Lesson plan Grade 10 Engineering & Design

Dovegate Cast-Iron Mains: Pipe Lining or Open-Cut Replacement

A grade 10 lesson comparing cured-in-place lining with open-cut replacement for a fictional town's 1920s water mains, using annualized cost and a break-even price.

Grade: Grade 10
Subject: Engineering & Design
Material: Lesson plan
Time: 60 min
Prep: Low prep
Materials: Printed Student Handout, one per student, Calculators, Board space for the worked example

Standards

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Students write and solve a one-variable equation for the lining price per foot at which both options cost the same each year.

All materials for HSA-CED.A.1 →

HSG-MG.A.1 Common Core Math

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

How this material aligns: Question 8 models the main as a cylinder and compares its cross-sectional area before and after the liner narrows the inside diameter.

All materials for HSG-MG.A.1 →

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: The lesson carries miles, feet, dollars per foot, and service life in years through a chain ending in dollars per year.

All materials for HSN-Q.A.1 →

HS-ETS1-3 NGSS

Evaluate a solution to a complex real-world problem based on prioritized criteria and trade-offs that account for a range of constraints, including cost, safety, reliability, and aesthetics as well as possible social, cultural, and environmental impacts.

How this material aligns: Questions 4 through 7 weigh open-cut replacement against cured-in-place lining on annualized cost and the break-even bid price, while questions 8 and 9 add the lost flow area and the days of street closure that the debrief asks students to cite when they recommend one option.

All materials for HS-ETS1-3 →

Download Word file

This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-09-03.

Objectives

Students will convert a length of water main from miles to feet and price two rehabilitation methods against it.
Students will annualize a capital cost by dividing it over the expected service life and compare the two results.
Students will solve for the break-even unit price at which two rehabilitation options cost the same each year.

Vocabulary

Key words: cured-in-place lining, open-cut replacement, annualized cost, service life, break-even price

Teacher overview

A utility replacing century-old cast-iron mains has more than one way to spend the money, and the cheapest option at bid opening is not always the cheapest option over the life of the pipe. This lesson has students price two rehabilitation methods for the fictional town of Dovegate, one that digs the old pipe out and replaces it and one that cures a resin liner inside it, then compare the two on cost per year rather than on total cost. The misconception this lesson targets is that a lower bid price is a lower cost; a liner that costs far less to install but lasts thirty fewer years can carry the higher annual cost, and students see that only when they divide by service life.

Procedure

Hook, 5 minutes. Show the class two numbers on the board without explanation, a total of about seven and a half million dollars and a total of about four and a half million dollars, and ask which project a town should choose. Take answers, then reveal that one lasts eighty years and the other fifty, and ask whether anyone wants to change their vote.
Direct instruction, the two methods, 10 minutes. Describe open-cut replacement, which excavates the trench, removes the old main, installs new ductile iron pipe, and restores the street. Then describe cured-in-place lining, which pulls a resin-saturated liner through the cleaned existing pipe and hardens it in place from a small number of access pits. Name the trade-offs out loud: lining is faster and disturbs less street, yet it leaves the old pipe in the ground and slightly narrows the flow path.
Worked example, teacher modeled, 10 minutes. Use a separate fictional project on the board, not the numbers on the handout. Suppose a 1,200-foot main can be replaced at $250 per foot for a 60-year life or lined at $160 per foot for a 40-year life. Work the chain aloud: replacement costs $300,000 total and $5,000 per year, while lining costs $192,000 total and $4,800 per year, so the cheaper installation is also the cheaper annual cost in that particular case. Point out that the comparison could flip if the liner's life were shorter.
Independent work, 22 minutes. Distribute the Student Handout and have students work questions 1 through 9 individually, then compare answers with a partner for the last five minutes. Circulate and watch for two errors: multiplying by service life instead of dividing, and squaring the diameter ratio incorrectly in question 8.
Formative check, 5 minutes. Ask every student to write their answer to question 6 on a scrap of paper and hold it up. If fewer than three quarters of the class is near $4,250 per year, rework questions 4 and 5 on the board before the debrief.
Debrief, 8 minutes. Ask which option students would fund and require the answer to cite question 6 for the annual cost and question 9 for the street closure. Then push on question 7: if a contractor bid the lining work above the break-even price, what would that mean for the recommendation? Close by asking what question 8 tells them about lining a main that is already near its flow capacity.

