Lesson plan Grade 8 Math
Sagebrush Basin Reverse Osmosis: Graphing Proportional Recovery Rates
A 50-minute grade 8 lesson comparing two fictional reverse osmosis membranes by graphing recovery rates as proportional relationships and reading slope as unit rate.
Objectives
- Students will graph two proportional relationships representing membrane recovery rates.
- Students will interpret the unit rate of a proportional relationship as the slope of its graph.
- Students will compare two proportional relationships to determine which is more water-efficient.
Vocabulary
- Reverse osmosis
- Reverse osmosis cleans water by pushing it through a membrane with holes so tiny that salt and most other dissolved things cannot fit through. It can even turn seawater into fresh water.
Also: permeate, recovery rate, proportional relationship, slope
Teacher overview
Sagebrush Basin (a fictional water district used for this lesson) is testing two reverse osmosis membranes side by side. In reverse osmosis, high-pressure pumps push feed water through a thin membrane, producing clean permeate on one side while a concentrated reject stream leaves the other; a membrane's recovery rate is the fraction of feed water it turns into usable permeate. This lesson has students graph each membrane's permeate-versus-feed data as a proportional relationship and compare the two lines' slopes. A common misconception is assuming the membrane with the steeper-looking graph line is always better without checking that both graphs use the same axis scale; students need to compare the actual slope values, not just the visual steepness of two differently scaled graphs.
Procedure
Hook (5 min): Ask students what they think happens to water pushed through a very fine filter: does all of it come out clean, or does some of it stay behind? Take a few guesses without confirming an answer yet.
Direct instruction (15 min): Explain that Membrane X recovers permeate at a fixed rate of 75% of whatever feed water enters it, a proportional relationship since permeate = 0.75 × feed. Model plotting Membrane X's data table:
| Feed water (gal) | Permeate (gal) |
|---|---|
| 0 | 0 |
| 100 | 75 |
| 200 | 150 |
| 300 | 225 |
Plot these four points, draw the line through them, and identify the slope as 0.75, matching the recovery rate. Note that the line passes through the origin, which is what makes this a proportional relationship rather than a general linear function.
Guided practice (15 min): In pairs, students plot Membrane Y's data on the same grid using a different color: feed water of 0, 100, 200, and 300 gallons producing permeate of 0, 60, 120, and 180 gallons. Students find Membrane Y's slope (0.60) and compare it to Membrane X's slope.
Independent practice (10 min): Each student answers two questions in writing: how much feed water each membrane needs to produce 300 gallons of permeate, and which membrane is more water-efficient and by how much (expressed as a difference in recovery rate).
Closure / exit ticket (5 min): Students answer one question on an index card: "A third membrane's data shows 100 gallons of feed producing 82 gallons of permeate. What is its recovery rate as a slope, and is it more or less efficient than Membrane X?" Collect and scan for the correct slope value (0.82) before dismissal.
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: 400 gal (Membrane X), 500 gal (Membrane Y).
To produce 300 gallons of permeate, Membrane X needs 300 ÷ 0.75 = 400 gallons of feed water, and Membrane Y needs 300 ÷ 0.60 = 500 gallons of feed water. - Answer: Membrane X, by 0.15.
Membrane X's slope of 0.75 is greater than Membrane Y's slope of 0.60, a difference of 0.15, so Membrane X recovers 15 percentage points more permeate per gallon of feed water and is more water-efficient. - Exit ticket answer: 0.82, more efficient than Membrane X.
The third membrane's slope is 82 ÷ 100 = 0.82, which is greater than Membrane X's 0.75, so the third membrane is more water-efficient.
Differentiation
Support
Provide Membrane X's graph already plotted and its slope already labeled, so students only need to plot Membrane Y's points and calculate its slope for comparison. Offer a printed grid with axes pre-scaled and labeled.
Extension
Ask students to find the feed water needed for a third, hypothetical membrane with a recovery rate of 0.90 to produce the same 300 gallons of permeate used in independent practice, and to explain in one or two sentences why a higher recovery rate means less feed water is wasted as reject.
ELL supports
Pre-teach "feed," "permeate," and "recovery rate" with a simple diagram of a membrane showing an arrow labeled "feed water" entering and two arrows labeled "permeate" and "reject" leaving. Provide a sentence frame for independent practice: "Membrane ___ is more efficient because its slope of ___ is greater than ___."