Place Values and Decimals
Read, write, compare, or partition each number using place value.
Measurement
Rounding to Decimal Points
Round each decimal to the required number of decimal places.
Multiplying and dividing by 10,100,1000
Calculate each using place value.
Scientific notation
Develop secure understanding of scientific notation.
Powers & Roots
Interpret, evaluate, compare, and apply positive, zero, and negative powers together with square and cube roots.
GEMA (formerly BEDMAS)
Develop secure understanding of order of operations (GEMA/BEDMAS).
Measurement prefixes
Develop secure understanding of measurement prefixes and common derived units.
Time
Develop secure understanding of time and time calculations.
Reading scales
Read values from scales, work out intervals, and place values accurately on scales.
Weight
Choose an appropriate metric unit of weight (grams, kilograms, or tonnes) for a real-world object, convert between these units using the hierarchy 1000 g = 1 kg and 1000 kg = 1 tonne, and order or combine weights to solve real-world problems.
Capacity
Convert between millilitres and litres using the metric relationships (1 cm cubed = 1 mL, 1000 mL = 1 L), and compare, order, and solve real-world capacity problems in mixed units.
Converting between length units
Convert measurements between metric length units, moving confidently between millimetres, centimetres, metres, and kilometres.
Estimating lengths from scale drawings and photos
Use a ratio scale of 1:n, converting both lengths to the same unit first, to find a real-world length from a scale drawing or photo, or a scale drawing/photo length from a real-world measurement.
Perimeter (shapes with straight edges only)
Find the perimeter of squares, rectangles, triangles, and regular polygons by summing their given side lengths, including converting the answer to a different unit or applying a stated change to each side.
Area of basic shapes only (squares, rectangles, triangles, parallelograms, kite)
Calculate the area of squares, rectangles, triangles, parallelograms, and kites using the standard area formulas, and work backwards from a given area to find a missing height, diagonal, or side.
Area of L-shapes and composite shapes
Decompose an L-shape or composite shape into rectangles, then find its total area using the addition method (adding the parts) or the subtraction method (subtracting a missing piece from a larger rectangle).
Area of trapezium
Calculate the area of a trapezium using the formula (average of the parallel sides) multiplied by the height, and explain how this formula relates to the area of a rectangle.
Circumference and area of circles
Calculate the circumference and area of a circle (and of semicircles and quarter circles) from its radius or diameter, using pi, and identify the parts of a circle.
Volume of cubes and cuboids
Calculate the volume of a cube using Volume = side cubed and of a cuboid using Volume = length x width x height, find a missing dimension given the volume, and convert between cubic units of measurement.
Volume of prism
Calculate and apply the volume formula for prisms, including cylinders, using base area and depth measurements.
Liquid volume
Apply the equivalence 1 cm cubed equals 1 millilitre (and 1000 cm cubed equals 1 litre) to convert between a solid's volume and its liquid capacity, and to calculate the liquid capacity of rectangular and triangular-prism containers.
Rounding to significant figures and applying precision to calculations
Count how many significant figures a number has, round a number to a stated number of significant figures, convert between ordinary numbers and scientific notation, and round the result of a calculation to a given number of significant figures.
Converting currency using an exchange rate, and working with ratios
Convert between currencies using a given exchange rate by the unitary method, and write, simplify, share, and apply ratios (including three-part ratios) to real quantities.
Perimeter and area of composite shapes involving circles
Calculate the perimeter and area of composite shapes built from rectangles or squares combined with semicircles or quarter circles, leaving answers in terms of pi where appropriate and rounding appropriately otherwise.
Drawing nets of basic shapes
Understand and draw nets of three-dimensional shapes, recognizing how 2D nets fold into 3D shapes and visualizing 3D shapes from their nets.
Surface area
Calculate the surface area of prisms and cylinders using nets and formulas; apply understanding to compound shapes.
Volume of other 3D shapes (cylinders, cones, pyramids, spheres, etc)
Calculate the volume of a cylinder, cone, pyramid, or sphere from given dimensions, and work backwards algebraically to find an unknown length when the volume is known.
Forming measurement expressions
Write algebraic expressions to represent measurements and use them to solve practical problems.
Solving measurement equations
Substitute known values into a measurement formula (perimeter, area, volume, or distance = speed x time), set it equal to a given value, and solve for the unknown, keeping units attached to the answer.
Advanced Power Rules
Evaluate and simplify expressions with negative and fractional indices, converting between index form, root form, and positive-index form.
Budgets
Read and construct a simple budget -- calculating amounts spent, saved, or remaining from income and expenses, finding a percentage of a budget for a category, and reasoning about surplus and deficit.
Rates and compound measures (speed, density, and best buys)
Calculate and compare rates and compound measures (speed, density, and unit price/best buys) using the product relationship top = left x right, and interpret distance-time and speed-time graphs.
Compound shapes with algebra
Form and fully simplify an algebraic expression for the area of a compound shape (an L-shape, a rectangle with a rectangular hole, a kite, or a trapezium) whose side lengths are algebraic terms or expressions, then solve for an unknown side length when a target area is given.
Metric Units and Conversions
Convert between adjacent metric units of length, mass, and capacity (e.g. mm/cm/m, g/kg, mL/L) using the prefix scale.
Reading and Using Timetables
Read a given timetable to identify a single stated departure or arrival time, and convert simple times between 12-hour and 24-hour notation.
Map Scales and Scale Drawings
Use a simple stated ratio scale (e.g. 1 cm : 5 km) to convert a single map distance into a real-world distance or vice versa.
Grid References and Map Reading
Read and state simple four-figure grid references for labelled points on a given map.
Vectors, Bearings and Grid References
Read and state three-figure bearings from a diagram, plot points using grid references, and write a simple displacement as a column/component vector.
Reading Measurement Scales and Interpreting Tables
Read a single value directly off a simple linear scale (ruler, jug, thermometer) and look up a single value from a one-way reference table.
Area, Perimeter and Circumference of 2D Shapes
Find the perimeter and area of simple rectangles, squares, and triangles from given dimensions using standard formulas.
Volume
Calculate the volume of a cube using Volume = side cubed and of a cuboid using Volume = length x width x height, find a missing dimension given the volume, convert between cubic units of measurement, and reason about volume in real-world, comparison, and algebraic contexts.
Converting units
Convert measurements of area between metric units using the linear conversion factor squared, convert measurements of volume between metric units using the linear conversion factor cubed, and calculate an area or volume from given dimensions and then convert the result into a different metric unit.
Algebra
Integers
Develop secure understanding of integers.
Substitution
Substitute numbers for variables in expressions and evaluate.
Forming expressions
Write an algebraic expression for each description.
Multiplying Expressions (no powers)
Simplify products of numbers and unlike variables, model them with area, and explain why powers are not used yet.
Adding and Subtracting Expressions (Like Terms)
Identify matching terms, collect their coefficients, and simplify algebraic and perimeter expressions accurately.
Multiplying Expressions (with powers)
Simplify products with the same base by adding exponents, apply the rule in algebra and area contexts, and explain what stays out of scope.
Dividing Expressions
Simplify quotients by dividing coefficients, cancelling common factors, and subtracting exponents for matching bases.
Simplifying Expressions with GEMA / BEDMAS
Evaluate or simplify numeric and algebraic expressions by applying the correct order of operations (GEMA -- Grouping symbols, Exponents, Multiplication/Division, Addition/Subtraction), including substituting given values, inserting brackets to reach a target value, and identifying errors in an incorrect order-of-operations argument.
Expanding single brackets
Expand and simplify each expression.
Factorising single brackets
Find the highest common factor of the terms and write the expression as a single bracket.
Defining Variables & Forming Equations
Define variables to represent unknown quantities and translate word problems or real-world situations into algebraic equations.
Solving 1-step equations
Solve each one-step equation using inverse operations.
Solving 2-step equations
Solve each two-step equation by undoing the operations in reverse order.
Solving equations with variables on both sides
Collect variable terms on one side and constants on the other, then solve each equation by undoing operations in reverse order.
Forming and solving equations from context
Form equations from word problems, solve them, and interpret solutions in context.
Solving inequalities
Solve one-step and two-step linear inequalities, including those requiring expanding brackets, and correctly reverse the inequality sign when multiplying or dividing by a negative number, applying this to real-world word problems.
Re-arranging formula
Rearrange a given formula to make a specified variable the subject, by undoing multiplicative, additive, fractional, bracketed, or squared/rooted operations in the correct inverse order.
