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Lesson plan Grade 12 Math

Vernbrook Reservoir: A Dry-Summer Water Budget Model

A 50-minute grade 12 lesson where students track a fictional reservoir's monthly storage with a water-budget equation and model its decline.

Objectives

  1. Students will apply a water-budget equation to find monthly change in reservoir storage.
  2. Students will model the storage decline with an average rate of change and project a critical-level month.

Vocabulary

Water budget
A water budget is a way of tracking all the water coming into an area, like rain and snowmelt, against all the water leaving it, like evaporation and streamflow. Scientists use water budgets to see whether a watershed or aquifer is gaining or losing water over time.
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: storage, average rate of change

Teacher overview

Vernbrook Reservoir is a fictional reservoir on a small river, and its managers must know how fast it is emptying during a dry summer. A water budget tracks that by balancing what comes in against what leaves: the change in storage equals inflow minus withdrawals minus evaporation. This lesson has students fill a monthly budget, compute the average rate of change across the summer, and use it to project when the reservoir would reach its minimum operating level. A common misconception is that a reservoir empties at a steady rate; the monthly figures show the decline speeding up as inflow shrinks, so a single average both summarizes the trend and hides the month-to-month change.

Procedure

Hook (5 min): Ask students what a city should watch to know whether its reservoir will last through a drought. Steer the discussion toward inflow, use, and evaporation, and note that Vernbrook is a fictional reservoir used to make the model concrete.

Direct instruction (10 min): Introduce the water-budget equation: change in storage = inflow - withdrawals - evaporation, with every term in acre-feet. Show that a negative change means storage is falling. Then define average rate of change as the total change in storage divided by the number of months.

Model the method (8 min): Work one month with the class using teacher-modeled numbers different from the handout so students learn the steps:

  • Inflow 800, withdrawals 1,300, evaporation 250, all in acre-feet.
  • Change = 800 - 1,300 - 250 = -750 acre-feet.
  • Starting from 7,000 acre-feet, end-of-month storage = 7,000 - 750 = 6,250 acre-feet.

Student Handout practice (20 min): Students complete the numbered handout in pairs, building the three-month budget table and then the projection. Circulate and check that students subtract both withdrawals and evaporation, a step often missed.

Closure and formative check (7 min): Students answer on a card: "Is the reservoir emptying faster or slower as the summer goes on, and how do you know from the numbers?" A correct answer cites the larger negative change in July and August than in June. Fewer than four of five correct signals a need to revisit the monthly changes.

Vernbrook Reservoir: A Dry-Summer Water Budget Model — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

Vernbrook Reservoir is a fictional reservoir that starts June at 8,000 acre-feet of storage. Its monthly inflow, withdrawals, and evaporation, all in acre-feet, are: June, inflow 900, withdrawals 1,400, evaporation 200; July, inflow 500, withdrawals 1,600, evaporation 300; August, inflow 400, withdrawals 1,500, evaporation 300. Use change in storage = inflow - withdrawals - evaporation.

  1. Find June's change in storage.
  2. Find the end-of-June storage, starting from 8,000 acre-feet.
  3. Find July's change in storage.
  4. Find the end-of-July storage.
  5. Find August's change in storage.
  6. Find the end-of-August storage.
  7. Find the average rate of change of storage from June 1 (8,000 acre-feet) to the end of August, over the three months.
  8. Write a linear equation for storage S after m months, starting at 8,000 acre-feet and falling at the rounded average rate from question 7, then use it to predict storage after 5 months.
  9. The reservoir's minimum operating level is 1,500 acre-feet. Using your equation, find how many months it would take to reach that level.
  10. Convert August's withdrawals of 1,500 acre-feet, spread over 31 days, to an average daily rate in MGD. Use 1 acre-foot = 325,851 gallons.

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: -700 acre-feet.
    June change = 900 - 1,400 - 200 = -700 acre-feet.

  2. Answer: 7,300 acre-feet.
    End-of-June storage = 8,000 + (-700) = 7,300 acre-feet.

  3. Answer: -1,400 acre-feet.
    July change = 500 - 1,600 - 300 = -1,400 acre-feet.

  4. Answer: 5,900 acre-feet.
    End-of-July storage = 7,300 + (-1,400) = 5,900 acre-feet.

  5. Answer: -1,400 acre-feet.
    August change = 400 - 1,500 - 300 = -1,400 acre-feet.

  6. Answer: 4,500 acre-feet.
    End-of-August storage = 5,900 + (-1,400) = 4,500 acre-feet.

  7. Answer: -1,167 acre-feet per month.
    Total change = 4,500 - 8,000 = -3,500 acre-feet over 3 months, so the average rate = -3,500 / 3 = -1,166.7, about -1,167 acre-feet per month.

  8. Answer: 2,165 acre-feet.
    The equation is S = 8,000 - 1,167m. At m = 5, S = 8,000 - 1,167(5) = 8,000 - 5,835 = 2,165 acre-feet.

  9. Answer: 5.6 months.
    Solve 8,000 - 1,167m = 1,500, so 1,167m = 6,500 and m = 6,500 / 1,167 = 5.57, about 5.6 months.

  10. Answer: 15.8 MGD.
    1,500 acre-feet x 325,851 gallons = 488,776,500 gallons over 31 days. Dividing, 488,776,500 / 31 = about 15,767,000 gallons per day, or about 15.8 MGD.

Differentiation

Support

Provide the budget table as a printed grid with the inflow, withdrawal, and evaporation rows filled in, so students focus on computing each change and running total. Pre-compute June for them as a model, and allow calculators.

Extension

Ask students to add a September row of their own choosing that would keep the reservoir above its minimum operating level through the fall, and to justify their inflow and withdrawal numbers with the budget equation.

ELL supports

Provide a labeled picture of a reservoir showing inflow, withdrawals, and evaporation as arrows in and out, glossing "storage," "water budget," and "average rate of change." Offer a sentence frame for the closure: "The reservoir empties ___ over the summer because ___."

Extensions

Have students find a real reservoir's public storage graph for a recent dry year, estimate its average rate of change over the summer months, and compare the shape of that curve with Vernbrook's steady-rate model.