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Lesson plan Grade 8 Math

Ravensmoor Lead Line Replacement: A Function from a Table

A grade 8 lesson in which students build a linear function from a fictional city's lead service line records and test whether the pace meets a ten-year deadline.

Objectives

  1. Students will write a linear function from a table showing a quantity that decreases at a constant rate.
  2. Students will interpret the rate of change and the initial value of that function in context.
  3. Students will solve for the yearly rate needed to finish a replacement program on a deadline.

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 function, rate of change, initial value, slope

Teacher overview

A service line is the small pipe that carries water from the main under the street into a single building, and millions of those pipes across the country are still made of lead. Federal rules now require systems to replace nearly all of them within ten years, which turns a public health problem into a rate problem that eighth graders can model. This lesson uses the fictional city of Ravensmoor, whose crews have replaced lines at a steady pace for three years, and asks whether that pace will finish the job in time. Teaching note: when the quantity in a table decreases, many students still write a positive rate of change and produce the function L = 4,800 + 400y, which shows the count growing and gives a negative finishing year. Ask each student to say out loud whether the count is going up or down before any rule is written.

Procedure

  1. Opening, 5 minutes. Show a labeled sketch of a street main, a service line, and a house. Explain that the city of Ravensmoor is fictional but the ten-year replacement requirement is real, and that lead service lines remain in water systems across the country.
  2. Frame the question, 5 minutes. Post the question the class will answer: at its current pace, will Ravensmoor finish replacing its lead service lines within ten years? Ask for predictions before any calculation.
  3. Read the table together, 8 minutes. Distribute the handout and read the four-row table aloud. Have students state in their own words what changed between year 0 and year 1.
  4. Guided modeling, 12 minutes. Work items 1 through 3 as a class. Record the rate of change as a negative number, and label the 4,800 as the count at the start rather than as a count replaced.
  5. Independent work, 15 minutes. Students complete items 4 through 9. Circulate and look for the sign error described in the Overview and for students who divide 4,800 by 400 without connecting the quotient to a year.
  6. Discussion, 8 minutes. Compare the answers to items 5, 6, and 7. Ask what the city would have to change to finish on time, and list the practical limits on that change, such as crew size and funding.
  7. Exit check, 2 minutes. Students write one sentence explaining what the number 480 in item 7 means for the fictional city.
Ravensmoor Lead Line Replacement: A Function from a Table — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

The city of Ravensmoor is fictional. Ravensmoor counted its lead service lines and began replacing them at a steady pace. The table shows the count remaining at the end of each year. Federal rules require systems to replace nearly all lead service lines within ten years.

Year Lead service lines remaining
0 4,800
1 4,400
2 4,000
3 3,600
  1. Find the rate of change shown in the table.
  2. Write a function rule for the lines remaining, L, after y years.
  3. What does the number 4,800 mean for the city of Ravensmoor?
  4. How many lines remain after 7 years at this pace?
  5. In what year does the count of remaining lines reach zero?
  6. Federal rules require nearly all lead service lines to be replaced within ten years. Does Ravensmoor's current pace meet that requirement?
  7. How many lines each year would Ravensmoor need to replace to finish all 4,800 in ten years?
  8. On one graph, sketch the current pace and the pace from item 7. Which line is steeper, and what does the steeper line mean?
  9. Suppose that in year 4 the crews replace 500 lines instead of 400. Is the relationship between year and lines remaining still linear? Explain.

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: -400 lines per year.
    The count drops from 4,800 to 4,400 to 4,000 to 3,600, a change of -400 for each one-year step.
  2. Answer: L = 4,800 - 400y.
    The starting count is 4,800 and the count falls 400 each year, so 400y is subtracted.
  3. Answer: 4,800 lines.
    It is the number of lead service lines in the fictional city before any replacement work began, which is the initial value of the function.
  4. Answer: 2,000 lines.
    L = 4,800 - 400(7) = 4,800 - 2,800 = 2,000 lines remaining.
  5. Answer: 12 years.
    Set L = 0: 4,800 - 400y = 0. Add 400y to both sides: 4,800 = 400y. Divide by 400: y = 12 years.
  6. Answer: No.
    The current pace finishes in 12 years, which is later than the ten-year requirement, so Ravensmoor would fall short by two years of work.
  7. Answer: 480 lines per year.
    Let r be the yearly rate. Solve 4,800 - 10r = 0, so 10r = 4,800 and r = 480 lines each year.
  8. Answer: The 480-per-year line.
    Both lines start at 4,800, but the 480-per-year line falls faster, so it is steeper and reaches zero sooner. A steeper line means more lines replaced each year.
  9. Answer: No.
    A linear relationship needs the same change for every equal step. Replacing 400 lines in some years and 500 in another makes the rate of change vary, so the points no longer sit on one straight line.

Differentiation

Support

Give students a partially completed table with a third column headed "change from the year before" and the first entry filled in as -400. Provide graph paper with the vertical axis already scaled from 0 to 5,200 in steps of 400, so item 8 becomes a plotting task rather than a scaling task.

Extension

Ask students to write a function for a neighboring fictional city that starts with 7,200 lines and must also finish in ten years, then compare the two rates and explain what the comparison suggests about crew size. Have them state one cost or staffing reason a city might not simply raise its rate.

ELL supports

Pre-teach service line, replacement, and rate of change with a labeled street diagram and a short cloze paragraph describing the work. Provide the sentence frame for items 3 and 8: "The number ___ means ___ because ___."

Extensions

Have students look up whether their own water system has published a lead service line inventory, which many systems now post online, and compare the reported count with the fictional Ravensmoor figure. Ask them to write two questions they would ask a utility about the pace of replacement work in their community.