Expanding double brackets
Expand a pair of binomial brackets using the distributive property (FOIL), and simplify the result by collecting like terms.
Factorising double brackets (with a leading coefficient of 1)
Factorise a quadratic expression of the form x squared + bx + c (with a leading coefficient of 1) into two brackets by finding two numbers that multiply to give c and add to give b.
Factorising special double brackets (perfect squares, difference between squares)
Recognise and factorise perfect-square trinomials and differences of two squares into double brackets.
Solving equations with fractions
Solve linear equations containing fractions -- including a fraction of the variable, a variable in the denominator, two fractional terms, or a fraction set equal to another fraction -- by clearing the fractions using the lowest common denominator.
Solving quadratics
Solve a quadratic equation with leading coefficient 1 by factorising it (including recognising differences of two squares and perfect-square trinomials), rearranging it to equal zero first if needed, and applying the zero product property, recognising that a quadratic can have zero, one, or two real solutions.
Simplifying algebraic fractions
Simplify algebraic fractions by factorising the numerator and denominator first and cancelling common factors, and apply this to adding, subtracting, multiplying, and dividing algebraic fractions.
Recognising equivalent algebraic expressions
Recognise when expressions are equivalent and justify the decision by collecting like terms, expanding, or using substitution.
Factorising quadratics where a ≠ 1
Factorise quadratics of the form ax^2 + bx + c where a is not 1, by taking out any common factor first and then splitting the middle term so the four resulting terms factorise in matching pairs.
Solving quadratics where a ≠ 1
Solve a quadratic equation ax^2 + bx + c = 0 where a is not 1, by factorising -- splitting the middle term after finding two numbers that multiply to a*c and add to b, then applying the zero-product property -- including equations already given in factorised form and equations that first need rearranging to zero.
Simplifying fractions with unfactorised quadratic in denominator or numerator
Simplify an algebraic fraction by first factorising the unfactorised quadratic in the numerator and/or the denominator, then cancelling the common factor and stating any restriction on the variable.
Solving Linear Equations
Solve linear equations by undoing operations in reverse order, collecting variable terms when the unknown appears on both sides, and leaving fraction or surd answers exact where they are not whole numbers.
Solving Linear Inequalities
Solve a linear inequality using the balance method, flipping the inequality sign when multiplying or dividing both sides by a negative number, and represent or read a solution set on a number line, including for compound (double) inequalities.
Surds -- simplifying, multiplying, dividing, and adding/subtracting
Simplify surds using the largest perfect-square factor, multiply and divide surds by combining under one root, add and subtract like surds, and rationalise a denominator.
Level 2 algebra review
Review and connect Level 2 algebra skills by manipulating expressions, solving equations, applying negative and fractional indices, and rearranging increasingly complex formulae.
Simplifying Expressions with GEMA / BEDMAS
Evaluate numerical and algebraic expressions using the correct order of operations (GEMA/BEDMAS) — grouping symbols, then exponents, then multiplication/division left to right, then addition/subtraction left to right.
Writing expressions
Translate a worded phrase or real-world context into an algebraic expression, and check a formed expression by substituting a test value.
Adding and Subtracting Expressions (Like Terms)
Simplify algebraic expressions by collecting like terms and like surds, distributing a single bracket before collecting, and naming the number property used at each step.
Expanding single and double brackets
Expand a single bracket using the distributive property, expand two linear brackets (FOIL) and collect like terms, and recognise the perfect-square and difference-of-squares patterns.
Linear number patterns and the nth term
Find the nth-term rule for a linear number pattern, use it to find any term or term number, and derive the rule from two given terms.
Introduction to Complex Numbers
Define the imaginary unit i, simplify powers of i, and add, subtract and multiply complex numbers in rectangular form a+bi.
Complex Number Arithmetic: Multiplying and Dividing
Multiply two complex numbers in rectangular form by expanding brackets and simplifying using $i^2=-1$, and divide by rationalising with the conjugate.
Complex Numbers in Polar Form
Find the modulus and argument of a given complex number and convert it between rectangular form a+bi and polar form r(\cos\theta+i\sin\theta).
Complex Roots of Polynomials
Use the factor theorem to verify a given real root of a cubic polynomial and divide to find the remaining quadratic factor.
De Moivre's Theorem: Powers and Roots of Complex Numbers
Convert a complex number between rectangular (a+bi) form and polar (modulus-argument) form, and state De Moivre's theorem for integer powers.
Loci in the Complex Plane
Plot and interpret a simple locus such as |z - a| = r as a circle on the Argand diagram, stating its centre and radius.
Ratios and Rates: Simplifying, Dividing and Comparing
Write and simplify ratios in the form a:b, and identify equivalent ratios using concrete diagrams or objects.
Capacity, Density and Time-Based Rates
Convert between common capacity units (mL, L, kL) and read/convert simple 24-hour clock times, and calculate a straightforward density given whole-number mass and volume values.
Ratios: Simplifying, Equivalent Ratios, and Sharing a Quantity
Write and simplify a ratio between two quantities, and share a small whole-number amount into two parts using a given ratio such as $3:2$.
Operations with Decimals
Add and subtract decimals with matching numbers of decimal places, including simple money and measurement contexts.
Solving Irrational (Square-Root) Equations
Solve equations with a single square-root term by isolating it and squaring once, then verify the solution satisfies the original equation.
Graphing Rational Functions / Rectangular Hyperbolae of the Form (ax+b)/(cx+d)
Identify the vertical asymptote (where cx+d=0) and horizontal asymptote (y=a/c) of a given function of the form (ax+b)/(cx+d) by inspection.
Standard Form (Scientific Notation)
Convert large whole numbers between ordinary form and standard form, ensuring the leading coefficient satisfies 1 ≤ a < 10.
Rounding Numbers to Decimal Places and Significant Figures
Round whole numbers and decimals to the nearest 10, 100, or one decimal place.
Estimating Measurements and Rounding to Significant Figures
Round given measurements to a stated number of significant figures and make sensible estimates of simple lengths, masses, or capacities using familiar benchmarks.
Standard Form and Estimation
Round whole numbers and decimals to a given number of decimal places or significant figures, and convert simple large/small numbers between ordinary and standard form.
Integer (Directed Number) Operations and Order of Operations
Add and subtract integers using a number line or counters, including simple negative results.
AS91027 Mixed Algebraic Procedures Practice (Achievement)
Mixed short-answer questions applying one algebraic procedure at a time (simplify, expand, or solve a one-step equation) with clear working expected.
Excellence-Level Algebra: Multi-Step Problem Solving with Justification
Combine two related algebraic procedures (e.g. expand then solve) in a guided multi-step problem with clear prompts for each step.
Merit-Level Algebraic Methods: Multi-Step Worked Problems
Combine two straightforward algebraic steps in sequence, such as expanding brackets and then collecting like terms to simplify an expression.
Solving Simultaneous Linear Equations by Substitution and Elimination
Solve simple simultaneous equations by substitution where one equation already gives a variable in terms of the other, or by elimination where coefficients already match.
AS91027 Exam-Style Algebra Review (Mixed Practice Exam)
Single-procedure exam-style questions (expand, factorise, simplify, or solve a one/two-step equation) matching typical Achievement-level marking-schedule expectations.
Solving Linear Inequalities
Solve one- and two-step linear inequalities with positive coefficients and represent the solution set on a number line.
Solving Linear Equations
Solve one-step and two-step linear equations with the pronumeral on one side, using inverse operations.
Graphing Linear Inequalities as a Region
Graph a single linear inequality such as $y > 2x - 1$ on a grid, correctly choosing a solid or dashed boundary line and shading the correct side.
Sketching a Straight Line from Its Equation
Complete a table of values for a simple linear equation and plot the points to draw a straight-line graph.
Graphical Models: Sketching and Interpreting Real-World Graphs
Read values, intercepts, and simple gradients off a given real-world graph (e.g. distance-time) to answer direct questions.
Linear Programming: Feasible Regions and Optimisation
Graph a single linear inequality (e.g. $y \le 2x + 3$) on a set of axes, shading the correct side of the boundary line.
Solving Cubic Equations
Solve cubic equations already given in fully factorised form using the null factor law.
Solving Quadratic Equations Using the Quadratic Formula
Substitute given integer values of a, b, and c directly into the quadratic formula to find two rational solutions.
