Resource Allocation Algebra
Treaty Settlement Economics · Linear Equations and Simultaneous Equations
Ngā Whāinga Akoranga · Learning Intentions
- Form and solve linear equations and inequalities from written constraints
- Solve simultaneous equations using substitution and elimination
- Interpret solutions in their economic and social context
- Apply algebra to intergenerational resource planning in an iwi context
Paearu Angitu · Success Criteria
- I can translate a word problem into a set of algebraic constraints
- I can solve a system of two simultaneous equations using elimination
- I can verify my solution by substituting back into both original equations
- I can interpret my algebraic result in terms of real iwi resource decisions
Hononga Marautanga · Curriculum Alignment
Level 5–7: form and solve linear equations and inequalities; solve simultaneous linear equations; apply algebraic methods in context.
Understand how resource allocation decisions reflect values and power; investigate how Treaty settlements affect Māori economic development.
Whakataukī
"Ehara taku toa i te toa takitahi, engari he toa takitini"
My strength is not the strength of one, but the strength of many.
When the Crown restores resources to iwi through Treaty settlements, those resources must be allocated across competing needs: housing, education, health, business development, cultural programmes, and conservation. This is fundamentally an algebra problem — expressing constraints as equations and finding the solution that best serves the community.
Wāhi 1 · Whakatauwehe Ōrite · Forming Linear Equations from Constraints
- Housing (h) must be at least twice the cultural programmes allocation (c): h ≥ 2c
- Education (e) + health (a) together must equal exactly $9 million
- Business development (b) gets $3 million more than cultural programmes: b = c + 3
- The total must equal exactly $28 million: h + e + a + b + c = 28
- Write all four constraints as algebraic expressions or equations. Number each clearly.
- Given that cultural programmes c = $3.5 million, solve for all other allocations. Check that each satisfies its constraint.
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If the board decides education should be exactly twice health, set up and solve the simultaneous equations to find exact values for e and a.
e + a = 9 e = 2a → Substitute e = 2a into the first equation and solve
- Graph the inequality h ≥ 2c: place c on the x-axis and h on the y-axis. Draw the boundary line h = 2c. Shade the feasible region. Plot the specific solution from Question 2 as a point and verify it falls in the feasible region.
Wāhi 2 · Whārite Ōrite · Simultaneous Equations — Revenue Streams
Last year: 120 quota units + 80 visitor-days = $860,000.
This year: 90 quota units + 110 visitor-days = $790,000.
Both revenue rates F and T remain constant. Find F and T.
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Write the system of simultaneous equations:
Equation 1: 120F + 80T = 860,000Solve using elimination. Show every algebraic step. Include what you multiplied each equation by.
Equation 2: 90F + 110T = 790,000 - Verify your solution by substituting F and T back into both original equations.
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Next year, the trust projects 200 total units. They must maintain at least 50 quota units (legal requirement) and at least 30 visitor-days (operational minimum). What combination of quota units and visitor-days maximises revenue? Show your working.
Maximise: Revenue = F × quota + T × visitor_days
Subject to: quota + visitor_days ≤ 200, quota ≥ 50, visitor_days ≥ 30 - Graph the feasible region for the Q3 constraints (quota on x-axis, visitor-days on y-axis). Identify the corner points of the feasible region and test each to confirm which gives maximum revenue.
Wāhi 3 · Whakahekenga 7 · The Seven-Generation Budget
Māori governance operates on the principle of intergenerational equity — decisions today should benefit the next seven generations (~175 years). An iwi trust has $50 million in investment assets.
The sustainable annual distribution from $50M = $50M × 0.062 = $3.1 million/year
- Confirm: what is the maximum annual distribution from a $50 million fund while maintaining long-term sustainability? Express as both a dollar amount and a percentage.
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If the fund distributes $3 million per year, calculate the fund value at Years 5, 10, 25, and 50. Create a table. Use the formula:
F(t) = 50 × (1.062)ᵗ − 3 × [(1.062)ᵗ − 1] ÷ 0.062
- The iwi wants the fund to reach $100 million while still distributing $2.5 million/year. Using trial-and-error or algebraic rearrangement, find the approximate year when F(t) = 100.
- Values-based algebra: The board is debating — distribute $4.5M/year now (above the sustainable threshold, to address urgent housing) or $2.5M/year to grow the fund. At what year does the conservative strategy's fund value overtake the aggressive strategy's depleted fund? Set up the comparison algebraically, then write 3 sentences on which strategy you think better reflects kaitiakitanga.
Aronga Mātauranga Māori
Algebra is not neutral. The equations we write reflect the values we hold. When an iwi board sits down to allocate Treaty settlement resources, they are solving a system of equations — but each constraint in that system represents a value decision: How much to the young? How much to elders? How much to the land itself?
The concept of whakapapa as structure is present in these equations: each allocation is in relationship to every other allocation, just as whakapapa links every person to every other through kinship. A solution that violates the constraints — that undermines one group to benefit another — is not a solution at all. Te wero: research a real Treaty settlement (Ngāi Tahu 1998, Tainui 1995, or Te Arawa 2004). Express the total amount and allocation categories as a system of algebraic constraints. Does the allocation meet intergenerational equity criteria?
Ngā Rauemi Tautoko · Support Materials
Resources already provided:
- This handout with all three problem scenarios and formula reference boxes
- Calculator (essential — especially for compound growth calculations in Part 3)
- Graph paper (for the feasible region diagram in Part 2, question 4)
- Treaty settlement reference: Manatū Māori (maoriaffairs.govt.nz) for settlement details
Aronga Rerekē · Differentiated Pathways
Tīmata · Entry Level
Complete Part 1 questions 1–3 only. Use the given value of c = $3.5M and teacher support for setting up the simultaneous equation.
Paerewa · On Level
Complete Parts 1 and 2 fully. Attempt Part 3 questions 1 and 2. Show all algebraic working.
Tūāpae · Extension
Complete all three parts including the graphical linear programming in Part 2 Q4 and the values-based comparison in Part 3 Q4. Research an actual Treaty settlement and express its allocation as your own system of equations with a written analysis of whether it meets the constraints you would set for intergenerational equity.
📋 Teacher Planning Snapshot
Ngā Whāinga Ako — Learning Intentions
Students will develop algebraic thinking and pattern recognition (tātai tauira) through te ao Māori contexts, connecting mathematical reasoning to cultural and real-world problem-solving in Aotearoa.
Ngā Paearu Angitū — Success Criteria
- ✅ Students can identify, describe, and extend patterns using algebraic notation.
- ✅ Students can explain their mathematical reasoning and connect it to real-world contexts.
Differentiation & Inclusion
Scaffold support: Provide concrete materials and visual representations before moving to abstract notation. Offer entry-level tasks using number patterns, and extension challenges involving proof or generalisation for capable learners.
ELL / ESOL: Pre-teach key mathematical vocabulary (variable, expression, equation, pattern). Allow diagrams and tables as alternate representations. Bilingual glossaries recommended.
Inclusion: Neurodiverse learners benefit from structured step-by-step templates and multiple representations (visual, numeric, algebraic). Avoid time pressure on procedural tasks.