Worksheet Grade 9 Math
Thornridge's Reservoir Allocation: Acre-Feet to MGD Unit Conversions
A grade 9 worksheet where students chain acre-foot, cubic-foot, and gallon conversions to size a fictional town's reservoir allocation in MGD and gpm.
Objectives
- Students will convert a volume given in acre-feet into gallons, gallons per day, MGD, and gallons per minute using a chain of unit conversions.
- Students will explain, using units, why skipping a stage in a multi-step conversion chain can produce an incorrect result.
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: million gallons per day (MGD), dimensional analysis, unit rate
Teacher overview
The fictional town of Thornridge is finalizing a new annual reservoir allocation, and its planning department needs that allocation expressed in several different units before the plant, the state, and the town council can each use it. Water rights are usually recorded in acre-feet, a large volume unit tied to land area, while a treatment plant's daily operations are tracked in gallons and in million gallons per day (MGD). This worksheet has students carry a single number through that entire chain of units, the same dimensional-analysis discipline that appears on operator-certification exams.
A common misconception in this kind of problem is treating each conversion as an isolated fact to memorize rather than as a chain where the output of one step becomes the input of the next. Students who lose track of which unit they are currently working in often multiply by the wrong constant partway through, producing an answer that is off by several orders of magnitude without anyone noticing until the final number looks unreasonable.
Student Worksheet
Use these conversion facts for every problem below:
- 1 acre-foot = 325,851 gallons
- 1 acre-foot = 43,560 cubic feet
- 1 cubic foot = 7.48 gallons
- 1 MGD (million gallons per day) = 694 gallons per minute (gpm)
The fictional town of Thornridge has a new reservoir allocation of 500 acre-feet per year. Convert 500 acre-feet to gallons.
Convert the same 500 acre-feet to cubic feet, then convert that cubic-foot answer to gallons using the 7.48 gallons-per-cubic-foot conversion. Compare this result to your answer from problem 1. Do the two methods agree closely, and if there is a small difference, what is the likely cause?
If Thornridge draws down its 500-acre-foot allocation evenly across a 365-day year, how many gallons per day is that? Round to the nearest whole gallon.
Convert your answer from problem 3 into MGD (million gallons per day), rounded to three decimal places.
Using the conversion 1 MGD = 694 gpm, convert Thornridge's daily draw (your rounded MGD answer from problem 4) into gallons per minute. Round to the nearest whole gallon per minute.
Thornridge's treatment plant is rated to treat up to 0.75 MGD. Is the new reservoir allocation's daily draw (from problem 4) within the plant's rated capacity? By how much, in MGD?
A neighboring system, Bellmere Junction, has a 1,200-acre-foot annual allocation. Assuming the same even 365-day draw, convert Bellmere Junction's allocation into gallons per day and then into MGD, rounded to three decimal places. Show every conversion step.
Explain, using units rather than just the final numbers, why converting acre-feet directly into gallons per minute in a single step, without tracking the gallons and MGD stages in between, could lead to an error.
Thornridge's planning board wants to describe next year's allocation to the public using whichever quantity is easiest to picture: total gallons for the year, MGD, or gallons per minute. Choose one of these three quantities and justify your choice in two to three sentences.
Reservoir engineers are also considering a smaller emergency allocation of 50 acre-feet. Without repeating the full conversion chain, use proportional reasoning from your problem 4 answer to estimate the daily MGD draw for this smaller allocation.
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: 162,925,500 gal.
500 acre-ft × 325,851 gal/acre-ft = 162,925,500 gallons.Answer: 162,914,400 gal.
500 acre-ft × 43,560 cu ft/acre-ft = 21,780,000 cu ft. Then 21,780,000 cu ft × 7.48 gal/cu ft = 162,914,400 gallons. This is close to problem 1's answer (a difference of 11,100 gallons, well under one hundredth of one percent), because 7.48 gallons per cubic foot is itself a rounded constant. The two methods agree within the rounding built into the conversion factors.Answer: 446,371 gal/day.
162,925,500 gal ÷ 365 days = 446,371.23 gallons per day, which rounds to 446,371 gallons per day.Answer: 0.446 MGD.
446,371.23 gal/day ÷ 1,000,000 = 0.446371 MGD, which rounds to 0.446 MGD.Answer: 310 gpm.
0.446 MGD × 694 gpm/MGD = 309.524 gpm, which rounds to about 310 gallons per minute.Answer: yes, by 0.304 MGD.
The plant is rated for 0.75 MGD, and the new allocation only requires 0.446 MGD. 0.75 − 0.446 = 0.304 MGD of unused rated capacity remains.Answer: 1.071 MGD.
1,200 acre-ft × 325,851 gal/acre-ft = 391,021,200 gallons per year. 391,021,200 ÷ 365 = 1,071,290.96 gallons per day, which is 1,071,290.96 ÷ 1,000,000 = 1.071 MGD when rounded to three decimal places.Answer: Skipping stages risks multiplying by a constant meant for a different pair of units.
Each conversion constant (325,851 gal/acre-ft, 7.48 gal/cu ft, 694 gpm/MGD) is defined for a specific pair of units. If a student multiplies acre-feet directly by 694, the number produced is not gallons per minute at all, because 694 converts MGD to gpm, not acre-feet to gpm. Tracking the unit at each stage catches this kind of mismatch before the wrong number gets reported.Answer: answers vary; must justify the choice.
A strong answer might choose MGD because plant operators and regulators already think in that unit, or gallons per minute because it connects to a familiar rate like a garden hose, or total gallons because it is the largest, most attention-getting number for a public announcement. Full credit requires a clear reason tied to the audience.Answer: about 0.045 MGD.
50 acre-feet is one-tenth of the 500-acre-foot allocation used in problem 4. Scaling problem 4's answer of 0.446 MGD by one-tenth gives approximately 0.045 MGD.
Differentiation
Support (below grade level)
Provide a printed conversion ladder listing all four conversion facts in the order students will need them (acre-feet to gallons, gallons to gallons-per-day, gallons-per-day to MGD, MGD to gpm), and allow students to check off each stage as they complete it. Work problem 1 together as a class before releasing students to problems 2 through 10.
Extension (above grade level)
Ask students to derive the acre-foot-to-MGD conversion as a single combined constant (325,851 gallons per acre-foot ÷ 365 days ÷ 1,000,000, which gives the MGD for each acre-foot of annual allocation), then use that single constant to solve problem 7 in one step and confirm it matches their multi-step answer.
ELL supports
Provide a labeled diagram showing the conversion chain as a series of boxes and arrows (acre-feet → gallons → gallons per day → MGD → gallons per minute), with the conversion factor written on each arrow. Allow students to show their work as a filled-in diagram rather than full sentences for problems 1 through 7.
Extensions
Contact a local water utility or planning department and ask what units they use internally when discussing a reservoir's supply, a plant's capacity, and a customer's usage. Compare their answer to the reasoning students gave in problem 9, and discuss whether the utility's choice matches or differs from the class's predictions.