Water for Schools
Menu

Lesson plan Grade 9 Math

Thrushmere Well Field Permit: Pumping Limits as Inequalities

A 50-minute grade 9 lesson where students write a system of inequalities for two wells' permit caps and test proposed pumping plans for viability.

Objectives

  1. Students will translate a demand requirement and two permit caps into a system of linear inequalities.
  2. Students will graph the constraint boundaries on axes labeled in million gallons per day.
  3. Students will decide whether a proposed pumping plan is viable under every stated constraint.

Vocabulary

Aquifer
An aquifer is an underground layer of rock, sand, or gravel that holds water in the tiny spaces between its grains. Wells reach down into aquifers to bring water up to the surface.

Also: constraint, viable solution, permit cap, million gallons per day

Teacher overview

The fictional Thrushmere Water District draws all of its supply from two wells, Beacon Hill and Kilnfield, and a state groundwater permit limits how much each well may pump. In this lesson students turn three plain-English requirements into a system of inequalities, graph the boundaries, and then judge whether particular pumping plans satisfy every requirement at once. The misconception this lesson is built to correct is a common one in first-year algebra: students solve the demand condition as the equation x + y = 1.8 and then accept any point on that line, forgetting that a point can satisfy the demand while still violating a well's individual cap. A second, related error is reversing the direction of an inequality when the constraint is a maximum rather than a minimum.

Procedure

Launch (7 min): Tell the class that the fictional Thrushmere district must deliver at least 1.8 million gallons per day and that each of its two wells has a separate legal ceiling. Ask students, without writing any algebra, to propose a pair of pumping rates that would work. Record two or three proposals on the board and hold them for later testing.

Building the constraints (13 min): Define x as Beacon Hill's output and y as Kilnfield's output, both in million gallons per day. Work with the class to write four conditions for this fictional district: x + y is at least 1.8, x is at most 1.2, y is at most 0.8, and both x and y are at least 0. Stress that "at least" produces the greater-than-or-equal symbol while "at most" produces the less-than-or-equal symbol, and have students say each inequality aloud in words before writing it.

Graphing the region (12 min): Model graphing the four boundary lines on axes scaled from 0 to 1.5 on the horizontal axis and 0 to 1.0 on the vertical axis, both labeled in million gallons per day. Shade the region satisfying all four conditions and identify its corner points. Return to the proposals recorded during the launch and plot them, so students can see which ones fall inside the shaded region.

Independent practice (13 min): Students complete questions 1 through 8 on the handout, working with the same fictional district. Circulate and check that students are writing units on every answer, since a bare number such as 0.8 is ambiguous between a rate and a total.

Closure and formative check (5 min): Students answer handout question 9 on an index card. Collect the cards and sort them into two piles before dismissal. If more than a quarter of the class misjudges whether the plan is viable, reopen the individual well caps at the start of the next session rather than moving on.

Thrushmere Well Field Permit: Pumping Limits as Inequalities — Grade 9 Math
Name: ____________________________ Date: ______________

Student Handout

Every number below describes the fictional Thrushmere Water District. Let x stand for Beacon Hill Well's output and y stand for Kilnfield Well's output, both measured in million gallons per day (MGD). Suppose the district must deliver at least 1.8 MGD in total, Beacon Hill may pump at most 1.2 MGD, and Kilnfield may pump at most 0.8 MGD.

  1. Write the inequality stating that the two wells together must supply at least 1.8 MGD.
  2. Beacon Hill is pumping 1.0 MGD. Find the smallest amount Kilnfield must pump so the two wells together reach 1.8 MGD.
  3. Beacon Hill drops to 0.9 MGD. Find the amount Kilnfield would have to pump so the two wells together reach 1.8 MGD.
  4. Kilnfield's permit ceiling is 0.8 MGD. Is the amount you found in question 3 allowed under that ceiling? Answer yes or no.
  5. Find the largest total the two wells can produce in one day when each pumps at its own ceiling.
  6. Suppose demand rises to 2.1 MGD. Using your answer to question 5, state whether any pumping plan can meet that demand. Answer yes or no.
  7. Convert the 1.8 MGD demand to gallons per minute, using 1 MGD equal to 694 gpm.
  8. Suppose Beacon Hill's annual permit allows 350 million gallons in a year. Running at its ceiling of 1.2 MGD, find the number of whole days the well can pump before reaching that annual total.
  9. A shift supervisor proposes running Beacon Hill at 1.1 MGD and Kilnfield at 0.75 MGD. Find the combined output of that plan and decide whether the plan is viable.

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: x + y is greater than or equal to 1.8.
    The phrase "at least 1.8" places the combined output at or above 1.8 MGD, so the inequality points the greater-than-or-equal direction rather than the less-than-or-equal direction.

  2. Answer: 0.8 MGD.
    Solving 1.0 + y greater than or equal to 1.8 gives y greater than or equal to 0.8, so Kilnfield's smallest acceptable output is 0.8 MGD. That value sits exactly on Kilnfield's ceiling, so the plan barely works.

  3. Answer: 0.9 MGD.
    Solving 0.9 + y greater than or equal to 1.8 gives y greater than or equal to 0.9 MGD.

  4. Answer: No.
    Kilnfield's ceiling is 0.8 MGD, and 0.9 MGD is above that ceiling, so the combination in question 3 is not a viable solution even though it satisfies the demand condition.

  5. Answer: 2.0 MGD.
    The largest viable point is the corner where both wells sit at their ceilings, giving 1.2 + 0.8 = 2.0 MGD.

  6. Answer: No.
    The maximum total is 2.0 MGD, which falls short of 2.1 MGD, so no combination of x and y satisfies all four conditions at that demand level.

  7. Answer: 1,249 gpm.
    Multiplying gives 1.8 MGD x 694 gpm per MGD = 1,249.2 gpm, which rounds to about 1,249 gpm.

  8. Answer: 291 days.
    Dividing gives 350 million gallons divided by 1.2 million gallons per day = 291.67 days, and only 291 whole days fit inside the annual permit.

  9. Answer: 1.85 MGD.
    Adding gives 1.1 + 0.75 = 1.85 MGD. Both wells stay under their ceilings and the total clears 1.8 MGD, so this plan is viable.

Differentiation

Support

Give these students a partially completed constraint list for the fictional district, with the demand inequality already written and the two ceiling inequalities left blank. Provide a two-column word bank pairing "at least" with one symbol and "at most" with the other, and let students graph on pre-scaled axes rather than setting up their own scale.

Extension

Ask students to add another constraint to the fictional district: Kilnfield's pump wears faster than Beacon Hill's, so the district wants Kilnfield to carry no more than 40 percent of the total output. Have them write that condition as an inequality in x and y, add it to the graph, and describe how the viable region changes.

ELL supports

Pre-teach "at least," "at most," "ceiling," and "viable" with a simple number-line picture for each phrase, since the mathematics here depends entirely on reading those phrases correctly. Provide sentence frames for questions 4 and 6, such as "This plan is not allowed because Kilnfield's limit is ___ MGD."

Extensions

Have students research their own state's groundwater withdrawal permitting rules and report on whether permits in that state are written as daily limits, annual limits, or both. Students can then rewrite the fictional Thrushmere constraints in whichever form their state actually uses and compare the two viable regions.