Solving Quadratics Using the Quadratic Formula and Completing the Square (Combined)
Substitute coefficients into the given quadratic formula to solve simple monic quadratics with rational or surd solutions.
Determining the Nature of the Roots of a Quadratic (Discriminant)
Calculate the discriminant $b^2-4ac$ for a given quadratic and state whether it is positive, negative, or zero.
Solving Exponential Equations
Solve equations where one side is already a power of the same base as the other, such as 2^x = 16.
Logarithm Laws and Solving Exponential Equations Using Logarithms
Convert between exponential and logarithmic form and evaluate simple logarithms (e.g. base 10 and base 2) without a calculator.
Solving Exponential and Logarithmic Equations
Solve exponential equations by expressing both sides with the same base and equating exponents, and evaluate simple logarithmic equations of the form $\log_a x = b$.
Logarithmic Functions
Convert simple expressions between exponential and logarithmic form and evaluate logarithms of common bases by hand.
Exponential Graph Transformations and Modelling
Sketch a graph of the form $y = a \cdot b^x$ from given values of $a$ and $b$, marking the y-intercept and horizontal asymptote.
Transformations of the Parabola
Identify and sketch simple translations of y=x^2 given equations in vertex form with a=1, stating the shift left/right and up/down.
Parabolas in Completed-Square (Turning-Point) Form
Complete the square for monic quadratics (a=1) and state the resulting turning point coordinates.
Completing the Square
Complete the square for monic quadratics with integer coefficients, e.g. rewriting $x^2+bx+c$ as $(x+p)^2+q$.
Identifying Key Features of a Parabola from Expanded Form
Read the y-intercept directly from an expanded quadratic and substitute x-values to plot points.
Identifying Features of Graphs (Algebraic, Non-Calculus)
Read x- and y-intercepts and describe basic shape features directly off given linear and quadratic graphs.
Finding and Classifying Maximum and Minimum Points Using Calculus
Find stationary points of a simple polynomial by solving f'(x)=0 and read off their coordinates.
Anti-Differentiation: Reversing the Power Rule
Anti-differentiate single power terms such as $ax^n$ using the reverse power rule, writing the general antiderivative with $+c$.
Integration of Polynomials
Integrate single polynomial terms of the form $x^n$ using the power rule, correctly including the constant of integration.
Integrating Exponential Functions
Find the indefinite integral of simple exponential functions of the form $e^{kx}$ using the rule $\int e^{kx}\,dx = \frac{1}{k}e^{kx}+c$.
Integrating Expressions of the Form (ax+b)/x
Split a simple expression like (x+3)/x into 1 + 3/x and integrate termwise to get x + 3\ln|x| + c.
Integrating Rational Functions of the Form (ax+b)/(cx+d)
Rewrite a simple rational function (ax+b)/(cx+d) using division to isolate a constant term plus a fraction with constant numerator, then integrate directly to obtain a linear term and a $\ln|cx+d|$ term.
Integrating Rational Functions of the Form f'(x)/f(x)
Integrate simple rational functions where the numerator is exactly the derivative of the denominator, e.g. recognising forms like \int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c.
Integrating Trigonometric Products
Integrate single trig functions and simple linear-argument cases such as cos(3x) using standard antiderivative rules.
Integration of Trigonometric Functions
Find indefinite integrals of single trig terms such as sin(ax) and cos(ax) using standard antiderivative rules.
Integration by Substitution
Integrate a simple composite function such as (ax+b)^n or e^{ax+b} using a given or obvious linear substitution u = ax+b.
Numerical Integration (Trapezium and Simpson's Rules)
Apply the trapezium rule with a given number of equal-width strips to estimate the area under a simple curve from a table of values.
Area Under a Curve (Integration)
Estimate the area under a simple curve between two given x-values using grid-square counting or a basic trapezium approximation, and evaluate a definite integral of a single power-of-x term.
Definite Integrals
Evaluate definite integrals of simple polynomial functions using antiderivatives and the fundamental theorem of calculus.
Particular Solutions of Differential Equations
Find the general solution (including +C) by integrating a simple polynomial derivative function.
Linear Graphs from Algebraic Rules (Table of Values)
Use a linear rule written as f(x) or y to calculate outputs, complete a table of values, write ordered pairs, plot the points, and draw and interpret the straight-line graph.
Finding the Rule for Quadratic Number Patterns
Find the rule y = ax^2 + bx + c from a table using first and second differences, then check the rule.
Solving Differential Equations by Separation of Variables
Separate and integrate simple differential equations of the form dy/dx = f(x), such as dy/dx = 4x, to find a general solution.
Solving Second-Order Differential Equations
Find the auxiliary (characteristic) equation of a given second-order homogeneous differential equation and solve it for real, distinct roots to write the general solution.
Factorising quadratics where a ≠ 1: slide and divide
Factorise ax^2 + bx + c (a ≠ 1, no common factor) by the slide-and-divide method: slide a across to multiply c, factorise the a = 1 look-alike, divide both bracket numbers by a, simplify, then slide any remaining denominator back in front of x.
Factorising quadratics where a ≠ 1: box method
Factorise ax^2 + bx + c (a ≠ 1, no common factor) with a 2-by-2 box: put ax^2 top-left and c bottom-right, split bx into two terms whose product is ac x^2, fill the other two cells, then read the brackets from the HCF of each row and column.
Factorising quadratics where a ≠ 1: cross method
Factorise ax^2 + bx + c (a ≠ 1, no common factor) with the cross method: write a factor pair of a down the left and a factor pair of c down the right, multiply along the two diagonals of the cross, and when the cross-products add to b read each bracket across a row.
Factorising quadratics with a common factor first
Factorise a quadratic fully by taking out the highest common factor of all three terms first (including -1 when the leading coefficient is negative), then factorising the quadratic left inside the bracket.
Introduction to logarithms
Convert between index form and logarithm form by holding the base and swapping the index and the answer across the equals sign, evaluate simple logarithms without a calculator, solve a single-logarithm equation by converting it, and prove the log laws from the index laws.
Number
Forming Fractions
Write a fraction (numerator over denominator) to represent a "part out of whole" description or a shaded diagram, including forming the complementary fraction for what remains, without needing to simplify (simplifying is a separate downstream LO).
Equivalent Fractions
Identify, generate, and justify equivalent fractions by multiplying or dividing the numerator and denominator by the same scale factor, including completing missing-value equivalents and using stacked bar/circle diagrams to see the same shaded length or area cut into finer or coarser parts.
Simplifying Fractions
Divide the numerator and denominator of a fraction by their highest common factor (HCF) to write it in simplest form, recognise when a fraction is already fully simplified, and use a simplify-bar diagram to see the same shaded length re-cut into coarser groups.
Converting between improper fractions and mixed numbers
Convert between improper fractions and mixed numbers.
Adding and subtracting fractions
Add or subtract each fraction. Simplify where you can.
Multiplying Fractions
Multiply each fraction. Simplify where you can.
Reciprocals
Find the reciprocal of each value.
Dividing fractions
Divide each fraction. Simplify where you can.
Finding fractions of quantity
Find each fraction of the quantity.
Multiples & Lowest Common Multiples
Find the multiples or lowest common multiple.
Factors, Highest Common Factors & Primes
List factors, identify primes, find HCFs, and use prime factor decompositions in product and index form.
Irrational numbers
Classify numbers — including surds, decimals, and π — as rational (can be written as a fraction of two integers) or irrational (cannot), justifying the classification.
Mixed Numbers
Convert between mixed numbers and improper fractions, and add, subtract, compare, and apply mixed numbers in real-world contexts such as recipes and measurement, simplifying results where needed.
Prime Factors
Decompose a number into prime factors using a factor tree, then express the result in product form and index form.
Basic Power Rules
Simplify expressions involving powers of the same base by adding the exponents when multiplying, subtracting them when dividing, and multiplying them when raising a power to another power.
Simplifying algebraic fractions with an unfactorized quadratic
Simplify each algebraic fraction by factorising the unfactorised quadratic side into two brackets, then cancelling the bracket that matches the other side.
Factorising into single and double brackets
Factorise an expression by extracting its highest common factor into a single bracket, factorise a trinomial $x^2+bx+c$ into two brackets, and recognise the perfect-square and difference-of-squares patterns.
Negative and Fractional Indices
Apply the zero-index and negative-index rules to rewrite numerical and simple algebraic expressions with positive indices only.
