Worksheet Grade 9 Math
Saltmarsh Bend's RO Membrane: Reading a Permeate-Flow Decline Table
A grade 9 worksheet where students read a reverse-osmosis membrane's permeate-flow data table, finding key features and average rate of change during fouling.
Objectives
- Students will identify the intercept, the decreasing interval, and the average rate of change from a table of permeate-flow readings.
- Students will use a data table's average rate of change to predict when a quantity will cross a stated minimum value.
Vocabulary
Key words: permeate, fouling, intercept, average rate of change
Teacher overview
At the fictional Saltmarsh Bend desalination plant, high-pressure pumps push water through a reverse-osmosis membrane, a thin barrier that holds back salt, producing clean "permeate" water on the other side. Over weeks of operation, tiny particles build up on the membrane's surface in a process called fouling, which gradually reduces how much permeate flow the membrane can produce until the plant runs a cleaning procedure to restore it. This worksheet has students read a table of one membrane's weekly flow readings during a single fouling cycle and answer questions about the pattern the same way an operator monitoring the plant's dashboard would.
A common misconception is assuming that any declining number must be an exponential decay, when in this membrane's data the flow actually drops by the same amount each week, a linear pattern. Students should look at the differences between consecutive readings, not just the fact that the numbers are getting smaller, before deciding what kind of pattern they are looking at.
Student Worksheet
The table below shows the fictional Saltmarsh Bend plant's permeate flow, in gallons per minute (gpm), for one membrane module during the seven weeks following its most recent cleaning.
| Week | Permeate flow (gpm) |
|---|---|
| 0 | 500 |
| 1 | 485 |
| 2 | 470 |
| 3 | 455 |
| 4 | 440 |
| 5 | 425 |
| 6 | 410 |
The plant's minimum acceptable permeate flow before a cleaning-in-place (CIP) procedure is required is 420 gpm.
What is the permeate flow at Week 0? In the context of this table, what does the Week 0 value represent?
Over which weeks is the permeate flow decreasing? Describe the pattern between each pair of consecutive weeks in one sentence.
Calculate the average rate of change in permeate flow from Week 0 to Week 6, in gallons per minute per week.
Using the table, in which week does the permeate flow first drop below the plant's minimum acceptable flow of 420 gpm?
In words, describe what a graph of permeate flow versus week would look like from Week 0 through Week 6. Include whether it rises, falls, or stays flat, and whether it curves or forms a straight line.
After Week 6, the plant performs a cleaning-in-place procedure and the flow rises back up before a new decline cycle begins. Explain why it makes sense to treat "Week 0 through Week 6" as the domain of one complete function describing this fouling cycle, rather than extending the same equation past Week 6.
Compare the flow at Week 3 to the flow at Week 0. What percent of the original Week 0 flow remains at Week 3? Round to the nearest whole percent.
Suppose the plant schedules its cleaning-in-place procedure at Week 5 instead of waiting until Week 6. How many gpm above the 420 gpm minimum does the plant still have in reserve at Week 5? Compare this to how far below the minimum the plant would already be if it waited until Week 6.
A second membrane module also starts at 500 gpm in Week 0, but it fouls more slowly, declining by only 10 gpm per week instead of 15. In which week would this second membrane's flow first reach 420 gpm?
Explain how knowing the average rate of change from problem 3 could help plant staff schedule a cleaning-in-place procedure ahead of time, instead of waiting until the flow has already dropped below the minimum.
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: 500 gpm.
The Week 0 value is the permeate flow measured right after the membrane's most recent cleaning, before any fouling has reduced it. This is the table's intercept, the starting value of the pattern.Answer: decreasing every week, Week 0 through Week 6.
Each week's flow is 15 gpm lower than the week before (500, 485, 470, 455, 440, 425, 410), so the flow decreases steadily across the entire table.Answer: -15 gpm per week.
(410 gpm − 500 gpm) ÷ (6 weeks − 0 weeks) = -90 ÷ 6 = -15 gpm per week.Answer: Week 6.
The table shows 425 gpm at Week 5, still above the 420 gpm minimum, and 410 gpm at Week 6, which is below it. Week 6 is the first week the flow drops below the minimum.Answer: a straight line sloping downward.
The graph would start at 500 gpm on the vertical axis at Week 0 and fall in a straight line to 410 gpm at Week 6, because the flow decreases by the same 15 gpm every week rather than curving.Answer: each fouling cycle is its own separate pattern.
Once the plant cleans the membrane, the flow resets to a new starting value and a new decline begins, so the straight-line pattern found for Weeks 0 through 6 does not apply to the weeks after cleaning. Restricting the domain to Week 0 through Week 6 keeps the model accurate to the one cycle it actually describes.Answer: 91 percent.
455 gpm ÷ 500 gpm = 0.91, or 91 percent of the original flow remains at Week 3.Answer: 5 gpm in reserve at Week 5, versus 10 gpm below the minimum at Week 6.
At Week 5, the flow is 425 gpm, which is 5 gpm above the 420 gpm minimum. At Week 6, the flow is 410 gpm, which is 10 gpm below the minimum. Scheduling the cleaning at Week 5 keeps the plant above its minimum the whole time, a 15 gpm difference in outcome compared with waiting until Week 6.Answer: Week 8.
500 gpm − 10 gpm per week × 8 weeks = 500 − 80 = 420 gpm, so the second membrane first reaches the 420 gpm minimum in Week 8.Answer: answers vary; must reference proactive scheduling.
A strong answer explains that once staff know the flow drops by a predictable 15 gpm each week, they can calculate ahead of time which future week the flow will first fall below the minimum and schedule the cleaning-in-place procedure before that week arrives, rather than discovering the problem only after flow has already dropped too low.
Differentiation
Support (below grade level)
Provide a printed grid with the axes already labeled and scaled for this fictional plant's data (Week 0-6 on the horizontal axis, 400-500 gpm on the vertical axis) so students can plot the table's points directly instead of estimating a scale themselves. Walk through problem 3's rate-of-change calculation as a class example before releasing problems 4 through 10.
Extension (above grade level)
Ask students to write the equation of the line describing this membrane's flow as a function of week number (using their answer from problem 3 as the slope and the Week 0 value as the starting point), then use that equation to predict the flow at Week 4.5, a value not shown directly in the table.
ELL supports
Pair the vocabulary terms "permeate," "fouling," and "intercept" with a simple labeled diagram of a membrane with clean water passing through one side and particles building up on the other. Allow students to answer problems 2, 5, and 6 with a hand-drawn sketch plus short phrases instead of full sentences.