Mountain Navigation Trigonometry
Mountain Navigation Trigonometry · Years 9–11
Ngā Whāinga Akoranga · Learning Intentions
- Investigate a significant question using evidence from multiple sources
- Analyse and evaluate information to form and support a reasoned position
- Connect learning to real-world contexts, including Aotearoa New Zealand settings
- Communicate understanding clearly and accurately for a specific audience
Paearu Angitu · Success Criteria
- I use at least two sources and can evaluate their credibility
- My position is clearly stated and supported by specific evidence
- I can connect my learning to at least one real-world Aotearoa context
- My communication is clear, organised, and appropriate for the audience
Mountain Navigation & Trigonometry
🌋 Trigonometry through Maunga — measuring what you cannot reachTūhoe and other iwi navigated the vast ranges of Te Urewera and beyond without GPS — using stars, rivers, ridgelines, and their mathematical understanding of angle and distance. This handout connects SOHCAHTOA trigonometry and bearings to that tradition. You will calculate the heights of real NZ mountains and navigate using compass bearings, exactly as tīpuna did — but now with algebraic tools to match.
Part 1 — Te Poutama: The Staircase Model of Learning
Te poutama — the staircase — is a Māori concept that describes learning as a series of ascending steps, each building on the last. In trigonometry, we do exactly this: right triangles → trig ratios → bearings → 3D navigation.
sides & angles
the three ratios
given an angle
using inverse trig
navigation
🔑 SOHCAHTOA — The Three Trig Ratios
Memory hook: "Some Old Heron Caught A Heron — Treading On Asparagus"
Key Vocabulary — Te Kuputaka
| English Term | Te Reo Māori / Meaning | Definition |
|---|---|---|
| Bearing | Ahunga / direction | Angle measured clockwise from North (000° to 360°) |
| Angle of elevation | Koki teitei | Angle from the horizontal up to an object |
| Hypotenuse | Tapa roa | The longest side, opposite the right angle |
| Trigonometry | Ine-topeka | The study of the relationship between angles and sides |
| Maunga | Maunga | Mountain — a taonga tuku iho (treasure passed down) |
Part 2 — Ngā Maunga o Aotearoa: Real Mountain Data
Every mountain in Aotearoa has a name — and that name carries whakapapa. To Māori, mountains are ancestors (tīpuna maunga). Here is data for three real maunga:
| Maunga (Mountain) | Iwi Connection | Height (m) | Distance to Base (km) | Angle of Elevation |
|---|---|---|---|---|
| Aoraki / Mt Cook | Ngāi Tahu — ancestor of South Island | 3,724 | 10 km from Hooker Valley floor | 20.5° |
| Taranaki / Mt Egmont | Taranaki iwi — exiled ancestor | 2,518 | 8 km from Stratford township | 17.5° |
| Tongariro | Ngāti Tūwharetoa — gifted to NZ 1887 | 1,978 | 6 km from National Park Village | 18.3° |
📐 Worked Example — Calculating the Height of Taranaki
From 8 km away (horizontal distance), the angle of elevation to the summit is 17.5°. Find the height.
(Small difference from 2,518 m because the viewing point is not at sea level — tino pai, close enough!)
Your Turn — Practice Problems
-
Aoraki / Mt Cook: Standing 10 km (10,000 m) from the base, the angle of elevation is 20.5°.
Use
tan(θ) = Opposite / Adjacentto calculate the height. Show your working.
Space for working: -
Tongariro: You are 6 km from the base and the angle of elevation is 18.3°. Calculate the
height.
Space for working: -
Unknown maunga: A mountain is 1,500 m tall. You measure it from 4 km away. What is the angle
of elevation?
Use
tan⁻¹(opposite/adjacent).
Space for working:
Part 3 — Bearings: The Mathematics of Direction
A bearing is a clockwise angle measured from North. Tīpuna navigators used stars and landscape features; today we use the same angles with a compass or GPS. Bearings are always written as three digits (e.g., 045°, not 45°).
Cardinal Bearings
| Direction | Bearing |
|---|---|
| North (Raro) | 000° |
| East (Rāwhiti) | 090° |
| South (Tonga) | 180° |
| West (Uru) | 270° |
🧭 Navigation Scenario
A tramping group leaves their camp on the Tongariro Crossing. They walk 3 km on a bearing of 065° to reach the Red Crater, then 2 km on a bearing of 145° to reach the Emerald Lakes.
