Australian Curriculum v9 / ACiQ Year 7 Mathematics - Unit 0 - Number patterns and rules

Number patterns and rules

Describe visually growing and numerical patterns, formulate rules and use them to find terms.

Updated 2026-07-24 - 5 min read

Describe visually growing and numerical patterns, formulate rules and use them to find terms. This draft teaching resource is aligned to the cited Year 7 content descriptions and still requires named human curriculum approval.

Define each variable before operating. Every symbol, table entry and graph coordinate should retain one consistent meaning and unit.

1. Generalise growing patterns

For a constant-change pattern, the change becomes the coefficient of n. A constant adjustment makes the rule fit term 1.

Decision check: Apply this lesson's rule to the worked example: Test a proposed rule at n = 1 and at least two later terms. Identify the exact value, label or property that satisfies the rule.

Worked examples with complete reasoning

Worked example 1

Question: Find the next term in 4, 7, 10, 13, ...

  1. The constant change is 3.
  2. Apply it once more to get 16.

Answer: 16

Check: Compare consecutive terms.

Worked example 2

Question: Write a rule for term n of the pattern 5, 8, 11, 14, ...

  1. Use the constant change as the coefficient of n.
  2. Adjust the constant so n = 1 gives the first term: 3n + 2.

Answer: 3n + 2

Check: Start with change × n, then compare its first value with the actual first term.

Worked example 3

Question: A pattern has rule 4n + 1. Find term 12.

  1. Substitute n = 12.
  2. Calculate to obtain 49.

Answer: 49

Check: Use the term number as n.

Method map for this topic

  1. Generalise growing patterns: Test a proposed rule at n = 1 and at least two later terms.

Common mistakes and how to correct them

  • pattern change: Not yet. Compare consecutive terms. Recheck the pattern change step, then try again.
  • nth term rule: Not yet. Start with change × n, then compare its first value with the actual first term. Recheck the nth term rule step, then try again.
  • rule substitution: Not yet. Use the term number as n. Recheck the rule substitution step, then try again.
  • inverse pattern: Not yet. Treat the required term number as the unknown in an equation. Recheck the inverse pattern step, then try again.
  • visual pattern: Not yet. Check that the rule gives the stated figure 1 before using it. Recheck the visual pattern step, then try again.
  • rule verification: Not yet. Test several term numbers, not just one. Recheck the rule verification step, then try again.

Practice

  1. Find the next term in 4, 7, 10, 13, ...
  2. Find the next term in 21, 17, 13, 9, ...
  3. Write a rule for term n of the pattern 5, 8, 11, 14, ...
  4. Write a rule for term n of the pattern 2, 7, 12, 17, ...
  5. A pattern has rule 4n + 1. Find term 12.
  6. A pattern has rule 6n - 2. Find term 9.

<details> <summary>Answers and reasoning</summary>

  1. 16 — The constant change is 3. Apply it once more to get 16.
  2. 5 — The constant change is -4. Apply it once more to get 5.
  3. 3n + 2 — Use the constant change as the coefficient of n. Adjust the constant so n = 1 gives the first term: 3n + 2.
  4. 5n - 3 — Use the constant change as the coefficient of n. Adjust the constant so n = 1 gives the first term: 5n - 3.
  5. 49 — Substitute n = 12. Calculate to obtain 49.
  6. 52 — Substitute n = 9. Calculate to obtain 52.

</details>

Checklist and vocabulary

| Skill | Observable evidence | | --- | --- | | identify constant pattern change | Identify and describe the constant change in a linear pattern. | | formulate nth term rule | Formulate an algebraic rule for a numerical or visually growing pattern. | | use verify pattern rule | Use and verify a pattern rule for specified term numbers. |

Retrieval prompts

  • State the defining relationship for each method above without looking.
  • Give one example where two visually similar representations mean different things.
  • Diagnose one misconception from the list and explain the first corrective step.
  • Name a check that is independent of simply repeating the same calculation.

Teach-back and self-check

Close the worked examples and explain number patterns and rules as if you were helping another Year 7 student. Begin by naming the clue that tells you which relationship, representation or operation is needed. Then show one complete example with labels and units where they matter. At each line, explain what stayed equivalent and why the next step is allowed.

Use this four-part check before calling the method secure:

  1. Identify: What information is given, and what must be found or justified?
  2. Represent: Which number line, diagram, table, expression, graph or organised list makes the relationship visible?
  3. Reason: Which rule connects the representation to the required answer?
  4. Verify: Can you check by estimation, substitution, an inverse operation, a second representation or the original context?

If the explanation depends on “I just knew”, return to the method map and name the missing decision. A durable method is one you can explain, check and use in a changed context after a delay.

Transfer task

Model a changing school or household quantity using number patterns and rules. Define every variable and unit, test at least two inputs, and explain what the outputs mean.

Sources