Waka Construction Geometry
Ratios, Symmetry, and Scale Drawing · Mātauranga Māori in Mathematics
Ngā Whāinga Akoranga · Learning Intentions
- Calculate and compare length-to-width ratios across different waka types
- Apply symmetry to understand structural requirements in waka design
- Create a scale drawing of a waka using a given scale (1 cm = 1 m)
- Calculate percentage of length using proportional reasoning
Paearu Angitu · Success Criteria
- I can calculate a ratio and express it in simplest form
- I can identify and draw a line of symmetry on a 2D shape
- I can use a scale factor to convert real dimensions to drawing dimensions
- I can calculate a percentage of a total length and show my working
Hononga Marautanga · Curriculum Alignment
Level 4–5: calculate ratios and proportions; apply symmetry; create and interpret scale drawings; use measurements in context.
Recognise mathematical knowledge embedded in traditional Māori craft and technology; understand that waka construction reflects sophisticated STEM knowledge.
Whakataukī
"He waka eke noa"
A canoe which we are all in with no exception.
The construction of a waka is one of the greatest engineering achievements of the Māori world. Tohunga waka (master builders) used sophisticated geometry — symmetry, ratios, and precise measurement — developed over centuries of practice. This mathematics was implicit, embedded in practice, yet every bit as rigorous as what we formalise on paper.
Wāhi 1 · Āhuatanga Waka · Waka Dimensions and Ratios
A traditional waka taua (war canoe) could be up to 25 metres long. Calculate the length-to-width ratio for each waka type below using the midpoint of each range.
| Waka type | Typical length | Beam (width) | Ratio (length ÷ width) | Simplified |
|---|---|---|---|---|
| Waka tīwai (single, small) | 5–8 m | 0.6–0.8 m | ||
| Waka hourua (double-hulled) | 15–20 m | 5–7 m | ||
| Waka taua (war canoe) | 20–25 m | 1.2–1.5 m |
1. Calculate the ratio for each row. Show your working for one calculation here:
2. What do you notice about the ratios? Why might the waka taua have a very different ratio to the waka hourua?
Wāhi 2 · Hangarite · Symmetry and Balance
A waka must be perfectly symmetrical — if it isn't, it will tip or travel in circles. Tohunga waka worked with lines of symmetry long before written geometry existed in Aotearoa.
Draw a simple top-down view of a waka hull in the box below. Draw and label its line of symmetry.
1. If the left side of a waka hull curves outward 15 cm at its widest point, how far out does the right side curve? Why?
2. A waka is 18 metres long. The bow and stern are each carved sections 2.5 m long. What percentage of the waka's length is carved? Show your working.
3. Sketch what you think the hull cross-section (viewed from the front) looks like. Label: waterline, keel, gunwale (top edge).
Wāhi 3 · Āhua Āwhata · Scale Drawing Challenge
Tohunga waka used a rākau (measuring stick) calibrated to body measurements — a form of scale drawing. Using a scale of 1 cm = 1 m, draw a side view of a 15-metre waka taua in the space below.
Your drawing must include:
- The hull shape (longest at the waterline, curving up at bow and stern)
- The tauihu (prow carving) and taurapa (stern post) at each end
- 12 evenly-spaced paddle positions along the hull
- Your scale label and all key dimensions annotated
1. What is the distance (in your drawing) between each paddle position? Show how you calculated the spacing.
2. (Extension) A waka hourua like Hōkūleʻa can sail to Hawai'i — approximately 4,000 km. At an average speed of 8 knots (≈15 km/h), how many hours would the crossing take? How many days?
Aronga Mātauranga Māori
Waka construction was not just technology — it was a deeply cultural and spiritual practice. The tohunga waka held knowledge passed down through generations, encoded in karakia, specific vocabulary for each part of the hull, and the physical memory of the craft itself. The symmetry calculations, ratio knowledge, and measurement systems embedded in waka construction represent a sophisticated indigenous mathematical tradition that predated European contact by centuries.
Today, waka hōro (waka ama outrigger) racing is one of the fastest-growing sports in Aotearoa. The same principles of hull geometry — length-to-weight ratios, hydrodynamics, symmetry — apply in modern waka design. Mātauranga and modern engineering are not opposites; they are in dialogue.
Ngā Rauemi Tautoko · Support Materials
Resources already provided:
- This handout with dimension data and drawing spaces
- Ruler (essential for scale drawing)
- Calculator (permitted for ratio and percentage calculations)
- Coloured pencils (optional — use to distinguish different waka parts)
Aronga Rerekē · Differentiated Pathways
Tīmata · Entry Level
Complete the ratio table using the midpoint values (teacher can provide these). Complete the symmetry drawing in Part 2. Answer questions 1 and 2 in Part 2.
Paerewa · On Level
Complete all three parts. Draw the scale plan accurately with all required features. Answer all questions including the cross-section sketch.
Tūāpae · Extension
Complete all sections. Research Te Aurere or Ngāhiraka Mai Tawhiti (NZ voyaging waka) — find their actual dimensions and compare their ratios to your table. Explain what the design priorities reveal about purpose (speed vs stability vs capacity). Include the Hōkūleʻa calculation.
📋 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.