Dovegate Cast-Iron Mains: Pipe Lining or Open-Cut Replacement — Grade 10 Engineering & Design

Name: ____________________________ Date: ______________

Student Handout

Dovegate is a fictional town created for this lesson. Its downtown has 4.6 miles of 8-inch cast-iron water main installed in the 1920s. The utility is choosing between two rehabilitation methods. Open-cut replacement with new ductile iron pipe costs $310 per foot and is expected to last 80 years. Cured-in-place lining costs $185 per foot and is expected to last 50 years. Lining leaves the old pipe in place and reduces the inside diameter from 8.0 inches to 7.6 inches. One mile equals 5,280 feet. Show the full chain for every answer.

Convert the 4.6 miles of main into feet.
At $310 per foot, what is the total cost of open-cut replacement for the full length from question 1?
At $185 per foot, what is the total cost of cured-in-place lining for the full length from question 1?
Divide the open-cut cost from question 2 by its 80-year service life to find the annualized cost in dollars per year.
Divide the lining cost from question 3 by its 50-year service life to find the annualized cost in dollars per year.
How many dollars per year lower is the lining option than the open-cut option?
Write and solve an equation for the lining price per foot at which the lining option would have exactly the same annualized cost as the open-cut option, still using the 50-year life and the length from question 1.
The liner narrows the inside diameter from 8.0 inches to 7.6 inches. Because cross-sectional area is proportional to the square of the diameter, what percent of the original cross-sectional area is lost?
Open-cut work closes the street for 12 working days per 500 feet, and lining closes it for 4 working days per 500 feet. How many working days of street closure does the lining option require for the full length from question 1?

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: 24,288 feet.
Multiplying 4.6 miles by 5,280 feet per mile gives 24,288 feet of main.
Answer: $7,529,280.
Multiplying 24,288 feet by 310 dollars per foot gives 7,529,280 dollars for open-cut replacement.
Answer: $4,493,280.
Multiplying 24,288 feet by 185 dollars per foot gives 4,493,280 dollars for cured-in-place lining.
Answer: $94,116 per year.
Dividing 7,529,280 dollars by the 80-year service life gives 94,116 dollars per year.
Answer: $89,865.60 per year.
Dividing 4,493,280 dollars by the 50-year service life gives 89,865.60 dollars per year.
Answer: $4,250.40 per year.
Subtracting the two annualized figures: 94,116 minus 89,865.60 equals 4,250.40 dollars per year in favor of lining. The gap is far smaller than the difference in total cost, which is what makes the comparison worth doing.
Answer: $193.75 per foot.
Let p be the lining price per foot. The equation is 24,288 multiplied by p, divided by 50, equals 94,116. Multiplying both sides by 50 gives 24,288p equals 4,705,800, and dividing by 24,288 gives p equals 193.75 dollars per foot. Any bid above that price makes open-cut replacement the cheaper option per year.
Answer: about 9.75 percent.
The diameter ratio is 7.6 divided by 8.0, which is 0.95, and squaring it gives 0.9025. The liner therefore leaves 90.25 percent of the original area, so the loss is 1 minus 0.9025, which is 0.0975, or about 9.75 percent.
Answer: about 194 working days.
Dividing 24,288 feet by 500 feet gives 48.576 sections, and multiplying by 4 working days per section gives 194.3, which rounds to about 194 working days. The same calculation for open-cut work gives about 583 days, which is the disruption the town avoids by lining.

Differentiation

Support

Give these students the handout with question 1 answered and questions 2 and 3 set up as labeled multiplication frames, so their first independent decision is whether to multiply or divide in question 4. Provide a reminder card stating that annualized cost equals total cost divided by service life, and let them skip question 7 until the rest is complete.

Extension

Ask these students to add a maintenance term to both options for the fictional Dovegate project, using a stated annual repair cost of their own choosing for the lined main and none for the new main, and to find the repair cost at which the recommendation flips. Have them also explain why a utility might still choose open-cut replacement even when lining is cheaper per year.

ELL supports

Post a labeled cross-section showing the old cast-iron pipe, the resin liner inside it, and the narrowed opening, beside a second sketch of an open trench with new pipe. Provide a vocabulary strip pairing annualized with per year, service life with how long it lasts, and break-even with the point where two costs are equal, and allow question 7 to be explained orally before it is written.

Extensions

Have students look up the water main replacement rate their own utility reports, in miles per year, and calculate how many years it would take that utility to replace its whole system once at the current pace. Students can also research a third method, pipe bursting, and add a column to the fictional Dovegate comparison with a stated price per foot and service life, then rank all three on annual cost and on street disruption.