Multiplying Fractions
Multiply a proper fraction by a whole number or by another simple proper fraction using visual area models or repeated addition.
Operations with Algebraic Fractions
Add or subtract two algebraic fractions that already share a common monomial denominator, and multiply or divide simple algebraic fractions with single-term numerators and denominators.
Simplifying and Solving with Algebraic Fractions
Factorise simple numerators and denominators (common factor or difference of squares) and cancel to simplify a single algebraic fraction.
Solving Equations with Algebraic Fractions (Rational Equations)
Solve equations containing a single algebraic fraction set equal to a whole number or another simple fraction, using cross-multiplication with one denominator.
Expanding and Factorising Algebraic Expressions
Expand a single bracket and collect like terms in simple linear expressions.
Parabola Factorised Form (a != 1)
Identify the x-intercepts and axis of symmetry directly from a factorised-form equation y=a(x-p)(x-q) with a given numeric value of a.
Graphing Higher-Order Polynomials from Factorised Form
Read off the x-intercepts and y-intercept from a given factorised cubic or quartic equation and plot them on a set of axes.
The Remainder and Factor Theorems
Evaluate a polynomial P(x) at a given value and use the remainder theorem to state the remainder when P(x) is divided by a linear factor (x-a), without performing full division.
Expanding and Factorising Using the Distributive Law (Foundational)
Expand a single bracket multiplied by a positive numeric coefficient, e.g. simplifying $3(x+2)$, and combine with simple like terms.
Solving Quadratic Equations by Factorising (Null Factor Law)
Solve simple quadratic equations already given in factorised form, or that factorise directly with a coefficient of 1, using the null factor law.
Solving Factorised Equations (Null Factor Law)
Solve two-factor equations of the form $(x-a)(x-b)=0$ by applying the null factor law directly.
Parabolas: Vertical Scale Factor
Sketch and compare y = x^2 with y = ax^2 for simple positive integer values of a, describing the parabola as narrower or wider.
Integrating 1/x and Related Reciprocal Functions
Recognise that $\int \frac{1}{x}\,dx = \ln|x| + C$ and apply it to simple constant-multiple integrals.
Statistics
Relationship between percentages, fractions, and decimals
Convert fluently between percentages, fractions, and decimals, and use each form to compare and order quantities.
Forming percentages
Express one quantity as a percentage of another (part divided by whole, times 100), converting units first where needed, and work backwards from a percentage to find a missing amount or total.
Percentage of a given quantity
Find a percentage of a quantity (using the decimal-multiplier method), and apply this to financial contexts including GST and simple interest.
Increasing and decreasing by a given percentage
Find the new amount after increasing or decreasing a quantity by a given percentage, using the decimal-multiplier method (1 plus or minus the rate).
Investigation types
Classify a statistical question as a summary, comparison, relationship, or time-series investigation, and classify the data involved as qualitative/quantitative, discrete/continuous, and primary/secondary.
Types of data
Classify data as qualitative or quantitative, discrete or continuous, and primary or secondary to support statistical investigation and data collection decisions.
Calculating averages and range
Calculate the mean, median, and mode (the three measures of centre) and the range of a dataset, work backwards from a given mean to find a missing value, and judge which measure of centre best represents a dataset (especially when an outlier is present).
Quartiles
Order a dataset and split it into quarters to find the lower quartile (Q1), the median (Q2), and the upper quartile (Q3), calculate the interquartile range (IQR = Q3 minus Q1) as a measure of spread, and read the five-number summary (minimum, Q1, median, Q3, maximum) from a box-and-whisker graph.
Analysing time series graphs - TSN
Analyse time series graphs using the TSN framework -- identifying the long-term Trend, Seasonality, and Noise present in a dataset, and explaining what each component reveals.
Analysing scatter plots (i.e. bivariate) - TARSOG
Analyse bivariate scatter plots using the TARSOG framework -- describing the Trend, Association, Relationship strength, Scatter, Outliers, and Groupings present in the data.
Comparing box-and-whisker graphs - CSSI
Compare distributions shown as box-and-whisker plots using the CSSI framework -- Centre (median, mean), Shape (left skew, right skew, symmetric/bell curve), Spread (IQR, range), and Interesting features (outliers, groups).
Misleading graphs
Identify and explain what makes a graph misleading -- truncated or broken axes, inconsistent scales, area-distorted pictograms, and other techniques -- and describe the five features of a good, honest graph.
Planning and collecting multivariate data
Plan and execute data collection for studies involving multiple variables, making decisions about sampling methods, study design, and bias reduction to ensure valid and representative results.
Reverse percentage
Given an amount after a percentage increase or decrease, find the original amount by dividing by the multiplier (1 plus or minus the rate), working backwards through the change.
Linear model for scatter plots
Form the equation of a line of best fit (y = mx + c) from its gradient and y-intercept, or from two points on it, and use the equation to predict values, distinguishing interpolation from extrapolation and discussing confidence in the prediction.
Sampling variation
Understand that repeated random samples from the same population give different estimates (sampling variation), and that increasing the sample size (e.g. from 30 to 100 to 1000) reduces the amount of variation and gives a more reliable estimate.
Sampling methodology
Describe and evaluate the pros and cons of five sampling methodologies -- simple random, systematic, stratified, cluster, and convenience -- and identify which method a given scenario or diagram uses.
Interpreting data from Time series - TSN
Read and quantify values from time-series graphs, then use Trend, Seasonality and Noise (TSN) to describe patterns, identify cycles and unusual points, calculate change, and make a justified prediction.
Expected outcome (expected value)
Predict how many times an event will occur over n trials using Expected value = n x p, where p is the theoretical probability; emphasise this is a prediction/average, not a guaranteed result, and connect to the Law of Large Numbers (a larger n brings the experimental proportion closer to the theoretical probability).
Frequency tables
Construct a frequency table from raw data or a tally chart (ungrouped, one row per discrete value, or grouped into class intervals), then read a completed table to find the mode, the total number of data points, and threshold-style counts that require summing frequencies across several rows (e.g. "how many scored 5 or more").
Frequency trees
Read a frequency tree (a two-stage tree diagram labelled with frequencies/counts rather than probabilities), find a missing frequency by subtracting known branches from the relevant stage total, and convert a frequency to a probability or proportion by dividing by the relevant total (the grand total for an unconditional probability, or a single branch's total for a conditional proportion).
Interpreting data from box and whisker - CSSI
Read and compare box-and-whisker plots using the CSSI framework (Centre, Spread, Shape, Interesting features), and apply the 75%/50% rule for informal inference -- if one group's median sits outside the interquartile range (the middle 50%) of the other group, that is evidence of a real difference between the groups. Distinct from the existing lo-comparing-box-and-whisker-graphs-cssi, which already reads/compares centre and spread from diagrams but never teaches the median-outside-IQR informal-inference rule; every tier here drills that specific rule.
Interpreting data graphs
Read values off pie charts, bar graphs, histograms, and pictograms, and use them to draw conclusions (comparisons, totals, percentages) -- not to construct the displays, which sits in the more granular pie/pictogram and bar/histogram LOs. Includes an introductory sample-versus-census discussion as a lead-in to why data is displayed graphically.
Measures of centre/spread
Read a dot plot to find the mode (tallest stack), median (middle dot), and range (first-to-last occupied value), and calculate the mean, for a data set -- including working backwards from a given mean to find a missing value, and judging which measure of centre best represents a data set that contains an outlier. Discourage the word "average" in favour of naming the specific measure. Distinct from the existing lo-calculating-averages-and-range, which is entirely computation-only (a list of raw numbers, no diagram anywhere); this LO's task decks embed a live \PicStatisticsDotPlot diagram in every card, so students read three of the four measures directly off a genuine display before calculating the mean.
Interpreting data from Scatter plot - TARSOG
Read and quantify data directly off a bivariate scatter plot, then use the TARSOG framework (Trend, Association, Relationship, Scatter, Outliers, Grouping) to describe what it shows -- including naming the explanatory/response variable, using the line of best fit to predict a value, and judging whether that prediction is interpolation or extrapolation. Distinct from the existing lo-analysing-scatter-plots-i-e-bivariate-tarsog, which mixes verbal TARSOG-vocabulary recall with some diagram reads; this LO's task decks embed a live \PicStatisticsScatter diagram in every card and always require a numeric prediction, an outlier's quantified gap from the trend, or an explanatory/response judgement, not just a TARSOG label.
