Classroom activity Grade 10 Math
Wistmere Drought Board: A Four-Station Constraint Rotation
A 55-minute grade 10 activity where teams collect one constraint per station, build a feasible region for a fictional town, and test four proposed plans.
Objectives
- Students will translate four stated limits into inequalities in two variables.
- Students will graph a feasible region and classify proposed allocations as viable or non-viable.
- Students will solve a system of two linear equations to locate a corner of the feasible region.
Vocabulary
- Recycled water
- Wastewater that has been cleaned and used again, often to water parks, golf courses, and crops instead of being thrown away. Purple pipes usually carry it.
Also: constraint, feasible region, viable solution, allocation
Teacher overview
This is a four-station rotation for teams of three or four students, built around the fictional town of Wistmere and a summer water shortage. Each station holds one card stating a single limit the town faces, and a team leaves the station with that limit written as an inequality, so the system is assembled piece by piece rather than handed over complete. The activity maps onto the mathematics directly, because a team that has visited only two stations can still shade a region and will find that it wrongly admits plans the remaining stations rule out. The rotation ends with four proposed allocations, each of which fails or passes for a different reason.
Activity Directions
Setup (5 min): Print the four station cards and place one at each corner of the room. Card 1 reads: the town must supply at least 900 acre-feet this summer. Card 2 reads: the state permit allows no more than 700 acre-feet from the reservoir. Card 3 reads: the recycled water plant can deliver no more than 400 acre-feet. Card 4 reads: reservoir water costs 300 dollars per acre-foot, recycled water costs 800 dollars per acre-foot, and the summer budget is 460,000 dollars. Assign teams of three or four and give each team graph paper and one handout. Announce the two variables: x is acre-feet taken from the reservoir and y is acre-feet from the recycled water plant.
Station rotation (32 min, 8 min per station): Teams start at different stations and rotate on your signal. At each station a team writes the card's limit as an inequality on the handout, then graphs its boundary line on the shared set of axes with x from 0 to 1,000 and y from 0 to 600. Teams shade the side of each boundary that satisfies the limit. Require every team to label each boundary line with the station number so the four constraints stay distinguishable on one graph.
Testing proposals (10 min): Once all four constraints are on the graph, teams complete handout items 5 through 14, which test four proposed allocations from the fictional town board and locate one corner of the feasible region by solving a system. Teams should mark each proposal as a point on their graph before checking it algebraically, and then confirm that the point's position matches the verdict their arithmetic gives.
Debrief (8 min): Ask each team to name which single constraint rules out Plan B and which rules out Plan C, and confirm that the two answers differ. Then ask the closing question: the cheapest viable plan leans as hard as the permit allows on reservoir water, so what would have to change before recycled water became the cheaper choice? Take two or three answers and connect them to the cost figures on Card 4.
Student Handout
The town of Wistmere is fictional, and every limit and price below was set for this activity. Let x be the acre-feet supplied from the reservoir this summer, and let y be the acre-feet supplied from the recycled water plant.
Station 1 states that the town must supply at least 900 acre-feet. Write that limit as an inequality in x and y.
Station 2 states that no more than 700 acre-feet may come from the reservoir. Write that limit as an inequality.
Station 3 states that the recycled water plant can deliver no more than 400 acre-feet. Write that limit as an inequality.
Station 4 gives prices of 300 dollars per acre-foot for reservoir water and 800 dollars per acre-foot for recycled water, with a budget of 460,000 dollars. Write that limit as an inequality.
Plan A takes 600 acre-feet from the reservoir and 300 acre-feet from the recycled plant. Find its cost in dollars.
Is Plan A viable under all four constraints? Answer yes or no.
Plan B takes 750 acre-feet from the reservoir and 150 acre-feet from the recycled plant. Which single constraint does it violate?
Plan C takes 450 acre-feet from the reservoir and 400 acre-feet from the recycled plant. Which single constraint does it violate?
