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Math problem set Grade 8 Math

Snowbrake Basin Snowmelt: Equations for Two Reservoirs

A grade 8 problem set on solving linear equations with variables on both sides, set in a fictional mountain basin where one reservoir rises as another falls.

Objectives

  1. Students will write a linear equation for a reservoir depth that changes at a constant daily rate.
  2. Students will solve linear equations with variables on both sides and check the solution.
  3. Students will identify equations with no solution and equations true for every value.

Vocabulary

Reservoir
A human-made lake, usually created by a dam, that stores water for later. Reservoirs supply cities and farms and help control floods.

Also: snowmelt, rate of change, solution, distributive property

Teacher overview

Snowbrake Basin is a fictional mountain watershed. Snow piles up all winter, then melts each spring and runs down into two reservoirs on the Elderglen River. The upper reservoir gains water from the melt while the lower one loses water to the town below, so the two depths move toward each other and the day they match can be found by solving a single equation. Teaching note: many eighth graders assume every linear equation has exactly one solution, so problems 9 and 10 are built to break that assumption. Watch also for students who subtract a variable term from one side only, which is the most common source of a wrong day count on problem 4.

Snowbrake Basin Snowmelt: Equations for Two Reservoirs — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

Snowbrake Basin is fictional, and so are the two reservoirs described below. Let t stand for the number of days since the melt began.

On day 0 the Upper Elderglen Reservoir is 12 feet deep and gains 0.75 foot each day. On day 0 the Lower Elderglen Reservoir is 30 feet deep and loses 0.5 foot each day.

  1. Write an equation for the depth u of the Upper Elderglen Reservoir after t days.
  2. On what day does the Upper Elderglen Reservoir reach 21 feet deep?
  3. On what day does the Lower Elderglen Reservoir drop to 21 feet deep?
  4. On what day are the two reservoirs exactly the same depth?
  5. How deep is each reservoir on the day you found in problem 4?
  6. On what day is the Upper Elderglen Reservoir 4 feet deeper than the Lower Elderglen Reservoir?
  7. The upper reservoir spills over its dam at 32 feet. On what whole day does it first spill?
  8. Two snow gauges sit on opposite slopes of the basin. After t days the north gauge reads 2(t + 6) inches of melted snow and the south gauge reads 3t + 4 inches. On what day do the two gauges read the same amount?
  9. Melt model P predicts 5t + 10 acre-feet of runoff and melt model Q predicts 5t + 3 acre-feet. On what day do the two models agree? Explain what your work shows.
  10. Melt model R predicts 2(3t + 5) acre-feet and melt model S predicts 6t + 10 acre-feet. On what days do these two models agree? Explain what your work shows.

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: u = 12 + 0.75t.
    The depth starts at 12 feet and rises 0.75 foot each day, so the rate of change multiplies t and the starting depth is added.
  2. Answer: 12 days.
    12 + 0.75t = 21. Subtract 12 from both sides: 0.75t = 9. Divide both sides by 0.75: t = 12 days. Check: 12 + 0.75(12) = 12 + 9 = 21 feet.
  3. Answer: 18 days.
    30 - 0.5t = 21. Subtract 30 from both sides: -0.5t = -9. Divide both sides by -0.5: t = 18 days. Check: 30 - 0.5(18) = 30 - 9 = 21 feet.
  4. Answer: 14.4 days.
    12 + 0.75t = 30 - 0.5t. Add 0.5t to both sides: 12 + 1.25t = 30. Subtract 12 from both sides: 1.25t = 18. Divide both sides by 1.25: t = 14.4 days.
  5. Answer: 22.8 feet.
    Upper: 12 + 0.75(14.4) = 12 + 10.8 = 22.8 feet. Lower: 30 - 0.5(14.4) = 30 - 7.2 = 22.8 feet. The two agree, which confirms problem 4.
  6. Answer: 17.6 days.
    12 + 0.75t = (30 - 0.5t) + 4, so 12 + 0.75t = 34 - 0.5t. Add 0.5t to both sides: 12 + 1.25t = 34. Subtract 12: 1.25t = 22. Divide by 1.25: t = 17.6 days.
  7. Answer: 27 days.
    12 + 0.75t = 32. Subtract 12: 0.75t = 20. Divide by 0.75: t = 26.67 days. The reservoir is still below the spill point on day 26, so it first spills on day 27.
  8. Answer: 8 days.
    2(t + 6) = 3t + 4. Distribute on the left: 2t + 12 = 3t + 4. Subtract 2t from both sides: 12 = t + 4. Subtract 4: t = 8 days. Check: 2(8 + 6) = 28 and 3(8) + 4 = 28.
  9. Answer: No solution.
    5t + 10 = 5t + 3. Subtract 5t from both sides: 10 = 3, which is false. The two models rise at the same daily rate but start 7 acre-feet apart, so the gap never closes and no day makes them agree.
  10. Answer: Infinitely many solutions.
    2(3t + 5) = 6t + 10. Distribute on the left: 6t + 10 = 6t + 10. Subtract 6t from both sides: 10 = 10, which is always true. The two models are the same rule written two ways, so they agree on every day.

Differentiation

Support

Give students a two-column recording sheet with the headings "what I did to the left side" and "what I did to the right side," so each move is written twice. Provide problems 2 and 3 with the first step already taken, and let students check every answer by putting the day back into both original expressions.

Extension

Ask students to find the day on which the upper reservoir holds twice the depth of the lower reservoir, ignoring the spill level for this question, then to explain why that day comes later than the day found in problem 6. Have them compare their answer with problem 7 and explain what the spill would change. Have them also write a melt model that would agree with model P on exactly one day.

ELL supports

Pre-teach reservoir, snowmelt, spill, and gauge with a labeled cross-section sketch showing water rising behind one dam and falling behind another. Provide the sentence frame for problems 9 and 10: "The equation became ___, which is ___, so the models agree on ___ days."

Extensions

Have students graph both reservoir depths on the same coordinate plane and mark the point where the lines cross. Ask them to explain how the crossing point on the graph matches the day found by solving the equation, and what the graph shows about days before and after that point.