Math problem set Grade 7 Math
Sable Ridge's Reverse-Osmosis Pilot: A Proportionality Problem Set
A grade 7 problem set where students test a reverse-osmosis pilot's data for proportionality, find its constant of proportionality, and write an equation.
Objectives
- Students will test a data table for a proportional relationship and identify the constant of proportionality.
- Students will write and use an equation to predict values outside a given data table.
Vocabulary
Key words: proportional relationship, constant of proportionality, permeate, reject water
Teacher overview
Sable Ridge (a fictional small water system used for this problem set) is piloting a reverse-osmosis unit that pushes brackish well water through a membrane to produce clean permeate water for the community system. Operators recorded how many gallons of permeate the pilot unit produced at four different run times, and this problem set has students check whether that data forms a proportional relationship, find the unit's constant of proportionality, and use an equation to predict output at times the operators did not directly measure. A common misconception in this math is assuming any steadily increasing data set must be proportional; a relationship is only proportional when every pair in the table reduces to the same constant ratio, which students must check rather than assume.
Problems
Sable Ridge is testing a new reverse-osmosis unit. This is a fictional pilot project used for this problem set. The unit cleans salty well water for the town. Operators measured how many gallons of clean water, called permeate, came out at four different run times. The table below shows their data.
| Feed time (minutes) | Permeate produced (gallons) |
|---|---|
| 5 | 9 |
| 10 | 18 |
| 15 | 27 |
| 20 | 36 |
- For each pair in the table, divide gallons of permeate by feed time in minutes. Show all four calculations.
- Based on your answers to problem 1, is the relationship proportional? Explain how you know.
- Write an equation relating permeate gallons (g) to feed time in minutes (m). Use the constant of proportionality you found.
- Use your equation to predict the gallons of permeate made in 50 minutes.
- The pump can run for at most 180 minutes (3 hours) before a filter change. Use your equation to find the gallons of permeate made in one full 180-minute run.
- A larger unit makes permeate at a rate of 2.5 gallons per minute. How many minutes would it need to make the same total gallons as problem 5?
- Not all the water becomes permeate. Some becomes "reject" water that carries away leftover salts. Each minute, the pilot takes in 3 gallons of feed water and makes 1.8 gallons of permeate. Find the reject-water rate in gallons per minute.
- Use your reject-water rate from problem 7. Find the gallons of reject water made during the 180-minute run in problem 5.
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: 1.8 gal/min.
9 ÷ 5 = 1.8; 18 ÷ 10 = 1.8; 27 ÷ 15 = 1.8; 36 ÷ 20 = 1.8. Every pair reduces to the same ratio, 1.8 gallons per minute. - Answer: yes, proportional.
The relationship is proportional because every gallons-to-minutes ratio in the table equals the same constant, 1.8, and the table implies a starting point of 0 gallons at 0 minutes. - Answer: g = 1.8m.
The constant of proportionality is 1.8 gallons per minute, so permeate gallons equal 1.8 times the feed time in minutes. - Answer: 90 gal. g = 1.8 × 50 = 90 gallons.
- Answer: 324 gal. g = 1.8 × 180 = 324 gallons.
- Answer: 129.6 min.
324 ÷ 2.5 = 129.6 minutes. - Answer: 1.2 gal/min.
Total feed (3 gal/min) minus permeate (1.8 gal/min) equals reject water: 3 − 1.8 = 1.2 gallons per minute. - Answer: 216 gal.
1.2 gallons per minute × 180 minutes = 216 gallons.
Differentiation
Support (below grade level)
Provide the four ratio calculations from problem 1 already completed, so students can focus on interpreting whether the relationship is proportional and on writing the equation. Give a printed equation template, "g = ___ × m," with a blank for the constant.
Extension (above grade level)
Ask students to combine the permeate equation (g = 1.8m) with a reject-water equation they write themselves (r = 1.2m) into a single equation for total feed water (f = g + r), and use it to check that the total feed at 180 minutes matches 3 gallons per minute times 180 minutes.
ELL supports
Provide a labeled diagram of the reverse-osmosis unit showing feed water entering and both permeate and reject water leaving, with the words "feed," "permeate," and "reject" defined next to each arrow. Allow students to complete problem 1's table calculations directly in the table cells rather than in full sentences.
Extensions
Have students graph the four data points from the table on a coordinate plane, draw a line through them and the origin, and check that the line's steepness matches the constant of proportionality they calculated in problem 1.