Classroom activity Grade 11 Math
Eastmere Bay Nutrient Trading Floor: Cutting a Load at Least Cost
A 60-minute grade 11 activity where three teams trade phosphorus reduction projects across a watershed cap and then find the least expensive package that meets it.
Objectives
- Students will compute a cost per unit removed and rank options by that derived rate.
- Students will express a load cap as an inequality and test packages against it.
- Students will find the least expensive package that satisfies a stated constraint.
Vocabulary
Key words: nutrient load, phosphorus, load cap, cost per pound, constraint
Teacher overview
When a watershed has to cut its phosphorus load, the cheapest pounds and the largest pounds rarely sit with the same party, so the region pays less when a sewage plant funds a farmer's cover crops than when every party fixes only its own discharge. This activity turns that idea into a trading floor for the fictional Eastmere Bay, where three teams hold six projects between them and a fourth group acts as the fictional Kestrelby County cap authority. The activity maps onto the mathematics because a trade is only accepted when the buying team can show the cap authority a package that clears the load cap, which makes the inequality the rule of play rather than a step in a worksheet. Teams that buy their own cheapest projects first usually find in the audit that a different package would have cost less.
Activity Directions
Setup (8 min): Print the six fictional project cards and deal them by owner: two to the Oxholt Water Reclamation Facility team, two to the Merrowby Farm Co-op team, and two to the Stanbury Township team. Appoint three to five students as the fictional Kestrelby County cap authority. Post the region's current load of 12,000 pounds of phosphorus per year and its cap of 8,400 pounds per year on the board. Announce that the region pays for whatever is built, so the goal is one package that clears the cap for the lowest total cost.
Cost-per-pound round (10 min): Each team computes the cost per pound removed for both of its own projects and writes the two figures on the board beside the project letters. No trading yet. The cap authority checks each posted figure and rejects any that is not the annual cost divided by the annual pounds removed.
Trading round 1 (10 min): Teams may propose a package of any projects, whoever owns them. A proposal is written on the board as a list of letters, a total of pounds removed, and a total annual cost. The cap authority marks each proposal viable or not viable by testing the pounds against the cap. Do not correct the cost arithmetic yet.
Trading round 2 (10 min): Teams may only propose packages that cost less than the cheapest viable proposal currently posted. Any team may challenge a posted cost, and the cap authority rules on the challenge. Erase proposals that are not viable so the board holds only the live options.
Trading round 3 (10 min): Announce that no new projects may be invented and that the round ends when three minutes pass with no cheaper viable proposal posted. Circle the cheapest viable package on the board.
Package audit (6 min): Individually, students complete handout questions 1 through 12 using the project table, working from the printed data rather than the board.
Debrief (6 min): Ask the class handout question 12 aloud, then ask the cap authority to state which proposals failed and why. Close by asking what would change if each team paid only for its own projects.
Student Handout
The fictional Eastmere Bay watershed now sends 12,000 pounds of phosphorus per year to the bay. The fictional Kestrelby County cap allows 8,400 pounds per year. Six projects are available, and each is all or nothing.
| Project | Owner | Pounds removed per year | Annual cost |
|---|---|---|---|
| A | Oxholt plant, chemical phosphorus removal | 2,000 | $240,000 |
| B | Oxholt plant, effluent filters | 800 | $160,000 |
| C | Merrowby co-op, cover crops | 1,200 | $54,000 |
| D | Merrowby co-op, streamside buffers | 600 | $42,000 |
| E | Stanbury township, street sweeping | 400 | $60,000 |
| F | Stanbury township, bioretention basins | 900 | $99,000 |
- Find the pounds of phosphorus per year the watershed must remove to meet the cap.
- Find the cost per pound removed for project A.
- Find the cost per pound removed for project C.
- Find the cost per pound removed for project F.
- List all six projects in order from lowest to highest cost per pound.
- Find the total pounds removed by the three projects with the lowest cost per pound.
- State whether that package meets the cap.
- Consider packages that leave out project A. Find the total annual cost of the only such package that meets the cap.
- Find the lowest total annual cost of any package that meets the cap.
- Find the pounds removed per year by the package you found in question 9.
- Find that package's average cost per pound removed, to the nearest cent.
- Explain in two or three sentences why the least expensive package is not simply the three projects with the lowest cost per pound.
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: 3,600 lb per year.
Subtracting gives 12,000 - 8,400 = 3,600 pounds of phosphorus per year.Answer: $120 per lb.
Dividing gives 240,000 divided by 2,000 = 120 dollars for each pound removed per year.Answer: $45 per lb.
Dividing gives 54,000 divided by 1,200 = 45 dollars per pound.Answer: $110 per lb.
Dividing gives 99,000 divided by 900 = 110 dollars per pound.Answer: C, D, F, A, E, B.
The rates are 45 for C, 70 for D, 110 for F, 120 for A, 150 for E, and 200 for B, all in dollars per pound removed per year.Answer: 2,700 lb per year.
Adding the three cheapest per pound gives 1,200 + 600 + 900 = 2,700 pounds per year.Answer: No.
The package removes 2,700 pounds per year, which is 900 pounds short of the 3,600 the cap requires.Answer: $415,000 per year.
Without project A the five remaining projects remove 800 + 1,200 + 600 + 400 + 900 = 3,900 pounds per year. Dropping any one of them removes at least 400 more pounds from that total and leaves 3,500 or fewer, below the required 3,600, so all five must be built. Their costs total 160,000 + 54,000 + 42,000 + 60,000 + 99,000 = 415,000 dollars per year.Answer: $336,000 per year.
Any package with project A still needs 1,600 more pounds. The cheapest set reaching that is C and D together at 1,800 pounds for 96,000 dollars, since C and E cost 114,000 and every other qualifying set costs more. Adding project A gives 240,000 + 96,000 = 336,000 dollars per year for projects A, C, and D. That beats the 415,000 found in question 8, so it is the overall lowest.Answer: 3,800 lb per year.
Adding gives 2,000 + 1,200 + 600 = 3,800 pounds per year, which clears the 3,600 required.Answer: $88.42 per lb.
Dividing gives 336,000 divided by 3,800 = 88.42 dollars per pound removed per year.Answer: The cheapest rates do not remove enough pounds.
Ranking by cost per pound answers which project is the best value, not which combination clears the cap, and the three best-value projects together fall 900 pounds short. Meeting the cap requires at least one large project, and once the largest one is bought the remaining shortfall is small enough that the two cheapest projects finish the job.
Differentiation
Support
Give these students the project table with the cost-per-pound column already computed for the fictional Eastmere projects, so questions 2 through 5 become a sort rather than six divisions. Provide a checklist of four candidate packages for questions 8 and 9, and ask only that they test each one against the cap and compare totals.
Extension
Ask these students to find, for the fictional Eastmere watershed, the lowest (strictest) cap the county could set that the three cheapest-per-pound projects would satisfy on their own. Then have them redo question 9 under a rule that lets a project be built at any fraction of its size for the same fraction of its cost, and explain why the answer changes.
ELL supports
Pre-teach "load," "cap," and "cost per pound" with a picture of a bay fed by three pipes labeled plant, farm, and street, each with a valve. Provide a bid card template with the headings "projects," "pounds removed," and "annual cost," and sentence frames such as "Our package removes ___ pounds, which is ___ than the cap."
Extensions
Students can research whether their own state runs a nutrient trading program and describe, in words alone, who verifies that a farm practice actually removes what it claims. A second task is to explain why a trading rule usually forbids concentrating every reduction in one part of a watershed, even when doing so costs the least.