Lesson plan Grade 10 Math
Pentyre Gorge Hydro Plant: Energy Density of Stored Water
A 50-minute grade 10 lesson where students compute energy per acre-foot and per surface acre for two fictional basins, then pick the quantity that fits.
Objectives
- Students will convert a stated energy rate per thousand gallons into energy per acre-foot.
- Students will compute energy density per unit volume and per unit surface area for two storage basins.
- Students will justify which quantity best answers a stated planning question about evaporation losses.
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.
- Evaporation
- Evaporation happens when liquid water, often warmed by the sun, turns into an invisible gas called water vapor that rises into the air. It is how water leaves oceans, lakes, and puddles.
Also: energy density, kilowatt-hour, surface acre
Teacher overview
Pentyre Gorge, the Wrenfell River, and the two storage basins in this lesson are fictional, as is the stated plant output of 1.20 kWh for every thousand gallons sent through the turbine. Students turn that rate into an energy density per acre-foot using the standard conversion of 325,851 gallons per acre-foot, then build a second density based on water surface area. The misconception addressed here is that the basin losing the most water to evaporation must also be the one giving up the largest share of its storage; evaporation scales with surface area rather than with volume, so the shallower fictional basin loses a far larger share of what it stores even though it loses fewer total acre-feet.
Procedure
Hook (5 min): Ask which of two fictional basins loses more to evaporation over a summer: one holding 8,400 acre-feet across 260 surface acres, or one holding 3,600 acre-feet across 200 surface acres. Take a quick vote, record the tally on the board, and leave it unresolved.
Direct instruction (12 min): Define energy density as energy per unit of something else, and work a short teacher example that is not on the handout: a fictional plant producing 2.00 kWh per thousand gallons yields 2.00 × 325.851, or about 651.7 kWh per acre-foot. Show the same figure expressed per surface acre for a fictional 100 acre-foot pond covering 10 acres, and name the units aloud at every step so students hear that the denominator changes meaning.
Guided practice (12 min): Distribute the handout and work items 1 through 4 together. Pause after item 2 so pairs can confirm that they divided the gallons per acre-foot by one thousand before multiplying. Ask each pair to write one sentence predicting which basin will show the larger energy per surface acre before computing items 5 and 6.
Independent practice (14 min): Students complete items 5 through 12 alone or in pairs. Circulate and watch for the common error of multiplying the evaporation depth by the volume instead of by the surface acreage.
Closure and formative check (7 min): Return to the hook tally. Read items 10 and 11 aloud and ask students to signal thumbs up if the vote should change. If fewer than two-thirds of the class now picks the smaller fictional basin as the bigger relative loser, rework item 11 on the board before the next class.
Student Handout
The Pentyre Gorge plant and both basins are fictional. The plant produces 1.20 kWh for every thousand gallons released through its turbine. One acre-foot equals 325,851 gallons. Sedgemill Basin holds 8,400 acre-feet of usable storage across 260 surface acres. Cattermole Basin holds 3,600 acre-feet across 200 surface acres.
How many gallons are in one acre-foot?
Find the plant's energy density in kWh per acre-foot. Round to the nearest whole kWh and use that rounded value in every item that follows.
Find the energy stored in Sedgemill Basin's 8,400 acre-feet.
Find the energy stored in Cattermole Basin's 3,600 acre-feet.
Sedgemill Basin covers 260 surface acres. Find its energy per surface acre, rounded to the nearest whole kWh.
Cattermole Basin covers 200 surface acres. Find its energy per surface acre.
Evaporation removes 3.5 acre-feet of water for every surface acre over one fictional summer. How many acre-feet does Sedgemill Basin lose?
How many kWh does Sedgemill Basin's evaporation loss represent?
Cattermole Basin loses 700 acre-feet the same summer. How many kWh is that?
What percent of Sedgemill Basin's stored energy is lost to evaporation? Round to the nearest tenth.
What percent of Cattermole Basin's stored energy is lost to evaporation? Round to the nearest tenth.
The fictional district asks which basin loses more of what it stores. Which quantity answers that question, total kWh lost or percent of stored energy lost? Explain in two or three sentences.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: 325,851 gallons. This is the standard conversion for one acre-foot.
Answer: 391 kWh per acre-foot. One acre-foot is 325,851 ÷ 1,000 = 325.851 thousand-gallon units, and 325.851 × 1.20 = 391.02, which rounds to 391 kWh per acre-foot.
Answer: 3,284,400 kWh. Multiplying gives 8,400 × 391 = 3,284,400 kWh.
Answer: 1,407,600 kWh. Multiplying gives 3,600 × 391 = 1,407,600 kWh.
Answer: 12,632 kWh per acre. Dividing gives 3,284,400 ÷ 260 = 12,632.3, which rounds to 12,632 kWh per acre.
Answer: 7,038 kWh per acre. Dividing gives 1,407,600 ÷ 200 = 7,038 kWh per acre.
Answer: 910 acre-feet. The loss is 260 surface acres × 3.5 acre-feet per surface acre = 910 acre-feet.
Answer: 355,810 kWh. Multiplying gives 910 × 391 = 355,810 kWh.
Answer: 273,700 kWh. Multiplying gives 700 × 391 = 273,700 kWh.
Answer: 10.8 percent. Dividing gives 355,810 ÷ 3,284,400 = 0.1083, which is 10.8 percent.
The same result comes from 910 ÷ 8,400, because the energy density cancels.Answer: 19.4 percent. Dividing gives 273,700 ÷ 1,407,600 = 0.1944, which is 19.4 percent, or equivalently 700 ÷ 3,600.
Answer: Percent of stored energy lost. Total kWh lost points to Sedgemill Basin, at 355,810 kWh, as the larger loss, yet that basin also stores more than twice as much energy. The question asks about the share of storage given up, so the percent figures of 10.8 and 19.4 answer it, and they show the smaller fictional basin surrendering nearly twice the share.
Differentiation
Support
Give students the conversion in item 2 already completed, along with a labeled diagram of each fictional basin showing volume in acre-feet on one side and surface acres on the other, so items 5 through 9 become a matter of choosing which labeled number goes in the denominator. Provide a worked model of item 7 with the units written under each factor.
Extension
Ask students to find the surface acreage that would make the two fictional basins lose the same percent of their stored energy, then explain what that says about depth. A second challenge is to compute each basin's average depth in feet from the volume and surface acreage given, and to connect the deeper basin to the smaller relative loss.
ELL supports
Provide a card showing the phrase "per acre-foot" beside a drawing of a cube of water and the phrase "per surface acre" beside a drawing of a flat lake surface, so the two denominators stay visually distinct. Offer a sentence frame for item 12: "The better quantity is ___ because the question asks about ___, not about ___."
Extensions
Have students look up the surface area and usable storage of a real reservoir their teacher selects and repeat items 5 through 11 with those figures in place of the fictional Pentyre Gorge values, keeping the same stated plant rate. A short written comparison of the real basin's relative evaporation loss with the two fictional basins makes a useful bridge to a later lesson on reservoir siting.