Tree diagrams
Construct and read a probability tree diagram for two consecutive events, with each branch labelled by a probability (not a count -- distinct from LO2's frequency tree); find a combined-event probability by multiplying along the branches leading to it, and distinguish "with replacement" (independent events, unchanged second-stage probabilities) from "without replacement" (dependent events, second-stage probabilities change because an item is removed).
Basic Probability
Introduce probability as a value between 0 (impossible) and 1 (certain), using everyday likelihood language before calculation; distinguish theoretical from experimental probability; and use a sample space (dice, spinners, cards, counters) to find a basic probability as favourable outcomes over total outcomes.
How we get data
Explain where statistical data comes from, decide when a sample or a census is appropriate, and classify how a variable is collected -- by counting, by measuring, or by observing and placing it in a category.
Statistical investigation types
Classify a statistical question as a summary, comparison, relationship, or time-series investigation, and match each investigation type to the graph or display best suited to it.
Selecting appropriate displays
Match a data-collection scenario to the most appropriate statistical display (bar graph, pie chart, pictogram, histogram, line graph / time series, dot plot, scatter plot, or box-and-whisker graph) by reasoning about the nature of the data (categorical vs numerical, individual values vs grouped intervals, a single variable vs a relationship between two, a snapshot vs change over time) and justify the choice.
Interpreting pie charts & pictograms
Interpret a given pie chart or pictogram -- read percentages directly, convert a sector's percentage or a pictogram's icon count into an actual amount using a stated total or key, and (for higher-ability students) calculate or work backwards from a sector's angle using angle = fraction of the data x 360 degrees.
Interpreting bar graphs & histograms
Interpret a given bar graph or histogram -- read values directly, add frequencies across categories or intervals, calculate a weighted average across groups of different sizes, and (for higher-ability students) estimate a mean or median from grouped data, read a cumulative frequency table, and explain why comparing histogram bar heights directly requires equal class widths.
Writing a statistical paragraph
Write statistical paragraphs that clearly communicate findings from data, using evidence-based statements supported by specific numerical data and contextual explanation.
Constructing and Selecting Statistical Displays
Construct a simple bar graph or pictograph from a given frequency table, including a title, labelled axes, and an even scale.
Dot Plots and Stem-and-Leaf Plots
Construct a dot plot from a small numeric data set and read off simple features such as the mode and range.
Grouped Frequency Tables
Complete a grouped frequency table by tallying a small raw data set into given equal-width class intervals.
Combined Probability: Tree Diagrams and Two-Way Tables
List all outcomes of two combined simple events (e.g. tossing a coin and rolling a die) using a two-way table and find basic probabilities from it.
Multiplicative Principle for Combined Probability Events
Apply the multiplicative principle to find the total number of outcomes or the probability of two simple independent events occurring together, such as a coin flip and a die roll.
Comparing Data Sets with Box Plots and Measures of Spread
Order a small data set, find the median, quartiles and range, and construct a simple box plot from given values.
Making an Inference from Sample Data
Compare two sample box plots and state which group has the higher median, using correct statistical language.
Calculating and Interpreting Standard Deviation
Use a calculator or given formula to find the standard deviation of a small data set and state what it measures.
Inverse Normal Distribution
Find a z-score from a given area/probability using an inverse normal table or calculator function, for simple upper-tail and lower-tail cases.
The Standard Normal Distribution
Students read and sketch a normal curve, identify the mean and standard deviation from a context, and use the empirical (68-95-99.7) rule to estimate probabilities.
Evaluating Statistical Claims and Misleading Reports
Identify basic features of a graph or report (title, axis labels, scale, source) and spot an obviously exaggerated or truncated axis.
Understanding and Calculating Risk
Calculate a simple risk as a proportion or percentage from a given frequency out of a total population or sample.
Percentage Change: Increase, Decrease and Reverse Percentages
Calculate a given percentage of a whole-number amount and convert fluently between simple percentages, fractions, and decimals.
Percentage Applications: GST, Interest and Depreciation
Calculate GST-inclusive or GST-exclusive prices and simple interest amounts using a given rate and time period.
The Statistical Enquiry Cycle (PPDAC)
Name and correctly order the five PPDAC stages, and match short investigation scenarios to the stage they describe.
Samples and the Population
Define population and sample, and identify which is which in simple given scenarios.
Probability
Tools of chance
Use coins, dice, and playing cards to find theoretical probabilities, using the language of chance (sample space, outcome, favourable outcome, mutually exclusive, independent/dependent, certain/impossible).
F/T
Calculate experimental probability (relative frequency) as favourable trials over total trials, using sample sizes of 30 and 1000, and compare it to theoretical probability via the Law of Large Numbers.
Probabilities from experiments and statistical displays
Read and complete frequency tables (and tally charts) recording the results of a chance experiment, use them to find experimental probabilities (relative frequencies), and compare the recorded distribution across all outcomes to a theoretical model to judge whether an experiment looks fair.
Trees/Combined Events
Read and complete probability tree diagrams for two consecutive (combined) events -- filling in missing branch probabilities and leaf outcomes -- then find a combined-event probability by multiplying along a pathway (AND) or adding qualifying pathways (OR), for both independent (with replacement) and dependent (without replacement) events.
Two-way tables
Read and complete two-way tables (categorised counts with row and column totals), then use them to find probabilities -- simple, AND (intersection), OR (union), and informal conditional ("given") probabilities -- and compare conditional rates across groups.
Expected Number
Use Expected Number = probability x number of trials to predict a result, and explain that this is an average or estimate -- the actual outcome may differ slightly (or, for a small number of trials, considerably) due to chance.
Venn Diagrams
Shade and interpret two- and three-set Venn diagrams, reading and calculating cardinalities (n(A), n(A union B), n(A intersect B), n(A')), working backwards from a stated total to find a missing region, and combining Venn diagrams with basic algebra to find an unknown region.
Set notation
Use set notation as the symbolic language for describing Venn diagram regions -- roster notation {}, membership (element/not an element), the universal set and empty set, subset, union, intersection, complement, and cardinality n(A) -- translating fluently between a shaded Venn diagram region and its set notation expression in both directions.
Probability assumptions
Understand and critically evaluate the core assumptions behind theoretical probability (fairness, equal likelihood, randomness) and recognise when these assumptions fail in real-world contexts.
Conditional and independent events
Distinguish independent events (P(A and B) = P(A) x P(B), one event does not affect the other) from conditional/dependent events (P(B given A), one event changes the other's probability, typically from drawing without replacement), and test independence by comparing P(B) with P(B given A) using a tree diagram, a Venn diagram, or a two-way table.
Patterns and Graphs
Extrapolating linear, quadratic, and exponential patterns
Extend linear, quadratic, and exponential sequences term by term using constant first differences, constant second differences, or constant ratios.
Identifying linear, quadratic, and exponential patterns
Identify whether a pattern is linear, quadratic, or exponential by analysing the differences or ratios between consecutive terms.
Spatial patterns (including drawing next pattern and converting to pattern)
Draw, continue, and represent spatial patterns; identify the change from one term to the next; and use the pattern rule to find specific terms.
Determining nth term of pattern or spatial pattern
Find the nth-term rule for a linear sequence and use it to find any term in the sequence or determine whether a number appears in the pattern.
Understanding Cartesian plane and co-ordinates
Read and write ordered pairs on a Cartesian plane; identify axes, origin, quadrants, and axis points; and compare points using shared coordinates.
Distance between two points (Pythagoras)
Use Pythagoras' theory to find the distance between two points on a coordinate plane by treating the horizontal and vertical changes as the two shorter sides of a right-angled triangle.
Drawing straight-line from given equation
Draw a straight-line graph from its equation by either building a table of values and plotting the points, or by starting at the y-intercept and using the gradient to plot the rest of the line.
Gradient
Calculate and interpret the gradient of a straight line from a graph, rise and run, two coordinates, a table, or a linear pattern.
Finding x- and y-intercepts
Read the x- and y-intercepts of a straight line from a graph, and find them algebraically by setting y=0 or x=0 respectively.
Forming equations of given straight line graph (using y=mx+c)
Form the equation of a supplied non-vertical straight-line graph in the form y = mx + c by reading or calculating its gradient and y-intercept.
Forming equations of horizontal and vertical lines
Write the equation of a horizontal line (y equals a constant) or a vertical line (x equals a constant) through a given point, and describe the gradient of each.
