Project Grade 9 Math
Aldergate's Aging Pipes: A Water-Main Break Trend and Budget Project
A grade 9 project where students model a fictional town's rising water-main break trend and propose a pipe-replacement plan within budget and construction limits.
Objectives
- Students will graph a multi-year data trend with labeled, scaled axes and write a linear equation modeling that trend.
- Students will evaluate a proposed plan against two stated constraints, a budget limit and a construction-capacity limit, and identify which constraint is binding.
Vocabulary
Key words: trend, linear model, constraint, binding constraint
Teacher overview
Nationally, the ASCE 2025 Infrastructure Report Card gave United States drinking water a grade of C-, and a water main breaks somewhere in the country roughly every two minutes, a reminder of how widespread aging pipe networks have become. This project brings that national picture down to the scale of a single fictional town, Aldergate, where student teams act as engineering interns analyzing four years of water-main break records, building a linear model of the trend, and proposing which segments of pipe to replace first within a fixed budget and a real construction limit. The final deliverable is a written recommendation with a labeled graph, presented to the class in the role of Aldergate's town council.
Project Brief
Driving question: How can Aldergate's public works department plan a pipe-replacement schedule that slows its rising rate of water-main breaks? The plan must fit a fixed annual budget and a limited construction capacity.
Scenario: Aldergate's public works director has asked a team of student engineering interns for help. A teacher or student can play this role during the final presentation. The team's job is to analyze four years of the fictional town's water-main break records, build a model that projects the trend forward, and recommend how many miles of old pipe to replace next year.
Data: Aldergate's recorded water-main breaks over its four most recent years:
| Year | Breaks |
|---|---|
| 1 | 12 |
| 2 | 15 |
| 3 | 18 |
| 4 | 21 |
Constraints:
- Budget: $150,000 is available per year for main replacement. This is a fixed budget set for this project.
- Replacing one mile of old pipe costs $50,000. Each replaced mile is estimated to reduce that segment's expected annual breaks by 2 breaks per year. This is a stated planning assumption for this project.
- Aldergate can only close one arterial road at a time for construction. This limits replacement work to at most 2 miles of pipe per year, no matter what the budget allows.
Final deliverable: A written recommendation of one to two pages. It must include a labeled and scaled graph of the four years of break data, plus the team's linear model. It must also state the number of miles the team recommends replacing next year, check that choice against both constraints, and project the effect on next year's break count.
Milestones
Session 1 — Build the model. Teams plot the four years of break data on graph paper with labeled, scaled axes. Year goes on the horizontal axis, and Breaks goes on the vertical axis. Teams identify the pattern as linear, with a constant increase of 3 breaks per year, and write the equation B(y) = 3y + 9. Here y is the year number, and B(y) is the projected number of breaks. Output: a completed, labeled graph and a written equation.
Session 2 — Project forward and check the constraints. Using their equation, teams project the number of breaks expected in Year 5 and Year 8 if no changes are made. Teams then calculate how many miles of pipe the $150,000 budget affords, by dividing by the $50,000 per-mile cost. They also calculate how many miles the one-road-at-a-time construction rule allows. Teams identify which of the two limits is more restrictive. Output: a completed constraint table showing both limits and a one-sentence statement of which limit is binding.
Session 3 — Propose a plan and calculate its effect. Teams choose how many miles of pipe to replace next year, up to the binding limit identified in Session 2. They calculate the total cost of that choice and confirm it fits the budget. They calculate the total reduction in expected breaks, using miles replaced times 2 breaks per mile. Finally, they calculate the resulting projected Year 5 break count with their plan in place, and compare it to the no-intervention projection from Session 2. Output: a completed proposal worksheet showing miles replaced, total cost, break reduction, and before-and-after Year 5 projections.
Session 4 (final) — Write and present the recommendation. Teams write their one-to-two-page recommendation. It includes the labeled graph, the equation, the chosen number of miles, the total cost, and the projected effect on breaks. Teams then present the graph and the recommendation to the class in a short presentation. The rest of the class acts as Aldergate's town council receiving the proposal. Output: a completed written recommendation and a 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 |
|---|---|---|---|---|
| Trend model | Graph or equation is missing, mislabeled, or does not match the data | Graph and equation are present but axes are not clearly labeled or scaled, or the equation does not match the data pattern | Graph has labeled, scaled axes and the equation B(y) = 3y + 9 correctly matches the four years of data | Graph and equation are correct and the team also correctly projects Year 5 and Year 8 break counts using the model |
| Constraint analysis | Recommendation does not check the budget or construction limits | Recommendation checks one constraint (budget or construction capacity) but not both | Recommendation checks both constraints and correctly identifies the construction limit as binding | Recommendation checks both constraints, correctly identifies the binding constraint, and explains in plain language why the less restrictive constraint (budget) does not fully determine the plan |
| Presentation & recommendation | Recommendation is incomplete or difficult to follow | Recommendation states a plan but does not clearly connect it to the projected break reduction | Recommendation clearly states the miles replaced, total cost, and projected break reduction, tied to the team's model | Recommendation clearly connects the model, both constraints, and the projected outcome, and anticipates a likely question from the town council, such as what happens in future years if funding stays the same |
Differentiation
Support
Provide the completed equation B(y) = 3y + 9 already derived for Session 1, so teams can spend their time interpreting and projecting with it rather than deriving it from the data. Provide a partially completed constraint table for Session 2 with the budget-miles calculation already worked out, leaving only the construction-limit comparison for teams to complete.
Extension
Ask teams to calculate, using their model, in what year Aldergate's break count would first exceed 30 if no replacement work is ever done, and to propose a multi-year replacement schedule (still respecting the 2-mile-per-year construction limit) that would keep the break count from ever reaching that level.
ELL supports
Provide a labeled diagram showing a pipe segment, a road closure sign, and a cost tag, illustrating the connection between "one mile replaced," "one road closed," and "$50,000 spent" in this fictional town's project, before Session 2's constraint work. Allow the written recommendation in Session 4 to include labeled diagrams and a data table alongside shorter sentences rather than full paragraphs.