Project Grade 9 Math
Thurlby's Stormwater Fee: Setting a Rate and a Credit Schedule
A five-session grade 9 project where teams measure hard surfaces, build a revenue equation for a town stormwater fee, and defend a credit schedule to a council panel.
Objectives
- Students will convert measured hard-surface areas into a town's stormwater billing units.
- Students will write and graph an equation relating a monthly rate to annual program revenue.
- Students will express a credit program as an inequality and test schedules against it.
Vocabulary
- Impervious surface
- An impervious surface is any hard surface, like a road, roof, or parking lot, that rain cannot soak through. The more impervious surface a town has, the more stormwater rushes off it.
Also: billing unit, stormwater fee, credit, constraint
Teacher overview
Rain that falls on a roof or a parking lot cannot soak in, so it runs to a storm drain and becomes the town's problem to move and treat. Many towns pay for that work with a stormwater fee based on how much hard surface a property has rather than on how much water it uses, so setting the fee begins with measuring area. In this five-session project, teams design the fee for the fictional town of Thurlby, which must raise $480,000 a year. Teams compute hard-surface areas from a parcel data sheet, scale a sample of eight properties up to the whole town, build a revenue equation, write a credit program as an inequality, and defend the whole package to a panel acting as the fictional town council. Address early the misconception that a stormwater bill should scale with lot size, which is easy to fall into because each parcel's lot dimensions are listed first on the data sheet. A large lot that is mostly lawn sends less water to the storm drain than a small lot paved end to end, and the billing unit in this project counts only the hard surface. Everything teams need is printed in the Project Brief below, and the teacher reference figures sit at the end of the Rubric section.
Project Brief
Driving question. How should the fictional town of Thurlby charge for stormwater service so that the program is funded and the charge is defensible?
Your client. The fictional Thurlby town council has voted to start a stormwater utility. The council has hired your team to design the fee. Your work goes to a council panel in the final session.
The budget constraint. The program must raise $480,000 each year. Revenue below that leaves work unfunded. Revenue far above it will draw questions the council does not want.
The billing unit. Thurlby bills by the equivalent residential unit, written ERU. One ERU is 2,400 square feet of hard surface, meaning roofs, driveways, walks, patios, courts, and paved parking. Lawns, fields, and gardens do not count. Every home is billed a flat 1 ERU. Every other property is billed by its measured hard surface, rounded up to the next whole ERU, so 2,500 square feet is billed as 2 ERU.
The town. Thurlby has 3,200 homes. It also has 140 other properties, listed by type in the property roll below. The town engineer has measured eight of those properties, and their dimensions are on the Parcel Data Sheet. Use those eight to estimate the ERUs for all 140.
The credit rule. A property owner who installs an approved practice, such as a rain garden or permeable pavement, may earn a credit of up to 40 percent off that property's bill. Your credit schedule must still leave total revenue at or above the budget once the owners in the credit applicant table below receive their credits.
What you turn in. A written proposal of two to three pages that contains a monthly rate per ERU, a labeled graph of annual revenue against that rate, a credit schedule, three sample bills, and one paragraph naming the fairness objection you expect and your response to it. Each team also gives a five-minute presentation to the council panel.
Thurlby property roll (other than homes)
| Property type | Number in Thurlby | Sampled on the Parcel Data Sheet |
|---|---|---|
| Small commercial (shops, offices, restaurants, gas stations) | 126 | Parcels A, B, and C |
| Large commercial (supermarkets, large stores, motels) | 10 | Parcel D |
| Churches | 2 | Parcels E and F |
| School | 1 | Parcel G |
| Factory | 1 | Parcel H |
| Total | 140 | 8 parcels |
Parcel Data Sheet
All dimensions are in feet and every surface is a rectangle unless the table says otherwise. The lot is the whole property, including lawn. List each hard surface separately and do not count the lot itself as hard surface.
