metalanguage (Years 7–8); Analyses images, film and multimodal texts using their codes and conventions (Years 7–8)

Understanding and responding to texts B

1. Evaluates how arguments are constructed and evidencedBuilds on: Analyses how arguments are constructed and evidenced (Years 7–8)
2. Analyses how context shapes texts, readers and languageNot written yetBuilds on: Explains how cultural context shapes a text and its language (Years 7–8); Explores how culture and time shape language in texts (Years 7–8)
3. Analyses style, from personal signature to literary movementNot written yetBuilds on: Describes the stylistic and aesthetic qualities of a text (Years 7–8)
4. Analyses voice and how the authority of a text is negotiatedNot written yetBuilds on: Analyses personal voice and how a text builds authority (Years 7–8); Evaluates how arguments are constructed and evidenced
5. Evaluates how texts position audiences and how they can be readNot written yetBuilds on: Explains how texts position readers to accept a point of view (Years 7–8); Analyses how context shapes texts, readers and language
6. Analyses how themes hold a text togetherNot written yetBuilds on: Identifies themes and explains how they develop across a text (Years 7–8); Analyses the main ideas and thematic concerns of a text

Understanding and responding to texts C

1. Analyses how genre shapes values and how genres evolveNot written yetBuilds on: Analyses genre conventions and how texts exhibit or challenge them (Years 7–8); Analyses narrative conventions across genres, modes and contexts
2. Analyses adaptation, appropriation and connections between textsNot written yetBuilds on: Analyses connections between texts, including adaptations and appropriations (Years 7–8); Analyses how genre shapes values and how genres evolve
3. Evaluates how audiences negotiate the value of a textNot written yetBuilds on: Explains why texts are valued differently in different contexts (Years 7–8); Evaluates how texts position audiences and how they can be read

Imaginative, discursive and poetic composition

1. Uses the structural conventions of discursive texts to present complex and nuanced ideasNot written yetBuilds on: Discusses a central idea from personal and objective positions (Years 7–8)
2. Experiments with a range of poetic forms to explore ideas and personal perspectivesNot written yetBuilds on: Selects poetic forms and features to express ideas and personal perspectives (Years 7–8)
3. Creates engaging and authentic temporal and spatial settingsNot written yetBuilds on: Creates imaginative texts that drive plot, develop character and build place and atmosphere (Years 7–8)
4. Experiments with the process of transformation to create texts with new meaningNot written yetBuilds on: Reshapes aspects of texts to create new meaning (Years 7–8)
5. Shapes complex complications and crafts authentic characters to depict ideasNot written yetBuilds on: Develops thematic elements around a central complication or conflict (Years 7–8); Creates engaging and authentic temporal and spatial settings

Polished writing

1. Adapts codes and conventions across analytical, informative, persuasive, discursive and imaginative writingNot written yetBuilds on: Controls the structural and grammatical conventions of imaginative, informative and persuasive writing (Years 7–8)
2. Uses the structural conventions of informative texts to build a field of facts and perspectivesNot written yetBuilds on: Composes informative texts that summarise conceptual information (Years 7–8)
3. Uses the structural conventions of persuasive texts to develop expanding argumentsNot written yetBuilds on: Composes persuasive texts with a defining thesis, elaboration paragraphs and a synthesising conclusion (Years 7–8)
4. Uses rhetorical language strategically and subtly to convince an audienceNot written yetBuilds on: Uses rhetorical language to shape ideas and express a perspective (Years 7–8)
5. Sustains a considered thesis and cohesive analytical progression across a full textNot written yetBuilds on: Writes a structured analytical text with introduction, supporting points and conclusion (Years 7–8); Develops a coherent thesis for extended analytical and persuasive texts (Years 7–8)
6. Uses tense accurately and applies narrative voice to depict complex ideasNot written yetBuilds on: Controls active and passive voice and tense for particular purposes (Years 7–8); Chooses first, second or third person narrative voice for effect (Years 3–4)
7. Develops a personal and informed voice and experiments with tone, atmosphere and moodBuilds on: Uses imagery, figurative language and modality to create tone, atmosphere and mood (Years 7–8); Selects topic-specific vocabulary that contributes to the stylistic features of writing (Years 7–8)
8. Expresses ideas in structures that reflect an emerging personal styleBuilds on: Develops a personal and informed voice and experiments with tone, atmosphere and mood

Representing and multimodal composition

1. Composes visual and multimodal texts with aesthetic qualities that influence viewpointsNot written yetBuilds on: Composes visual and multimodal texts that create tone, atmosphere and mood (Years 7–8)
2. Experiments with a combination of modes for specific effect and impactNot written yetBuilds on: Composes visual and multimodal texts with aesthetic qualities that influence viewpoints

