Growing Patterns and Rational-number Rules

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

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.

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.

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.

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.

Australian Curriculum v9.0

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

The Australian Curriculum code and wording are exact. International teachers can use the underlying mathematical concept while matching the lesson to their local grade or year outcomes.

• Australian Curriculum Version 9.0
• NSW Curriculum
• Victorian Curriculum F–10