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Worksheet Grade 9 Math

Wrayford Beck Watershed: Rain Depth to Reservoir Yield

A grade 9 worksheet on a fictional watershed where students write a runoff equation and convert an annual yield to gallons, MGD, and cubic feet per second.

Objectives

  1. Students will convert a rainfall depth and a drainage area into an annual runoff volume measured in acre-feet.
  2. Students will write and graph a linear equation relating annual runoff to annual precipitation.
  3. Students will carry units through a conversion chain from acre-feet to gallons, to MGD, and to cubic feet per second.

Vocabulary

Watershed
A watershed is all the land where rain that falls will drain to the same stream, river, or lake. Everyone lives in a watershed, no matter how far from the water they live.
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: drainage area, runoff coefficient, yield

Teacher overview

A watershed is the land area that drains to one stream or reservoir. Only part of the rain that falls on a watershed reaches the stream; the rest soaks in, evaporates, or is taken up by plants. Planners estimate the fraction that reaches the stream with a runoff coefficient, then convert that volume into the units a utility actually uses. This worksheet works through that chain for the fictional Wrayford Beck watershed, and a common error it targets is treating the rainfall depth in inches as though it could be multiplied directly by an area in acres without first converting inches to feet.

Wrayford Beck Watershed: Rain Depth to Reservoir Yield — Grade 9 Math
Name: ____________________________ Date: ______________

Student Worksheet

Use these values for the fictional Wrayford Beck watershed in every item. Show the units in each step.

  • Drainage area: 12,000 acres
  • Average annual precipitation: 30 inches
  • Runoff coefficient: C = 0.25 (the fraction of precipitation that reaches the beck)
  • 1 acre-foot = 325,851 gallons
  • 1 million gallons per day (MGD) = 1.547 cubic feet per second (cfs)
  • Average daily demand of the town of Wrayford: 3.2 MGD
  1. Convert the average annual precipitation of 30 inches to feet.
  2. Find the total volume of precipitation that falls on the watershed in an average year. Give your answer in acre-feet.
  3. Multiply by the runoff coefficient to find the annual runoff that reaches the beck, in acre-feet.
  4. Let p stand for the annual precipitation in inches. Write an equation for the annual runoff R in acre-feet, with the area and the runoff coefficient held fixed.
  5. Graph your equation for p from 0 to 40 inches. Label each axis with its quantity and its unit, and choose a scale that fits the values.
  6. State what the coefficient of p in your equation represents. Include its units.
  7. Convert the average-year runoff from item 3 into gallons.
  8. Divide your item 7 answer by 365 days to find the average daily yield in million gallons per day. Round to two decimal places.
  9. Convert the daily yield from item 8 into cubic feet per second. Round to two decimal places.
  10. Wrayford withdraws 3.2 MGD on an average day. Find that withdrawal as a percent of the yield you found in item 8. Round to the nearest tenth of a percent.
  11. Use your equation from item 4 to find the annual precipitation, in inches, that would make the yield exactly equal the town's 3.2 MGD demand. Round to the nearest tenth of an inch.
  12. New rooftops and streets raise the runoff coefficient from 0.25 to 0.30. Find the new annual runoff in acre-feet, with precipitation still at 30 inches.
  13. Explain why the larger runoff you found in item 12 is not automatically good news for the town's water supply.

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: 2.5 ft. Divide by 12 inches per foot: 30 / 12 = 2.5 feet.
  2. Answer: 30,000 acre-ft. Volume = 12,000 acres x 2.5 feet = 30,000 acre-feet. An acre-foot is one acre covered one foot deep, so acres times feet gives acre-feet directly.
  3. Answer: 7,500 acre-ft. Runoff = 0.25 x 30,000 = 7,500 acre-feet.
  4. Answer: R = 250p. Runoff = 0.25 x 12,000 acres x (p / 12) feet = (3,000 / 12) p = 250p acre-feet.
  5. Answer: A straight line from the origin rising to the point (40, 10000). The horizontal axis is annual precipitation in inches, scaled 0 to 40; the vertical axis is annual runoff in acre-feet, scaled 0 to 10,000. The point (30, 7500) sits on the line and matches item 3.
  6. Answer: 250 acre-feet of runoff per inch of precipitation. It is the slope, so each extra inch of rain over the whole watershed adds 250 acre-feet of runoff.
  7. Answer: 2,443,882,500 gallons. 7,500 acre-feet x 325,851 gallons per acre-foot = 2,443,882,500 gallons.
  8. Answer: 6.70 MGD. 2,443,882,500 / 365 = 6,695,568.5 gallons per day, which is 6.6956 million gallons per day, or 6.70 MGD rounded.
  9. Answer: 10.36 cfs. 6.70 MGD x 1.547 cfs per MGD = 10.3649, which rounds to 10.36 cubic feet per second.
  10. Answer: 47.8 percent. 3.2 / 6.70 = 0.4776, and 0.4776 x 100 = 47.76, which rounds to 47.8 percent of the average yield.
  11. Answer: 14.3 inches. A yield of 3.2 MGD is 3,200,000 x 365 = 1,168,000,000 gallons per year. Divide by 325,851 gallons per acre-foot to get 3,584.5 acre-feet. Solve 250p = 3,584.5, so p = 14.34, which rounds to 14.3 inches.
  12. Answer: 9,000 acre-ft. Runoff = 0.30 x 30,000 = 9,000 acre-feet, an increase of 1,500 acre-feet over item 3.
  13. Answer: More of the rain now leaves as fast surface runoff instead of soaking into the ground. The reservoir can only hold what it captures, so much of the extra water passes through as a short high flow, while less water recharges the groundwater that keeps the beck flowing in dry months. Faster runoff also raises flood peaks and carries more sediment into the reservoir.

Differentiation

Support

Give students a partly completed conversion chain with the units already written in and only the numbers missing, so the sequence acre-feet to gallons to gallons per day to MGD to cfs is visible before they compute. Work items 1 through 3 together as a class, and let students use the class result for item 3 even if their own arithmetic differs.

Extension

Ask students to write a second equation giving annual runoff as a function of the runoff coefficient with precipitation fixed at 30 inches, graph both equations, and explain why one line is steeper than the other when the axes use the same scale.

ELL supports

Pre-teach "watershed," "drainage area," and "runoff coefficient" with a simple sketch of a hillside draining to a stream, with arrows for the water that soaks in and the water that runs off. Provide a sentence frame for item 13: "More runoff is not always helpful because the water ___, so the reservoir ___."

Extensions

Have students find the drainage area of the creek or river nearest the school on a published watershed map, then apply the same fictional runoff coefficient of 0.25 and the local average annual precipitation to estimate its yield in acre-feet. Ask them to name two reasons their estimate could be off, such as a different soil type or a large share of paved surface.