Math problem set Grade 7 Engineering & Design
Nettleton Rain Barrel Array: A Proportional Relationship Set
Seven problems where students test a rain barrel array for proportionality, write its equation, and use percent to see how much of a storm it can capture.
Objectives
- Students will test a data set for proportionality and identify the constant of proportionality.
- Students will write an equation for a proportional relationship and use it to solve percent-based design problems.
Vocabulary
Key words: proportional, constant of proportionality, equation
Teacher overview
A grade 7 engineering-design class is helping Nettleton Elementary, a fictional school used in this problem set, plan a rain barrel array that will catch roof runoff from a covered walkway for the school's garden. Each barrel in the design holds exactly 55 gallons, and this problem set has students test whether total storage grows proportionally with the number of connected barrels, write an equation for that relationship, and then use percent reasoning to size the array against a real storm total. A common misconception in this math is rounding down when a design question asks for the minimum whole number of barrels needed to reach a goal, when rounding up is required because a partial barrel cannot actually be built.
Problems
For this scenario, each rain barrel holds 55 gallons, and barrels can be connected so their totals add together.
Complete this table showing total storage for an array of 1, 2, 3, and 4 connected barrels.
Barrels Total gallons 1 2 3 4 Is the relationship between the number of barrels and the total gallons proportional? State the constant of proportionality and what it represents.
Write an equation for total gallons (g) in terms of the number of barrels (b), using the constant of proportionality from problem 2.
Using your equation from problem 3, find the total storage for an array of 7 barrels without extending the table by hand.
The covered walkway roof produces 700 gallons of runoff during a typical storm. Using your 7-barrel array's total capacity from problem 4, what percent of the storm's runoff can the array capture? Round to the nearest whole percent.
The school wants to capture at least 80 percent of a typical storm's runoff. How many whole barrels are needed to reach that goal?
Challenge. The school's grant covers up to 10 barrels. What is the maximum total storage possible under that limit, and what percent of the 700-gallon storm would that capture? Round to the nearest whole percent.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: 55, 110, 165, 220. 1 barrel = 55 gal; 2 barrels = 110 gal; 3 barrels = 165 gal; 4 barrels = 220 gal.
Answer: Yes, proportional. The constant of proportionality is 55 gallons per barrel, since each additional barrel adds exactly 55 gallons to the total.
Answer: g = 55b, where g is total gallons and b is the number of barrels.
Answer: 385 gallons. g = 55 × 7 = 385 gallons.
Answer: 55%. 385 ÷ 700 × 100 = 55%.
Answer: 11 barrels. The array needs at least 80% of 700 gallons, which is 560 gallons. 560 ÷ 55 ≈ 10.18, which must round up to 11 barrels, since 10 barrels (550 gallons) falls short of 560 gallons, and 11 barrels (605 gallons) meets the goal.
Answer: 550 gallons, about 79%. 10 barrels × 55 gal = 550 gallons. 550 ÷ 700 × 100 ≈ 78.57%, which rounds to about 79%, just short of the school's 80 percent goal.
Differentiation
Support
Provide the completed table from problem 1 already filled in, and let students begin at problem 2, focusing on identifying the constant of proportionality rather than computing it from scratch. Allow a calculator for every problem.
Extension
Ask students to find the smallest number of barrels that would let the array capture more than 100 percent of a smaller, 400-gallon storm, and to explain what capturing "more than 100 percent" of a storm would actually mean for the design in practice (for example, unused storage capacity left over).
ELL supports
Provide a simple diagram of connected barrels with gallon labels building up (55, 110, 165) before problem 1, so the proportional pattern is visible. Allow students to complete problem 2 by pointing to or circling the pattern in the diagram rather than writing a full sentence.