| Chapter | Subtopic | Year(s) | Strand | ACARA-style content wording | One-line justification | Match level |
|---|---|---|---|---|---|---|
| 1 | Rigorous definition of arithmetic basics | Years 7–8 | Number and Algebra | Apply and explain properties of operations (commutativity, associativity, distributivity) and place-value reasoning for whole numbers and decimals. | ACARA Years 7–8 focus on operational properties and number sense; a rigorous definition supports that learning. | Direct match |
| 1 | Clever applications of arithmetic rules to make seemingly-complicated calculations simple | Years 7–9 | Number and Algebra | Solve problems efficiently using number properties, estimation and arithmetic strategies to simplify calculations. | Problem-solving with arithmetic strategies appears across Years 7–9 though the book extends beyond routine items. | Partial match |
| 2 | Exponentiation and powers | Year 8 | Number and Algebra | Use index notation for whole number powers and interpret multiplication as repeated addition and powers as repeated multiplication. | ACARA Year 8 introduces index notation and use of powers. | Direct match |
| 2 | Exponent laws | Years 8–9 | Number and Algebra | Apply index laws for multiplication and division of powers with same base and use these rules in algebraic contexts. | Index laws are covered in Year 8 and extended with algebraic manipulation in Year 9. | Direct match |
| 2 | Zero as an exponent | Year 8 | Number and Algebra | Recognise and justify that any non-zero number to the zero power equals 1 using index laws. | Zero exponent follows naturally from index laws typically taught in Year 8 and is consistent with ACARA index notation curriculum. | Direct match |
| 2 | Negative exponents | Years 9–10 | Number and Algebra | Interpret negative integer indices as reciprocals and apply index laws to simplify expressions with negative exponents. | Negative indices are usually introduced later than positive indices; this extends Year 8 content into Years 9–10 algebra. | Partial match |
| 3 | Multiples and divisibility | Year 7 | Number and Algebra | Investigate multiples, factors and divisibility rules and use them to solve problems. | ACARA Year 7 includes factors, multiples and divisibility reasoning. | Direct match |
| 3 | Primes, composites, and prime factorization | Years 7–8 | Number and Algebra | Identify prime and composite numbers and represent integers by their prime factorisation. | Prime factorisation and classification of numbers appear in Years 7–8. | Direct match |
| 3 | Fundamental Theorem of Arithmetic | Years 9–10 | Number and Algebra | Recognise the unique prime factorisation of integers and use it for proofs or problem solving. | Uniqueness of prime factorisation is more formal than standard Years 7–8 expectations and is usually considered an extension for deeper number-theory study. | Extension / exceeds year expectations |
| 3 | Least common multiple (LCM) and greatest common divisor (GCD) | Year 7 | Number and Algebra | Use prime factorisation and divisibility to find greatest common divisors and least common multiples to solve problems. | LCM and GCD via factorisation is expected in middle years. | Direct match |
| 4 | Rigorous definition of fractions | Years 7–8 | Number and Algebra | Understand fractions as numbers representing parts of a whole, points on the number line and ratios; interpret and represent equivalence. | ACARA emphasises fraction understanding in Years 7–8; a rigorous approach supports curriculum aims. | Direct match |
| 4 | Arithmetic with fractions and mixed numbers | Years 7–8 | Number and Algebra | Add, subtract, multiply and divide fractions and mixed numbers, converting where necessary. | Standard Year 7–8 content includes fraction operations and mixed numbers. | Direct match |
| 4 | Fraction comparison and simplification | Year 7 | Number and Algebra | Compare and order fractions, and express fractions in simplest terms using factorisation. | Comparing and simplifying fractions is a core Year 7 topic. | Direct match |
| 4 | Challenging word problems with fractions | Years 8–9 | Number and Algebra | Solve multi-step problems involving fractions, mixed numbers and proportion, using appropriate strategies and notation. | Complex fraction problems require multi-step reasoning that goes beyond routine Year 7 tasks. | Partial match |
| 5 | Expressions and equations | Years 7–8 | Number and Algebra | Use algebraic notation to generalise number properties and form and evaluate expressions and simple equations. | Algebraic expressions and simple equations are taught in Years 7–8. | Direct match |
| 5 | Linear equations | Years 8–9 | Number and Algebra | Solve linear equations including those with one variable and apply algebraic techniques to rearrange equations. | Solving linear equations is a Year 8–9 expectation, varying by complexity. | Direct match |
| 5 | Applications of linear equations to word problems | Years 8–9 | Number and Algebra | Model real problems with linear equations, solve and interpret solutions in context. | Equation modelling is emphasised in Years 8–9; word-problem complexity can extend Year expectations. | Direct match |