Finding point of intersection of two lines
Find the point of intersection of two straight lines, either by reading it directly from a graph or by working it out algebraically from their equations.
Drawing parabola from given equation
Draw the graph of a quadratic equation by completing a two-column substitution table, matching each x-value with its y-value, and plotting the resulting parabola.
Using tables to explore patterns and graphs
Use visual patterns, tables, formulae, and graphs to explore linear relationships, complete values, and make and test predictions.
Applying straight line graphs to solve problems
Model real-world situations with straight-line relationships in the form $y = mx + c$, use the equation to solve problems by substituting values, and interpret the meaning of the gradient and y-intercept within the context of the problem.
Sketching a parabola from factorised form (a=1)
From a parabola in factorised form y = (x - p)(x - q) with a = 1, read off the x-intercepts, find the axis of symmetry and vertex, and sketch the upward-opening curve.
Sketching a parabola from vertex form (a=1)
From a parabola in vertex form y = (x - h)^2 + k with a = 1, read off the vertex and axis of symmetry directly, and find the y-intercept and any x-intercepts.
Using algebra to find the point of intersection of two lines
Find the exact point of intersection of two given straight lines algebraically, and verify that the coordinate satisfies both equations.
Forming parabola equations: x-intercept form
Read a parabola's two x-intercepts from a large grid graph, then substitute one more marked point to find the scale factor a and write the equation in x-intercept form y = a(x - r1)(x - r2).
Solving contextual problems with parabolas (a=1)
Once a parabola's equation is known, read off its x-intercepts, y-intercept, axis of symmetry, and vertex, then interpret what each feature means in a real-world context, using the axis of symmetry to find a mirrored point without needing to resolve the full equation.
Drawing exponential graphs
Build a table of values for an exponential rule y = ab^x, then plot the points and draw a smooth curve, identifying its horizontal asymptote.
Factorising quadratics ax^2 + bx + c where a is not 1
Factorise quadratics of the form ax^2 + bx + c where a is not 1, by taking out any common factor first and then splitting the middle term so the four resulting terms factorise in matching pairs.
Solving quadratics ax^2 + bx + c = 0 where a is not 1, by factorising
Solve a quadratic equation ax^2 + bx + c = 0 where a is not 1, by factorising -- splitting the middle term after finding two numbers that multiply to a*c and add to b, then applying the zero-product property -- including equations already given in factorised form and equations that first need rearranging to zero.
Reading and forming parabolas in vertex form y = a(x - h)^2 + k where a is not 1
Read the vertex, axis of symmetry, opening direction, and width from a parabola's equation in vertex form y=a(x-h)^2+k where a is not 1, and work backwards from given features such as a vertex, points, or roots to write the equation.
Exponential Rules from Tables
Build tables of values and identify patterns for exponential rules y = a·b^x + c, reading off the common ratio, growth or decay, y-intercept, and horizontal asymptote, and deriving the rule from a table or from given starting/growth information.
Forming parabola equations: vertex form
Read a parabola's vertex from a large grid graph, then substitute one more marked point to find the scale factor a and write the equation in vertex form y = a(x - h)^2 + k.
Forming parabola equations: x-intercept form (a = ±1)
Read a parabola's two x-intercepts from a large grid graph, decide from its direction whether a = 1 or a = -1, and write the equation in x-intercept form y = (x - r1)(x - r2) or y = -(x - r1)(x - r2).
Forming parabola equations: vertex form (a = ±1)
Read a parabola's vertex from a large grid graph, decide from its direction whether a = 1 or a = -1, and write the equation in vertex form y = (x - h)^2 + k or y = -(x - h)^2 + k.
Finding and interpreting the intersection of two linear graphs
Plot two linear relationships on the same grid using tables of values, identify the point of intersection as an ordered pair, and interpret what that point means in context.
Sketching parabolas from intercepts (a = ±1)
Given a parabola's two x-intercepts and its y-intercept (or the factorised equation y = a(x - r1)(x - r2), a = 1 or -1), plot the three intercepts, find the axis of symmetry midway between the x-intercepts, place the vertex on that axis, and sketch the smooth symmetric curve.
Sketching parabolas from vertex form (a = ±1)
Given y = a(x - h)^2 + k with a = 1 or a = -1, sketch the parabola on a labelled grid: plot the vertex (h, k), decide the direction from the sign of a, step out the y = x^2 shape from the vertex (across 1 up/down 1, across 2 up/down 4), then check with the y-intercept.
Gradient from two coordinates
Calculate the gradient of a straight line from two coordinate points, then extend the method to find y = mx + c.
Geometry
Naming angles
Identify the vertex of an angle and name marked angles using correct three-letter notation.
Angles around a point
Use the fact that angles around a point sum to 360 degrees to find missing angles, including algebraic and ratio problems, and justify the reasoning used.
Angles on a straight line
Use the fact that angles on a straight line sum to 180 degrees to find missing angles, including algebraic and ratio problems, and justify the reasoning used.
Vertically opposite angles
Use the rule that vertically opposite angles formed by two intersecting straight lines are always equal to find unknown angles, including when angles are given as algebraic expressions or combined with the rule that adjacent angles on a straight line are supplementary.
Interior angles of a triangle
Use the fact that the interior angles of any triangle sum to 180 degrees, together with the angle properties of isosceles and equilateral triangles, to find unknown angles and to judge whether a set of angles can form a triangle.
Parallel line rules
Use the corresponding, alternate, and co-interior angle rules formed when a transversal crosses two parallel lines to find unknown angles, including algebraic problems and deciding whether a pair of lines is parallel.
Regular/irregular polygon
Determine whether a polygon is regular or irregular, and use the interior-angle-sum rule 180(n - 2) to find a polygon's angle sum, one interior angle of a regular polygon, the number of sides of a polygon, or a missing angle in an irregular polygon.
Pythagoras' theory
Use Pythagoras' theorem to find a missing hypotenuse (the long side) in a right-angled triangle.
Trigonometric Ratios: Finding a Side
Use the trigonometric ratios (SOH CAH TOA) to find a missing side in a right-angled triangle, given one angle and one other side.
Transformation (Translation, reflection, enlargement)
Apply and describe translations, reflections, and enlargements of points and shapes on a coordinate grid, including finding image coordinates, translation vectors, and enlargement scale factors, and explain which properties each transformation preserves.
Representing and constructing 3D shapes
Identify rectangular prisms, triangular prisms, and square-based pyramids from their numbers of faces, edges, and vertices, and construct or interpret their nets and plan/front/side views.
Compass direction
Use the 8-point compass and three-figure bearings to name, read, and calculate directions -- including opposite/turned directions, converting between compass points and bearings, reverse bearings, and the angle turned between two bearings.
Similar shapes
Identify and work with similar and congruent shapes, understanding scale factors and corresponding sides.
Introduction to Circle Geometry
Learn and apply four basic circle facts — the angle in a semicircle is 90 degrees, the angle between a tangent and a radius is 90 degrees, angles in the same segment are equal, and the angle at the centre is twice the angle at the circumference on the same arc — to find missing angles in circle diagrams.
Construction and Loci
Use a ruler and compass to construct the perpendicular bisector of a segment and the bisector of an angle, and describe or shade the locus of points satisfying a given distance condition -- including compound loci formed by combining two conditions.
Circle theorem
Apply the five core circle theorems -- tangent-radius, angle in a semicircle, angles in the same segment, angle at the centre versus the circumference, and opposite angles in a cyclic quadrilateral -- to find missing angles, including multi-step problems that chain two theorems together.
Algebra with geometry
Read a labelled geometry diagram, write the relevant angle, perimeter, area, or scale-factor fact as an equation, then solve the resulting linear or quadratic equation.
Applying Pythagoras’ theory and trigonometry to 3D shapes
Apply Pythagoras' theorem and trigonometric ratios together to solve problems involving distances, angles, and heights in three-dimensional shapes, recognising when to use two-step methods.
Volume of pyramids, cones, and spheres
Find the volume of pyramids, cones, and spheres by substituting into the correct formula, including reverse problems that rearrange the formula to find a missing radius, height, or base length, and scale-factor questions.
Using the Sine Rule to find missing sides
Use matching opposite side-angle pairs in the Sine Rule to calculate missing side lengths in non-right-angled triangles.
Using the Sine Rule to find missing angles
Use matching opposite side-angle pairs in the Sine Rule and inverse sine to calculate missing angles in non-right-angled triangles.
