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Math problem set Grade 8 Math

Copperline Township Lead Testing: Equations for Corrosion Control

Eight problems modeling a fictional town's declining lead readings with linear equations, comparing results against the real current and 2027 federal action levels.

Objectives

  1. Students will write a linear equation modeling a quantity that decreases at a constant rate over time.
  2. Students will solve linear equations in one variable to find when a quantity reaches a target value.
  3. Students will compare two linear models to find when they produce the same result.

Vocabulary

Corrosion control
Treatment that keeps water from slowly eating away at the inside of pipes. It helps stop metals like lead from getting into drinking water.
Action level
A trigger number in lead and copper testing. If too many homes test above this level, the utility must take steps like improving treatment and telling the public.

Also: linear equation, rate of decrease

Teacher overview

Copperline Township (a fictional water system used for this problem set) installed corrosion control treatment after lead sampling found levels above the federal action level, and each testing round afterward shows the lead reading dropping by a constant amount as the treatment stabilizes scale inside older pipes. This set has students model that decline with a linear equation and compare it against the real action levels: 15 parts per billion (ppb) today, dropping to a stricter 10 ppb in 2027 under the federal Lead and Copper Rule Improvements. A common misconception is subtracting the decrease rate from the wrong side of the equation, writing something like "2t − 22" instead of "22 − 2t," which produces a level that rises instead of falls; encourage students to check that their equation predicts a smaller number as t increases.

Copperline Township Lead Testing: Equations for Corrosion Control — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

All testing figures below are fictional, for made-up systems Copperline Township and Birch Fork. Only the 15 ppb and 10 ppb action levels are real regulatory figures.

  1. Copperline's first post-treatment testing round shows a lead level of 22 ppb. Each round after that, the level drops by a constant 1 ppb. Write a linear equation for the lead level, L, in ppb, after t testing rounds.
  2. Using your equation from problem 1, find the lead level after 3 rounds.
  3. Solve your equation from problem 1 for t. After how many rounds will Copperline's level first reach the current action level of 15 ppb?
  4. Solve your equation from problem 1 for t. After how many rounds will Copperline's level first reach the stricter 2027 action level of 10 ppb?
  5. Compare your answers from problems 3 and 4. How many more rounds does the stricter 2027 standard need, compared to the current standard?
  6. A neighboring fictional system, Birch Fork, starts corrosion control at a higher level of 30 ppb. It decreases faster, by 3 ppb per round. Write Birch Fork's linear equation for lead level after t rounds.
  7. Set Copperline's equation (problem 1) equal to Birch Fork's equation (problem 6). Solve for t. Then find the lead level both systems share at that round.
  8. Explain whether Birch Fork's declining lead level (problem 6) could keep falling forever if the equation went on and on. Describe the real-world limit that would actually stop the decline.

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: L = 22 − t.
    The level starts at 22 ppb and decreases by 1 ppb for each round, t.
  2. Answer: 19 ppb.
    L = 22 − 3 = 19 ppb after 3 rounds.
  3. Answer: 7 rounds.
    22 − t = 15 → −t = −7 → t = 7 rounds.
  4. Answer: 12 rounds.
    22 − t = 10 → −t = −12 → t = 12 rounds.
  5. Answer: 5 rounds.
    The stricter 2027 standard (12 rounds) takes 12 − 7 = 5 more rounds to reach than the current standard (7 rounds).
  6. Answer: L = 30 − 3t.
    Birch Fork starts at 30 ppb and decreases by 3 ppb per round, t.
  7. Answer: t = 4 rounds, shared level 18 ppb.
    Setting 22 − t = 30 − 3t gives 22 − t + 3t = 30, or 22 + 2t = 30, so 2t = 8 and t = 4. Substituting into either equation: Copperline's L = 22 − 4 = 18 ppb, and Birch Fork's L = 30 − 3(4) = 30 − 12 = 18 ppb, confirming the two match.
  8. Answer: no, it would not make sense.
    A linear equation extended far enough eventually predicts a negative lead level, which is not physically possible. In reality, the decline would level off as it approaches some low background level rather than continuing in a straight line forever, since a linear model like this one is only realistic within the range of testing rounds it was built from.

Differentiation

Support (below grade level)

Provide a printed table showing Copperline's predicted lead level for rounds 0 through 12 already calculated, so students can check their equation against the table instead of solving every step from scratch. Allow a calculator for all arithmetic.

Extension (above grade level)

Ask students to find a decrease rate (in place of Copperline's 1 ppb per round) that would let the fictional system reach the 2027 action level of 10 ppb in exactly 6 rounds instead of 12, starting from the same 22 ppb, by writing and solving an equation for the unknown rate.

ELL supports

Pre-teach "action level" and "corrosion control" with a simple diagram showing a declining line on a graph with the two action-level lines (15 ppb and 10 ppb) marked. Provide a sentence frame for problem 5: "The 2027 standard takes ___ more rounds than the current standard because ___."