Subject
Mathematics — Pāngarau (Geometry strand)
Pāngarau / Mathematics · Geometry · Years 5–8
Tukutuku Patterns & Mathematics
Ngā Tauira Tukutuku · Symmetry, transformation, and coordinates through Māori lattice weaving — mathematics embedded in toi (art).
Free class-ready resource, premium progression path
This handout is ready to use. For additional patterns, assessment tasks, or integration with te reo Māori geometry vocabulary, Te Wānanga can generate a full sequence.
- Add niho taniwha, patiki, or pātiki pattern extensions.
- Generate bilingual geometry vocabulary cards.
- Save adapted geometry sequences in My Kete.
Ngā Whāinga Akoranga · Learning Intentions
- We are learning to identify line symmetry and rotational symmetry in patterns.
- We are learning to describe transformations (translation, reflection, rotation) in tukutuku.
- We are learning to use a coordinate grid to design and describe a geometric pattern.
- We are learning to connect Māori art with mathematical concepts.
Paearu Angitu · Success Criteria
- I can draw and label lines of symmetry on a tukutuku pattern.
- I can identify which transformations each pattern uses.
- I can design my own pattern on a coordinate grid using at least one transformation.
- I can explain the mathematical properties of tukutuku patterns in my own words.
Hononga Marautanga · Curriculum Alignment
NZC Geometry and Measurement
- Symmetry: line and rotational
- Transformations: translation, reflection, rotation
- Tessellation: repeating without gaps
- Coordinate grids and grid references
Te Ao Māori integration
- Tukutuku as a vehicle for geometric thinking
- Wharenui (meeting house) as a mathematical space
- Cultural significance of each pattern
- Toi Māori as rigorous mathematical practice
Horopaki · What are Tukutuku?
Tukutuku are decorative lattice panels found inside wharenui (meeting houses). They are created by weaving kiekie or pīngao strips through a wooden frame to produce geometric patterns. Each pattern carries a name, a meaning, and a whakapapa (genealogy) of knowledge. Tukutuku demonstrates that Māori have always practised sophisticated mathematics — long before formal schooling arrived in Aotearoa.
The weavers who create tukutuku use symmetry, ratio, and transformation intuitively, counting rows and columns with precision. Every panel is a proof that mathematical thinking is culturally embedded.
Ngā Tauira · Three Tukutuku Patterns
Poutama — Stairway
Represents the stairway of knowledge and achievement — each step upward is a new level of learning.
Mathematical ideas: diagonal translation, arithmetic sequence (number of red cells increases by 1 each row).
Kaokao — Rib
Represents the ribs of the body and the structure of the wharenui — strength and protection through careful, interlocking design.
Mathematical ideas: 2 lines of reflective symmetry, order-2 rotational symmetry.
Niho Taniwha — Shark's Tooth
A zigzag pattern named for the teeth of the taniwha (water guardian) — representing alertness, danger, and the power of natural forces.
Mathematical ideas: translational symmetry (slides horizontally), glide reflection.
Mahi 1 · Activity 1: Symmetry Identification
Hangarite — Symmetry
Line (reflective) symmetry means one side is a mirror image of the other. Rotational symmetry means the shape looks the same after a rotation less than 360°.
Complete the table for each tukutuku pattern shown above.
| Pattern / Tauira | Number of lines of symmetry | Has rotational symmetry? | Order of rotational symmetry |
|---|---|---|---|
| Poutama | |||
| Kaokao | |||
| Niho Taniwha |
Choose one pattern and describe where its line(s) of symmetry sit (e.g. "vertical line through the centre").
Mahi 2 · Activity 2: Design on a Coordinate Grid
Use the 12 × 12 coordinate grid below to design your own tukutuku-inspired pattern. Label the x-axis (across) and y-axis (up) with numbers 0–12. Use at least two colours and include at least one transformation (translation, reflection, or rotation).
Describe your pattern: Which transformation(s) did you use and where?
