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Lesson plan Grade 12 Math

Haleboro's Aging Mains: Replacement Rate and Backlog

A grade 12 lesson where students compute a water system pipe replacement rate, cycle, funding gap, and backlog, then link the results to an infrastructure grade.

Objectives

  1. Students will calculate a water system's pipe replacement rate, replacement cycle, and annual funding gap.
  2. Students will interpret how a utility's replacement rate connects to its infrastructure condition and cost.

Vocabulary

Key words: replacement rate, replacement cycle, backlog, useful life, funding gap

Teacher overview

Most water mains are buried and out of sight, so a utility that replaces too little pipe each year can build a long, expensive backlog before anyone notices. This lesson has students work the replacement math for the fictional Haleboro water system, then connect a slow replacement rate to a national infrastructure grade. A common misconception to correct is reading a replacement rate of a couple of miles a year as "plenty"; when it is set against the size of the system and the life of the pipe, that same rate can imply a replacement cycle many times longer than the pipe will last.

Procedure

Hook (5 min): Ask students how long they think a buried water main lasts and how many miles of pipe their town might have. Tell them the fictional Haleboro water system has 240 miles of main and replaces less than one mile a year, and ask what could go wrong with that.

Direct instruction and model (12 min): Define useful life, replacement rate, and replacement cycle. Model one calculation on the board using a different fictional system: if a system has 180 miles of pipe and targets a 90-year useful life, dividing 180 by 90 gives a target replacement rate of 2.0 miles per year. Show that dividing the total miles by the actual replacement rate gives the replacement cycle in years. Keep the modeled example separate from the handout so students solve the handout themselves.

Guided practice (13 min): With the class, work the replacement cycle for that system at an actual 0.6 miles per year (180 / 0.6 = 300 years), and compare it to its 90-year target life. Do a quick check for understanding: ask students to show thumbs up if a 300-year cycle is longer than the pipe's life and thumbs down if it is shorter, and re-explain if fewer than about 80 percent answer correctly.

Independent practice (18 min): Students complete the Student Handout, computing Haleboro's replacement rate, cycle, backlog clearing time, replacement costs, and funding gap, then answering the interpretation question. Circulate and check that students divide in the correct order for the cycle.

Closure and exit ticket (7 min): Each student answers on an index card: "In one sentence, why does a replacement rate that sounds small still matter for a water system?" Collect the cards and check that at least 80 percent connect the rate to the system size or the pipe's life before dismissal.

Haleboro's Aging Mains: Replacement Rate and Backlog — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

The Haleboro water system is fictional. Use these figures for every question.

  • Total distribution main: 240 miles
  • Target useful life of pipe: 100 years
  • Current replacement rate: 0.8 miles per year
  • Backlog already past useful life: 30 miles
  • Replacement cost: $1,200,000 per mile
  • Approved annual replacement budget (not fully spent in recent years): $1,500,000
  1. At the 100-year target life, how many miles per year must Haleboro replace to renew all 240 miles on schedule?
  2. At the current 0.8 miles per year, how many years would it take to replace all 240 miles once (the replacement cycle)?
  3. The replacement cycle from question 2 is how many times longer than the 100-year target life?
  4. To clear the 30-mile backlog at 0.8 miles per year, and assuming no new backlog forms, how many years would it take?
  5. At the current 0.8 miles per year and $1,200,000 per mile, what is Haleboro's annual main-replacement spending?
  6. What annual budget would meet the target rate from question 1 at $1,200,000 per mile?
  7. With its full approved $1,500,000 budget at $1,200,000 per mile, how many miles per year can Haleboro actually replace?
  8. By how many miles per year does the budget-supported rate from question 7 fall short of the target rate from question 1?
  9. Express the budget-supported rate from question 7 as a percent of the target rate from question 1. Round to the nearest tenth.
  10. The ASCE 2025 Report Card gave US drinking water a C-. Using your results, explain in two or three sentences why a low grade like that fits a system with replacement numbers like Haleboro's.

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: 2.4 miles per year. Dividing 240 miles by the 100-year target life gives 2.4 miles per year.
  2. Answer: 300 years. Dividing 240 miles by 0.8 miles per year gives a 300-year replacement cycle.
  3. Answer: 3 times. The 300-year cycle divided by the 100-year target life is 3, so the cycle is three times the pipe's life.
  4. Answer: 37.5 years. Dividing the 30-mile backlog by 0.8 miles per year gives 37.5 years.
  5. Answer: $960,000. Multiplying 0.8 miles per year by $1,200,000 per mile gives $960,000 per year.
  6. Answer: $2,880,000. Multiplying the 2.4-mile target rate by $1,200,000 per mile gives $2,880,000 per year.
  7. Answer: 1.25 miles per year. Dividing the $1,500,000 budget by $1,200,000 per mile gives 1.25 miles per year.
  8. Answer: 1.15 miles per year. The target rate of 2.4 minus the budget-supported 1.25 leaves a shortfall of 1.15 miles per year.
  9. Answer: 52.1 percent. Dividing 1.25 by 2.4 gives 0.5208, or about 52.1 percent of the target rate.
  10. Answer: Haleboro replaces pipe far slower than the pipe wears out, so its mains keep aging and its backlog grows. A national grade like C- reflects many systems in exactly this position, replacing only a third of what they should each year, and even the full budget would cover only about half, letting condition slip.

Differentiation

Support

Give students who need it the first two handout answers already set up (the divisions written but not evaluated) so they can focus on interpreting the replacement cycle. Provide a labeled example of the fictional system with rate and cycle marked on a timeline.

Extension

Ask students to find what replacement rate would clear Haleboro's backlog and keep pace with the target renewal at the same time, then estimate the annual budget that rate would require and compare it to the current budget.

ELL supports

Pre-teach "replacement," "cycle," "backlog," and "useful life" with a simple timeline showing pipe installed, pipe worn out, and pipe replaced. Provide the sentence frame "The replacement cycle is ___ years because ___" for question 2.

Extensions

Have students look up their own utility's reported miles of main and annual replacement figures where available, compute its replacement cycle, and compare it with the fictional Haleboro result, discussing what the comparison suggests about local investment.