Worksheet Grade 11 Math
Fernwall Digester Startup: A Biogas Average Rate-of-Change Worksheet
A grade 11 worksheet computing average rate of change from a fictional digester's biogas startup table and interpreting how a maturing digester levels off.
Objectives
- Students will calculate the average rate of change of biogas production over stated intervals from a data table.
- Students will interpret a shrinking rate of change as a digester approaching steady operation.
Vocabulary
- Anaerobic digester
- An anaerobic digester is a sealed tank where bacteria break down sewage sludge without any oxygen, shrinking its volume and producing methane gas as a byproduct. Some treatment plants capture that gas and burn it to generate their own electricity or heat.
Also: biogas, average rate of change, steady state
Teacher overview
An anaerobic digester is a sealed tank where bacteria break down sludge without oxygen and give off biogas, which a treatment plant burns for heat and electricity. When a digester is first started, its bacteria population grows and biogas output climbs, then levels off as the tank reaches steady operation. This worksheet uses a fictional startup record from the Fernwall Water Reclamation Facility so students can practice average rate of change on a real-shaped biological trend. A common misconception is that a shrinking rate of change means production itself is falling; the point to reinforce is that a positive but shrinking rate of change still means production is going up, just more slowly.
Student Worksheet
The Fernwall Water Reclamation Facility (a fictional plant used for this worksheet) started a new anaerobic digester and recorded its biogas production every two weeks. The table shows production in cubic feet per day.
| Week | Biogas production (cubic feet per day) |
|---|---|
| 0 | 500 |
| 2 | 2,300 |
| 4 | 5,600 |
| 6 | 9,200 |
| 8 | 11,000 |
| 10 | 11,600 |
- Find the average rate of change of biogas production from week 0 to week 4. State the units.
- Find the average rate of change from week 4 to week 8.
- Find the average rate of change from week 8 to week 10.
- Using your answers to items 1 through 3, state over which interval production grew fastest.
- Find the average rate of change across the whole startup, from week 0 to week 10.
- Explain, in one or two sentences, what the small rate of change after week 8 tells you about the digester approaching steady operation.
- Biogas from this digester is about 60 percent methane. How many cubic feet of methane per day does the week-10 biogas contain?
- Between week 2 and week 6, production rose from 2,300 to 9,200 cubic feet per day. Find the average rate of change over that interval.
- A technician assumes production will keep rising by 1,350 cubic feet per day each week after week 10. Using the week-10 value, estimate production at week 12 under that assumption, then state whether the assumption fits the earlier trend.
- Based on the numbers, would a graph of biogas versus week look like a straight line or a curve that rises and then flattens? Explain your choice.
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: 1,275 cubic feet per day per week. (5,600 - 500) divided by (4 - 0) = 5,100 / 4 = 1,275 per week.
- Answer: 1,350 cubic feet per day per week. (11,000 - 5,600) divided by (8 - 4) = 5,400 / 4 = 1,350 per week.
- Answer: 300 cubic feet per day per week. (11,600 - 11,000) divided by (10 - 8) = 600 / 2 = 300 per week.
- Answer: Weeks 4 to 8. Its rate of change, 1,350 per week, is larger than 1,275 per week (weeks 0 to 4) and 300 per week (weeks 8 to 10).
- Answer: 1,110 cubic feet per day per week. (11,600 - 500) divided by (10 - 0) = 11,100 / 10 = 1,110 per week.
- Answer: The rate of change has dropped to 300 per week, so production is still rising but only slightly. That small, positive rate signals the digester is nearing steady operation, where output holds roughly constant.
- Answer: 6,960 cubic feet per day. 0.60 x 11,600 = 6,960 cubic feet of methane per day.
- Answer: 1,725 cubic feet per day per week. (9,200 - 2,300) divided by (6 - 2) = 6,900 / 4 = 1,725 per week.
- Answer: 14,300 cubic feet per day. 11,600 + 2 x 1,350 = 14,300. The assumption does not fit the trend, because the rate of change was already falling toward 300 per week by week 10, so a steady 1,350 per week is unlikely to continue.
- Answer: A curve that rises and then flattens. Production climbs quickly early on and then adds less each interval, so the graph increases while bending toward a level line rather than staying straight.
Differentiation
Support
Give students a partially completed table with the first rate of change (weeks 0 to 4) already worked out as a model, and provide the average-rate-of-change formula with the interval endpoints labeled. Allow calculators and accept answers stated without units for the first two items before requiring units.
Extension
Ask students to sketch the six data points, draw a smooth curve, and estimate the week at which the rate of change first drops below 500 cubic feet per day per week, justifying the estimate from the spacing of the points.
ELL supports
Pre-teach "digester," "biogas," and "steady operation" with a simple labeled diagram of a sealed tank releasing gas. Provide a sentence frame for items 6 and 10: "The rate of change is ___, so production is ___."