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Lesson plan Grade 8 Math

Selbridge Valve Crews: A System With No Solution

A grade 8 lesson in which students set up systems for three fictional valve-replacement crews and find a pair whose totals are never equal.

Objectives

  1. Students will write a linear equation for each crew's running total from a stated starting count and daily rate.
  2. Students will solve a pair of simultaneous equations and interpret the solution as a workday and a valve count.
  3. Students will explain why a pair of equations with equal rates and unequal starting counts has no solution.
  4. Students will distinguish no solution from infinitely many solutions by reading the statement left when the variable cancels.

Vocabulary

Distribution system
A distribution system is the network of pipes, pumps, valves, and storage tanks that carries treated water from a treatment plant to homes, schools, and businesses. Most of it is buried underground beneath streets.

Also: system of equations, no solution, rate of change, gate valve

Teacher overview

Buried gate valves let a utility shut off one block of pipe instead of a whole neighborhood, and older systems often have valves that no longer close. In our scenario the fictional town of Selbridge has hired three crews to replace them, each crew starting with a different number already finished and working at its own steady pace. Students write an equation for each crew and then ask when two running totals are equal. A misconception this lesson targets directly is that every system of equations must have an answer; when the variable cancels, students often report a number anyway instead of reading the false statement that remains as a sign that the totals never meet.

Procedure

Hook (5 min): Ask students whether a slower runner who starts ahead can always be caught. Take two or three answers, then ask the harder version: what if both runners move at exactly the same speed? Leave the question open and turn to the crew records for the town of Selbridge.

Direct instruction (14 min): Introduce the three crews on the board, naming the starting count and the daily rate for each, and build the North crew equation together as N = 6d + 40, where d is workdays. Do the same for the South crew as S = 6d + 12. Set the two expressions equal and work through the algebra in full: subtracting 6d from both sides leaves 40 = 12, which is false for every value of d. Name what that means in the scenario, which is that the two crews are always 28 valves apart. Contrast it with a pair whose rates differ, using a quick sketch of two lines that cross and two lines that are parallel.

Guided practice (12 min): As a class, set the North crew equal to the East crew, which starts at 12 valves and replaces 10 each workday. Solve 6d + 40 = 10d + 12 step by step to reach d = 7. Ask a check-for-understanding question before going further: "What does the number 7 mean here, a count of valves or a count of days?" Expect workdays. If more than a few students say valves, substitute back into both equations in front of the class and show that the valve count is 82.

Independent practice (14 min): Students complete the Student Handout, which repeats the three crews with new questions and adds a fourth crew whose record matches the North crew exactly. Circulate and watch for students who write 0 = 28 and then answer "zero" instead of "no solution."

Closure and exit ticket (5 min): On an index card, students answer one question: "Two crews both replace 9 valves a day. One has finished 30 and the other has finished 55. Will their totals ever be equal? Explain in one sentence." Collect and sort for the reasoning, not just the yes or no.

Selbridge Valve Crews: A System With No Solution — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

Three crews are replacing old gate valves in the fictional town of Selbridge. Let d stand for the number of workdays from today.

  • North crew: 40 valves already replaced, 6 more each workday, so N = 6d + 40.
  • South crew: 12 valves already replaced, 6 more each workday, so S = 6d + 12.
  • East crew: 12 valves already replaced, 10 more each workday, so E = 10d + 12.
  1. How many valves will the North crew have replaced after 5 workdays?
  2. How many valves will the South crew have replaced after 5 workdays?
  3. Set the North and South expressions equal and solve 6d + 40 = 6d + 12. Report what the algebra tells you about this pair of equations.
  4. After 20 workdays, how many more valves has the North crew replaced than the South crew?
  5. Solve the pair N = 6d + 40 and E = 10d + 12. On which workday are the two totals equal?
  6. How many valves has each of those two crews replaced on that workday?
  7. A fourth crew, the West crew, has also replaced 40 valves and also replaces 6 each workday. Set the North and West expressions equal and solve. Report what the algebra tells you about this pair.
  8. Explain in one or two sentences why two crews working at the same daily rate can never have equal totals unless they started with the same number of valves.

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: 70 valves. Substitute d = 5 into N = 6d + 40, so N = 6(5) + 40 = 30 + 40 = 70 valves.
  2. Answer: 42 valves. Substitute d = 5 into S = 6d + 12, so S = 6(5) + 12 = 30 + 12 = 42 valves.
  3. Answer: No solution. Subtracting 6d from both sides of 6d + 40 = 6d + 12 leaves 40 = 12, which is false for every value of d, so no workday makes the two totals equal.
  4. Answer: 28 valves. After 20 workdays the North crew has 6(20) + 40 = 160 valves and the South crew has 6(20) + 12 = 132 valves, and 160 - 132 = 28 valves. The gap is the same on any workday.
  5. Answer: 7 workdays. Set 6d + 40 = 10d + 12. Subtract 6d from both sides to get 40 = 4d + 12, then subtract 12 to get 28 = 4d, then divide by 4 to get d = 7 workdays.
  6. Answer: 82 valves. Substitute d = 7 into either equation. North gives 6(7) + 40 = 42 + 40 = 82 valves, and East gives 10(7) + 12 = 70 + 12 = 82 valves, which confirms the solution.
  7. Answer: Infinitely many solutions. The West crew's equation is W = 6d + 40, the same as the North crew's. Setting them equal gives 6d + 40 = 6d + 40, which reduces to 40 = 40, a true statement for every value of d, so the two totals match on every workday.
  8. Answer: Equal daily rates mean the two totals grow by the same amount each day, so whatever gap exists at the start never shrinks. The difference between the two expressions is a constant, and a constant that is not zero can never equal zero.

Differentiation

Support

Give these students a filled table of running totals for the North and South crews at days 0, 5, 10, 15, and 20 in the town of Selbridge, so the constant 28-valve gap is visible before any algebra happens. Assign items 1 through 5 and provide the prompt "the variable disappeared, so I look at the numbers that are left" beside item 3.

Extension

Ask students to invent a fifth crew whose record makes a system with the East crew that has no solution, and a sixth crew whose record makes a system with the East crew that has exactly one solution on workday 12. They must state each crew's starting count and daily rate and show the algebra for both claims.

ELL supports

Pre-teach gate valve, crew, and replaced with a simple street diagram showing valves at intersections. Post the three outcomes as a reference card with one example each: one solution, no solution, and infinitely many solutions. Provide the sentence frame for item 8: "The two crews can never be equal because ___, so the difference stays ___."

Extensions

Connect the scenario to the work itself. Utility crews that locate, exercise, and replace valves are usually led by licensed distribution operators, and becoming an operator typically requires a high school diploma, on-the-job training, and a state license earned by exam, while the mapping side of the same project is often handled by a technician with a certificate or an associate degree and by engineers with four-year degrees. Ask students to write two questions they would ask a crew supervisor in the town of Selbridge about how the replacement schedule was planned, then compare those questions with the assumptions built into the equations, such as a rate that never changes.