Math City · For Teachers
Teacher Guide
Facilitation notes, answer keys, and printables — plus the full multi-week Volume City project.
📥 Printables
The lesson plan covers objectives, standards alignment, and a minute-by-minute flow; the License Kit is the Lesson 1 practice worksheet. Generated right here; nothing is uploaded anywhere.
About this unit
Math City is project-based: students play urban planner, using volume, capacity, and cost math to design buildings — and, in later lessons, whole cities against a $125,000 budget. Unit 3 has four lessons — all four are ready.
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Lesson 1 — City Hall Training Day
Goal: students compute floor space (w × d × floors), capacity (÷ 25 m² per person), and cost (× $5/m²); discover that doubling two dimensions quadruples capacity; and design a building meeting two constraints at once.
Answer key — License Exam
- 30 × 20 × 10 → 6,000 m² · capacity → 240 people.
- Wide (50×50×4 = 10,000 m²) vs. tower (20×20×30 = 12,000 m²) → the tower.
- Doubling question → ×4 (each doubling multiplies: 2 × 2).
Answer key — License Kit Part 1
- A) 4,000 m² · 160 people · $20,000 B) 6,000 m² · 240 · $30,000 C) 6,000 m² · 240 · $30,000.
- The planted surprise: cost per person is $125 for all three — in this simple model, cost and capacity both scale with floor space, so the ratio never budges. Ask: what real-world factors would break the tie (land price, elevators, taller = pricier)? Lesson 2 adds them.
Look-fors: students state formulas in words; in the Building Designer, compare classmates' different winning designs — "many right answers" is the point; the doubling insight said multiplicatively ("times four"), not additively.
Discussion prompts: Why do cities build up instead of out? · What does 25 m² per person feel like — pace it out! · What would a building for 10,000 people look like?
Lesson 2 — Build Your City
Goal: plan under three simultaneous constraints (10,000 people, $125,000, full services); apply ceiling division in context; compare unit rates.
- Economics: Tower $18k/2,500 ($7.20/pp) · Apartment $5.5k/600 ($9.17/pp) · Cottage $1.2k/100 ($12/pp). Services: Fire $6k per 4,000 · School $5k per 3,500 · Park $2k per 2,500.
- One winning build: 4 towers + 3 fire + 3 schools + 4 parks = 10,000 people, $113,000 ✓. Many others work — compare mixes as a class.
- Ceiling question key: 9,000 ÷ 3,500 = 2.57 → 3 schools (leftover kids still need school). Value check → the tower.
Lesson 3 — The Three Cities
Goal: compute and compare cost per person; reason about trade-offs; support a verdict with cited numbers.
- Bureau key: Skyline $12.00 · Sprawl $15.50 · Balanced $11.50 per person.
- Trade-offs key: fastest fire response → Skyline (2 min); most green/yard space → Sprawl; strongest case against Sprawl → highest cost per person AND 2,000 fewer people housed.
- The verdict is graded on the justification, not the choice — a numberless answer isn't done.
Lesson 4 — Pitch to the Mayor
Goal: distinguish evidence from opinion, and structure a pitch as claim → evidence → call to action. The Official Proposal PDF is the unit's summative artifact.
- Evidence Locker key: the data statements are #1 ($11.50/person), #3 (3-minute response), #5 (10,000 housed, $10k under budget). The rest are opinions.
- Mayor's Test key: the sentence with specific numbers ($11.50, 10,000, $10,000 reserve).
- Have students actually deliver the pitch — a partner, a group, or the family "city council."
📦 The Volume City project (2–3 weeks)
The original deep-dive version of this unit: three complete city designs on graph paper, comparative analysis, and a written justification. Use it after Lesson 1 (or the full unit) as the capstone.