| 5 | Principles of inequalities | Year 9 | Number and Algebra | Understand inequality notation, solve simple linear inequalities and represent solutions on number lines. | Inequalities appear in the senior middle years; core understanding matches Year 9. | Direct match |
| 5 | Linear inequalities | Year 9 | Number and Algebra | Solve and graph linear inequalities and use them in modelling contexts. | Graphing and solving linear inequalities are part of Years 9–10 algebra content. | Direct match |
| 6 | Definition of decimal notation | Years 7–8 | Number and Algebra | Use decimal notation to represent fractions and apply place-value concepts for decimals. | Decimals and their place-value interpretation are central in Years 7–8. | Direct match |
| 6 | Rigorous exploration of arithmetic with decimals | Years 7–8 | Number and Algebra | Perform operations with decimals, justify procedures and estimate results to assess reasonableness. | Decimal operations and estimation are Year 7–8 priorities. | Direct match |
| 6 | Decimal comparison and approximation | Years 7–8 | Number and Algebra | Compare and order decimals and round or approximate decimals to specified degrees of accuracy. | Comparing and rounding decimals are part of middle-year curriculum. | Direct match |
| 6 | Conversion between fractions and decimals | Years 7–8 | Number and Algebra | Convert between fractions, decimals and percentages and use these forms interchangeably in problem solving. | Conversion among forms is expected in Years 7–8. | Direct match |
| 6 | Rational numbers and their decimal representation | Years 8–9 | Number and Algebra | Describe rational numbers in terms of their decimal expansions (terminating, recurring) and use them in computations. | Understanding decimal representations of rationals is introduced and explored across Years 8–9. | Direct match |
| 7 | Definition of ratio and proportion | Year 7 | Number and Algebra | Use ratio language to compare quantities and solve simple proportion problems using multiplicative thinking. | Ratios and proportion are explicitly in Year 7 curriculum. | Direct match |
| 7 | Proportional thinking | Years 7–8 | Number and Algebra | Recognise and use multiplicative relationships in tables, graphs and equations to solve proportional problems. | Proportional reasoning is a strand across Years 7–8. | Direct match |
| 7 | Problem-solving with part-to-part and part-to-whole ratios | Years 7–8 | Number and Algebra | Solve part-to-part and part-to-whole ratio problems, including scaling and mixtures. | These ratio problem types align with middle-years content. | Direct match |
| 7 | Variables in ratios and proportions | Years 8–9 | Number and Algebra | Express proportional relationships using algebraic notation and solve for unknowns in proportion equations. | Introducing variables into proportional relationships extends Year 7 work toward Year 9 algebra. | Partial match |
| 7 | Unit conversion with conversion factors | Year 7 | Measurement and Geometry | Use and apply metric and common conversion factors to convert units of length, mass, capacity and time. | Unit conversion is part of measurement content in middle years. | Direct match |
| 7 | Relationship between speed, distance, and time | Years 8–9 | Measurement and Geometry | Apply relationship distance = speed × time in problem solving and unit-consistent calculations. | Speed-distance-time problems are standard Year 8–9 measurement applications. | Direct match |
| 7 | Joint work, relative speed, and average speed | Years 9–10 | Number and Algebra / Measurement | Solve advanced problems involving combined work rates, relative movement and average speed using algebraic models. | Joint work and advanced rate problems typically exceed routine middle-years expectations. | Extension / exceeds year expectations |
| 8 | Definition of percent | Year 7 | Number and Algebra | Understand percentages as 'per 100' and represent percentages as fractions and decimals. | Percent concepts are introduced in Year 7 ACARA content. | Direct match |
| 8 | Relationships among percents, fractions, and decimals | Years 7–8 | Number and Algebra | Convert between percentages, fractions and decimals and use these conversions in computations and comparison. | Conversion relationships are core Year 7–8 content. | Direct match |
| 8 | Percents of numbers | Years 7–8 | Number and Algebra | Calculate percentages of quantities and use percentage operators in problem contexts. | Computing percentages of amounts is explicitly taught in the middle years. | Direct match |
| 8 | Percent word problems | Years 8–9 | Number and Algebra | Model and solve real-life situations involving percentages (e.g., discounts, interest, parts to whole). | Application problems appear across Years 8–9; complexity in the book may go beyond basic problems. | Partial match |
| 8 | Percent increase and decrease | Years 8–9 | Number and Algebra | Calculate and interpret percentage increase and decrease and apply multiplicative factors to repeated percent change. | Percent change is covered in middle/senior middle years and multiplicative interpretation is expected. | Direct match |
