- explain the central idea: Explore functions and relations with digital tools, test conjectures against evidence and counterexamples, and generalise patterns carefully.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Year 10 Mathematics · AC9M10A05
Functions, Relations and Digital Conjectures
Use graphs, tables and digital tools systematically: make a conjecture, test it across cases and boundaries, refine it, and state what the evidence actually supports
Ready to project and teach
Learning goalsSay it simply
Key conceptTeach from the board
Digital tools can reveal patterns quickly, but a screenshot is not a mathematical argument. A strong investigation varies one feature deliberately, records evidence, checks boundary/counterexamples and then generalises only as far as the evidence allows.
Conjecture cycle
Observe → conjecture → test → refine → generalise → justify.
Intersection
At an intersection, both relations have the same ordered pair/output.
Bisection
If a continuous function changes sign over an interval, a root lies between; repeatedly halve the interval.
Digital-tool discipline
Change scale/zoom, test more than one case and record enough precision to justify the conclusion.
Worked examplesWe do
Worked examples
AC9M10A05 - Functions, Relations and Digital Conjectures
Example 1
Conjecture cycle Observe → conjecture → test → refine → generalise → justify.
Example 2
Intersection At an intersection, both relations have the same ordered pair/output.
Example 3
Bisection If a continuous function changes sign over an interval, a root lies between; repeatedly halve the interval.
Example 4
Digital-tool discipline Change scale/zoom, test more than one case and record enough precision to justify the conclusion.
Curriculum examplesCopied content
Australian Curriculum: AC9M10A05 — Year 10 Algebra. Explore functions and relations with digital tools, test conjectures against evidence and counterexamples, and generalise patterns carefully.
Victoria: VC2M10A11 and VC2M10A16 support the investigation; optional Level 10A VC2M10AA10 closely mirrors the digital-conjecture intent
NSW: Stage 5 Path — Functions and graphs; Working mathematically
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10A05 | VC2M10A11 and VC2M10A16 support the investigation; optional Level 10A VC2M10AA10 closely mirrors the digital-conjecture intent | Stage 5 Path — Functions and graphs; Working mathematically |
| Guided and independent practice | Builds fluency and application for AC9M10A05 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10A05: experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns.
- E1: use zoom/refined intervals to approximate function intersections such as x²=2ˣ. Example 1.
- E2: apply bisection to approximate quadratic intercepts such as f(x)=2x²−3x−7. Example 2.
- E3: transform x²+y²=1. Example 3.
- E4: find intersections of linear graphs with quadratics/circles. Examples 4–5.
- E5: identify intervals where quadratics are positive/negative. Example 6.
- E6: use tables to locate when exponential growth/decay crosses a threshold. Example 7.
- E7: investigate functions/relations as mathematical foundations of machine learning, including transformations, models and optimisation. Example 10.
Questions and answersWith answers
- Explain why a graph intersection solves f(x)=g(x).
- Because both outputs are equal at that ordered pair.
- For f(x)=x²−5, use signs at x=2 and x=3 to trap a positive root.
- f(2)=−1 and f(3)=4, so root lies in (2,3).
- Describe the transformation from x²+y²=1 to (x/3)²+y²=1.
- Horizontal stretch by factor 3; x-intercepts ±3.
Practice and reviewReady for practice
- Trusting one graph window without changing the scale.
- Calling a visual intersection exact when the tool only gives an approximation.
- Generalising from one or two examples.
- Failing to test a boundary/counterexample.
- Changing several parameters at once and then not knowing which caused the effect.
Curriculum alignmentStart here
- explain the central idea: Explore functions and relations with digital tools, test conjectures against evidence and counterexamples, and generalise patterns carefully.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
Closest US CCSS alignment includes HSF-IF.C.7, HSF-BF.B.3 and HSA-REI.D.10–11; closest UK GCSE/NZ/Canadian/Victorian/NSW alignments cover functions, graphs, transformations, intersections and digital investigation.
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