| Parcel | Lot | Hard surfaces |
|---|---|---|
| A. Birch Street Corner Market | 100 by 120 | Store roof 50 by 50; paved parking 30 by 50; front walk 4 by 25 |
| B. Main Street Diner | 80 by 150 | Restaurant roof 40 by 55; paved parking 60 by 70; concrete dumpster pad 10 by 20 |
| C. Brook Road Gas Station | 120 by 140 | Store roof 30 by 40; paved pump area 80 by 100; pump canopy 40 by 60, which stands over part of the paved pump area |
| D. Hollis Grocery | 300 by 250 | L-shaped store roof drawn as a 160 by 150 rectangle with a 40 by 100 notch cut from one corner; paved parking 200 by 150; truck dock 40 by 60 |
| E. St. Brendan's Church | 200 by 250 | Church roof 60 by 100; parish hall roof 40 by 50; paved parking 80 by 120; walks 6 by 150 |
| F. Thurlby Community Church | 150 by 200 | Church roof 50 by 80; paved parking 70 by 100; walk 5 by 80 |
| G. Thurlby School | 600 by 500 | L-shaped classroom roof drawn as a 300 by 200 rectangle with a 200 by 120 notch cut from one corner; gym roof 100 by 120; bus loop and parking 250 by 140; basketball courts 60 by 100; walks 8 by 400; the rest of the lot is grass playing fields |
| H. Pell Box Company (factory) | 400 by 400 | Factory roof 200 by 250; paved truck yard 150 by 200; employee parking 100 by 180 |
Credit applicant table
The town engineer has estimated which owners will apply for a credit in the first year and which practice each will install. Use these numbers when you test a credit schedule.
| Practice | Expected applicants | ERUs billed to those applicants |
|---|---|---|
| Rain barrel or downspout disconnection | 200 homes | 200 |
| Rain garden | 12 small commercial properties at 3 ERU each, plus both churches (Parcels E and F) | 49 |
| Permeable pavement | 2 large commercial properties at 22 ERU each, plus the school (Parcel G) | 83 |
| Detention pond | The factory (Parcel H) | 41 |
| Total | 373 |
Milestones
Session 1, count the units. Use the Parcel Data Sheet to find each parcel's hard-surface area in square feet. Break each L-shaped roof into rectangles or subtract the notch, and count a surface only once even when one structure stands over another. Divide each total by 2,400 and round up to get the parcel's ERUs, then build a table of all eight. Draw one parcel to scale on graph paper, choosing your own scale, and shade its hard surfaces. Then use the property roll to estimate the ERUs for all 140 non-home properties and add 1 ERU for each of the 3,200 homes. Turn in the table, the scale drawing, the town's total ERUs, and two or three sentences explaining how you scaled eight parcels up to 140 and why your method fits the property roll.
Session 2, build the revenue equation. Let r be the monthly charge per ERU in dollars. Write annual revenue R as an equation in r, using your total ERU count and 12 months. Graph R against r on axes you label and scale yourself. Mark the point where revenue meets the $480,000 budget and read the rate from the graph. Then solve the equation for that rate and round it to a whole cent that still meets the budget, showing a check of the revenue at your rounded rate. Turn in the equation, the graph, and the rate.
Session 3, design the credit. Decide how large a credit each practice in the credit applicant table earns, up to the 40 percent limit. Choose an operating rate at or above your Session 2 rate, since a rate set exactly at the budget leaves almost no room for credits. Write an inequality stating that revenue after credits must stay at or above the budget, using a variable for each practice's credit fraction. Test two schedules at your operating rate, one that satisfies the inequality and one that does not, and show the arithmetic for both. Then find the lowest rate that would keep the budget even if every applicant received the full 40 percent. Turn in the credit table, both tests, and that lowest rate.
Session 4, draft and check. Write the proposal. Compute three sample bills: a home, the school (Parcel G), and the factory (Parcel H). Compute each one at your operating rate twice, once with no credit and once with the credit your schedule gives that property's practice, and show both the monthly and the annual amount. Check that your operating rate still clears the budget with your credits in place. Turn in the draft and the three worked bills.
