Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Growth, decay and model selection

Growth, decay and model selection

Compare linear and exponential models, use them for growth or decay and state limitations of model predictions.

Updated 2026-07-26 - 11 min read

Growth, decay and model selection is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

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.

Select linear or exponential models

Use constant differences for linear relationships and constant ratios for exponential relationships. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by select linear or exponential models 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: Model selection depends on how quantities change and on context, not appearance alone.

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

Outputs increase 12, 17, 22, 27. Which model type fits?

Step 1 - identify the governing idea: Use constant differences for linear relationships and constant ratios for exponential relationships.

Step 2 - apply it to this evidence: The first difference is constantly 5.

Result: Linear

The relationship is visible in the working: The first difference is constantly 5. 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:

  • Exponential — It does not agree with the required relationship: The first difference is constantly 5.
  • Quadratic for certain — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No model — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A value changes 100, 90, 81, 72.9. Which model type fits?

Step 1 - identify the governing idea: Use constant differences for linear relationships and constant ratios for exponential relationships.

Step 2 - apply it to this evidence: Each step multiplies by 0.9.

Result: Exponential decay

The relationship is visible in the working: Each step multiplies by 0.9. 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 decrease — It does not agree with the required relationship: Each step multiplies by 0.9.
  • Exponential growth — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Constant — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A savings balance gains exactly $25 each month. Which model is appropriate?

Step 1 - identify the governing idea: Use constant differences for linear relationships and constant ratios for exponential relationships.

Step 2 - apply it to this evidence: Equal time steps add a constant dollar amount.

Result: Linear

The relationship is visible in the working: Equal time steps add a constant dollar amount. 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:

  • Exponential — It does not agree with the required relationship: Equal time steps add a constant dollar amount.
  • Inverse — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Random only — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Apply growth and decay models

Use initial amount times a repeated multiplier raised to the number of equal periods. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by apply growth and decay models 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: A decrease retains 94%, so the multiplier is 0.94.

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

A $1200 value grows 5% for 3 years. Which expression gives the value?

Step 1 - identify the governing idea: Use initial amount times a repeated multiplier raised to the number of equal periods.

Step 2 - apply it to this evidence: Each year retains 100% and adds 5%, giving factor 1.05.

Result: 1200(1.05³)

The relationship is visible in the working: Each year retains 100% and adds 5%, giving factor 1.05. 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:

  • 1200 + 0.05³ — It does not agree with the required relationship: Each year retains 100% and adds 5%, giving factor 1.05.
  • 1200(0.95³) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1200 + 3(5) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A population of 500 declines 8% per year. Estimate after 2 years.

Step 1 - identify the governing idea: Use initial amount times a repeated multiplier raised to the number of equal periods.

Step 2 - apply it to this evidence: 500(0.92²) = 423.2.

Result: 423.2

The relationship is visible in the working: 500(0.92²) = 423.2. 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:

  • 420 — It does not agree with the required relationship: 500(0.92²) = 423.2.
  • 580 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 460.8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A quantity triples every 4 hours from 6. What is it after 12 hours?

Step 1 - identify the governing idea: Use initial amount times a repeated multiplier raised to the number of equal periods.

Step 2 - apply it to this evidence: Twelve hours contains 3 growth periods, so 6 × 3³ = 162.

Result: 162

The relationship is visible in the working: Twelve hours contains 3 growth periods, so 6 × 3³ = 162. 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:

  • 54 — It does not agree with the required relationship: Twelve hours contains 3 growth periods, so 6 × 3³ = 162.
  • 216 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 60 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Evaluate model limitations

Check domain, data range, assumptions and plausible long-term behaviour before using a model for interpolation or extrapolation. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by evaluate model limitations 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: Patterns can change outside observed conditions; model predictions inherit assumptions and uncertainty.

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 bacteria model assumes unlimited doubling. What is a key long-term limitation?

Step 1 - identify the governing idea: Check domain, data range, assumptions and plausible long-term behaviour before using a model for interpolation or extrapolation.

Step 2 - apply it to this evidence: Unlimited exponential growth eventually conflicts with environmental constraints.

Result: Resources and space are finite

The relationship is visible in the working: Unlimited exponential growth eventually conflicts with environmental constraints. 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 model uses numbers — It does not agree with the required relationship: Unlimited exponential growth eventually conflicts with environmental constraints.
  • Doubling is linear — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The initial value is positive — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Data cover 0–10 days. Which prediction is more defensible?

Step 1 - identify the governing idea: Check domain, data range, assumptions and plausible long-term behaviour before using a model for interpolation or extrapolation.

Step 2 - apply it to this evidence: Interpolation within the observed range requires less assumption than far extrapolation.

Result: A prediction at day 8

The relationship is visible in the working: Interpolation within the observed range requires less assumption than far extrapolation. 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 prediction at day 100 — It does not agree with the required relationship: Interpolation within the observed range requires less assumption than far extrapolation.
  • Both are equally certain — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Neither can ever be made — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Two models fit current data similarly. What should guide selection?

Step 1 - identify the governing idea: Check domain, data range, assumptions and plausible long-term behaviour before using a model for interpolation or extrapolation.

Step 2 - apply it to this evidence: Fit alone is not enough; mechanism and errors matter.

Result: Context, residual pattern and future assumptions

The relationship is visible in the working: Fit alone is not enough; mechanism and errors matter. 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:

  • Choose the most complicated — It does not agree with the required relationship: Fit alone is not enough; mechanism and errors matter.
  • Choose the largest prediction — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Average equations without checking — 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 value changes 100, 90, 81, 72.9. Which model type fits?
  2. A population of 500 declines 8% per year. Estimate after 2 years.
  3. Data cover 0–10 days. Which prediction is more defensible?

Answers

  1. Exponential decay — Each step multiplies by 0.9.
  2. 423.2 — 500(0.92²) = 423.2.
  3. A prediction at day 8 — Interpolation within the observed range requires less assumption than far extrapolation.

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