📚 Unit Plans → 🌍 Hamilton Zoo Advanced Analytics - Phase 4
Years 9–10 Pāngarau / Mathematics

⚗️ Hamilton Zoo Advanced Analytics 📊 Tātari Arō Kōpere Kirikiriroa

Zoo data shows visitor movement follows predictable linear patterns between major attractions.

🪶 Mātauranga Māori — handoff required (te reo accuracy). This page previously used Whakatōhea as if it were a word for a concept. Te Whakatōhea is an iwi of Ōpōtiki, and that false attribution has now been removed rather than replaced — substituting a guessed kupu would repeat the original error with better spelling. Still open for kaiako Māori or kaumātua: the te reo in this page's own title strings — “Tātari Arō Kōpere Kirikiriroa” for “Advanced Analytics”, “Rangahau Pāngarau Arā”, and “Kua Oti te Angitū Arā!” — reads as machine-generated and has not been verified. Ask: what kupu was intended, and does anything here imply a connection to Te Whakatōhea that is not real?
NZ Curriculum Phase 4 (Years 9-10)

⚗️ Hamilton Zoo Advanced Analytics 📊 Tātari Arō Kōpere Kirikiriroa

Four-Quadrant Coordinate Geometry & Mathematical Modelling

🎯 Advanced Mathematical Investigation | Rangahau Pāngarau Arā

Welcome to professional-level spatial analysis! You'll use advanced coordinate geometry and mathematical modelling to solve complex zoo management challenges.

📊 Advanced Mathematical Concepts:

🔢 Core Mathematics:

  • Four-quadrant coordinate system
  • Distance & midpoint formulas
  • Linear equations & gradients
  • Geometric transformations
  • Optimisation techniques

🔬 Applied Analysis:

  • Mathematical modelling
  • Data interpretation
  • Constraint optimisation
  • Statistical relationships
  • Professional problem-solving
d = √[(x₂-x₁)² + (y₂-y₁)²]  |  m = (y₂-y₁)/(x₂-x₁)  |  M = ((x₁+x₂)/2, (y₁+y₂)/2)

Extended coordinate plane: Scale 1 unit = 1 grid square = 80 m | Origin at zoo geometric centre

Hamilton Zoo grid map: a 6 by 6 grid with columns lettered A to F across the top and rows numbered 1 to 6 down the side. The giraffe is at A3, the entrance and car park at F3, the Oasis cafe and toilets at C3, the ponds at D3, and the savannah holding zebra and ostrich at C1.
Hamilton Zoo grid map — a simplified schematic for classroom use. Positions are approximate, placed from Hamilton Zoo’s published visitor map (November 2025); it is not to scale and not a survey. One grid square = 80 metres. For the current official map see hamiltonzoo.co.nz.

Investigation 1: Four-Quadrant Coordinate Mapping

Research Question: How can we use an extended coordinate system to optimise zoo layout and operations across all quadrants?

📍 Extended Coordinate System Setup:

Origin (0, 0): Zoo geometric centre

Quadrant I (+, +): Main exhibits and visitor areas

Quadrant II (-, +): Research facilities and breeding programs

Quadrant III (-, -): Infrastructure (parking, maintenance, storage)

Quadrant IV (+, -): Future expansion zone and conservation areas

Scale Factor: 1 coordinate unit = 1 grid square = 80 metres

Two layers, stated plainly: the Hamilton Zoo map you already know answers where — letter-number squares (A–F across, 1–6 down). The quadrant model on this page answers a different question — what kind of land use — so it divides the site by operations, not by map letters: main exhibits and visitor areas in Quadrant I, research and breeding in II, infrastructure in III, future expansion in IV. To place a facility in the model, read its map square for direction and rough distance from the site centre, then choose a signed coordinate pair that satisfies its zone's sign conditions — a modelling decision you state and defend, like a surveyor choosing axes. Converting a map position such as B5 into a defended (x, y) pair is a skill the investigations below will repeatedly ask of you.

The sign conditions built into each answer box are the point, not a hint: a coordinate pair only counts if it satisfies its zone's quadrant signs. Choose values, then check them against the conditions.

