Water for Schools
Menu

Math problem set Grade 8 Engineering & Design

Kessington Row Lead Lines: Comparing Two Crew Rates

A grade 8 set on a lead service line replacement program where students write two linear equations, solve for the week the crews match, and interpret a no-solution case.

Objectives

  1. Students will write a linear equation for each crew's cumulative replacement count.
  2. Students will solve a pair of linear equations to find the week the two counts are equal.
  3. Students will interpret an equation with no solution in terms of the crews' work rates.

Vocabulary

Service line
A service line is the small pipe that connects the water main under the street to the plumbing inside one home or building. It is the last leg of the trip water makes to your tap.

Also: linear equation, rate of change, no solution

Teacher overview

Under the federal Lead and Copper Rule Improvements, the lead action level drops from 15 ppb to 10 ppb in 2027, and water systems must replace nearly all of their lead service lines within ten years. Kessington Row is a fictional town working through that requirement with two replacement crews that started at different times and work at different speeds. Students write a linear equation for each crew, solve the pair to find when the two crews have finished the same number of lines, and then examine what happens to that equation if both crews work at the same rate. The common error here is assuming that any equation with a variable on both sides must have exactly one solution, so problem 10 is written to make the no-solution case visible.

Kessington Row Lead Lines: Comparing Two Crew Rates — Grade 8 Engineering & Design
Name: ____________________________ Date: ______________

Problems

The town of Kessington Row has two crews replacing lead service lines. Let w stand for the number of weeks from today.

  • Crew A has already finished 180 lines and replaces 12 more lines each week.
  • Crew B has already finished 60 lines and replaces 20 more lines each week.
  1. Write an equation for a, the total number of lines Crew A has finished after w weeks.
  2. Write an equation for b, the total number of lines Crew B has finished after w weeks.
  3. Set the two expressions equal and solve for w, the week when both crews have finished the same number of lines.
  4. How many lines has each crew finished in that week? Show the substitution.
  5. In that same week, how many lines have the two crews finished together?
  6. The town has 900 lead service lines in all. After the week you found in problem 3, how many lines still need to be replaced?
  7. Working together, the two crews replace 32 lines each week. How many more full weeks do they need to finish the lines remaining from problem 6?
  8. Write and solve an equation to find the week when Crew B reaches exactly 500 finished lines.
  9. Crew B started with far fewer lines finished. Explain, in terms of the two rates, how Crew B is able to catch up to Crew A.
  10. Suppose Crew A adds a second truck and now replaces 20 lines each week instead of 12, keeping its 180 finished lines. Solve 20w + 180 = 20w + 60 and explain what your result says about the two crews.

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: a = 12w + 180.
    The 180 lines already finished form the starting value, and the crew adds 12 lines for each week that passes.
  2. Answer: b = 20w + 60.
    The 60 lines already finished form the starting value, and the crew adds 20 lines for each week that passes.
  3. Answer: 15 weeks.
    Set 12w + 180 = 20w + 60. Subtract 12w from both sides to get 180 = 8w + 60. Subtract 60 from both sides to get 120 = 8w. Divide both sides by 8 to get w = 15.
  4. Answer: 360 lines.
    Crew A: 12(15) + 180 = 180 + 180 = 360. Crew B: 20(15) + 60 = 300 + 60 = 360. Both crews stand at 360 finished lines, which is the point where the two graphs would cross.
  5. Answer: 720 lines.
    360 + 360 = 720 lines finished by the two crews together.
  6. Answer: 180 lines.
    900 - 720 = 180 lines still need replacement.
  7. Answer: 6 weeks.
    180 ÷ 32 = 5.625 weeks. Because the crews finish lines during a week rather than all at once at the end, the work spills into a sixth week, so 6 full weeks are needed.
  8. Answer: 22 weeks.
    Set 20w + 60 = 500. Subtract 60 from both sides to get 20w = 440. Divide both sides by 20 to get w = 22.
  9. Answer: Crew B has the greater rate of change.
    Crew B starts 120 lines behind, yet it replaces 8 more lines each week than Crew A does. That gap of 8 lines per week closes the 120-line difference in 15 weeks, after which Crew B moves ahead.
  10. Answer: No solution.
    Subtracting 20w from both sides leaves 180 = 60, which is false for every value of w. The two crews now work at the same rate, so the 120-line head start never closes and the totals are never equal.

Differentiation

Support

Give students a two-column table with the weeks 0, 5, 10, 15, and 20 listed and the first row filled in for both crews, so they can see the totals converge before they write the equations. Provide the solving steps for problem 3 as a scaffold with one step left blank.

Extension

Ask students to graph both equations on the same axes, label the intersection point, and then write a third equation for a crew that would finish all 900 lines in exactly 30 weeks starting from zero. Have them explain where that third line would sit relative to the other two.

ELL supports

Provide sentence frames for the explanation items: "Crew B catches up because it replaces ___ more lines each week than Crew A." Pre-teach "service line," "rate," and "solution" using a simple labeled drawing of a pipe running from a street main to a house.