Basic angle rules
Apply the four core angle rules — angles on a straight line, angles around a point, vertically opposite angles, and the triangle angle sum — to find missing angles.
Rotational symmetry and order of symmetry
Decide whether a shape or picture has reflective (line) symmetry, count how many axes of symmetry it has, and state its order of rotational symmetry -- then describe all of a figure's symmetries together in one answer.
Isometric drawing of 3D shapes
Read an isometric drawing of a solid built from unit cubes to count how many cubes it contains including the hidden ones, name the solid and its faces, edges and vertices, and work out its volume and surface area.
Nets
Recognise, fold and draw the nets of cubes, cuboids, prisms, pyramids and cylinders, deciding which flat arrangement of faces folds into a given solid and which faces meet when it does.
Measuring Angles
Estimate, measure, and construct angles up to 360 degrees with a protractor, reading from the zero of whichever scale lines up with one arm, and classify each angle as acute, right, obtuse, straight, or reflex.
Trigonometric Ratios: Finding an Angle
Use the inverse trigonometric ratios (sin^-1, cos^-1, tan^-1) to find a missing angle in a right-angled triangle, given two known sides.
Measurement applications: simple shapes
Solve substantial, realistic measurement problems that chain a perimeter or area calculation of a rectangle, triangle, parallelogram, trapezium, or circle through a unit conversion to a quantity of material, a discrete purchase, and a cost, and make and justify a decision from the result.
Measurement applications: compound shapes
Solve substantial, realistic measurement problems that decompose a compound 2D shape (an L-shaped floor plan or deck, a rectangle with an excluded region, a shape combining a rectangle/square/parallelogram with a triangle/semicircle/quarter-circle) into simpler component shapes, find a total area or perimeter, then chain the result through a unit conversion, a quantity of material, a discrete purchase, and a cost, making and justifying a decision from the result.
Measurement applications: surface area and paint
Solve substantial, realistic measurement problems that find the area of a surface to be painted or clad -- a wall, room, shed, tank, or box -- after excluding a door, window, or other unpainted region, then chain the result through a unit conversion, a paint-coverage/coats calculation, whole-container rounding, wastage, and a cost, making and justifying a decision from the result.
Measurement applications: compound 3D shapes
Solve substantial, realistic measurement problems about objects assembled from two familiar solids -- a cylinder and a cone, a cylinder and a hemisphere, or a cuboid and a pyramid -- covering volume, capacity, mass from density, material quantities and cost, always making explicit whether a quantity is the solid's internal volume/capacity or its exposed surface area.
Measurement applications: Pythagoras
Solve substantial, realistic measurement problems -- ladders, loading ramps, diagonal braces, TV/screen diagonals, diagonal paths and roof rafters -- where a missing length of a right-angled triangle is found with Pythagoras' theorem and then used to complete a later perimeter, area, volume, material-quantity, cost, or decision calculation, including problems where students must decide for themselves which given length is the hypotenuse.
Measurement applications: SOHCAHTOA
Solve substantial, realistic right-triangle measurement problems -- roof pitches, wheelchair and loading ramps, extension-ladder safety angles, zip-line descents, and a surveyor's sight-line to a building's top -- using SOHCAHTOA to find a missing side (given an angle and one side) or a missing angle (given two sides), then using that trig result to complete a later compliance check, comparison, material-quantity, or cost calculation.
Measurement applications: sine rule
Solve substantial, realistic non-right-angled-triangle measurement problems -- a surveyor triangulating land from a baseline, coastguard bearings to a vessel, a radio mast's guy-wires, a reserve boundary needing fencing, a narrow garden bed needing edging, and a pedestrian bridge truss -- using the sine rule (a/sin A = b/sin B = c/sin C) to find a missing side (given two angles and one side) or a missing angle (given two angles, via angle sum, or given two sides and a safely unambiguous angle, via the sine rule directly), then using that result to complete a later cost, material- quantity, compliance, or comparison calculation.
Measurement applications: cosine rule
Solve substantial, realistic non-right-angled-triangle measurement problems -- fencing a new lifestyle-block boundary, finding a tramper's direct route home, fabricating a carport frame's diagonal brace, checking a park's triangular corner, confirming a vineyard boundary corner, and a canopy frame's diagonal and post height -- using the cosine rule (a^2=b^2+c^2-2bc cos A) to find a missing side (given two sides and the included angle, SAS) or a missing angle (given all three sides, SSS, via the rearranged form cos A=(b^2+c^2-a^2)/(2bc)), then using that result to complete a later cost, material-quantity, compliance, or comparison calculation.
Measurement applications: sine area rule
Solve substantial, realistic non-right-angled-triangle measurement problems -- valuing a triangular building lot, costing a triangular sail's fabric, costing a roof gable's paint/cladding, turfing a lawn with an adjoining triangular garden bed, valuing a triangular rural block, and paving a courtyard around a triangular garden-bed corner -- using the sine area rule (A=half a b sin C) to find the area of a non-right-angled triangle given two sides and the angle between them, then using that area (directly, or as one component added to or subtracted from a larger compound region) to complete a later material-quantity, cost, or compliance/decision calculation.
Measurement applications: arcs and sectors
Solve substantial, realistic measurement problems involving arcs and sectors -- a garden-centre roundabout's kerb edging and native planting, a pie-slice-shaped park sign, a curved garden path's edging, a quarter- circle-cornered deck's decking material, an arched window's glazing and framing, and a sector-shaped garden bed set against a rectangular lawn -- using the arc length formula (theta/360 x 2 pi r) to find the length of a curved edge and the sector area formula (theta/360 x pi r squared) to find the area of a pie-slice-shaped region, then using that length or area (directly, or as one component added to or subtracted from a larger compound region containing a familiar polygon) to complete a later material-quantity, cost, or compliance/decision calculation.
Measurement applications: limits of accuracy
Solve substantial, realistic measurement problems involving error- bound propagation -- a paved patio's area, a paddock's fencing perimeter, a wall's paintable area around a window, a concrete slab's volume, a decking area with a rounded quarter-circle corner, and a triangular land survey's boundary via Pythagoras' theorem -- using the convention that a measurement "given to the nearest X" has an implied tolerance of plus-or-minus half of X, recomputing each calculation using the most extreme plausible inputs to find the minimum and maximum possible final answer, and correctly choosing which extreme (upper or lower) of each input pushes the final answer toward the overall minimum or maximum given that specific formula's structure, then discussing whether that uncertainty range actually changes the real-world decision being made.
Measurement applications: design and justify
Solve open-ended measurement design problems -- a garden bed, a backyard deck around a fixed hot-tub footprint, a rectangular water tank, a wheelchair ramp, a triangular land parcel, and a lean-to carport roof -- where the student is given CONSTRAINTS rather than fixed dimensions, must invent specific numbers that satisfy every constraint simultaneously, state and justify the chosen dimensions, compute the resulting measurement using a technique already built earlier in the series, verify every constraint against an explicit checklist, and -- at the higher tiers -- reason about a genuine trade-off or compare two valid designs to justify which better meets a cost-minimisation goal. There is no single correct answer -- any dimension set that genuinely satisfies every stated constraint is acceptable.
SOHCAHTOA and composite shapes involving circles
Use SOHCAHTOA to find a shared length in a right-angled triangle, then use that length to calculate the area or exposed perimeter of a composite shape involving a semicircle or quarter circle.
Using SOHCAHTOA with compound shapes
Use SOHCAHTOA to derive shared measurements, then calculate areas and exposed perimeters of varied compound shapes by adding or subtracting components.
Triangle construction (SSS, SAS, ASA)
Use a ruler and compass to construct a triangle from three given sides (SSS), two sides and the included angle (SAS), or two angles and the included side (ASA), and identify which construction method applies to a given set of measurements.
NCEA Numeracy Co-Requisite: Everyday Problem Solving
Extract the relevant numbers from a short real-life scenario and perform a single calculation (e.g. addition, multiplication, unit conversion) to answer it.
Geometric Reasoning with Parallel Lines and Polygons (Combined)
Identify and name angle pairs formed at a point, on a straight line, and by a transversal crossing parallel lines (vertically opposite, alternate, corresponding, co-interior).
Cyclic Quadrilaterals
Identify a cyclic quadrilateral and state that opposite interior angles sum to 180 degrees, using this to find one missing angle.
