Project Grade 10 Math
Tarnbury Pressure Zones: Designing for 40 to 80 psi on a Hillside
A five-session grade 10 project where teams divide a fictional hillside town into pressure zones, size each zone's tank, and defend the plan against a capital budget.
Objectives
- Students will convert a pressure requirement in psi into a range of allowable water-surface elevations.
- Students will divide a range of ground elevations into pressure zones that satisfy that requirement.
- Students will size a cylindrical tank for each zone and price the plan against a fixed capital budget.
Vocabulary
- Pressure zone
- A pressure zone is a section of a water system that a utility manages separately because it sits at a different elevation than the rest of the town. Utilities create pressure zones so that water pressure stays safe and steady for everyone, whether they live at the bottom of a hill or near the top.
Also: hydraulic grade line, head, psi, storage
Teacher overview
Tarnbury is a fictional hillside town, and its elevations, household counts, and costs are set for this project. A single water tank cannot serve a town whose homes sit hundreds of feet apart in elevation, because the same water surface that gives a hilltop house adequate pressure would put damaging pressure on the plumbing of a house at the bottom of the hill. Teams divide the town into pressure zones, choose a water-surface elevation for each zone, size a tank for each, and price the plan against a capital budget before presenting it to a fictional water board. The driving mathematics is the relationship between feet of head and pounds per square inch, which turns a pressure requirement into a limit on how much elevation one zone can span.
Project Brief
Driving question: How should the fictional town of Tarnbury divide its hillside into pressure zones so that every home receives water between 40 and 80 psi, and can the plan be built for the money the town has?
Client: The Tarnbury Water Board, played by classmates during the final session.
The town: Tarnbury is fictional. Its homes sit between elevation 140 feet and elevation 460 feet. The town has surveyed its 2,370 households into eight elevation bands, each 40 feet tall:
| Elevation band (ft) | Households |
|---|---|
| 140 to 180 | 340 |
| 180 to 220 | 395 |
| 220 to 260 | 430 |
| 260 to 300 | 360 |
| 300 to 340 | 305 |
| 340 to 380 | 250 |
| 380 to 420 | 175 |
| 420 to 460 | 115 |
Rules and constants for this project:
- Pressure at a home equals the water-surface elevation of its zone's tank minus the home's ground elevation, multiplied by 0.433 psi per foot.
- Every home must receive at least 40 psi and no more than 80 psi. Distribution systems commonly run in that range, and most states set a 20 psi floor to keep contamination from entering a main.
- A zone may not split an elevation band. Each band belongs to exactly one zone.
- Each zone's tank must store one day of average demand for the households in that zone, at 300 gallons per household per day.
- The lowest zone already has a tank and a pump station, so no new facility cost applies to it.
- Every other zone needs a new station and site at $900,000, plus tank construction at $1.80 for each gallon of storage.
- The total capital cost must not exceed $4,200,000.
- Tanks are cylinders. Volume in cubic feet equals 0.785 times the diameter squared times the depth, and 1 cubic foot holds 7.48 gallons.
Final deliverable: A zone map on grid paper with the bands colored by zone, a pressure table, a storage and cost sheet, a one-page written recommendation to the board, and a presentation of no more than five minutes.
Milestones
Session 1, pressure rules. Convert 40 psi and 80 psi into feet of head, then state the largest elevation span a single zone can serve. Output: a completed conversion table and one sentence giving the maximum span in feet, with the arithmetic shown.
Session 2, zone boundaries. Assign all eight elevation bands to zones without splitting any band, using the maximum span from Milestone 1. Output: a colored zone map and a table giving each zone's elevation range and household count, with the counts summing to 2,370.
Session 3, hydraulic grade lines. Choose a water-surface elevation for each zone and check the pressure at the highest and the lowest home in that zone. Output: a table with one row per zone showing the chosen elevation and both pressures, every value between 40 and 80 psi, plus the two inequalities the team used.
Session 4, storage and cost. Size each new tank in gallons, choose a diameter and compute the depth, then build a cost sheet. Output: a tank table with diameter, depth, and volume for each new zone, and a cost total compared against the budget.
Session 5, recommendation. Present the plan to the board and submit the written page. Output: a five-minute presentation and a one-page recommendation naming the number of zones, the total cost, and one tradeoff the team made.
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 |
|---|---|---|---|---|
| Pressure constraint | Pressures are missing, or some homes fall outside 40 to 80 psi | Pressures are computed for some zones but not for both the highest and lowest home | Every zone shows both pressures inside the required range, with the 0.433 conversion shown | Both pressures are shown for every zone and the team states how much margin it left at each end and why |
| Zone layout and storage | Bands are split, or household counts do not sum to 2,370 | Zones are formed but at least one exceeds the maximum elevation span | All bands are assigned, no zone exceeds the maximum span, and each tank's storage matches its household count | The layout meets every rule and the team explains why it chose those boundaries rather than another valid set |
| Tank geometry | Tank sizes are missing or given without dimensions | A volume is stated but the diameter and depth do not produce it | Each new tank has a diameter and depth whose cylinder volume holds the required gallons | Dimensions are correct and the team explains how it balanced diameter against depth for this fictional hillside site |
| Cost and budget | The cost sheet is incomplete or exceeds the $4,200,000 limit without comment | Costs are itemized but the station and tank components are not separated | An itemized cost at or below the budget is shown, with stations and tank gallons priced separately | The itemized cost is under budget and the team shows why one more zone would push the plan over the limit |
| Recommendation and presentation | The recommendation omits the zone count or the total cost | The page states the plan but does not connect it to the constraints | The page and presentation report zones, pressures, and cost against every stated rule | The board's likely follow-up question is anticipated and answered with a number from the team's own tables |
Differentiation
Support
Give these teams the Milestone 1 conversions already completed, so they begin with the maximum elevation span in hand. Provide a partly filled zone map with the first zone's two bands already colored, and a tank table with the diameters supplied so students compute only depth. A worked example for one zone of the fictional town, using bands the class does not need, keeps the model visible without giving the answer away.
Extension
Ask these teams to test whether a five-zone plan could ever fit the $4,200,000 budget, showing the arithmetic that settles it. A second challenge is to find the tank diameter that would hold the largest zone's storage in exactly 20 feet of depth, and a third is to recompute the plan if the board raised the minimum pressure from 40 psi to 45 psi.
ELL supports
Provide a labeled hillside diagram showing a tank, a water surface line, and two homes at different elevations, with the words head, elevation, and pressure placed on the drawing. Supply sentence frames for the recommendation page, such as "Our plan uses ... zones because ..." and "The total cost is ... , which is ... the budget." Allow the pressure table to be submitted with values and units only, without full sentences.
Extensions
Have students find the elevation range of their own community using a public topographic map, then apply the Milestone 1 rule to estimate how many pressure zones it would need. A local distribution operator invited to the final presentations can describe how many zones the real system actually uses and what set its boundaries, which rarely matches the clean bands of the fictional Tarnbury exercise.