Session 5, present and defend. Give a five-minute presentation to the council panel with your graph displayed. State your rate, your credit schedule, and the fairness objection you expect. Answer two questions from the panel. Turn in the final proposal after the presentation.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Rubric
Score each criterion on the four levels below.
| Criterion | 4, exceeds | 3, meets | 2, approaching | 1, beginning |
|---|---|---|---|---|
| Area and billing units | Every area is computed correctly with units shown, rounding to whole ERUs follows the stated rule, and the scaling method treats the school, the factory, and both churches as single properties rather than averaging them across all 140. | Areas and ERU counts are correct with minor rounding slips, and the scaling method is stated, even if it averages all eight parcels together. | Some areas are correct, but the ERU rule is applied inconsistently, lot area is counted as hard surface, or the scaling method is missing. | Areas are largely incorrect or the hard-surface total is not converted into billing units. |
| Revenue equation and graph | The equation is correct in two variables, the graph carries labeled and sensibly scaled axes, and the budget rate is read from the graph, solved exactly, and rounded up to a cent with a revenue check. | The equation is correct and the graph is labeled, with a minor scale, reading, or rounding error. | An equation is present but omits the twelve months or the ERU total, or the graph lacks labels. | No usable equation, or the graph does not relate rate to revenue. |
| Credit schedule as a constraint | The budget rule is written as a correct inequality with a variable for each practice, both schedules are tested with arithmetic that supports the verdict, and the lowest rate that covers full 40 percent credits is found. | The inequality is correct and both schedules are tested, with a small arithmetic error. | An inequality appears but is stated loosely, or only one schedule is tested. | Credits are proposed with no test against the budget. |
| Proposal and defense | The proposal contains every required element, the three sample bills are correct with and without credit, and the fairness objection is stated fairly and answered with the team's own numbers. | All required elements appear, sample bills are correct, and the objection is answered. | Some elements are missing, or the sample bills do not match the proposed rate. | The proposal is incomplete and the presentation does not explain the rate. |
Teacher reference
Teams may reach a different town total if they scale differently, so judge their later work against their own total. The figures below follow from the Parcel Data Sheet and the stratified scaling method.
| Parcel | Hard-surface area (sq ft) | Area divided by 2,400 | ERUs billed |
|---|---|---|---|
| A. Corner Market | 2,500 + 1,500 + 100 = 4,100 | 1.71 | 2 |
| B. Diner | 2,200 + 4,200 + 200 = 6,600 | 2.75 | 3 |
| C. Gas station | 1,200 + 8,000 = 9,200 | 3.83 | 4 |
| D. Hollis Grocery | 20,000 + 30,000 + 2,400 = 52,400 | 21.83 | 22 |
| E. St. Brendan's | 6,000 + 2,000 + 9,600 + 900 = 18,500 | 7.71 | 8 |
| F. Community Church | 4,000 + 7,000 + 400 = 11,400 | 4.75 | 5 |
| G. School | 36,000 + 12,000 + 35,000 + 6,000 + 3,200 = 92,200 | 38.42 | 39 |
| H. Pell Box Company | 50,000 + 30,000 + 18,000 = 98,000 | 40.83 | 41 |
The Hollis Grocery roof is 160 x 150 - 40 x 100 = 24,000 - 4,000 = 20,000 square feet, and the school classroom roof is 300 x 200 - 200 x 120 = 60,000 - 24,000 = 36,000 square feet. At the gas station the canopy stands over pavement already counted, so a team that adds its 2,400 square feet gets 11,600 square feet and 5 ERU, which double-counts the same ground. The school lot is 300,000 square feet and the factory lot is 160,000 square feet, yet the two are billed 39 and 41 ERU, which is a useful example against the lot-size misconception.
Stratified scaling (expected for a 4). Small commercial averages (2 + 3 + 4) / 3 = 3 ERU, so 126 x 3 = 378 ERU. Large commercial uses Parcel D, so 10 x 22 = 220 ERU. The two churches, the school, and the factory are the only ones of their kind, so they enter directly as 8 + 5 + 39 + 41 = 93 ERU. Non-home total: 378 + 220 + 93 = 691 ERU.
Answer: 3,200 + 691 = 3,891 ERU for the town.
Plain average (acceptable for a 3). The eight parcels total 124 ERU, an average of 15.5, which gives 15.5 x 140 = 2,170 non-home ERU and 5,370 ERU for the town. This overstates the town because the sample contains the only school, the only factory, and both churches. A team using it would solve for a rate of $7.45, and at that rate the town's actual 3,891 ERU would bring in 12 x 3,891 x 7.45 = $347,855.40, which is $132,144.60 short. Raise this in Session 2 if a team does not catch it.