| 9 | Definition of square root | Years 8–9 | Number and Algebra | Define square roots as the inverse of squaring and use them to solve simple equations and evaluate expressions. | Square roots arise in Year 8–9 topics (notably with Pythagoras and index notation). | Direct match |
| 9 | Equations with square roots | Years 9–10 | Number and Algebra | Solve equations involving square roots and justify solutions, including recognising extraneous roots when applicable. | Solving radical equations is usually Year 9–10 algebra content; checking extraneous roots is more advanced. | Partial match |
| 9 | Non-integer square roots | Years 9–10 | Number and Algebra | Approximate non-integer square roots and interpret their numerical and geometric significance. | Approximation and calculation of non-integer roots extends Year 8 square-root ideas into applied contexts. | Partial match |
| 9 | Simplification of square roots | Years 9–10 | Number and Algebra | Simplify expressions involving square roots using factorisation and indices where appropriate. | Simplifying surds is generally treated in later middle years or early senior years; more formal surd work is an extension. | Partial match |
| 9 | Arithmetic with square roots | Years 9–10 | Number and Algebra | Perform arithmetic on expressions containing square roots and use conjugates or rationalisation where needed. | Arithmetic with surds often exceeds routine Year 9 content and is typically Year 10 extension material. | Extension / exceeds year expectations |
| 10 | Angle measurement | Year 7 | Measurement and Geometry | Measure and construct angles using degrees and use angle properties to solve problems. | Angle measurement and use of protractors is Year 7 content. | Direct match |
| 10 | Parallel lines | Year 8 | Measurement and Geometry | Use properties of parallel lines and transversals to identify corresponding, alternate and interior angles and solve related problems. | Parallel line angle relationships are a Year 8 topic in ACARA. | Direct match |
| 10 | Angles in a triangle | Year 7 | Measurement and Geometry | Use angle sum properties of triangles and reasoning to solve for unknown angles. | Triangle angle-sum knowledge is part of Year 7–8 geometry. | Direct match |
| 10 | Angles in other polygons | Years 8–9 | Measurement and Geometry | Calculate interior and exterior angles for polygons and use angle-sum formulas to solve problems. | Interior/exterior angle relationships are taught in Years 8–9. | Direct match |
| 11 | Segments and perimeter | Year 7 | Measurement and Geometry | Calculate perimeters of plane figures and work with line segments and basic constructions. | Perimeter and segment measurement are Year 7 measurement topics. | Direct match |
| 11 | Triangle inequality | Years 9–10 | Measurement and Geometry / Number and Algebra | Use the triangle inequality to reason about possible side lengths and to justify geometric conclusions. | Triangle inequality is not usually emphasised in early middle years and is more formal reasoning expected later. | Partial match |
| 11 | Triangle area | Year 8 | Measurement and Geometry | Calculate the area of triangles using base and height and apply area formulas in problem contexts. | Triangle area is ACARA Year 8 measurement content. | Direct match |
| 11 | Circumference of a circle | Year 8 | Measurement and Geometry | Relate diameter, radius and circumference and use π to calculate circumference and arc lengths. | Circle measures and circumference are Year 8 topics. | Direct match |
| 11 | Area of a circle; Unusual areas | Years 8–10 | Measurement and Geometry | Calculate areas of circles and composite or non-standard regions using decomposition and formula application. | Circle area is Year 8; complex composite-area problems reach into Years 9–10 problem solving. | Partial match |
| 12 | The Pythagorean Theorem | Year 8 | Measurement and Geometry | Apply the Pythagorean theorem to determine side lengths in right-angled triangles and solve related problems. | Pythagoras is explicitly taught in Year 8 ACARA measurement content. | Direct match |
| 12 | Pythagorean triples | Years 8–9 | Number and Algebra / Measurement | Recognise integer solutions to a^2 + b^2 = c^2 and use triples in problem solving and proofs. | Triples are an enrichment topic that extend the standard Year 8 Pythagoras material. | Partial match |
| 12 | 30–60–90 and 45–45–90 triangles | Years 9–10 | Measurement and Geometry | Use known side ratios of special right triangles to calculate lengths and solve geometry problems. | Special-triangle ratios are typically beyond Year 8 basics and fit senior middle years geometry. | Extension / exceeds year expectations |
| 12 | Types of quadrilaterals | Year 7 | Measurement and Geometry | Classify quadrilaterals by properties (parallel sides, equal sides, right angles) and represent them geometrically. | Classification of polygons including quadrilaterals is core Year 7 geometry. | Direct match |
