Handout: Kōwhaiwhai Patterns

Finding the algebra in Māori art.

Pattern 1: The Koru Swirl

This pattern starts with one swirl and adds another one each time.

Stage 1: (

Stage 2: ( (

Stage 3: ( ( (

1. Draw Stage 4.

2. Complete the table:

Stage (n)	1	2	3	4	10
Swirls	1	2	3	?	?

3. What is the algebraic rule for this pattern? Rule: n

Pattern 2: The Pitau Branch

This pattern grows by adding two branches each time.

Stage 1: /\

Stage 2: /\_/\

Stage 3: /\_/\_/\

1. Draw Stage 4.

2. Complete the table:

Stage (n)	1	2	3	4	10
Branches	2	4	6	?	?

3. What is the algebraic rule for this pattern? Rule: _________

Challenge: The Rafter Pattern

This pattern starts with 3 lines and adds 2 more each time.

Stage 1: |||

Stage 2: |||||

Stage 3: |||||||

Can you find the algebraic rule for the number of lines in stage 'n'?

Rule: _________

🔑 Kaiako answer key

Pattern 1 — The Koru Swirl

Table: Stage 4 → 4 swirls; Stage 10 → 10 swirls.
Rule: n (printed on the sheet as a worked example).

Pattern 2 — The Pitau Branch

Table: Stage 4 → 8 branches; Stage 10 → 20 branches.
Rule: 2n. Check: 2(1) = 2, 2(2) = 4, 2(3) = 6 — matches the drawings.

Challenge — The Rafter Pattern

Rule: 2n + 1. Check: 2(1) + 1 = 3, 2(2) + 1 = 5, 2(3) + 1 = 7 ✓.

Watch for: the Rafter pattern is the first one on this sheet that does not start at the rule’s natural first value, so “add 2 each time” alone gives 2n and fails at Stage 1. Have ākonga test their rule at Stage 1 before they commit to it.

📋 Teacher Planning Snapshot

Ngā Whāinga Ako — Learning Intentions

A worksheet for Lesson 3. Ākonga analyse the growth structure of kōwhaiwhai-inspired designs and write the rule that generates each one.

Ngā Paearu Angitū — Success Criteria

✅ I can count how a drawn pattern grows from stage to stage.
✅ I can complete the table and write the rule as an expression in n.
✅ I can check my rule at Stage 1, not just at the stage I counted.

Differentiation & Inclusion

Pattern 1 has its rule printed as a worked example; use it as the model. Ākonga who need support should draw Stage 4 before filling the table. Extension: ask what stage would have 99 branches, which forces the rule to be used backwards.

🌿 Mātauranga Māori Lens

The designs on this sheet are kōwhaiwhai-inspired shapes drawn to make a growth structure countable. They are not kōwhaiwhai. Real kōwhaiwhai are painted on the rafters of a wharenui and carry meaning this handout does not touch. What ākonga are analysing here is repetition and growth — a mathematical property that the designs share, not the whole of what they are.