Facility Quadrant x-coordinate y-coordinate Coordinate Pair Real Distance from Origin (m)
🚗 Main Car Park III ( , )
🦒 Giraffe Exhibit I ( , )
🔬 Research Centre II ( , )
🌱 Future Conservation Zone IV ( , )

Investigation 2: Linear Modelling & Gradient Analysis

Research Question: How can linear relationships model visitor flow patterns and optimise pathway efficiency?

📊 Visitor Flow Analysis

Zoo data shows visitor movement follows predictable linear patterns between major attractions.

Task 1: Pathway Gradient Calculation

Calculate the gradient of the main pathway from Car Park to Giraffe Exhibit:

Car Park: ( , )

Giraffe Exhibit: ( , )

m = (y₂ - y₁)/(x₂ - x₁) = ( - )/( - ) =

Interpretation: A gradient of means for every units horizontally, the path rises units vertically.

Task 2: Linear Equation Modelling

Develop a linear equation for the main visitor pathway using point-slope form:

y - y₁ = m(x - x₁)

Using Car Park as reference point:

y - ( ) = (x - ( ))

Pathway Equation: y = x +

Investigation 3: Geometric Transformations

Research Question: How can coordinate transformations optimise zoo exhibit layouts for seasonal changes and special events?

🔄 Transformation Matrices

Apply transformation matrices to relocate the temporary petting zoo exhibit:

Original Petting Zoo Vertices:

A(2, 1), B(4, 1), C(4, 3), D(2, 3)

Translation Vector:

Move 3 units left, 2 units down

T = [ ]
[ ]

Rotation Matrix (90°):

Rotate 90° counterclockwise about origin

R = [ ]
[ ]

Scaling Matrix:

Scale by factor of 1.5

S = [ ]
[ ]

Composite Transformation: T ∘ R ∘ S

Calculate the final position of vertex A(2, 1) after all transformations:

Step 1 - Scale: S × [2, 1]ᵀ = [ , ]ᵀ

Step 2 - Rotate: R × result = [ , ]ᵀ

Step 3 - Translate: T + result = [ , ]ᵀ

Final position of A: ( , )

Investigation 4: Optimisation & Constraint Analysis

Research Question: Given multiple constraints, what is the optimal location for a new educational centre?

🎯 Multi-Constraint Optimisation Problem

Requirements for New Educational Centre:

  • Must be within 5 coordinate units (400 m) of the main visitor pathway
  • Cannot be more than 7.5 coordinate units (600 m) from any existing major exhibit
  • Must maintain a minimum 3-coordinate-unit (240 m) buffer from animal enclosures
  • Should minimise total visitor walking distance
  • Must be in Quadrant I for accessibility

Mathematical Constraint Modelling:

Constraint 1: Distance from pathway y = mx + c ≤ 5 units (400m)

Point-to-line distance: d = |ax + by + c|/√(a² + b²)

Constraint 2: Maximum distance from exhibits

∀ exhibit E: √[(x - xₑ)² + (y - yₑ)²] ≤ 7.5 units (600m)

Constraint 3: Minimum buffer from enclosures

∀ enclosure F: √[(x - xf)² + (y - yf)²] ≥ 3 units (240m)

Objective Function: Minimise Σ dᵢ where dᵢ = distance to visitor attraction i

Proposed Solution:

Educational Centre Coordinates: ( , )

Constraint Verification:

Constraint Required Calculated Value Satisfied? ✓/✗
Distance from pathway ≤ 5.0 units units
Max distance to exhibits ≤ 7.5 units units
Min buffer from animals ≥ 3.0 units units

Total visitor walking distance: units = metres

Investigation 5: Statistical Modelling & Prediction

Research Question: How can coordinate-based analysis predict visitor patterns and optimise resource allocation?