Angles in Polygons
Use the interior angle sum formula (n-2)\times180^\circ to find the total interior angle sum of a named polygon and calculate one missing interior angle.
Tessellation
Identify which regular polygons (e.g. equilateral triangles, squares, hexagons) tessellate on their own and explain why, using angle sums at a vertex.
Angle Properties of Parallel Lines
Identify and name corresponding, alternate, and co-interior angle pairs formed when a transversal crosses two parallel lines shown in a diagram.
Enlargement (Transformations)
Enlarge a simple shape on a grid using a given centre and a positive whole-number scale factor, correctly plotting and labelling the image vertices.
Transformations and Congruence: Reflection, Rotation, Translation and Symmetry
Perform and identify single transformations (reflection, rotation, translation) of simple shapes on a grid, and identify lines of symmetry in given figures.
Reflection: Lines of Reflection, Invariant Points and the Mirror Line
Reflect simple shapes or points in the x-axis, y-axis, and lines such as $x=a$ or $y=b$, reading coordinates off a grid.
Analysing Symmetry in Patterns
Identify and draw lines of symmetry and state the order of rotational symmetry for simple regular shapes and everyday patterns.
Performing and Describing Rotation (including Combined Transformations)
Rotate a shape by a given angle about a marked centre point using a grid or tracing paper, and identify any invariant points.
Combined Transformations (Sequences of Transformations)
Apply two given transformations (e.g. a translation then a reflection) to a shape in sequence and draw the final image.
Isometric Drawing, Plan Views, and Nets of Solids
Draw simple solids such as cubes and cuboids accurately on isometric dot paper and match a given net to the simple prism or pyramid it folds into.
Geometry of Conic Sections
Identify and sketch parabolas, ellipses, hyperbolas and circles from their standard-form equations, locating the vertex, centre, axes and foci.
Rearranging Formulae (Changing the Subject)
Rearrange a simple one-step formula to make the required letter the subject, where that letter appears once with no fraction, bracket, or power involved.
Using Formulae: Substituting and Rearranging (Combined)
Substitute given numeric values into a simple one- or two-step formula (e.g. perimeter, speed, cost) and evaluate.
Rearranging Formulae and Multi-Step Algebraic Problem Solving
Rearrange a simple one-step formula to make a given letter the subject.
Similarity and Congruence in Geometric Figures
Identify corresponding sides and angles in pairs of congruent or similar shapes and state whether two given shapes are congruent or similar by inspection.
Symmetry and Similarity of Shapes
Identify and draw lines of symmetry and state the order of rotational symmetry for common 2D shapes and polygons.
Geometric Proof and Reasoning
State a single geometric property (angle rule, triangle property, or circle theorem) as the reason for one missing angle or length in a diagram.
Substituting Values into Formulae
Substitute whole-number values into simple one- or two-step formulae (e.g. perimeter, area) and evaluate correctly using order of operations.
Line (Reflective) Symmetry
Identify and draw the line(s) of symmetry on simple, familiar 2D shapes such as isosceles triangles, rectangles, and letters.
Introduction to Trigonometry: Finding an Angle
Given a right-angled triangle with two known sides labelled, identify the correct ratio and use the inverse trig function to find the marked angle to one decimal place.
Properties and Nets of 3D Solids
Name common 3D solids (prisms, pyramids, cylinders, cones, spheres) and identify/count their faces, edges, and vertices.
Applying the Sine and Cosine Rules to Non-Right-Angled Triangles
Use the sine rule to find an unknown side or angle in a non-right-angled triangle given two angles and one side, or two sides and a non-included angle.
Angles Between Lines and Planes in 3D Solids
Identify right-angled triangles hidden within simple 3D solids such as cuboids and pyramids, and use Pythagoras' theorem to find an unknown edge or diagonal length.
Hamiltonian Circuits and Paths in Networks
Identify whether a given route through a small network visits every vertex exactly once, distinguishing a Hamiltonian path from a Hamiltonian circuit.
Angle Properties of Circles
Name the parts of a circle (centre, radius, chord, tangent, arc, segment) and state a single circle theorem such as the angle at the centre being twice the angle at the circumference.
Measurement applications: integrated projects
Solve sustained real-world measurement projects by interpreting one shared diagram, selecting and coordinating length, area, volume and rate calculations, combining component costs, and evaluating assumptions and limitations.
Trigonometry
Pythagoras theory finding short and long sides
Use Pythagoras' theorem to find a missing hypotenuse or a missing shorter side in a right-angled triangle.
Applying Pythagoras’ theory
Recognise a hidden right-angled triangle inside a real-world shape or situation, then use Pythagoras' theorem to solve the problem.
Checking whether a triangle is right-angled
Use Pythagoras' theorem to check whether a triangle with given side lengths is right-angled.
Pythagorean triples
Recognise, verify, complete, and generate Pythagorean triples using Pythagoras' theorem and its converse.
Mixed revision of Pythagoras' theory
Solve right-angled triangle problems by selecting and applying the correct Pythagoras move — finding a missing side, checking for a right angle, or solving a real-world application.
Multi-step Pythagoras' theory
Multi-step Pythagoras questions require students to find an intermediate length first (such as a hidden height or diagonal), then use that result to calculate a second value (such as area, perimeter, or comparison).
Perimeter of compound shapes using Pythagoras' theory
Finding the perimeter of compound shapes using Pythagoras' theorem requires students to first identify the missing length (often a sloped edge or diagonal), then calculate it using a² + b² = c², and finally add all the sides to find the total perimeter.
Applications of Trigonometric Ratios: Finding Lengths
Model a real-world situation with a right-angled triangle, choose the appropriate trigonometric ratio, and calculate an unknown height, distance, or length.
Applications of Trigonometric Ratios: Finding Angles
Model a real-world situation with a right-angled triangle, choose the appropriate inverse trigonometric ratio, and calculate an unknown angle of elevation, depression, inclination, or slope.
Applying Pythagoras’ Theory to 3D representations
Apply Pythagoras' theorem to find face diagonals and space diagonals of cuboids and other 3D shapes, recognising when multiple applications are needed.
Other
Internal and external
Find missing interior and exterior angles of triangles and polygons using the angle sum formula (n-2) x 180 degrees, the linear-pair relationship between an interior angle and its adjacent exterior angle (they sum to 180 degrees), and the fact that the exterior angles of any polygon always sum to 360 degrees; apply these both forwards (find an angle) and backwards (find the number of sides of a regular polygon from a given angle).
Perimeter (including circles)
Calculate perimeters of shapes, including circles and composite shapes, applying formulas for circumference and combining measurements of different shape types.
Area (including circles and composite shapes)
Calculate the area of rectangles, triangles, parallelograms, trapeziums, circles, and composite shapes using appropriate formulas.
Drawing nets of 3D
Draw, label, and identify accurate flat nets for cuboids, cubes, triangular prisms, and square-based pyramids.
Using measurement tools
Read accurate length measurements from a ruler and angle measurements from a protractor.
Comparing sequences: nth terms, graphs and intersection
Find the nth term of two linear sequences, plot both sequences as points and lines on the same axes, and find where they intersect or where one overtakes the other, graphically and algebraically.
Probabilities from experiments and statistical
Find the experimental (relative-frequency) probability of an outcome from recorded trial results, compare it against the theoretical probability of a fair model, judge how reliable an estimate is given the number of trials, and critique claims made from small or biased samples of experimental data.
Calculus
Deriving functions
Find the derived function f'(x) by differentiating from first principles -- substitute into f'(x) = lim(h->0) [f(x+h) - f(x)]/h, expand, simplify, divide by h, then let h approach zero.
Limits
Find the value a function approaches as x tends to a number or to infinity, using direct substitution first, factorising and cancelling when substitution gives the indeterminate form 0/0, and comparing leading terms when x tends to infinity.
Differentiating power functions
Differentiate power functions using the power rule, multiplying by the exponent and reducing it by one, and first rewrite products, algebraic fractions and roots as powers of x so every term can be differentiated.
Gradient of tangents on curves
Estimate the gradient of a curve at a point from the tangent line on its graph, and use whether that gradient is positive, negative, or zero to describe where the curve rises and falls and to classify its turning points.
Sketching functions from derivative graphs
Use the graph of f'(x), together with a known value of f(x), to sketch a possible graph of f(x).
Sketching the derivative of a function
Use the graph of f(x) to sketch the graph of its derivative f'(x) by interpreting tangent gradients.