Plan D takes 500 acre-feet from the reservoir and 400 acre-feet from the recycled plant. Find its cost in dollars.
Is Plan D viable under all four constraints? Answer yes or no.
Solve the system x + y = 900 and 300x + 800y = 460,000 for y.
Find the value of x that goes with your item 11 answer.
Among plans that supply exactly 900 acre-feet and satisfy all four constraints, find the least possible cost in dollars.
How many acre-feet come from the reservoir in that least-cost plan?
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: x + y ≥ 900. The two sources together must reach the fictional town's stated summer need of 900 acre-feet.
Answer: x ≤ 700. The permit caps reservoir withdrawals, and it says nothing about the recycled plant.
Answer: y ≤ 400. The recycled water plant's capacity caps y alone.
Answer: 300x + 800y ≤ 460,000. Each acre-foot from the reservoir costs 300 dollars and each from the recycled plant costs 800 dollars, and the total must stay within the fictional budget.
Answer: $420,000. The cost is 300(600) + 800(300) = 180,000 + 240,000 = 420,000 dollars.
Answer: Yes. Plan A supplies 600 + 300 = 900 acre-feet, keeps x at 600 and y at 300 within both caps, and costs 420,000 dollars, which is under the 460,000 dollar budget.
Answer: The reservoir permit. Plan B sets x at 750 acre-feet, above the 700 acre-foot cap in item 2. It meets the demand at 900 acre-feet, stays under the recycled cap, and costs 345,000 dollars, so no other constraint fails.
Answer: The demand constraint. Plan C supplies 450 + 400 = 850 acre-feet, short of the 900 acre-feet required in item 1. Its cost of 455,000 dollars is within budget and both caps hold.
Answer: $470,000. The cost is 300(500) + 800(400) = 150,000 + 320,000 = 470,000 dollars.
Answer: No. Plan D meets the demand and both caps, yet its 470,000 dollar cost exceeds the 460,000 dollar budget from item 4.
Answer: 380 acre-feet. Substituting x = 900 − y into the budget equation gives 300(900 − y) + 800y = 460,000, so 270,000 + 500y = 460,000, then 500y = 190,000 and y = 380 acre-feet.
Answer: 520 acre-feet. Since x + y = 900 and y = 380, x = 520 acre-feet. Checking, 300(520) + 800(380) = 156,000 + 304,000 = 460,000 dollars.
Answer: $370,000. With x + y = 900, the cost becomes 300x + 800(900 − x) = 720,000 − 500x, which falls as x rises, so the cheapest allowed plan pushes x to the permit cap of 700, giving 300(700) + 800(200) = 370,000 dollars.
Answer: 700 acre-feet. The least-cost plan takes the full permitted 700 acre-feet from the reservoir and the remaining 200 acre-feet from the recycled plant.
Differentiation
Support
Give teams a graph with the four boundary lines for the fictional town already drawn and labeled by station number, so the station work is limited to writing each inequality and shading the correct side. Provide items 5 and 9 as partly completed cost calculations with one product filled in.
Extension
Ask teams to add a fifth constraint of their own that would make Plan A non-viable while leaving the least-cost plan from item 13 untouched, then justify it as a policy the fictional board might adopt. A second challenge is to find the recycled water price at which the cheapest plan would shift away from the permit cap on reservoir water.
ELL supports
Provide each station card in a two-column layout, with the sentence on the left and a matching symbolic frame on the right, such as "x ___ 700," so the translation step is a choice among symbols rather than a blank page. Offer sentence frames for items 7 and 8: "This plan fails the ___ constraint because ___ is ___ than ___."
Extensions
Have teams rebuild the fictional Wistmere system with a third source added, such as a purchase from a neighboring district at a stated price per acre-foot, and describe in writing why a third variable makes the graph on two axes no longer sufficient. A short written comparison of the four-station system with a three-variable version makes a useful bridge to a later lesson on matrices.