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Lesson plan Grade 11 Math

Sedgemount Orchards' Two Wells: Rational Equations for Fill Time

A 50-minute grade 11 lesson where students write and solve rational equations for the time two wells need to fill an irrigation pond, then add a leak to the model.

Objectives

  1. Students will write a rational equation representing two sources filling one container together.
  2. Students will solve rational equations for an unknown fill time and an unknown drain rate.
  3. Students will convert an irrigation depth over an acreage into gallons and then into pumping hours.

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: fill rate, rational equation, irrigation pond, leak rate

Teacher overview

Two wells feeding one pond do not fill it in the sum of their separate times, and they do not fill it in the average of those times either, although both answers are common first guesses. The reliable route is to add rates rather than times, which turns the question into a rational equation. This lesson uses the fictional Sedgemount Orchards on Tarbrook Ridge, where two wells feed a 360,000-gallon irrigation pond, and it extends the same model to a leaking pond, where the leak enters the equation with a negative sign. The misconception this lesson targets directly is the addition of times, and the launch is built to surface it before any algebra begins.

Procedure

Launch (6 min): State the fictional Sedgemount situation, that one well fills the pond in 15 hours and the other in 30 hours, and ask for a prediction of the time both together would need. Record every distinct prediction, which commonly includes 45 hours and 22.5 hours, and leave them on the board for the rest of the period.

Building the rate model (12 min): Convert each well's time into a share of the pond filled per hour, one-fifteenth and one-thirtieth. Show that these shares add while the times do not, and write the equation one-fifteenth plus one-thirtieth equals one over t. Solve it, then check the answer against the flows in gallons per minute so students see the two routes agree. Return to the board and cross out the predictions the model rules out.

Guided practice (10 min): In pairs, students set up the equation for a third well paired with the first, where the pair fills the pond in 6 hours and the third well's solo time is unknown. Circulate and check that pairs are placing the unknown in a denominator rather than solving for a rate directly.

Independent practice (17 min): Students complete questions 1 through 11 on the handout, including the leak in questions 7 and 8 and the irrigation demand in questions 9 and 10.

Closure and formative check (5 min): Students answer handout question 11 on an index card, explaining why the two solo times cannot simply be added. Collect the cards and count how many describe rates rather than times, with fewer than three-quarters doing so signaling that the next session should reopen the rate model before new work begins.

Sedgemount Orchards' Two Wells: Rational Equations for Fill Time — Grade 11 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Sedgemount Orchards on Tarbrook Ridge fills a 360,000-gallon irrigation pond from two wells.

  • Well A alone fills the empty pond in 15 hours.
  • Well B alone fills the empty pond in 30 hours.
  1. Find Well A's flow in gallons per minute.
  2. Find Well B's flow in gallons per minute.
  3. Write and solve a rational equation for the time t, in hours, that the two wells running together need to fill the empty pond.
  4. Find the combined flow of the two wells in gallons per minute, and check that it agrees with your answer to question 3.
  5. A new Well C, running together with Well A, fills the empty pond in 6 hours. Write and solve a rational equation for the time Well C alone would need.
  6. Find Well C's flow in gallons per minute.
  7. The pond develops a leak. With the leak open, Wells A and B together take 12 hours to fill the pond. Write and solve a rational equation for the time the leak alone would take to empty the full pond.
  8. Find the leak's rate in gallons per minute.
  9. The orchard applies 1.2 inches of water across 60 acres. Find that volume in gallons, using 1 acre-foot equals 325,851 gallons.
  10. With the leak repaired, find the number of hours Wells A and B together need to supply the volume from question 9, to one decimal place.
  11. Explain in one or two sentences why the two solo times cannot be added to find the time the wells need together.

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: 400 gpm.
    Filling 360,000 gallons in 15 hours is 360,000 divided by 900 minutes, which is 400 gallons per minute.

  2. Answer: 200 gpm.
    Filling 360,000 gallons in 30 hours is 360,000 divided by 1,800 minutes, which is 200 gallons per minute.

  3. Answer: 10 hours.
    The equation is one-fifteenth plus one-thirtieth equals one over t. The left side is two-thirtieths plus one-thirtieth, or three-thirtieths, which is one-tenth, so t = 10 hours.

  4. Answer: 600 gpm.
    Adding gives 400 + 200 = 600 gallons per minute, and 360,000 divided by 600 = 600 minutes, which is the same 10 hours found in question 3.

  5. Answer: 10 hours.
    The equation is one-fifteenth plus one over x equals one-sixth. Subtracting gives one over x = five-thirtieths minus two-thirtieths = three-thirtieths = one-tenth, so x = 10 hours.

  6. Answer: 600 gpm.
    Filling 360,000 gallons in 10 hours is 360,000 divided by 600 minutes, which is 600 gallons per minute.

  7. Answer: 60 hours.
    The equation is one-fifteenth plus one-thirtieth minus one over L equals one-twelfth. The first two terms give one-tenth, so one over L = one-tenth minus one-twelfth = six-sixtieths minus five-sixtieths = one-sixtieth, and L = 60 hours.

  8. Answer: 100 gpm.
    Draining 360,000 gallons in 60 hours is 360,000 divided by 3,600 minutes, which is 100 gallons per minute.

  9. Answer: 1,955,106 gallons.
    Applying 1.2 inches across 60 acres gives 72 acre-inches, which is 72 divided by 12 = 6 acre-feet, and 6 x 325,851 = 1,955,106 gallons.

  10. Answer: 54.3 hours.
    Dividing gives 1,955,106 divided by 600 = 3,258.5 minutes, and 3,258.5 divided by 60 = 54.3 hours.

  11. Answer: Rates add, but times do not.
    Each well fills a share of the pond in an hour, and those shares add to give the pair's hourly share, so the joint time is the reciprocal of the sum of the reciprocals rather than the sum of the times. Adding the times would say two wells are slower than one, which the flows in gallons per minute immediately contradict.

Differentiation

Support

Give these students a filled-in table for the fictional Sedgemount pond showing gallons added per hour by each well, so the rate idea arrives before the fraction notation does. Provide questions 3, 5, and 7 with the equation already written and only the solving left to do, and let them check every answer against the gallons-per-hour column.

Extension

Ask these students to find, for the fictional Sedgemount pond, the largest leak the two wells together could overcome and still fill the pond within 24 hours, and to explain why a leak beyond a certain size makes the equation unsolvable in context. Then have them rewrite question 5 with the pair time as the unknown and the third well's solo time given, and describe which form is easier to solve.

ELL supports

Pre-teach "fill rate," "leak," and "acre-foot" with a labeled sketch of the pond showing two inflow pipes and one outflow hole. Provide a worked model of a simpler two-source problem with small numbers, and sentence frames for question 11 such as "We add the ___ and not the ___ because each well fills ___ of the pond in one hour."

Extensions

Students can rewrite the whole fictional Sedgemount analysis with the pond volume left as a variable and show that the joint time does not depend on it, which explains why the reciprocal method works without knowing the pond's size. A second task is to describe how an orchard would measure a leak in practice, and what a measured drop in pond level over a night would tell an operator.