Lesson 4: Gradient & Slope — The Mathematics of Zoo Pathways
Understanding gradient as rise over run and applying it to design accessible, safe zoo pathways — connecting algebra (linear equations) to real-world civil engineering.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Gradient = rise/run = (y2-y1)/(x2-x1); how to calculate gradient between two coordinate points; how gradient affects wheelchair accessibility and stroller navigation at the zoo.
Calculate gradients for 5 zoo pathway segments using coordinates; determine which pathways meet accessibility standards (gradient ≤ 1:12); design a new accessible pathway.
🎥 Media Anchor & Mathematical Scaffold
Mathematics in Context: Hamilton Zoo
Watch the full clip — 6:48. Source: Cognito. The zoo context is ours; the clip teaches the method, then the lesson applies it to real enclosure coordinates.
🧠 1. Before Viewing
Why do wheelchair ramps need to be less steep than normal slopes? The NZ standard for accessible ramps is a gradient of 1:12 — that means for every 12m horizontal, you rise maximum 1m. How would you check if a zoo path meets this standard?
👁️ 2. During Viewing
- Early in the clip: Calculate the gradient of the pathway from (1,0) to (7,2): gradient = (2-0)/(7-1) = 2/6 = 1/3. Is this steeper or flatter than the 1:12 standard?
- Midway: A gradient of 1:12 in our coordinate system means for every 12 x-units, y increases by 1. What coordinates would describe a 50m ramp at exactly 1:12 gradient?
- Later in the clip: How does gradient connect to the slope in the equation y = mx + c? What does 'm' represent in terms of zoo pathways?
🗣️ 3. After Viewing — Kaiako Move
Kaiako Move: Display a zoo pathway map with elevation data. Work through gradient calculations for three pathways. Discuss: which paths would be challenging for wheelchair users? How would you redesign them?
Task: Accessibility Audit: Calculate gradients for 5 zoo pathways. Check each against the 1:12 standard. Identify which paths need redesigning and propose coordinate-based alternative routes.
⚡ Whakaoho | Do Now (5 mins)
Quick check: which path is steeper — one that rises 3m over 15m horizontal, or one that rises 4m over 24m horizontal? Show the gradient calculation for each.
📐 Activity 1: Guided Mathematical Exploration (25 mins)
Pathway redesign: one zoo path has a gradient of 1:6 (too steep for wheelchairs). Using coordinates, redesign it to meet the 1:12 standard. Show the new coordinate endpoints.
✍️ Activity 2: Application & Reflection (20 mins)
Gradient real-world connection: find one example outside of a zoo where gradient calculation matters (road design, ski slopes, roof design, ramps). Explain what gradient value would be appropriate.
🏫 Kaiako Planning & Pedagogy Notes
Curriculum Alignment: Te Mātaiaho Mathematics Phase 3/4 — Algebra and Geometry: calculate gradient between coordinate pairs; apply to accessibility and design contexts.
Hamilton Zoo Context: All tasks use authentic zoo spatial data. Print the map with any contour or level information; project it to discuss which paths look flat and are not.
Cultural Connection: Gradient decides which zoo paths are usable by a wheelchair, a pushchair or an older visitor. That makes slope an access question before it is an algebra question.