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Lesson plan Grade 10 Math

Emberton's Non-Revenue Water: Loss, Cost, and Payback

A grade 10 math lesson where students turn a fictional utility's unbilled water into a loss percent, an annual dollar figure, and a leak-program payback.

Objectives

  1. Students will calculate non-revenue water as a volume and as a percent of production.
  2. Students will find annual lost revenue and the payback period of a leak-detection program.

Vocabulary

Non-revenue water
Water that a utility treats and pumps but never gets paid for. Most of it escapes through leaks in old pipes, and some is lost to broken meters or theft.
Leak detection
Finding hidden leaks in buried water pipes before they become big breaks. Crews listen for the sound of escaping water using sensitive equipment.

Also: payback period, revenue

Teacher overview

Every utility treats and pumps more water than it manages to bill, and the gap is called non-revenue water. Some is real leakage from aging pipes, and some is metering error, but all of it costs money that customers ultimately cover. This lesson uses a fictional utility, Emberton Municipal Water, to turn that gap into a percent, an annual dollar figure, and a payback period for a leak-detection program. A common misconception this lesson addresses is that lost water is free because it came from the utility's own source; treated water carries real chemical, energy, and pumping costs whether or not it is billed.

Procedure

Hook (5 min): Ask students to guess what share of treated water a typical utility never bills for. Record guesses, then explain that the gap has a name, non-revenue water, and that today they will price it out.

Direct instruction (12 min): Define non-revenue water as produced water minus billed water. Define payback period as cost divided by yearly savings. Write the two working relationships on the board: percent lost = lost volume divided by produced volume, times 100; and annual revenue = gallons divided by 1,000, times the rate per thousand gallons. This treats every recovered gallon as if it could be sold; utilities usually value repaired leaks at the lower cost of producing the water, which lengthens the payback.

Guided modeling (13 min): Model the method with a fictional district that produces 2.0 MGD, bills 1.8 MGD, and charges $5.00 per 1,000 gallons. Together, find the loss of 0.2 MGD, the 10 percent loss share, the yearly lost volume of 73,000,000 gallons, and the lost revenue of $365,000 per year. Label every unit as you go. This worked example is for instruction only; students do not turn it in.

Independent practice (15 min): Students complete the Student Handout for Emberton Municipal Water. Circulate and check item 2's loss percent before students move past it, since a wrong percent there carries through the later items.

Closure and formative check (5 min): On an index card, students write one sentence explaining why water a utility never bills for still costs money. Collect the cards and scan for the idea that treatment and pumping are already paid for before the water leaks away.

Emberton's Non-Revenue Water: Loss, Cost, and Payback — Grade 10 Math
Name: ____________________________ Date: ______________

Student Handout

Emberton Municipal Water is a fictional utility, and its numbers are set for this handout. Show your work for each item.

Utility data:

  • Water produced: 4.0 MGD; water billed: 3.4 MGD
  • Water rate: $4.00 per 1,000 gallons
  • Leak-detection program cost: $60,000; expected water recovered: 0.3 MGD
  1. Find the non-revenue water in MGD (produced minus billed).
  2. Find the non-revenue water as a percent of production.
  3. Find the water lost in one year, in gallons. Use 365 days.
  4. Find the value of the lost water per year at the $4.00 retail rate per 1,000 gallons.
  5. The leak program recovers 0.3 MGD. Find the value of the recovered water per year at that rate.
  6. Find the payback period of the $60,000 program, in years.
  7. After recovering 0.3 MGD, find the new non-revenue water percent.
  8. Explain why non-revenue water still costs the utility money, even though the water came from its own source.

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: 0.6 MGD.
    Non-revenue water = 4.0 MGD - 3.4 MGD = 0.6 MGD.
  2. Answer: 15 percent.
    0.6 MGD / 4.0 MGD x 100 = 15 percent of production is non-revenue water.
  3. Answer: 219,000,000 gal.
    0.6 MGD is 600,000 gallons per day. Then 600,000 x 365 = 219,000,000 gallons per year.
  4. Answer: $876,000.
    219,000,000 gallons / 1,000 = 219,000 thousand-gallon units. Then 219,000 x $4.00 = $876,000 per year.
  5. Answer: $438,000.
    0.3 MGD is 300,000 gallons per day, or 109,500,000 gallons per year. Then 109,500,000 / 1,000 x $4.00 = 109,500 x 4 = $438,000 per year.
  6. Answer: about 0.14 years.
    Payback = cost / yearly savings = $60,000 / $438,000 = 0.137 years, which is roughly 1.6 months.
  7. Answer: 7.5 percent.
    Recovering 0.3 MGD leaves 0.6 - 0.3 = 0.3 MGD lost. Then 0.3 / 4.0 x 100 = 7.5 percent.
  8. Answer: Treated water already cost money to pump and treat before it was lost.
    Chemicals, energy, and pumping are spent on every gallon produced. When that water leaks away unbilled, those costs are never recovered, so all customers pay more.

Differentiation

Support

Provide the two working relationships pre-written on the handout and a filled example of item 1, so students start from a clear model. Allow calculators and a units checklist that pairs each number with MGD, gallons, or dollars.

Extension

Have students find the water rate at which the leak program would still pay for itself within one year if it recovered only half as much water, and explain how a lower recovery changes that break-even rate.

ELL supports

Pre-teach "non-revenue water," "revenue," and "payback" with a labeled diagram showing water produced, water billed, and the gap between them. Provide a sentence frame for item 8: "Lost water still costs money because ___."