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Lesson plan Grade 8 Social Studies & Civics

Extending Water Mains to Carrick Row: A Halverford Hearing

A grade 8 civics lesson on a city water-service extension policy, in which students solve a system of equations to compare two ways of paying for the pipe.

Objectives

  1. Students will describe the steps a city follows before extending water service beyond its boundary.
  2. Students will solve a system of two linear equations to compare two payment plans for one project.
  3. Students will explain how the choice of payment plan shifts costs between new customers and existing customers.

Vocabulary

Key words: annexation, service extension, connection fee, unincorporated, public hearing

Teacher overview

Cities decide where their water pipes stop, and that decision is made through a process with several steps: a petition from property owners, a staff cost estimate, a public hearing, and a council vote. This lesson walks students through that process using a fictional neighborhood outside a city boundary, then puts the financial question in front of them as a system of two linear equations. The teaching note worth making explicit is that many students assume the plan with no up-front fee must be the cheaper plan. Under the second plan here, the higher monthly charge overtakes the connection fee after about eight years, so students who compare only the first year of payments will recommend the plan that costs a long-term resident more. The mock hearing at the end asks students to say who should carry the cost, which is a judgment the mathematics informs without settling.

Procedure

Framing the problem (8 min): Describe the fictional situation. Carrick Row is a neighborhood of 40 homes just outside the city of Halverford. The homes use private wells, and several wells have failed. Residents have petitioned the city to extend a water main to the neighborhood and to annex it. Ask students to name every group with a stake in the decision before you continue.

Direct instruction (12 min): Walk through the four steps in Halverford's fictional service extension policy: property owners representing at least half the lots must sign the petition; utility staff prepare a cost estimate; the council holds a public hearing; and the council votes on annexation and on how the extension will be paid for. Explain the difference between a connection fee paid once and a monthly charge paid for as long as the household takes service.

Guided practice (12 min): In pairs, students complete handout items 1 through 4. Circulate to confirm that pairs are writing both equations before comparing plans. As a formative check, ask each pair to hold up a card with the letter of the plan that is cheaper at 24 months. Every pair should show Plan B. Any pair showing Plan A has almost certainly left the $2,400 connection fee out of its Plan A total.

Independent practice (12 min): Students complete handout items 5 through 9 on their own, showing the solution steps for item 5.

Mock hearing and closure (11 min): Hand out role cards for a Carrick Row homeowner, a longtime Halverford ratepayer, the utility director, and two council members. Give each role two minutes to speak using at least one number from the handout, then take a council vote. Close by asking students to complete handout item 10 and turn in the handout.

Extending Water Mains to Carrick Row: A Halverford Hearing — Grade 8 Social Studies & Civics
Name: ____________________________ Date: ______________

Student Handout

Carrick Row is a fictional neighborhood of 40 homes just outside the city of Halverford. The homes rely on private wells, and several wells have failed. Residents have asked the city to extend a water main to the neighborhood.

City staff estimate the extension will cost $600,000 to build. The city has offered connecting households a choice of two payment plans:

  • Plan A: pay a connection fee of $2,400 once, then pay $32 each month for water service.
  • Plan B: pay no connection fee, then pay $56 each month for water service.

Halverford has 8,000 water accounts inside the city today.

  1. If the $600,000 cost were divided evenly among the 40 connecting homes, what would each home owe?
  2. Write an equation for the total amount a household pays under Plan A after t months.
  3. Write an equation for the total amount a household pays under Plan B after t months.
  4. After 24 months, which plan has cost the household less, and by how much?
  5. Solve the pair of equations from items 2 and 3 to find the number of months at which the two plans cost the same.
  6. About how many years is the answer to item 5? Round to the nearest tenth of a year.
  7. If all 40 households choose Plan A, how much do their connection fees raise toward the $600,000 cost?
  8. Suppose the city spreads the rest of the cost from item 7 across its 8,000 existing accounts. How much is that for each account?
  9. If that amount from item 8 is collected in equal monthly pieces over 60 months, what is the monthly increase for each existing account?
  10. Using at least one number from items 4 through 9, write two sentences explaining who you think should pay for the extension and why.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: $15,000 per home.
    600,000 divided by 40 homes = 15,000 dollars for each home, which is a large one-time cost for a single household.
  2. Answer: A = 2400 + 32t.
    The connection fee is the initial value and is paid once; the 32 dollar monthly charge is the rate of change.
  3. Answer: B = 56t.
    Plan B has no initial value, so its graph passes through the origin.
  4. Answer: $1,824.
    Plan A: 2400 + 32(24) = 2400 + 768 = 3,168 dollars. Plan B: 56(24) = 1,344 dollars. Plan B has cost 3,168 - 1,344 = 1,824 dollars less at 24 months.
  5. Answer: 100 months.
    Set 2400 + 32t = 56t. Subtracting 32t from both sides gives 2400 = 24t, so t = 100 months.
  6. Answer: about 8.3 years.
    100 months divided by 12 months per year = 8.33 years, which rounds to 8.3 years.
  7. Answer: $96,000.
    40 homes x 2,400 dollars = 96,000 dollars in connection fees.
  8. Answer: $63.00 per account.
    600,000 - 96,000 = 504,000 dollars left to cover. 504,000 divided by 8,000 accounts = 63.00 dollars for each account.
  9. Answer: $1.05 per month.
    63.00 divided by 60 months = 1.05 dollars each month for each existing account.
  10. Answer: Accept any recommendation supported by a number from items 4 through 9.
    A strong response names a group and a figure, such as arguing that Carrick Row households should pay the $2,400 connection fee because existing accounts would otherwise carry $63.00 apiece, or arguing that the city should spread the cost because $1.05 a month is a small amount next to a failed well. Reject responses that give no number or that state a preference without a reason.

Differentiation

Support

Give students a completed payment table for the fictional plans showing totals at 12, 24, 48, and 96 months, so the crossing point is visible before they solve algebraically. Provide item 5 with the first step written out, and allow calculators for items 7 through 9.

Extension

Ask students to find the connection fee that would make the two fictional plans cost the same at exactly 60 months, and to explain in writing why a city might prefer a fee that recovers its cost sooner even if fewer households agree to connect.

ELL supports

Pre-teach "annexation," "connection fee," "unincorporated," and "petition" with a simple map showing a city boundary and a neighborhood outside it. For the mock hearing, provide each role card with two sentence frames, such as "I represent ___, and this plan costs my household ___," and give students one minute of silent preparation before speaking.

Extensions

Students can look up whether their own community's water utility serves customers outside the city or town limits, and what those customers are charged compared with customers inside. Most utilities publish a rate schedule that states an outside-city rate, and comparing the two rates gives students a real version of the fictional decision they analyzed here.