Sentence-level grammar and punctuation

1. Maintains agreement in complex structures and controls active and passive voiceNot written yetBuilds on: Controls agreement, prepositions, articles, conjunctions, linking devices and pronouns for cohesion (Years 7–8); Controls active and passive voice and tense for particular purposes (Years 7–8)
2. Crafts elaborated noun and verb groups for effect, clarity or complexityNot written yetBuilds on: Selects noun groups and verb forms to control clarity and express modality (Years 7–8)
3. Applies punctuation to suit text purpose, support meaning and control reader responseNot written yetBuilds on: Applies a wide range of punctuation to control clarity, pace and reader response (Years 7–8)
4. Justifies varied sentence type, length and complexity, and crafts concise sentencesNot written yetBuilds on: Constructs varied sentence structures and controls the four sentence types for purpose (Years 7–8)

Word-level language and spelling

1. Uses grammatical features to describe relationships between complex ideasNot written yetBuilds on: Controls agreement, prepositions, articles, conjunctions, linking devices and pronouns for cohesion (Years 7–8)
2. Selects technical vocabulary and register appropriate to audience, purpose, form and contextNot written yetBuilds on: Selects topic-specific vocabulary that contributes to the stylistic features of writing (Years 7–8)
3. Uses etymology and reference resources to spell unfamiliar, complex and technical wordsNot written yetBuilds on: Spells unfamiliar, complex and technical words using phonological, orthographic and morphological knowledge (Years 7–8); Explains graphemes and roots by their etymology when spelling (Years 5–6)
4. Makes vocabulary choices that enhance style and shape meaning through connotationBuilds on: Selects topic-specific vocabulary that contributes to the stylistic features of writing (Years 7–8); Develops a personal and informed voice and experiments with tone, atmosphere and mood

Planning, monitoring, revising and reflecting

1. Produces co-constructed complex texts representing a diversity of ideas and valuesNot written yetBuilds on: Produces co-constructed texts that represent different ideas and values (Years 7–8)
2. Engages with model texts to develop and refine features, structures and stylistic approachesNot written yetBuilds on: Uses model texts to plan and to shape compositional choices (Years 7–8)
3. Selects collaborative drafting strategies and feedback processes to improve clarity, meaning and effectNot written yetBuilds on: Monitors accuracy and responds to verbal and written feedback (Years 7–8)
4. Researches, evaluates and synthesises sources to generate new ideasNot written yetBuilds on: Integrates information and perspectives from different sources into informed texts (Years 7–8); Assesses the reliability and authority of sources when researching (Years 5–6)
5. Evaluates own authorial decisions and the composition process itselfNot written yetBuilds on: Reflects on own composition using technical vocabulary (Years 7–8); Selects collaborative drafting strategies and feedback processes to improve clarity, meaning and effect
6. Develops an effective thesis based on critical thinking for extended textsNot written yetBuilds on: Develops a coherent thesis for extended analytical and persuasive texts (Years 7–8); Sustains a considered thesis and cohesive analytical progression across a full text
7. Plans a progressive sequence of arguments and sets goals at every levelNot written yetBuilds on: Plans a logical sequence of arguments and sets goals at conceptual, text and paragraph level (Years 7–8); Develops an effective thesis based on critical thinking for extended texts

Speaking

1. Crafts spoken, signed or communicated texts that convey complex ideas to specific audiencesNot written yetBuilds on: Creates spoken, signed or communicated texts that convey ideas and viewpoints to an audience (Years 7–8)
2. Selects rhetorical strategies to position an audience in formal argument and debateNot written yetBuilds on: Uses rhetorical strategies to engage an audience and evoke an emotional response (Years 7–8); Participates in informal, speculative and exploratory discussion about texts and ideas (Years 7–8)
3. Delivers spoken, signed or communicated texts with engaging use of intonation, emphasis, volume, pace and timingNot written yetBuilds on: Delivers a spoken text with control of intonation, emphasis, volume, pace and timing (Years 7–8)
4. Participates in and leads analytical, speculative and exploratory discussionNot written yetBuilds on: Participates in informal, speculative and exploratory discussion about texts and ideas (Years 7–8)
5. Signals the development of ideas through language, structure and presentational featuresNot written yetBuilds on: Selects multimedia components, visual displays and gestural features to bring clarity (Years 5–6); Crafts spoken, signed or communicated texts that convey complex ideas to specific audiences; Analyses images, film and multimodal texts using their codes and conventions (Years 7–8)

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