Year 6 Mathematics · AC9M6A01

Growing Patterns and Rational-number Rules

Connect visual growth, term values and direct or recursive rules

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Learning goalsSay it simply

Learning goal

Students identify what changes and what remains constant, represent patterns in tables and diagrams, use recursive and direct rules and test whether a rule generates every term.

Success criteria

  • I can represent or identify the concept.
  • I can explain the underlying relationship.
  • I can select an appropriate strategy or feature.
  • I can apply it in a new context.
  • I can justify and verify the response.

Teaching routine

  1. Represent
  2. Reason
  3. Calculate
  4. Interpret
  5. Verify
Curriculum focus: recognise and use rules that generate visually growing patterns and number patterns involving rational numbers
Key conceptTeach from the board

Connect a growing tile pattern to a rule

Use the visual model first. Ask students to identify the quantities, structure or conditions before calculating or explaining.

term n1234ntiles5811143n + 2growth+3+3+3constant +3

The recursive rule adds 3 each term. The direct rule 3n + 2 gives any term without generating all previous terms.

Extend patterns involving fractions and decimals

Connect the central relationship to a new context, then verify the conclusion with a second representation, estimate, inverse operation or reasonableness check.

0.5, 1.0, 1.5, 2.0add 0.5; term = 0.5n
3/4, 1, 1 1/4, 1 1/2add 1/4; term = n/4 + 1/2
visual border growthcount repeated section plus fixed corners
digital tabletest many terms after defining rule

A rule must fit the visual structure and all known terms. Equivalent forms of a rational rule may look different but produce the same values.

Clean visual examplesOne-page board

Clean one-page examples

AC9M6A01 - Growing Patterns and Rational-number Rules
Example 1

term n 1 2 3 4 n tiles 5 8 11 14 3n + 2 growth +3 +3 +3 constant +3

Example 2

0.5, 1.0, 1.5, 2.0 add 0.5; term = 0.5n

Example 3

: investigating patterns such as the number of tiles in a geometric pattern, or the number of dots or other shapes in successive repeats of a strip or border pattern; looking for patterns in the way the numbers increase/decrease

Example 4

: using a calculator or spreadsheet to experiment with number patterns that result from multiplying or dividing; for example, 1 ÷ 9, 2 ÷ 9, 3 ÷ 9…, 210 \times 11, 211 \times 11, 212 \times 11…, 111 \times 11, 222 \times 11, 333 \times 11…, or 100 ÷ 99, 101 ÷ 99, 102 ÷ 99…

Curriculum examplesCopied content

AC9M6A01: recognise and use rules that generate visually growing patterns and number patterns involving rational numbers

  • E1: investigating patterns such as the number of tiles in a geometric pattern, or the number of dots or other shapes in successive repeats of a strip or border pattern; looking for patterns in the way the numbers increase/decrease
  • E2: using a calculator or spreadsheet to experiment with number patterns that result from multiplying or dividing; for example, 1 ÷ 9, 2 ÷ 9, 3 ÷ 9…, 210 \times 11, 211 \times 11, 212 \times 11…, 111 \times 11, 222 \times 11, 333 \times 11…, or 100 ÷ 99, 101 ÷ 99, 102 ÷ 99…
  • E3: creating an extended number sequence that represents an additive pattern using decimals; for example, representing the additive pattern formed as students pay their \$2.50 for an incursion as 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 …
  • E4: investigating the number of regions created by successive folds of a sheet of paper: one fold, 2 regions; 2 folds, 4 regions; 3 folds, 8 regions, and describing the pattern using everyday language
  • E5: creating a pattern sequence with materials, writing the associated number sequence and then describing the sequence with a rule so someone else can replicate it with different materials; for example, using matchsticks or toothpicks to create a growing pattern of triangles using 3 for one triangle, 5 for 2 triangles, 7 for 3 triangles and describing the pattern as, “Multiply the number of triangles by 2 and then add one for the extra toothpick in the first triangle”

Use the central and application models above to connect each elaboration to the same underlying concept.

Questions and answersWith answers

Check understanding

  • Find the next two terms.
  • Write a recursive rule.
  • Test 3n + 2.
  • Describe fixed and growing parts.
  • Extend a decimal pattern.

Evidence of mastery

  • Represent or identify the concept
  • Explain the underlying relationship
  • Select an appropriate strategy or feature
  • Apply it in a new context
  • Justify and verify the response

Decision: continue when students can explain the model, apply it to a new example and justify their check. Otherwise return to the central model and reduce the numerical or representational load.

Practice and reviewReady for practice
Rule describes only the next termA direct rule should work for any term number.
Constant part omittedVisual patterns often contain repeated growth plus fixed pieces.
Rule accepted from two terms onlyTest several terms and the visual structure.
Term number confused with term valueKeep input n separate from output.
Curriculum alignmentStart here

Learning goal

Students identify what changes and what remains constant, represent patterns in tables and diagrams, use recursive and direct rules and test whether a rule generates every term.

Success criteria

  • I can represent or identify the concept.
  • I can explain the underlying relationship.
  • I can select an appropriate strategy or feature.
  • I can apply it in a new context.
  • I can justify and verify the response.

Teaching routine

  1. Represent
  2. Reason
  3. Calculate
  4. Interpret
  5. Verify
Curriculum focus: recognise and use rules that generate visually growing patterns and number patterns involving rational numbers
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