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Math problem set Grade 8 Math

Quillon Marsh Phosphorus Cleanup: Solving Linear Equations

A grade 8 problem set where students write and solve one-variable linear equations for the phosphorus left in a fictional marsh as a cleanup wetland removes it.

Objectives

  1. Students will write a linear equation for a quantity decreasing at a steady rate.
  2. Students will solve linear equations, including one with the variable on both sides.

Vocabulary

Key words: linear equation, phosphorus, nutrient

Teacher overview

Quillon Marsh is a fictional wetland with too much phosphorus, a nutrient that plants need but that causes algae problems when there is too much. A treatment wetland removes phosphorus at a steady rate each week, which makes the amount left a linear function of time. This set has students write that equation and solve for a week or an amount. A common mistake is subtracting the starting amount from the removal rate instead of subtracting the removed amount from the start; students should keep the starting pounds and take away the weekly removal times the number of weeks.

Quillon Marsh Phosphorus Cleanup: Solving Linear Equations — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

Use the values from the fictional Quillon Marsh cleanup. The marsh starts with 900 pounds of phosphorus, and the treatment wetland removes 30 pounds each week. Let w stand for the number of weeks and P for the pounds of phosphorus left.

  1. Write an equation for the phosphorus, P, left in the marsh after w weeks.
  2. Using your equation from item 1, how many pounds remain after 10 weeks?
  3. Solve for w: after how many weeks will 450 pounds remain?
  4. The cleanup goal is to reach 150 pounds. Solve for w to find how many weeks that takes.
  5. A second wetland is added, so the marsh now removes 60 pounds each week, still starting from 900 pounds. Write the new equation and solve for the weeks to reach 150 pounds.
  6. Nearby Marsh Y starts with 600 pounds and removes 10 pounds each week, so its equation is P = 600 - 10w. Set Quillon Marsh's original equation (from item 1) equal to Marsh Y's equation and solve for w.
  7. Using your answer from item 6, how many pounds does each marsh hold in that week?
  8. Explain your reasoning: Some phosphorus removed from the marsh is dried and reused as farm fertilizer. Explain why phosphorus can be both a pollutant and a resource.

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: P = 900 - 30w.
    Start with 900 pounds and take away 30 pounds for each week w, so P = 900 - 30w.
  2. Answer: 600 pounds.
    P = 900 - 30(10) = 900 - 300 = 600 pounds.
  3. Answer: 15 weeks.
    900 - 30w = 450. Subtract 900: -30w = -450. Divide by -30: w = 15 weeks.
  4. Answer: 25 weeks.
    900 - 30w = 150. Subtract 900: -30w = -750. Divide by -30: w = 25 weeks.
  5. Answer: 12.5 weeks.
    The new equation is P = 900 - 60w. Set 900 - 60w = 150. Subtract 900: -60w = -750. Divide by -60: w = 12.5 weeks.
  6. Answer: 15 weeks.
    Set 900 - 30w = 600 - 10w. Add 30w to both sides: 900 = 600 + 20w. Subtract 600: 300 = 20w, so w = 15 weeks.
  7. Answer: 450 pounds.
    Quillon Marsh: 900 - 30(15) = 900 - 450 = 450. Marsh Y: 600 - 10(15) = 600 - 150 = 450. Both hold 450 pounds.
  8. Answer: Phosphorus is a pollutant when there is too much in the water, because it feeds algae that harm the marsh, but it is a resource when it is captured and reused as fertilizer, because plants need it to grow.

Differentiation

Support

Provide the equation P = 900 - 30w already written for items 1 through 4 so students focus on solving, and model one solve step by step before item 3. Allow a calculator throughout.

Extension

Ask students to find how many pounds per week a single wetland would need to remove to reach the 150-pound goal in exactly 10 weeks, writing and solving the equation that answers it.

ELL supports

Pre-teach "phosphorus," "nutrient," and "remove" with a simple picture of a marsh losing a fixed amount each week. Provide a sentence frame for item 8: "Phosphorus is a pollutant when ___, and a resource when ___."