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Math problem set Grade 8 Math

Kestrel Ridge Rooftops: Comparing Rain Harvest Slopes

A grade 8 problem set where students model two school rooftops as proportional rain-harvest lines and compare their slopes, using the standard runoff conversion factor.

Objectives

  1. Students will find the unit rate of rain harvested per inch for two roofs.
  2. Students will graph each proportional relationship and compare the two slopes.

Vocabulary

Key words: proportional, slope, unit rate, origin

Teacher overview

A rooftop that sends rain into barrels collects gallons in direct proportion to the rain that falls, and the collection rate is a slope students can graph and compare. This set uses the fictional Kestrel Ridge School, whose gym and library roofs feed separate rain barrels. Students find each roof's gallons-per-inch rate with the standard conversion (roof area in square feet times rainfall in inches times 0.623 equals the gallons of rain falling on the roof; a real system collects somewhat less), write each as a proportional equation, and compare the two slopes. A common error is treating both roofs as collecting at the same rate; the slope depends on the roof's area, so the larger roof gives the steeper line. Another error is forgetting that both lines start at the origin.

Kestrel Ridge Rooftops: Comparing Rain Harvest Slopes — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

Kestrel Ridge School is fictional. Each inch of rain drops (roof area in square feet) × 0.623 gallons on a roof. A real barrel system collects somewhat less than this, but use the full amount for the problems below.

  1. The gym roof covers 4,000 square feet. How many gallons does each inch of rain collect from it?
  2. The library roof covers 2,500 square feet. How many gallons does each inch of rain collect from it?
  3. Write an equation for the gym's gallons g in terms of inches of rain r.
  4. Write an equation for the library's gallons g in terms of inches of rain r.
  5. If you graph both equations, which line is steeper, and what does the steeper slope mean?
  6. In a 2-inch rain, how many gallons does each roof collect?
  7. The gym's barrels hold 5,000 gallons. In a 2-inch rain, do they overflow?
  8. How many inches of rain would exactly fill the gym's 5,000-gallon barrels? Solve your gym equation for r and round to the nearest tenth.
  9. Compare the slopes. The gym collects how many times as much per inch as the library? Round to the nearest tenth.
  10. Explain why both the gym line and the library line pass through the point (0, 0).

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: 2,492 gallons per inch. 4,000 × 0.623 = 2,492 gallons for each inch of rain.
  2. Answer: 1,557.5 gallons per inch. 2,500 × 0.623 = 1,557.5 gallons for each inch of rain.
  3. Answer: g = 2,492r. The gallons equal the rate 2,492 gallons per inch times the inches of rain r.
  4. Answer: g = 1,557.5r. The gallons equal 1,557.5 gallons per inch times the inches of rain r.
  5. Answer: The gym line is steeper. Its slope, 2,492, is larger than the library's 1,557.5, which means the gym collects more gallons per inch of rain.
  6. Answer: Gym 4,984 gallons, library 3,115 gallons. Gym: 2,492 × 2 = 4,984. Library: 1,557.5 × 2 = 3,115.
  7. Answer: No. The gym collects 4,984 gallons, which is just under the 5,000-gallon capacity, so the barrels do not overflow.
  8. Answer: About 2.0 inches. Solve 2,492r = 5,000: r = 5,000 ÷ 2,492 = 2.006, which rounds to 2.0 inches.
  9. Answer: 1.6 times. 2,492 ÷ 1,557.5 = 1.60, so the gym collects about 1.6 times as much per inch as the library, which matches 4,000 ÷ 2,500 = 1.6.
  10. Answer: With 0 inches of rain, a roof collects 0 gallons, so each line passes through (0, 0). A proportional relationship always starts at the origin.

Differentiation

Support

Give students the two gallons-per-inch rates already computed (2,492 and 1,557.5) so they can focus on writing the equations and comparing slopes. Provide a partly filled table of gallons at 1, 2, and 3 inches for each roof.

Extension

Ask students to find the roof area that would collect exactly 2,000 gallons per inch of rain, then explain how that new line's slope would compare with the gym's and the library's on the same graph.

ELL supports

Pre-teach "slope," "steeper," and "origin" with a graph showing two lines of different steepness from the same corner. Provide a sentence frame for problem 5: "The ___ line is steeper because its slope is ___, which means it collects ___ gallons per inch."