| 12 | Quadrilateral area | Year 8 | Measurement and Geometry | Calculate areas of simple quadrilaterals using decomposition into triangles and rectangles. | Area of quadrilaterals via decomposition is part of Year 8 measurement work. | Direct match |
| 13 | Average (mean) | Year 7 | Statistics and Probability | Calculate and interpret mean for sets of data and use averages to compare data sets. | Computing and interpreting the mean is Year 7 statistics content. | Direct match |
| 13 | Averages as a balancing act | Years 8–9 | Statistics and Probability | Interpret mean as a balancing point and use this understanding to reason about distributions and effect of values. | Conceptual interpretation of mean is promoted, though the balancing metaphor extends conceptual depth. | Partial match |
| 13 | Median, mode, and range | Year 7 | Statistics and Probability | Calculate and compare median, mode and range and explain their use in describing data sets. | Median, mode and range are Year 7 statistics core items. | Direct match |
| 13 | Limits of basic statistics | Years 9–10 | Statistics and Probability | Discuss limitations of mean/median/mode in describing distributions and how outliers affect measures of centre and spread. | Critical interpretation of statistics is emphasised in senior middle years as reasoning about data. | Partial match |
| 13 | Types of graphs and charts | Year 7 | Statistics and Probability | Create and interpret tables, dot plots, column graphs and other representations to display data appropriately. | Graph types and basic interpretation are Year 7 content. | Direct match |
| 14 | Numbers in lists | Years 7–8 | Number and Algebra | Investigate number sequences and generate terms using rules, including arithmetic progressions and simple recursive definitions. | Pattern and sequence work is introduced in Years 7–8; advanced sequence theory is beyond scope. | Direct match |
| 14 | Venn diagrams | Years 8–9 | Statistics and Probability / Number and Algebra | Use Venn diagrams to represent sets, intersections and unions and solve counting problems involving overlapping sets. | Venn diagrams and set-based counting appear in Years 8–9 curriculum. | Direct match |
| 14 | Multiplication principle | Years 8–9 | Number and Algebra | Apply the counting (multiplication) principle to determine the number of outcomes for combined independent choices. | The multiplication principle is taught as basic counting in senior middle years. | Direct match |
| 14 | Casework | Years 8–10 | Number and Algebra / Probability | Solve counting problems by exhaustive case analysis and use systematic listing to ensure completeness. | Casework is a problem-solving strategy reinforced in middle to senior years; used in counting/probability tasks. | Direct match |
| 14 | Pairs | Years 8–9 | Number and Algebra | Count and reason about unordered and ordered pairs, including applications in Cartesian products and simple combinatorics. | Pair counting relates to combinatorics introduced in middle years; full combinatorics may exceed some Year 8 expectations. | Partial match |
| 14 | Introduction to probability | Years 7–8 | Statistics and Probability | Describe outcomes, sample spaces and calculate probabilities for simple events using equally likely outcomes. | Introductory probability is Year 7–8 ACARA content. | Direct match |
| 15 | Find a pattern | Years 7–9 | Number and Algebra | Use numerical and visual patterns to conjecture rules, express general terms and justify generalisations. | Pattern finding and generalisation is present across Years 7–9 as part of algebraic thinking. | Direct match |
| 15 | Make a list | Years 7–9 | Number and Algebra / Problem Solving | Use systematic listing as a strategy for problem solving in counting, probability and combinatorics. | Systematic listing is a recommended problem-solving technique in ACARA's Working Mathematically and counting contexts. | Direct match |
| 15 | Draw a picture | Years 7–10 | Working Mathematically / All strands | Use diagrams, models and drawings to represent and solve problems and to reason mathematically. | Using visual representations is a cross-curriculum problem-solving strategy emphasised in Working Mathematically. | Direct match |
| 15 | Work backwards | Years 7–10 | Working Mathematically / Number and Algebra | Solve multi-step problems by reasoning backwards from the desired goal to find required conditions or inputs. | Backwards reasoning is a core problem-solving strategy supported by ACARA's Working Mathematically goals. | Direct match |
Notes for Excel/printing: copy the HTML table into Excel (Paste Special > Text or open as HTML) or save as .html and open in Excel; each <tr> becomes a row and each <td> a cell. Rows list chapter, subtopic, suggested year(s), curriculum strand, an ACARA-style content statement, a one-line justification, and the recommended match level (Direct match / Partial match / Extension).
Reference: Prealgebra by Rusczyk, Patrick & Boppana, ISBN 978-1-934124-21-5 mapped to ACARA Years 7–10 (Number and Algebra; Measurement and Geometry; Statistics & Probability; Working Mathematically).