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Lesson plan Grade 12 Math

Shalefoot Coast RO Membranes: Linearizing a Flux Decline

A 60-minute grade 12 lesson where students take logarithms of membrane flux readings, fit a straight line to the transformed data, and predict a cleaning week.

Objectives

  1. Students will transform exponential data with natural logarithms and fit a line to the result.
  2. Students will construct an exponential model from the slope and intercept of that fitted line.
  3. Students will solve the model for the week a stated cleaning trigger is reached.

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: flux, natural logarithm, decay factor, residual

Teacher overview

Reverse osmosis membranes lose throughput as material collects on their surface, and operators track that loss as flux, the gallons a square foot of membrane produces per day. The decline is close to exponential, so plotting the readings themselves gives a curve that is hard to extend, while plotting the natural logarithm of the readings gives a line that a student can fit by hand. This lesson works with five weeks of flux records from the fictional Shalefoot Coast Water Authority and asks when the plant will hit its cleaning trigger. The misconception worth naming in advance is the assumption that a constant drop appears in the data itself: students look at readings falling by roughly 3, then 2.7, then 2.4 and try to fit a line to the raw numbers, when the constant difference lives in the logarithms rather than in the flux values.

Procedure

Launch (8 min): Project the five fictional Shalefoot readings and ask students to predict the flux at week 26 using any method they like, working alone for three minutes. Collect two or three predictions and the reasoning behind each. Most classes produce a linear extrapolation, which sets up the rest of the lesson.

Direct instruction (14 min): Show the two scatter plots side by side, flux against week and the natural logarithm of flux against week, using the fictional Shalefoot data. Name what the second plot shows: equal differences in the logarithm over equal time intervals, which is the signature of a constant multiplying factor. Derive the general move from F = ae^(kt) to ln(F) = ln(a) + kt, and identify the slope as k and the vertical intercept as ln(a).

Guided practice (13 min): Using a value that does not appear on the handout, work one prediction together. Ask the class for the flux the model gives at week 8, which is 18.0e^(-0.0405 times 8), or about 13.0 gallons per square foot per day. Then ask students to state, in one sentence, what the number 0.960 means as a weekly decay factor. Take three or four responses and correct any that describe it as a 96 percent loss rather than a 96 percent survival.

Independent practice (20 min): Students complete questions 1 through 10 on the handout, which uses the same fictional Shalefoot record. Watch for sign errors when students divide by a negative slope in the cleaning-week item, and for students who report the residual as predicted minus observed.

Closure and formative check (5 min): On an exit slip, students answer one question: why does taking the logarithm of the flux readings make a straight line appear? Collect the slips and sort them into responses that mention a constant multiplying factor and responses that do not. If fewer than half mention it, reteach the transformation before assigning further work with exponential models.

Shalefoot Coast RO Membranes: Linearizing a Flux Decline — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Shalefoot Coast Water Authority records the flux of one reverse osmosis membrane train every five weeks. Flux is reported in gallons per square foot per day, written gfd. The record since the last cleaning is below.

Week, t Flux, F (gfd)
0 18.0
5 14.7
10 12.0
15 9.6
20 8.0
  1. Compute the natural logarithm of the week 10 flux reading. Round to three decimal places.
  2. Compute the natural logarithm of the week 15 flux reading. Round to three decimal places.
  3. The transformed points at week 0 and week 20 are (0, 2.890) and (20, 2.079). Find the slope of the line through them, in units of per week. Round to four decimal places.
  4. Using the slope from question 3, compute the weekly decay factor e raised to that slope. Round to three decimal places.
  5. The fitted model is F(t) = 18.0e^(-0.0405t). Use it to predict the flux at week 26. Round to one decimal place.
  6. The fictional plant cleans a train when flux falls to 60 percent of the flux measured right after the previous cleaning. What flux triggers a cleaning here? Round to one decimal place.
  7. Solve 18.0e^(-0.0405t) for the value of t at which flux equals the trigger you found in question 6. Round to one decimal place.
  8. The fictional plant schedules cleanings on whole weeks and will not let a train run below the trigger. What is the last whole week the train may run before cleaning?
  9. The model predicts a flux of 9.8 gfd at week 15, and the recorded value was 9.6 gfd. Find the residual, using observed minus predicted.
  10. Explain in two or three sentences why a straight line in the plot of ln(F) against t is evidence that the flux decline is exponential rather than linear.

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: 2.485.
    The week 10 reading is 12.0 gfd, and ln(12.0) is about 2.4849, which rounds to 2.485.

  2. Answer: 2.262.
    The week 15 reading is 9.6 gfd, and ln(9.6) is about 2.2618, which rounds to 2.262.

  3. Answer: -0.0405 per week.
    The slope is (2.079 - 2.890) divided by (20 - 0), which is -0.811/20, or about -0.04055, rounding to -0.0405 per week. Accept -0.0406 from students who carry more decimal places in the logarithms.

  4. Answer: 0.960.
    The decay factor is e^(-0.0405), which is about 0.96031 and rounds to 0.960. The train keeps about 96 percent of its flux from one week to the next.

  5. Answer: 6.3 gfd.
    F(26) = 18.0e^(-0.0405 times 26) = 18.0e^(-1.053), and e^(-1.053) is about 0.3488, so the product is about 6.28, which rounds to 6.3 gfd.

  6. Answer: 10.8 gfd.
    Sixty percent of the 18.0 gfd measured right after the last cleaning is 0.60 times 18.0, or 10.8 gfd.

  7. Answer: 12.6 weeks.
    Setting 18.0e^(-0.0405t) = 10.8 gives e^(-0.0405t) = 0.600. Taking natural logs gives -0.0405t = ln(0.600) = -0.5108, so t = 12.61, which rounds to 12.6 weeks.

  8. Answer: Week 12.
    The model gives 11.1 gfd at week 12 and 10.6 gfd at week 13, so week 13 already sits below the 10.8 gfd trigger and week 12 is the last full week the train may run.

  9. Answer: -0.2 gfd.
    The residual is 9.6 minus 9.8, or -0.2 gfd. The negative sign means the recorded flux came in below what the fitted model predicted for that week.

  10. Answer: A straight line in the transformed plot means equal changes in ln(F) over equal time intervals.
    Equal differences in the logarithm correspond to a constant multiplying factor in the flux itself, which is what defines exponential decay. A linear decline would instead show equal subtractions in the flux values and would bend downward once transformed by a logarithm.

Differentiation

Support

Give these students the natural logarithm column already computed for all five fictional Shalefoot readings, so their work starts at the graph rather than at the calculator. Provide a worked reference showing the move from b = e^k to k = ln(b) on a neutral example, and let them answer questions 5 through 8 with a graphing calculator's table feature instead of by hand.

Extension

Have students fit the line to all five transformed points with a least-squares routine rather than using only the two endpoints, then report how much the predicted cleaning week from question 7 shifts. Ask them to explain why the week 15 residual pulls the least-squares slope slightly steeper than the endpoint slope in this fictional record.

ELL supports

Pre-teach "flux," "decay factor," "residual," and "trigger" with a labeled sketch of a membrane surface and a falling curve marked at the cleaning level. Provide sentence frames for questions 9 and 10, such as "The recorded value was ___ than the model predicted because ___," and allow a labeled graph in place of prose for question 10.

Extensions

Students can repeat the transformation on a second fictional train whose readings fall from 18.0 gfd to 11.0 gfd over the same 20 weeks, then compare the two decay factors and argue which train needs the shorter cleaning interval. A further session can introduce the cost side by giving a fictional cleaning cost and asking students to weigh more frequent cleanings against the production lost while a train is offline.