Revenue equation. R = 12 x 3,891 x r, so R = 46,692r. Setting R = 480,000 gives r = 10.2801..., and rounding to the nearest cent gives $10.28, which brings in only $479,993.76. The rate must be rounded up.
Answer: $10.29 per ERU per month, which brings in $480,460.68 a year.
Credit inequality. With a, b, c, and d as the credit fractions for rain barrels, rain gardens, permeable pavement, and detention ponds, each between 0 and 0.40, the constraint is 12r(3,891 - (200a + 49b + 83c + 41d)) ≥ 480,000. At $10.29 the margin is only $460.68, enough to cover about 3.7 credited ERU, so nearly every useful schedule fails at the break-even rate. This is why Milestone 3 asks for an operating rate.
Sample tests at an operating rate of $10.50, where 12 x 10.50 = 126 and the credited ERUs may not exceed 3,891 - 480,000 / 126, about 81.5 ERU.
Viable schedule. Rain barrels 10 percent, rain gardens 25 percent, permeable pavement 30 percent, detention 40 percent. Credited ERUs are 20 + 12.25 + 24.9 + 16.4 = 73.55, so revenue is 126 x (3,891 - 73.55) = 126 x 3,817.45.
Answer: $480,998.70, which meets the budget with $998.70 to spare.
Non-viable schedule. Rain barrels 20 percent and every other practice 40 percent. Credited ERUs are 40 + 19.6 + 33.2 + 16.4 = 109.2, so revenue is 126 x (3,891 - 109.2) = 126 x 3,781.8.
Answer: $476,506.80, which is $3,493.20 short of the budget.
Lowest rate covering full 40 percent credits. Credited ERUs are 0.40 x 373 = 149.2, leaving 3,741.8 billed ERU, so r ≥ 480,000 / (12 x 3,741.8) = 10.6900.... At $10.69 revenue is $479,998.10, which falls short.
Answer: $10.70 per ERU per month, which brings in $480,447.12.
Sample bills at $10.50 with the viable schedule above.
| Property | No credit, monthly | No credit, annual | With credit, monthly | With credit, annual |
|---|---|---|---|---|
| Home, 1 ERU, 10 percent credit | $10.50 | $126.00 | $9.45 | $113.40 |
| School, 39 ERU, 30 percent credit | $409.50 | $4,914.00 | $286.65 | $3,439.80 |
| Factory, 41 ERU, 40 percent credit | $430.50 | $5,166.00 | $258.30 | $3,099.60 |
Extension figures. If the credit cap rises to 50 percent and every applicant takes it, 186.5 ERU are credited and 3,704.5 are billed, so the budget floor rises to r ≥ 10.7977..., or $10.80 per ERU per month. A household ceiling of H dollars a year on an uncredited home sets r ≤ H / 12, so the range of workable rates disappears if a team sets H below 12 x 10.80 = $129.60.
Differentiation
Support
Give these teams the Parcel Data Sheet with the L-shaped roofs already split into two labeled rectangles, so the measurement step is addition of areas rather than decomposition. Provide the town total of 3,891 ERU from the teacher reference so their work begins at the revenue equation, and supply a partially built equation with the twelve months already in place.
Extension
Ask these teams to add a second constraint to the fictional Thurlby model: no home's annual bill may exceed a limit the team sets and justifies. Have them find the range of rates that satisfies both the budget floor and the household ceiling, and describe what happens to that range when the credit cap rises from 40 percent to 50 percent.
ELL supports
Provide a labeled drawing of Parcel A from the Parcel Data Sheet with the hard surfaces shaded and the words "roof," "walk," "parking," and "lawn" placed on the picture. Give a glossary card for "impervious," "billing unit," "credit," "revenue," and "round up," and supply presentation sentence frames such as "Our rate is ___ per ERU each month, which raises ___ a year."
Extensions
Students can look up whether their own city or county charges a stormwater fee, find the billing unit it uses, and compare that unit with the fictional 2,400 square feet used here. A second task is to write a short letter to a property owner explaining why two homes on the same street pay the same stormwater bill while two businesses on that street do not.