📈 Advanced Statistical Analysis

Work from the practice dataset below — ten sampled days of peak-hour visitor centroids on the signed coordinate system. It is an illustrative practice set for computation, not real Hamilton Zoo data; treat any conclusion as about the dataset, and say so in your write-up. Compute the mean centroid and the spread, then analyse spatial patterns:

DayVisitorsCentroid x̄Centroid ȳ
11,8400.80.5
22,3101.20.9
3960−0.6−0.4
41,4500.20.1
52,7501.61.1
61,120−0.90.3
71,9800.5−0.7
8830−1.4−1.0
92,2001.00.6
101,5600.1−0.2

Heat Map Analysis

Visitor density function: ρ(x,y) = Ae^(-((x-μₓ)² + (y-μᵧ)²)/(2σ²))

Where μₓ, μᵧ are mean coordinates and σ is standard deviation

Peak visitor coordinates: μ = ( , )

Spread parameter: σ = units

Maximum density: A = visitors/unit²

Correlation Analysis

Pearson correlation coefficient between daily visitors and centroid x̄:

r = Σ[(nᵢ - n̄)(xᵢ - x̄)] / √[Σ(nᵢ - n̄)² × Σ(xᵢ - x̄)²]

Calculated correlation: r =

Interpretation:

Predictive Modelling

Linear regression for visitor numbers based on coordinates:

V(x,y) = β₀ + β₁x + β₂y + β₃xy + ε

Regression coefficients:

β₀ (intercept) =

β₁ (x-coefficient) =

β₂ (y-coefficient) =

β₃ (interaction) =

R² value:

Prediction for coordinates (2.5, 1.8): visitors/hour

🎉 Advanced Mathematics Mastery Complete! | Kua Oti te Angitū Arā!

Outstanding! You've worked through coordinate geometry and mathematical modelling that stretches well past the Years 9-10 curriculum.

✅ Mathematical Skills Mastered:

  • 🎯 Four-quadrant coordinate systems
  • 📐 Advanced distance & gradient formulas
  • 🔄 Matrix transformations
  • ⚗️ Constraint optimisation
  • 📊 Statistical coordinate analysis
  • 🧮 Linear regression modelling

🚀 Career Pathways Unlocked:

  • 🏗️ Engineering & Architecture
  • 💻 Computer Science & Gaming
  • 📡 Geospatial Technology (GIS)
  • 📊 Data Science & Analytics
  • 🛰️ Aerospace Engineering
  • 🔬 Research & Development

📚 Where this maths goes next:

  • 🔢 Calculus & Advanced Mathematics
  • 📊 Statistics & Probability
  • 🎯 Linear Algebra
  • ⚗️ Operations Research
  • 💻 Computational Mathematics
  • 🌍 Applied Mathematics

🎯 Teacher Notes | Ngā Kōrero Kaiako

📚 NZ Curriculum Phase 4 Alignment:

📐 Geometry & Measurement:

  • Use four-quadrant coordinate systems fluently
  • Apply distance, midpoint, and gradient formulas
  • Understand and perform geometric transformations
  • Use coordinate geometry to solve optimisation problems
  • Connect geometric and algebraic representations

🔢 Algebra:

  • Form and solve linear equations in context
  • Use algebraic methods for constraint problems
  • Apply function notation and transformations
  • Model relationships using coordinate geometry

📊 Statistics:

  • Use statistical methods to analyse spatial data
  • Apply correlation and regression in coordinates
  • Interpret statistical measures in context
  • Make predictions using mathematical models

🔬 Mathematical Practices:

  • Develop mathematical arguments and proofs
  • Use mathematical modelling to solve problems
  • Connect mathematics to other disciplines
  • Communicate mathematical ideas professionally

🌿 Cultural & Career Integration:

Kaiako Planning Snapshot

Ngā Whāinga Akoranga — Learning Intentions

Paearu Angitu — Success Criteria

Teacher Planning Snapshot

Related Handouts & Worksheets

Curriculum alignment

🔗 Unit Progression & Next Steps

Six lessons that move from building a shared coordinate model to defending a proposal somebody else can audit. Each step adds one kind of evidence — position, then distance, then transformation, then a model and its limits, then the constraints that rule options out — so the final proposal is argued from accumulated working rather than asserted at the end.

The lesson files live in the companion unit Hamilton Zoo Analytics; this page is the Phase 4 planning view of the same sequence.