Place Values and Decimals
Explore Place Values and Decimals.
Measurement
34 objectives
Rounding to Decimal Points
Explore Rounding to Decimal Points.
Multiplying and dividing by 10,100,1000
Explore Multiplying and dividing by 10,100,1000.
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, and parallelograms using the standard area formulas, and work backwards from a given area and base to find a missing height.
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
Explore Forming measurement expressions.
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.
Algebra
32 objectives
Integers
Develop secure understanding of integers.
Substitution
Substitute numbers for variables in expressions and evaluate.
Forming expressions
Explore Forming expressions.
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
Explore Expanding single brackets.
Factorising single brackets
Find the highest common factor of the terms and write the expression as a single bracket.
Defining Variables & Forming Equations
Explore Defining Variables & Forming Equations.
Solving 1-step equations
Explore Solving 1-step equations.
Solving 2-step equations
Explore Solving 2-step equations.
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 co-efficient equal to 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 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.
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.
Number
16 objectives
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
Explore Converting between improper fractions and mixed numbers.
Adding and subtracting fractions
Explore Adding and subtracting fractions.
Multiplying fractions
Explore Multiplying fractions.
Reciprocals
Explore Reciprocals.
Dividing fractions
Explore Dividing fractions.
Finding fractions of quantity
Explore Finding fractions of quantity.
Multiples & Lowest Common Multiples
Explore Multiples & Lowest Common Multiples.
Factors, Highest Common Factors & Primes
Explore Factors, Highest Common Factors & Primes.
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.
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.
Statistics
33 objectives
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 9MAT-derived 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 9MAT-derived 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 9MAT-derived 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.
Probability
10 objectives
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
Calculate and estimate probabilities from experimental data and statistical displays, compare experimental and theoretical probabilities, and justify conclusions using evidence.
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
24 objectives
Extrapolating linear, quadratic, and exponential patterns
Extend patterns by identifying their rule and predicting future terms for linear, quadratic, and exponential sequences.
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)
Explore Spatial patterns (including drawing next pattern and converting to pattern).
Determining nth term of pattern or spatial pattern
Explore Determining nth term of pattern or spatial pattern.
Understanding Cartesian plane and co-ordinates
Explore Understanding Cartesian plane and co-ordinates.
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
Explore Gradient.
Finding x- and y-intercepts
Explore Finding x- and y-intercepts.
Forming equations of given straight line graph (using y=mx+c)
Explore Forming equations of given straight line graph (using y=mx+c).
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 constructing a table of values and plotting the resulting parabola, then label key features such as the turning point.
Using tables to explore patterns and graphs
Explore Using tables to explore patterns and graphs.
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
Explore Using algebra to find the point of intersection of two lines.
Forming equation of parabolas
Form the equation of a parabola (with a=1) from given features such as its x-intercepts (roots) or its vertex, writing it in factorised form y=(x-p)(x-q) or vertex form y=(x-h)^2+k.
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.
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.
Identifying geometric (exponential) number patterns and tables for y = a*b^x + c
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.
Coordinates and grid points
Read, plot, and reason with coordinates on a Cartesian plane using the existing legacy task decks.
Geometry
39 objectives
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
Explore Parallel line rules.
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 (which is NOT always "all lower bounds give the minimum," particularly for a formula containing a subtraction step), 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 (a range for one dimension, a minimum for another, a target area/volume/cost) rather than fixed dimensions, must invent specific numbers that satisfy every constraint simultaneously, state and justify the chosen dimensions, compute the resulting measurement (area, perimeter, volume, or cost) using a technique already built earlier in the series (rectangle area/perimeter, cuboid volume, Pythagoras' theorem), verify every constraint against an explicit checklist, and -- at the higher tiers -- reason about a genuine trade-off between competing constraints or compare two valid designs to justify which better meets a cost/material-minimisation goal. There is no single correct answer -- any dimension set that genuinely satisfies every stated constraint is an acceptable response.
Measurement applications: integrated projects
Coordinate area, Pythagoras, prism volume, rates and assumptions to build and evaluate a complete project quote.
Trigonometry
10 objectives
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
5 objectives
Internal and external
Explore Internal and external.
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.
Calculus
4 objectives
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.