List three coordinates (x, y) of coloured cells in your pattern:
Point 1: ( _____ , _____ ) Point 2: ( _____ , _____ ) Point 3: ( _____ , _____ )
Mahi 3 · Activity 3: Transformation Identification Table
For each tukutuku pattern, identify which transformations it uses. Tick all that apply and add a short explanation.
| Pattern | Translation (slide)? | Reflection (flip)? | Rotation (turn)? | Brief explanation |
|---|---|---|---|---|
| Poutama | ||||
| Kaokao | ||||
| Niho Taniwha | ||||
| My own pattern |
Extension: Which of the three named patterns tessellates? Explain why.
Aronga Mātauranga Māori
Tukutuku weaving is an expression of kaitiakitanga (guardianship) of knowledge. Every pattern contains whakapapa — the genealogy of its makers and the communities who use it. Māori weavers do not see geometry as a school subject: they see it as a living practice that connects past, present, and future. When ākonga engage with tukutuku patterns mathematically, they are participating in an ancient intellectual tradition. This is mātauranga Māori — knowledge that is embedded in practice, community, and environment rather than separated into abstract disciplines.
Ngā Rauemi Tautoko · Support Materials
Resources already provided:
- Descriptions and grid visuals for three tukutuku patterns (poutama, kaokao, niho taniwha)
- Symmetry identification table (line and rotational)
- 12 × 12 coordinate grid for student design activity
- Transformation identification table (translation, reflection, rotation)
- Bilingual headings and kupu Māori throughout
Aronga Rerekē · Differentiated Pathways
Entry-level support
Focus on the poutama pattern only. Use a printed 8 × 8 grid with squares to colour in. Ask students to fold the paper to check for line symmetry. Provide a vocabulary card with translation, reflection, rotation illustrated.
On-level
Complete all three activities independently. Use coordinate notation correctly. Identify symmetry properties for all three patterns and design an original pattern on the coordinate grid.
Extension / Whakaaro Hohonu
Research the patiki (flounder) or pūhoro (speed lines) pattern. Describe all its mathematical properties: symmetry, transformation type, whether it tessellates, and what coordinate rule generates it. Connect the pattern's meaning to its mathematical structure.
📋 Teacher Planning Snapshot
Ngā Whāinga Ako — Learning Intentions
Students will engage with this resource to build pāngarau (mathematical) understanding — developing number sense, pattern recognition, and mathematical reasoning through hands-on, culturally grounded activities that connect to tamariki's world.
Ngā Paearu Angitū — Success Criteria
- ✅ Students can explain their mathematical thinking using words, objects, drawings, or symbols.
- ✅ Students can apply the number or pattern concept in this resource to a real or everyday context.
Differentiation & Inclusion
Scaffold support: Use concrete materials (blocks, counters, fingers) for entry-level engagement before progressing to abstract representations. Offer extension challenges asking students to generalise a pattern, write their own word problem, or explain their strategy to a partner.
ELL / ESOL: Mathematical language is a discipline-specific barrier — pre-teach key terms (e.g., equals, more than, fewer, pattern, factor) using visual representations. Allow students to demonstrate mathematical understanding non-verbally or through drawing. Pair with a bilingual buddy where possible.
Inclusion: Embed choice in how students engage — oral, written, or diagrammatic responses are all valid. Neurodiverse learners benefit from short, chunked task sequences with immediate feedback loops. Avoid timed drills in favour of exploratory tasks that reward curiosity. Make the maths classroom a safe place to be wrong and try again.
Mātauranga Māori lens: Pāngarau is a living tradition in Te Ao Māori — from the geometric precision of tukutuku and kōwhaiwhai patterns to the navigational mathematics of waka hourua, and the seasonal calculations embedded in maramataka. Framing early number sense within these contexts shows tamariki that mathematics is a human, culturally rich endeavour — not a foreign import. Encourage students to see counting, measuring, and patterning as acts of knowing their world.
Prior knowledge: Designed for early learners. No prior formal mathematics knowledge required. Teachers should assess current number knowledge before selecting appropriate entry points.
Curriculum alignment
- Number and Algebra — Number Strategies: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
- Number and Algebra — Patterns and Relationships: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.