Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 2 - Linear and quadratic modelling

Linear and quadratic modelling

Choose, formulate and evaluate linear or quadratic models in financial and applied contexts.

Updated 2026-07-26 - 11 min read

Linear and quadratic modelling is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Choose a model family

A graph's appearance helps, but a table of differences and the mechanism in context give stronger evidence for model choice.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by choose a model family and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The type of change matters: constant rate suggests linear; changing rate with constant second differences suggests quadratic.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

A table has y-values 4,7,10,13 for consecutive x. Choose a model.

Step 1 - identify the governing idea: Linear models have constant first differences; quadratic models have constant second differences.

Step 2 - apply it to this evidence: First differences are constantly 3.

Result: Linear

The relationship is visible in the working: First differences are constantly 3. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Quadratic — It does not agree with the required relationship: First differences are constantly 3.
  • Exponential — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No relationship — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A table has y-values 1,4,9,16 for x=1,2,3,4. Choose a model.

Step 1 - identify the governing idea: Linear models have constant first differences; quadratic models have constant second differences.

Step 2 - apply it to this evidence: First differences 3,5,7 have constant second difference 2.

Result: Quadratic

The relationship is visible in the working: First differences 3,5,7 have constant second difference 2. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Linear — It does not agree with the required relationship: First differences 3,5,7 have constant second difference 2.
  • Constant — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Reciprocal — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A taxi charges a fixed fee plus the same rate per kilometre. Choose a model.

Step 1 - identify the governing idea: Linear models have constant first differences; quadratic models have constant second differences.

Step 2 - apply it to this evidence: Fixed start and constant rate match y=mx+c.

Result: Linear

The relationship is visible in the working: Fixed start and constant rate match y=mx+c. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Quadratic — It does not agree with the required relationship: Fixed start and constant rate match y=mx+c.
  • No model — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Inverse — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Formulate and solve a model

Units and domain restrictions are part of the mathematics. A numerical root or intersection is not complete until it is translated back to the situation.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by formulate and solve a model and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Define variables and interpret the result so another person can verify the model.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Plan A costs $20 plus $5 per visit. Write C for n visits.

Step 1 - identify the governing idea: A model needs defined variables, an equation connected to the context, a solution and an interpretation.

Step 2 - apply it to this evidence: The fixed fee is the intercept and $5 per visit is the gradient.

Result: C=5n+20

The relationship is visible in the working: The fixed fee is the intercept and $5 per visit is the gradient. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • C=20n+5 — It does not agree with the required relationship: The fixed fee is the intercept and $5 per visit is the gradient.
  • C=25n — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • C=5n — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A rectangle has width x and length x+3. Write its area.

Step 1 - identify the governing idea: A model needs defined variables, an equation connected to the context, a solution and an interpretation.

Step 2 - apply it to this evidence: Area is x(x+3).

Result: A=x²+3x

The relationship is visible in the working: Area is x(x+3). Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • A=2x+3 — It does not agree with the required relationship: Area is x(x+3).
  • A=x²+3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A=3x² — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

For C=5n+20, how many visits cost $70?

Step 1 - identify the governing idea: A model needs defined variables, an equation connected to the context, a solution and an interpretation.

Step 2 - apply it to this evidence: 70=5n+20 gives 50=5n and n=10.

Result: 10 visits

The relationship is visible in the working: 70=5n+20 gives 50=5n and n=10. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 14 visits — It does not agree with the required relationship: 70=5n+20 gives 50=5n and n=10.
  • 18 visits — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 250 visits — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Evaluate models and domains

Evaluation compares predictions with the mechanism, data and practical constraints. Whole-number counts, non-negative quantities and capacity limits often restrict the domain.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by evaluate models and domains and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Interpret and test the solution; mathematically valid values can still be impractical.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

A ticket model gives n=42.6 people. What should a report do?

Step 1 - identify the governing idea: A model is useful only within a sensible domain and under assumptions that match the context.

Step 2 - apply it to this evidence: The context requires an integer and the rounding direction depends on the purpose.

Result: Explain that people require a whole-number decision, likely 43 if capacity must be met

The relationship is visible in the working: The context requires an integer and the rounding direction depends on the purpose. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Report exactly 42.6 people — It does not agree with the required relationship: The context requires an integer and the rounding direction depends on the purpose.
  • Always round down to 42 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Reject the entire model — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A quadratic height model predicts −8 m after the object lands. What does this show?

Step 1 - identify the governing idea: A model is useful only within a sensible domain and under assumptions that match the context.

Step 2 - apply it to this evidence: The algebra may continue while the physical model stops at ground contact.

Result: The model is being used outside its sensible time domain

The relationship is visible in the working: The algebra may continue while the physical model stops at ground contact. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The object is underground in all contexts — It does not agree with the required relationship: The algebra may continue while the physical model stops at ground contact.
  • Quadratics are invalid — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Negative numbers are calculation errors — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A linear cost model assumes a constant unit price, but a bulk discount begins at 100 units. Best evaluation?

Step 1 - identify the governing idea: A model is useful only within a sensible domain and under assumptions that match the context.

Step 2 - apply it to this evidence: The constant-rate assumption changes at the discount threshold.

Result: Use a piecewise or revised model beyond 100 units

The relationship is visible in the working: The constant-rate assumption changes at the discount threshold. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Keep one line forever — It does not agree with the required relationship: The constant-rate assumption changes at the discount threshold.
  • Delete the fixed cost only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Change to a circle model — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. A table has y-values 1,4,9,16 for x=1,2,3,4. Choose a model.
  2. A rectangle has width x and length x+3. Write its area.
  3. A quadratic height model predicts −8 m after the object lands. What does this show?

Answers

  1. Quadratic — First differences 3,5,7 have constant second difference 2.
  2. A=x²+3x — Area is x(x+3).
  3. The model is being used outside its sensible time domain — The algebra may continue while the physical model stops at ground contact.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources