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Lesson plan Grade 7 Math

Chancery Flats Detention Basin: A Grade 7 Inequality Lesson

A 50-minute grade 7 math lesson where students write and solve inequalities for how long a fictional town's stormwater basin can fill before it overflows.

Objectives

  1. Students will find a net rate of change by subtracting an outflow rate from an inflow rate.
  2. Students will write an inequality that keeps a basin's volume at or below its capacity.
  3. Students will solve those inequalities for the number of hours a storm can run.

Vocabulary

Detention basin
A detention basin is a dug-out area that fills with stormwater during a storm and then slowly drains empty over hours or days. It holds the water back so streets and streams downstream do not flood.

Also: capacity, inflow, outflow, net rate, inequality

Teacher overview

A detention basin is a shallow bowl beside a road or a subdivision that catches storm runoff and lets it out slowly through a small pipe, so the stream downstream does not receive all of the rain at once. The basin fills whenever runoff arrives faster than the outlet can release it, and the difference between those two rates is what an engineer plans around. Students work with the fictional town of Chancery Flats, writing and solving inequalities that keep the basin at or below its capacity. The error to watch for is dropping the water already in the basin: a student who solves 6,000 times h is at most 180,000 gets 30 hours, while the correct inequality starts from the 24,000 cubic feet already standing in the basin and gives 26 hours.

Procedure

Hook (5 min): Ask students what happens to rain that lands on a parking lot, and where it goes if the storm drain leads to a small creek. Introduce the fictional town of Chancery Flats and its basin, and state that the basin holds a fixed amount and no more.

Direct instruction (13 min): Model a basin that is not on the handout. A basin holds at most 60,000 cubic feet and already holds 10,000 cubic feet. Runoff enters at 8,000 cubic feet per hour and the outlet releases 5,000 cubic feet per hour. Show the net rate as 8,000 minus 5,000, or 3,000 cubic feet per hour, then write 10,000 plus 3,000 times h is at most 60,000. Subtract 10,000 from both sides, divide both sides by 3,000, and read the result as h is at most about 16.67 hours. Name each part of the inequality aloud as you write it.

Guided practice (10 min): Distribute the handout. Complete items 1 through 3 with the class so students see the net rate, the inequality, and the solution in that order. Ask a pair to state what the variable h stands for before moving on.

Independent practice (14 min): Students complete items 4 through 9 on their own. Point out that item 6 changes only the outlet rate, that item 7 works backward from a volume limit to a rate, and that item 9 builds on the answer to item 8.

Closure and exit ticket (8 min): Ask students what a town could do if a storm was forecast to last longer than the basin's limit, and accept answers such as opening the outlet wider or adding storage. Then collect an exit ticket: "A second basin holds at most 96,000 cubic feet and is empty. Water enters at a net rate of 3,000 cubic feet per hour. For how many hours can the storm run before the basin is full?" If more than a quarter of the class misses the division, reteach it before the next class.

Chancery Flats Detention Basin: A Grade 7 Inequality Lesson — Grade 7 Math
Name: ____________________________ Date: ______________

Student Handout

Chancery Flats is a fictional town. Its detention basin holds at most 180,000 cubic feet of water. When a storm begins, the basin already holds 24,000 cubic feet left from an earlier rain. During the storm, runoff enters the basin at 9,500 cubic feet per hour and the outlet pipe releases 3,500 cubic feet per hour. Let h stand for the number of hours the storm has been running.

  1. What is the net rate at which the basin gains water, in cubic feet per hour?
  2. Write an inequality that keeps the basin at or below its capacity after h hours.
  3. Solve your inequality. For how many hours can the storm run before the basin is full?
  4. How much water is in the basin after 15 hours, in cubic feet?
  5. How much room is left in the basin after 15 hours, in cubic feet?
  6. Suppose the crew opens the outlet wider so it releases 5,500 cubic feet per hour, while runoff still enters at 9,500 cubic feet per hour. For how many hours can the storm run before the basin is full?
  7. In a later storm the basin starts empty. The town wants the basin to hold no more than 135,000 cubic feet after 20 hours. What is the greatest net inflow rate that meets this, in cubic feet per hour?
  8. Return to the first storm. An inspection crew must reach the basin before it holds 150,000 cubic feet. Solve an inequality to find how many hours they have.
  9. The crew needs 4 hours to set up before they arrive. Using your answer to item 8, how many hours after the storm starts is the latest they can begin setting up?

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: 6,000 cubic feet per hour.
    Subtract the outflow from the inflow: 9,500 minus 3,500 equals 6,000 cubic feet per hour.
  2. Answer: The inequality is 24,000 + 6,000h ≤ 180,000.
    The basin begins with 24,000 cubic feet and gains 6,000 cubic feet each hour, and the total may not pass the capacity of 180,000 cubic feet.
  3. Answer: 26 hours.
    Subtract 24,000 from both sides to get 6,000h is at most 156,000. Divide both sides by 6,000 to get h is at most 26.
  4. Answer: 114,000 cubic feet.
    Multiply the net rate by the hours and add the starting volume: 6,000 times 15 equals 90,000, and 90,000 plus 24,000 equals 114,000 cubic feet.
  5. Answer: 66,000 cubic feet.
    Subtract the volume in the basin from the capacity: 180,000 minus 114,000 equals 66,000 cubic feet.
  6. Answer: 39 hours.
    The new net rate is 9,500 minus 5,500, or 4,000 cubic feet per hour. The basin still has 156,000 cubic feet of room, and 156,000 divided by 4,000 equals 39 hours.
  7. Answer: 6,750 cubic feet per hour.
    The basin starts empty, so the net rate times 20 must be at most 135,000. Divide 135,000 by 20 to get 6,750 cubic feet per hour.
  8. Answer: 21 hours.
    Write 24,000 plus 6,000h is at most 150,000. Subtract 24,000 to get 6,000h is at most 126,000, then divide by 6,000 to get h is at most 21.
  9. Answer: 17 hours.
    The crew must finish by hour 21 and needs 4 hours, so h plus 4 is at most 21, which gives h is at most 17 hours.

Exit ticket answer: 32 hours. The basin is empty, so 3,000h is at most 96,000, and 96,000 divided by 3,000 equals 32.

Differentiation

Support

Give these students a partially written inequality with the starting volume and the capacity already in place, so they supply only the net rate and the variable term. Assign items 1 through 5 and provide the modeled basin from the direct instruction as a reference card.

Extension

Ask these students to find the outlet release rate that would let the basin run through a 48-hour storm without filling, given the same 9,500 cubic feet per hour of runoff and the same 24,000 cubic feet already in the basin. Have them state the answer as an inequality on the outlet rate and explain what a larger outlet would mean for the creek downstream.

ELL supports

Pre-teach "basin," "capacity," "inflow," and "outflow" with a labeled cross-section sketch showing an arrow in, an arrow out, and a line marking the top. Provide the frame "The basin gains ___ cubic feet each hour, so after h hours it holds ___" for items 1 and 2.