Lesson 3: Pathways & Navigation — Ngā Ara me te Whakatere

Students plan the most efficient visitor route through Hamilton Zoo, combining coordinate geometry with compass directions and connecting to Māori navigation traditions.

Phase 3 · Years 7–8 Mathematics · Pāngarau ⏱ 60 min

📎 Learning Intentions | Ngā Whāinga Ako

Know | Mōhio

  • That compass directions (N, S, E, W, NE, SW, etc.) describe movement direction on a grid.
  • That the horizontal distance between two points is the difference in their x-values; vertical distance is the difference in their y-values.
  • That the modern Pacific voyaging revival teaches wayfinding with a 32-house star compass, building on ancestral navigation by star paths and ocean swells — how those ancestors organised direction before European contact is a genuine research question.

Demonstrate | Whakaatu

  • Describe a route between two zoo enclosures using compass directions and grid steps.
  • Calculate the total grid distance of a planned zoo visit route (at least 5 stops).
  • Design an optimal 4-enclosure visit route that minimises total travel distance.

📐 Curriculum Alignment

NZC Mathematics and Statistics · Level 3 · Geometry and Measurement — Position and orientation
Use a co-ordinate system or the language of direction and distance to specify locations and describe paths.
Ministry of Education (2007). The New Zealand Curriculum. Mathematics and statistics achievement objectives, Level 3.

🌿 Whakatūwhera

Polynesian navigators crossed Te Moana-nui-a-Kiwa reading star paths, ocean swells, and bird flight as one system of direction. In today's Pacific voyaging revival, that knowledge is taught with a star compass that divides the horizon into 32 named houses. Precision in direction was survival at sea — the same reason we are precise with grid references and compass points on our zoo map.

🪶 Mātauranga Māori — handoff required (whakataukī sourcing). This opening previously carried an unattributed line presented as a whakataukī ("Ko Māui i hua ake i ngā whetū…") and a claim that today's compass rose "directly descends" from the Polynesian star compass. Both were removed — the line could not be sourced, and the descent claim is false (the Western compass rose has its own history). If kaiako Māori or kaumātua can offer a genuinely sourced whakataukī for wayfinding, it belongs here with its attribution.

🎤 Kaiako Hook (5 min)

Show the Polynesian Voyaging Society star compass image. Ask students: "How many directions are on this compass? Why would sailors need 32 points instead of just 4?" Connect to the idea that precision in direction reduces travel error — just like precision in coordinates.

🎬 Watch

Watch the full clip — 5:32. Source: TED-Ed. Wayfinding across the Pacific without instruments — star paths, swell and bird flight held as a system. Navigating a zoo map is the same skill at a much smaller scale, and the ancestors' version is the harder one.

📚 Lesson Content

Part 1: Compass Directions on the Zoo Grid (10 min)

Review the 8 main compass directions with students. Then show how they map onto grid movements:

🔑 Grid Distance Formula

Horizontal distance between (x₁, y₁) and (x₂, y₂) = |x₂ − x₁|

Vertical distance = |y₂ − y₁|

Total grid path distance = horizontal + vertical (moving along grid lines — no diagonal shortcuts yet!).

Part 2: Route Planning Challenge (25 min)

Working on the class zoo grid map (Lesson 1), each student plans a route from the main entrance (0, 0) that visits these five animals — Elephant (3, 4), Lion (4, 3), Giraffe (4, −2), Kiwi (−3, 4) and Meerkats (−4, −3) — and that:

  1. Visits all 5 animals in some order.
  2. Returns to the entrance at the end.
  3. Has the shortest possible total grid distance.

Students record: Starting point → Animal 1 → Animal 2 → ... → Entrance, listing compass direction and distance for each segment.

🔑 Kaiako answer key — Part 2 shortest route

The shortest possible round trip is 34 grid units (680 m at 1 unit = 20 m). One route that achieves it:

  1. Entrance (0, 0) → Giraffe (4, −2): 4 + 2 = 6
  2. Giraffe (4, −2) → Lion (4, 3): 0 + 5 = 5
  3. Lion (4, 3) → Elephant (3, 4): 1 + 1 = 2
  4. Elephant (3, 4) → Kiwi (−3, 4): 6 + 0 = 6
  5. Kiwi (−3, 4) → Meerkats (−4, −3): 1 + 7 = 8
  6. Meerkats (−4, −3) → Entrance (0, 0): 4 + 3 = 7

6 + 5 + 2 + 6 + 8 + 7 = 34 grid units. Walking the same loop in reverse (Meerkats first) is exactly as short — no other order of the five animals beats 34, and the longest order costs 62, so there is real ground for students to gain here.

Accept any order totalling 34. Distances are grid-path (along the lines), matching the rule in Part 1 — no diagonal shortcuts.

Part 3: Compare Routes (10 min)

Students share their routes with a partner. Whose route is shorter? By how many grid units? Why might a different order of visits change the total distance?

Connect to real-world logistics: delivery drivers, zoo keepers, and doctors doing rounds all solve this same problem — called the Travelling Salesman Problem — one of mathematics' most famous unsolved challenges for large numbers of stops.

✅ Assessment

🎟 Exit Ticket

  1. The Lion is at (4, 3) and the Giraffe is at (4, −2). What direction do you travel and how many grid units?
  2. Plan a 3-stop route: Entrance (0,0) → Kiwi (−3, 4) → Café (2, 4) → Entrance. What is the total grid distance?
  3. Name one way Māori navigators used a "coordinate system" without a map.
🔑 Kaiako answer key
  1. South, 5 grid units — same x-value (4), y drops from 3 to −2, so |−2 − 3| = 5.
  2. 18 grid units: (0,0)→(−3,4) is 3 + 4 = 7; (−3,4)→(2,4) is 5 + 0 = 5; (2,4)→(0,0) is 2 + 4 = 6; 7 + 5 + 6 = 18.
  3. Open response — e.g. star paths (ara whetū) rising at known horizon points, or ocean swells held at a constant angle to the waka.