Lesson plan Grade 12 Math
Marlowe Ferry's Main Replacement: Rate Surcharge or Revenue Bond
A grade 12 lesson where students compute and compare a pay-as-you-go rate surcharge and a revenue-bond surcharge to fund a fictional utility's capital program.
Objectives
- Students will calculate the monthly per-account surcharge needed to fund a stated capital program under a pay-as-you-go rate-surcharge model with no borrowing.
- Students will calculate total simple interest, total repayment, and the resulting monthly surcharge for the same capital program financed by a revenue bond.
- Students will compare the two financing methods' monthly cost and total lifetime cost to ratepayers and explain the resulting trade-off.
Vocabulary
Key words: capital program, principal, simple interest, revenue bond, rate surcharge, cumulative cost
Teacher overview
Marlowe Ferry Water Utility (a fictional utility used for this lesson) has identified $2,400,000 in main-replacement work on cast-iron pipe installed decades before current standards, work its board has decided must begin now rather than being deferred again. Across the US, a water main breaks roughly every two minutes, a reminder that aging distribution pipe is an ongoing, shared cost across the water sector rather than a problem unique to any one town. The utility's rate-analysis staff, the role students take on in this lesson, must compare two ways to pay for the work: a pay-as-you-go rate surcharge collected gradually with no borrowing, and a revenue bond that borrows the full amount immediately and repays it, with interest, over a longer term. A common misconception in this lesson is assuming that the option with the lower monthly bill is automatically the better deal for ratepayers; the lesson is built so students discover, from their own calculations, that the lower monthly option here actually costs Marlowe Ferry's customers more in total over the life of the financing.
Procedure
Hook (5 min): Present the fictional Marlowe Ferry Water Utility's $2,400,000 main-replacement need and its 4,000 residential accounts on the board. Ask students to guess, without calculating, whether households would pay more per month if the utility borrowed the money right away or collected it gradually with no borrowing, and record a few guesses without confirming them.
Direct instruction (18 min): Model Option 1, the rate surcharge: Marlowe Ferry collects the full $2,400,000 gradually, with no interest, over 10 years (120 months), spread evenly across its 4,000 accounts. Show two ways to find the monthly surcharge, as a check on each other. First, divide the total need by the number of accounts to get $600 per account over the full 10 years ($2,400,000 ÷ 4,000 = $600), then divide by 120 months to get $5.00 per account per month ($600 ÷ 120 = $5.00). Second, divide the total need directly by the total number of account-months: $2,400,000 ÷ (4,000 × 120) = $2,400,000 ÷ 480,000 = $5.00, confirming the same answer.
Model Option 2, the revenue bond: Marlowe Ferry borrows the full $2,400,000 immediately so the work can start now, and repays it over 20 years (240 months) at a simple annual interest rate of 4 percent. Calculate total interest using simple interest, principal times rate times time: $2,400,000 × 0.04 × 20 = $1,920,000. Add this to the principal to find the total repayment: $2,400,000 + $1,920,000 = $4,320,000. Divide by the number of accounts and months to find the monthly surcharge: $4,320,000 ÷ (4,000 × 240) = $4,320,000 ÷ 960,000 = $4.50 per account per month.
Point out the result directly: Option 2's monthly surcharge ($4.50) is lower than Option 1's ($5.00), even though Option 2 costs $1,920,000 more in total, because Option 2 spreads its larger total across twice as many months.
Guided practice (15 min): In pairs, students write a cumulative-collection equation for each option, using t for elapsed months and P for total dollars collected from all accounts combined: Option 1 is P = 20,000t for t from 0 to 120, after which collections stop and P stays flat at $2,400,000; Option 2 is P = 18,000t for t from 0 to 240, reaching $4,320,000 at t = 240. Students graph both equations on the same labeled, scaled axes (months on the horizontal axis from 0 to 240, cumulative dollars on the vertical axis from 0 to about $4,500,000), marking the point where Option 1's line goes flat. Pairs use their graph to estimate, then confirm algebraically, the total each option has collected by month 150: Option 1 has already collected its full $2,400,000 (flat since month 120), while Option 2 has collected 18,000 × 150 = $2,700,000, meaning Option 2 has actually collected more from ratepayers by that point despite its lower monthly rate.
Independent practice (7 min): Each student answers two questions individually.
- What is the total dollar amount Marlowe Ferry collects from all accounts combined under Option 1, and under Option 2? State the dollar difference between the two totals.
- By month 150, how much has each option collected in total, and which option has collected more by that point?
Closure / exit ticket (5 min): Students answer one question on an index card, continuing the numbered set:
- A neighboring fictional utility, Calloway Springs, faces the same $2,400,000 need with the same 4,000 accounts and wants the lowest possible monthly bill increase for residents, regardless of the total amount paid over time. Which financing option should Calloway Springs choose, and what is the exact monthly surcharge under that option?
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: $1,920,000 more collected under Option 2.
Option 1 collects $2,400,000 in total, since it charges no interest. Option 2 collects $4,320,000 in total: $2,400,000 principal plus $1,920,000 simple interest, since $2,400,000 × 0.04 × 20 = $1,920,000. The difference is $4,320,000 − $2,400,000 = $1,920,000.Answer: Option 2 has collected $300,000 more by month 150.
Option 1 finished collecting at month 120 and stays flat at $2,400,000 afterward. Option 2 is still collecting at month 150: 18,000 × 150 = $2,700,000. Option 2 has collected $2,700,000 − $2,400,000 = $300,000 more by month 150, even though it charges a lower monthly rate, because its surcharge is still active while Option 1's has already finished.Answer: Option 2, at $4.50 per account per month.
Option 2's monthly surcharge of $4.50 is lower than Option 1's $5.00, so Calloway Springs should choose Option 2 if minimizing the monthly bill increase is the only priority, even though Option 2 costs $1,920,000 more in total over the life of the financing.
Differentiation
Support
Provide the two monthly-surcharge calculations already worked through with each step labeled (total need, divided by accounts, divided by months for Option 1; total need, plus simple interest, divided by accounts and months for Option 2), so students focus on comparing the two results rather than performing every division from scratch. Provide the two cumulative-collection equations, P = 20,000t and P = 18,000t, already written, with their valid ranges of t marked, before guided practice begins.
Extension
Ask students to find the number of months at which Option 2's cumulative collections first exceed Option 1's final total, and to explain what that month represents in terms of the two financing plans' timelines.
ELL supports
Pre-teach "principal," "simple interest," and "surcharge" using a simple diagram of a single lump-sum jar (the principal) with a smaller second jar (the interest) added on top before the total is divided into monthly slices. Provide sentence starters for the independent-practice questions: "Option ___ collects a total of $, which is $ more/less than Option ___."
Extensions
Invite a local water utility's finance or billing staff member to describe, in general terms, how their utility actually chooses between pay-as-you-go funding and bond financing for a large capital project, and have students compare that real-world decision process to the trade-off they found between Marlowe Ferry's two options.