Project Grade 10 Math
Sundown Bay Water Budget: Conservation and Desalination Project
A 4-session grade 10 project where teams use a system of inequalities to design a feasible conservation-and-desalination plan within a fictional town's budget.
Objectives
- Students will write a system of two linear inequalities representing a real supply and budget constraint.
- Students will test candidate solutions for whether they satisfy both constraints.
- Students will graph a system of inequalities and identify the feasible region.
Vocabulary
Key words: system of inequalities, feasible region, constraint, viable solution
Teacher overview
Sundown Bay (a fictional coastal town used for this project) faces a growing gap between the water it has and the water it needs, and its town council must decide how to close that gap. Over four sessions, student teams build and graph a system of inequalities representing two options, household retrofits and a small desalination unit, then use the feasible region to recommend a specific, workable combination of both. The final deliverable is a graph and a written proposal presented to a fictional town council, the kind of quantitative recommendation real water planners prepare for public decision-makers.
Project Brief
Driving question: How can Sundown Bay close a 250,000-gallon-per-day water supply gap within a $2,000,000 capital budget, using some combination of household retrofits and a small desalination unit?
Scenario: Sundown Bay's town council has asked engineering teams (the student teams) to propose a plan to close the town's projected supply gap. The average US household uses about 300 gallons of water per day, and the council wants a plan that does not depend on rainfall to work.
Constraints (fictional values set for this project):
- Each household that receives a low-flow retrofit reduces its water use by 45 gallons per day, at a cost of $120 per household.
- A desalination unit can be built at any capacity, at a cost of $9 for every gallon per day of capacity.
- Let x = the number of households retrofitted and y = the desalination unit's capacity in gallons per day. The plan must close at least 250,000 gallons per day of gap: 45x + y ≥ 250,000.
- The plan's total cost may not exceed the $2,000,000 capital budget: 120x + 9y ≤ 2,000,000.
- No more than 4,000 households in Sundown Bay are available for retrofits (0 ≤ x ≤ 4,000).
Final deliverable: A labeled graph (on grid paper) showing both constraint lines with the feasible region shaded, plus a one-to-two-page written proposal to the town council recommending one specific (x, y) combination, with the math showing it meets both constraints.
Milestones
- Session 1 — Set up the constraints: Teams write two inequalities: one for the supply gap and one for the budget. Both use x (households retrofitted) and y (desalination capacity). Teams discuss what each side of each inequality means in the real scenario. Output: both inequalities written and explained in the team's own words.
- Session 2 — Test candidate plans: The teacher gives each team at least four candidate (x, y) pairs to test. Some rely mostly on retrofits. Others rely mostly on desalination. One pair is designed to fail a constraint. Teams substitute each pair into both inequalities and sort it as viable or non-viable. Output: a completed table showing each candidate, its gap closed, its cost, and whether it is viable.
- Session 3 — Graph the system: Teams graph both inequalities on one coordinate plane. The x-axis shows households retrofitted. The y-axis shows desalination capacity in gallons per day. Teams shade the feasible region, where both constraints hold true. They mark their chosen (x, y) recommendation inside that region. Output: a completed, labeled graph.
- Session 4 — Write and present the proposal: Teams write a one-to-two-page proposal. It explains their chosen combination, the total cost, and how the plan meets both constraints. Teams then present the graph and proposal to the class. One team member plays a Sundown Bay town council member who asks questions. Output: a completed written proposal and a short team presentation.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Rubric
| Criterion | Emerging | Developing | Proficient | Advanced |
|---|---|---|---|---|
| Constraint accuracy | Inequalities are missing or do not match the scenario's numbers | Inequalities are written but contain a sign or coefficient error | Both inequalities correctly represent the supply-gap and budget constraints | Both inequalities are correct and the team clearly explains what each term represents in the real scenario |
| Viable solution | Recommended (x, y) combination fails at least one constraint | Recommended combination satisfies one constraint but the team has not checked the other | Recommended combination satisfies both constraints, shown with substitution | Recommended combination satisfies both constraints with a clear margin, and the team explains a tradeoff they considered |
| Graph accuracy | Graph is missing or axes are unlabeled | Graph shows both lines but the feasible region is not shaded or is shaded incorrectly | Both lines and the correctly shaded feasible region appear on labeled, scaled axes | Graph is fully correct and the recommended point is plotted and clearly marked inside the feasible region |
| Presentation & proposal | Proposal is incomplete or difficult to follow | Proposal explains the recommendation but not how it meets both constraints | Proposal clearly explains how the recommendation meets both constraints | Proposal clearly explains both constraints and anticipates a likely question from the town council |
Differentiation
Support (below grade level)
Provide the two inequalities already written and correctly labeled before Session 2, so teams can focus their time on testing candidates and graphing, and provide a pre-drawn coordinate grid with axes already scaled from 0 to 4,000 (x) and 0 to 300,000 (y).
Extension (above grade level)
Ask teams to find the (x, y) combination that satisfies both constraints at the lowest possible total cost, by testing points along the boundary of the supply-gap inequality, and to explain why a point on that boundary line, rather than deep inside the feasible region, is likely to be the cheapest viable option.
ELL supports
Provide a labeled diagram showing a household retrofit (a faucet with an arrow showing reduced flow) and a desalination unit (a simple treatment-plant icon with an arrow showing added supply), and allow the written proposal to include labeled sketches and a completed table alongside shorter sentences.
Extensions
Have teams discuss (using only the numbers provided in this fictional project, not outside sources) what would happen to the feasible region if the town's budget were cut to $1,500,000, and have them redraw the graph to show how much smaller the feasible region becomes.