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Lesson plan Grade 7 Math

Cottersall's Aquifer and Reuse Plan: A Grade 7 Proportion Lesson

A 45-minute lesson where students find the constant linking pumping to aquifer drawdown, write its equation, and test whether a reuse plant closes the yearly deficit.

Objectives

  1. Students will identify the constant of proportionality linking pumping to aquifer drawdown.
  2. Students will write and use an equation for drawdown in terms of gallons pumped.
  3. Students will combine recharge and drawdown to evaluate a yearly water balance.

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.
Drawdown
Drawdown is how far the water level in a well drops when a pump pulls water out. The harder and longer you pump, the bigger the drawdown gets.
Recharge
Recharge is water soaking down through the ground to refill an aquifer, the way rain refills an underground sponge. Aquifers stay healthy only if recharge keeps up with pumping.

Also: water reuse

Teacher overview

Cottersall is a fictional town that pumps all of its drinking water from a shallow aquifer, and every measurement in this lesson was written for it. Monitoring wells show that the water level falls in proportion to the gallons pumped each month and recovers over the winter, so a year of records becomes a proportional relationship with a clear constant. The town is considering a small water reuse plant that would replace part of the summer pumping. The misconception to watch for is an inverted constant: students see 1.5 feet of drop from 10 million gallons and write the constant as 10 divided by 1.5 rather than 1.5 divided by 10, which gives about 6.7 and produces drawdowns far larger than any well could show.

Procedure

  1. Hook (5 min): Show a simple cross-section of the fictional Cottersall aquifer with a monitoring well and a marked water level. Ask what students expect to happen to that level during a hot, dry August, and record two predictions on the board. State the June record for the class: 10 million gallons pumped and a 1.5-foot drop.

  2. Model the constant (10 min): Using the June record, ask which quantity depends on which. Write the ratio 1.5 feet divided by 10 million gallons and simplify it to 0.15 feet per million gallons, then write d = 0.15g. Deliberately show the inverted version, 10 divided by 1.5, and ask the class to predict what drawdown it would give for 20 million gallons. When the wrong constant produces a drop of more than 130 feet, ask students what physical check would have caught the error.

  3. Guided practice (8 min): In pairs, students use d = 0.15g to find the drawdown for 14 million gallons and for 30 million gallons, then check that doubling the gallons doubles the drop. Ask one pair to state aloud why that doubling is a property of proportional relationships.

  4. Independent work (15 min): Distribute the Student Handout. Students complete items 1 through 11 on the fictional Cottersall records. Pause at the eight-minute mark to point out that item 6 mixes a positive recharge with a negative drawdown, and that the sign of the answer tells the town whether the aquifer gained or lost ground over the year.

  5. Formative check (4 min): Ask every student to write one number on a slip: the drawdown, in feet, that 48 million gallons of pumping would cause. The target answer is 7.2 feet. If more than a quarter of the class writes 320 or 7,200, reteach the constant from step 2 with a units check.

  6. Debrief (3 min): Take two answers to handout item 11, then state the general result: the reuse plant helps, yet it does not close the gap on its own, so Cottersall would need conservation or a second source as well.

Cottersall's Aquifer and Reuse Plan: A Grade 7 Proportion Lesson — Grade 7 Math
Name: ____________________________ Date: ______________

Student Handout

Cottersall is a fictional town, and all of these records were written for this lesson. Cottersall pumps its drinking water from an aquifer. In June the town pumped 10 million gallons and the water level dropped 1.5 feet.

  1. In July the town pumped 20 million gallons. If the drop is proportional to the gallons pumped, how far did the level drop in July?
  2. Using the June record of 1.5 feet for 10 million gallons, what is the constant of proportionality in feet of drop per million gallons?
  3. Let g stand for million gallons pumped and d stand for the drop in feet. Write an equation for d in terms of g.
  4. In August the town pumped 26 million gallons. Use your equation to find the August drop.
  5. Add the June, July, and August drops of 1.5, 3.0, and 3.9 feet. What is the total summer drop?
  6. Winter rain recharges the aquifer and raises the level 5.2 feet. Using the 8.4-foot summer drop, what is the net change in the water level for the whole year?
  7. Cottersall wants the net change for the year to be 0 feet. With 5.2 feet of winter recharge, how many feet of summer drop can the town allow?
  8. Use the equation d = 0.15g. How many million gallons of summer pumping would cause a 5.2-foot drop? Round to the nearest tenth.
  9. Cottersall pumped 56 million gallons over the summer. A new reuse plant would supply 8 million gallons of that. What would the new summer pumping total be?
  10. Use the equation d = 0.15g with the 48 million gallons from item 9. What summer drop would the town see?
  11. Compare your answers to items 7 and 10. Explain whether the reuse plant alone brings the yearly net change to zero, and what the town would still need to do.

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: 3.0 feet.
    Doubling the gallons doubles the drop in a proportional relationship: 20 million gallons is twice 10 million, so the drop is 2 x 1.5 = 3.0 feet.
  2. Answer: 0.15 feet per million gallons.
    Divide the drop by the gallons: 1.5 / 10 = 0.15 feet for each million gallons pumped.
  3. Answer: d = 0.15g.
    The drop equals the constant of proportionality times the gallons pumped.
  4. Answer: 3.9 feet.
    Substitute 26 for g: d = 0.15 x 26 = 3.9 feet.
  5. Answer: 8.4 feet.
    Add the three monthly drops: 1.5 + 3.0 + 3.9 = 8.4 feet.
  6. Answer: -3.2 feet.
    Combine the recharge and the drop: 5.2 + (-8.4) = -3.2 feet. The negative sign means the aquifer ended the year lower than it started.
  7. Answer: 5.2 feet.
    For a net change of zero, the summer drop must equal the winter recharge, so the allowed drop is 5.2 feet.
  8. Answer: 34.7 million gallons.
    Solve 0.15g = 5.2 by dividing both sides by 0.15: g = 34.666, which rounds to 34.7 million gallons.
  9. Answer: 48 million gallons.
    Subtract the reuse supply from the summer total: 56 - 8 = 48 million gallons still pumped from the aquifer.
  10. Answer: 7.2 feet.
    Substitute 48 for g: d = 0.15 x 48 = 7.2 feet.
  11. Answer: No, the reuse plant alone does not reach a net change of zero.
    Item 7 shows the town can afford only a 5.2-foot summer drop, while item 10 shows the reuse plant still leaves a 7.2-foot drop. The gap is 2.0 feet, which is about 13.3 million gallons of pumping at the same constant. Cottersall would need conservation, a second source, or a larger reuse plant to close it.

Differentiation

Support

Give students a labeled ratio table for the fictional Cottersall records with the columns "million gallons" and "feet of drop," and the June row filled in, so the constant is visible before item 2. Students who invert the constant can be asked to write the units under each number and check that the answer comes out in feet.

Extension

Ask students to find the size of reuse plant, in million gallons, that would bring the fictional town's yearly net change to exactly zero, then to write a short explanation of why the answer is larger than the 2.0-foot gap suggests at first glance.

ELL supports

Preteach "aquifer," "drawdown," and "recharge" with a cross-section sketch of the town of Cottersall showing the water level falling in summer and rising in winter, with arrows labeled in simple language. Provide the sentence frame for item 11: "The reuse plant is not enough because the drop is ____ but the town can only allow ____."

Extensions

Students can look up whether their own community uses groundwater, surface water, or both, and compare the fictional Cottersall drawdown pattern with a published monitoring-well record. A second follow-on asks students to graph d = 0.15g on a coordinate grid, mark the point that represents August, and explain what the steepness of the line means for a town deciding how much to pump.