Draw a scale diagram (1 cm = 0.5 km) and find their total displacement from camp:
Bearing Calculations
- A ranger hut is due East of the camp (090°) and 5 km away. The maunga summit is 4 km North of the hut. What is the bearing from camp to the summit? What is the straight-line distance?
- Point A is at the base of Taranaki. Point B is 12 km away on a bearing of 315° (North-West). Draw this and find how far north and how far west B is from A using trigonometry.
- Extension: Why do navigators always use three-digit bearings? What confusion could arise from writing "45°" instead of "045°"?
Part 4 — Whakaaro Māori: Critical and Cultural Thinking
🌿 Te Ao Māori & Mathematics
In 2013, the Whanganui River was granted legal personhood. In 2017, Tongariro and its peaks received formal recognition as a legal entity — an ancestor, not a resource. This reflects the Māori worldview in which maunga have mana (authority) and are not just geographical features to be measured.
Yet Māori also developed sophisticated measurement systems. The unit kūiti (a fist-width) and aho (a cord measure) were used in construction and navigation. The tohunga kōkōrangi (expert astronomers) calculated the risings and settings of stars as angles — effectively doing trigonometry without algebra.
- Historical Thinking: Māori navigators from Hawaiki found Aotearoa using stars. The star Matariki rises at approximately 63° North of East (bearing 027°). If ancient navigators sailed in this direction for 3,000 km, how far north and east would they have travelled? Calculate using trig.
- Cultural Analysis: Tongariro is named Te Kāhui Maunga — the sacred mountains. When surveying teams first measured Aoraki's height in the 19th century, they renamed it "Mount Cook" without consultation. How does this connect to the concept of mana whenua (authority over the land)? Write 3–4 sentences.
- Systems Thinking: GPS can now calculate mountain heights and bearings instantly. Does this technology make the mathematics of trigonometry less important, or more important? Argue your case with at least one specific example.
- Research Extension: Find the coordinates (latitude/longitude) of any two NZ mountains. Use the Haversine formula (or a simplified distance formula) to calculate the distance between them. Present your working and reflect on how tīpuna navigated this same landscape without GPS.
🏔️ Whakamutunga — Synthesis
Trigonometry was not invented in a European classroom. It stretches back to Mesopotamia, Egypt, India, and across the Pacific. Māori navigated the largest ocean on Earth — te Moana nui a Kiwa — using angle, pattern, and deep mathematical thinking. SOHCAHTOA is the algebra that names what they already knew in practice.
Self-check: Can you explain sin, cos, and tan to a classmate using a mountain as your example? Can you draw a bearing? If yes — you're climbing the poutama. 🌟
🌿 Ngā Rauemi Hono — Related Resources
Hononga Marautanga · Curriculum Alignment
Level 3–4: Investigate social, cultural, environmental, and economic questions; gather and evaluate evidence from diverse sources; communicate findings and reasoning clearly for different audiences and purposes.
Level 3–4: Read, interpret, and evaluate information texts; write clearly and purposefully for specific audiences; apply critical thinking skills to evaluate sources and construct well-reasoned responses.
Tuhia ōu whakaaro · Write Your Thoughts
Reflect on your learning. What was the most important idea? What question do you still have?
Aronga Mātauranga Māori
This resource sits within a kaupapa that recognises mātauranga Māori as a living knowledge system with its own frameworks, values, and ways of understanding the world. The New Zealand Curriculum calls for learning that reflects the bicultural partnership of Te Tiriti o Waitangi, which means every subject area has an obligation to engage authentically with Māori perspectives — not as cultural decoration but as substantive contributions to how we understand our topics. The concepts of manaakitanga (care for others), kaitiakitanga (guardianship), whanaungatanga (relationship and belonging), and tino rangatiratanga (self-determination) provide a values framework applicable across all learning areas, and all are relevant to the work in this handout.
Ngā Rauemi Tautoko · Resources already provided
This handout is designed to be used alongside other resources in the same unit. Related materials are linked in the unit planner. All content is provided — no additional preparation is required to use this handout in your classroom.
📋 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.