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Lesson plan Grade 12 Math

Quillsdale Canal Allotment: Choosing Acreage by Corner Points

A 55-minute grade 12 lesson where students model an irrigation allotment as a system of inequalities and test corner points to maximize a grower's net return.

Objectives

  1. Students will represent an irrigation water allotment and acreage limits as a system of linear inequalities.
  2. Students will solve boundary equations to locate the corner points of a feasible region.
  3. Students will evaluate an objective function at each corner point and interpret which constraint limits the plan.

Vocabulary

Acre-foot
The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.

Also: allotment, feasible region, corner point, objective function

Teacher overview

Irrigation districts deliver water by allotment, so a grower's planting decision is a constrained optimization rather than an open choice. This lesson uses the fictional Quillsdale Irrigation District, which delivers a fixed number of acre-feet down the Nornby Canal to a grower who must divide 400 acres between two crops with different water requirements and different returns. Students write the constraints as inequalities, locate the corner points by solving pairs of boundary equations, and evaluate net return at each. The misconception worth confronting is the assumption that the best plan uses every last acre-foot of the allotment: the corner that maximizes return here leaves 75 acre-feet undelivered, because land and a contract cap bind before water does.

Procedure

Launch (7 min): State the fictional Quillsdale situation with no equations on the board: 400 acres available, an allotment of 1,100 acre-feet, alfalfa needing 3.5 acre-feet per acre and processing tomatoes needing 2.0, and a processing contract that will not take more than 250 acres of tomatoes. Ask students to propose one planting plan they think is best and to say in one sentence why. Record two proposals for revisiting at closure.

Direct instruction (13 min): Translate each restriction into an inequality with the class, defining x as acres of alfalfa and y as acres of tomatoes. Graph the three boundary lines and shade the feasible region on the projector. Name the corner points as the only candidates a linear objective can maximize, and explain why an interior point can always be improved by moving toward a boundary.

Guided practice (12 min): Work the two corners that the handout does not ask about, so students see the full comparison. At the corner where no alfalfa is planted, the fictional grower plants 250 acres of tomatoes for a net return of $237,500. At the corner where no tomatoes are planted, the water line limits alfalfa to about 314 acres for a net return of about $220,000. Ask the class to predict, before computing, whether either of these can beat a plan that uses the full 400 acres.

Independent practice (18 min): Students complete questions 1 through 10 on the handout, which uses the same fictional Quillsdale figures. Watch for students who set the water constraint as an equation rather than an inequality, and for students who solve the boundary pair correctly but then report the point in the wrong order.

Closure and formative check (5 min): Return to the two launch proposals and ask each student to write on an exit slip which constraint is binding at the best plan and which is slack. Collect the slips and count how many name water as binding. If more than a third do, reopen the next session with the acre-foot accounting from question 10 before moving on.

Quillsdale Canal Allotment: Choosing Acreage by Corner Points — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Quillsdale Irrigation District delivers 1,100 acre-feet of water each season to one grower on the Nornby Canal. The grower farms 400 acres and will plant two crops. Alfalfa needs 3.5 acre-feet of water per acre and returns $700 per acre. Processing tomatoes need 2.0 acre-feet per acre and return $950 per acre, but the cannery contract will accept tomatoes from no more than 250 acres. Let x be the acres of alfalfa and y be the acres of tomatoes.

  1. Write the inequality that expresses the water allotment limit.
  2. Write the inequality that expresses the planted acreage limit.
  3. Write the objective function P for the grower's total net return in dollars.
  4. The land boundary x + y = 400 and the water boundary 3.5x + 2y = 1100 cross at one point. Solve the pair and report the acres of alfalfa at that point.
  5. At the same crossing point from question 4, report the acres of tomatoes.
  6. The land boundary x + y = 400 and the contract boundary y = 250 cross at a second corner. Report the acres of alfalfa at that corner.
  7. Evaluate the net return at the corner from questions 4 and 5.
  8. Evaluate the net return at the corner from question 6, where tomatoes cover 250 acres.
  9. Which of those two corners gives the larger net return for the fictional grower?
  10. At the corner you chose in question 9, how many acre-feet of the 1,100 acre-foot allotment go unused?

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: 3.5x + 2y is at most 1,100.
    Alfalfa draws 3.5 acre-feet per acre and tomatoes draw 2.0, so total water use is 3.5x + 2y, and the season allotment caps it at 1,100 acre-feet.

  2. Answer: x + y is at most 400.
    The two crops share 400 acres, so their acreages sum to no more than 400. Accept x + y less than or equal to 400 written with the inequality symbol.

  3. Answer: P = 700x + 950y.
    Each acre of alfalfa returns $700 and each acre of tomatoes returns $950, so total net return in dollars is 700x + 950y.

  4. Answer: 200 acres.
    Substituting x = 400 - y into 3.5x + 2y = 1100 gives 1400 - 1.5y = 1100, so y = 200 and x = 400 - 200 = 200 acres of alfalfa.

  5. Answer: 200 acres.
    From the same substitution, y = 200 acres of tomatoes. Water use there is 3.5(200) + 2(200) = 1,100 acre-feet, exactly the allotment.

  6. Answer: 150 acres.
    With y = 250 in x + y = 400, the alfalfa acreage is x = 400 - 250 = 150 acres. Water use is 3.5(150) + 2(250) = 1,025 acre-feet, which is within the allotment.

  7. Answer: $330,000.
    P = 700(200) + 950(200) = 140,000 + 190,000 = $330,000.

  8. Answer: $342,500.
    P = 700(150) + 950(250) = 105,000 + 237,500 = $342,500.

  9. Answer: The corner with 150 acres of alfalfa and 250 acres of tomatoes.
    Its return of $342,500 beats the $330,000 from the equal-split corner, and it also beats the $237,500 and roughly $220,000 returns at the two corners worked during the lesson.

  10. Answer: 75 acre-feet.
    That plan uses 3.5(150) + 2(250) = 525 + 500 = 1,025 acre-feet, and 1,100 - 1,025 leaves 75 acre-feet of the allotment undelivered.

Differentiation

Support

Provide these students with the feasible region already graphed and its five corners, including the origin, already marked, so their work is solving one boundary pair and evaluating the objective rather than constructing the whole picture. Supply a filled-in units table for the fictional Quillsdale scenario showing acre-feet per acre beside dollars per acre, which prevents the common mix-up of multiplying a water rate by a dollar figure.

Extension

Ask students to find how far the tomato return per acre would have to fall before the equal-split corner becomes the better plan, which means solving 700(150) + 250c = 700(200) + 200c for c (the equal-split return also depends on c) and interpreting the result. Then have them explain what happens to the best plan if the fictional district cuts the allotment to 900 acre-feet, and identify which constraint becomes binding.

ELL supports

Pre-teach "allotment," "acre-foot," "feasible," and "binding constraint" with a labeled diagram showing a rectangle of land beside a tank of water, each with its own limit written on it. Provide sentence frames for questions 9 and 10, such as "The better corner is ___ because its return is ___," and allow a labeled graph in place of prose wherever a written justification is requested.

Extensions

Students can add a third crop to the fictional Quillsdale problem, which pushes the model into three variables and makes a graphical solution impractical, then use a spreadsheet solver and report how the answer changes. A second follow-up asks students to research how irrigation districts in their own state publish allotments, and to compare the units used there with the acre-feet in this scenario.