Experimenting with Linear Functions and Relations — AC9M8A04

• Vary parameters systematically in linear functions and relations.
• Predict and test how gradient, intercept and inequality sign affect representations.
• Use tables and graphs as evidence for or against a conjecture.
• Generalise patterns only after testing enough relevant cases.

Recall straight-line graphs, coordinates, gradient as rate of change, intercepts, tables of values, equations and simple inequality regions.

A useful digital experiment controls variables. In y=mx+b, hold b fixed while changing m to isolate the effect of gradient; hold m fixed while changing b to isolate the intercept effect.

The conjecture must be stated before or during testing, then challenged with positive, zero and negative cases where relevant. One attractive graph is not proof of a general rule.

Digital tools accelerate variation and comparison, but reasoning remains mathematical: identify what changed, what stayed invariant and whether the evidence supports the claimed generalisation.

• One graph proves a general rule. Test multiple strategically different cases.
• Changing b changes gradient. In y=mx+b, b shifts the line vertically but does not change m.
• Digital tools replace explanation. They provide evidence; the conjecture and generalisation require reasoning.
• A visual intersection is automatically exact. Readings may be approximate; verify algebraically when exactness matters.

1. Graph y=2x+b for b=−3,0,4 and state what remains invariant.
2. Graph y=mx+1 for m=−2,0,3 and describe the effect of m.
3. Test which side of y=x+2 satisfies y>x+2.
4. List three integer solutions to x+2y=8.

1. Predict the effect of increasing b in y=4x+b, then test it.
2. Predict the effect of changing m from 2 to −2 with b fixed.
3. Decide whether y=5x−1 and y=5x+8 are parallel and justify.
4. Compare y=−x and y>−x on a graph.
5. Find four integer solutions to 2x+y=10.
6. Design a digital experiment to test “all lines with the same gradient are parallel”.
7. State what evidence would disprove your conjecture.

A student compares y=2x+1 with y=5x+7 and concludes “increasing m makes a line steeper.” Explain why the experiment is confounded, redesign it so only one parameter changes, and state the evidence needed for a reliable generalisation.

1. Why vary one parameter at a time? To isolate its effect rather than mixing causes.
2. What happens when b changes with m fixed? The line shifts vertically and remains parallel.
3. What does negative m do? The line decreases from left to right.
4. What makes a conjecture convincing? It survives systematic tests, including relevant boundary or contrasting cases.

1. Investigate y=mx+3 for m=−3,−1,0,2,4 and generalise the effect of m.
   Keep b fixed and compare direction and steepness.
2. Test the claim “lines y=4x+b are parallel for all b” with digital evidence and mathematical reasoning.
   Use several b-values and identify the invariant gradient.
3. Compare y=2x−1, y>2x−1 and y<2x−1, explaining boundary and solution regions.
   Test a point not on the line to identify each half-plane.

• I change one parameter at a time.
• I make a testable conjecture.
• I use graphs/tables as evidence.
• I distinguish evidence from generalisation.

Exit ticket: Why is changing both m and b at once a weak test of what m controls?

Australian Curriculum v9.0 — AC9M8A04: experiment with linear functions and relations using digital tools, making and testing conjectures and generalising emerging patterns.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8A05: Exact relationship. Victoria inserts a separate algorithms/error-correction descriptor at VC2M8A04, so the equivalent linear-functions descriptor is VC2M8A05.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-LIN-C-01; MAO-WM-01: Partial. NSW supports creating/displaying linear relationships and Working Mathematically, but does not reproduce the full national systematic digital experimentation descriptor as a single Stage 4 outcome.

• AC9M8A02 — linear relations, equations and inequalities
• AC9M8A03 — linear modelling
• AC9M8SP04 — design, test and refine algorithms

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Experiment with linear functions and relations using digital tools, making...

Mapped skill: experiment with linear functions and relations using digital tools, making and testing conjectures and generalising emerging patterns

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.