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Worksheet Grade 7 Math

Redmoss Basin Rain and Runoff: A Proportionality Worksheet

Grade 7 students test a gauge table for proportionality, name the constant, write the runoff equation, and solve a flood-threshold inequality for a fictional basin.

Objectives

  1. Students will decide whether a table of rainfall and runoff shows a proportional relationship.
  2. Students will identify the constant of proportionality and write it as an equation.
  3. Students will solve an inequality to find the rainfall that causes flooding.

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.

Also: runoff, constant of proportionality, gauge

Teacher overview

Redmoss Basin is a fictional 40-square-mile watershed drained by a stream called the Amber Fork, and every reading below was written for this worksheet. A stream gauge at the basin outlet records how much water leaves after each storm, which lets students test whether runoff is proportional to rainfall. The misconception worth naming for the class is the belief that any table whose numbers rise together is proportional. Students should check the quotient row by row, because a table can climb steadily and still fail the constant-ratio test.

Redmoss Basin Rain and Runoff: A Proportionality Worksheet — Grade 7 Math
Name: ____________________________ Date: ______________

Student Worksheet

Redmoss Basin is fictional, and all of its gauge readings were written for this worksheet. The gauge sits where the Amber Fork leaves the basin. It records the runoff from each storm in millions of gallons.

Rainfall (inches) Runoff (million gallons)
0.5 12
1.0 24
1.5 36
2.0 48
2.5 60
  1. Use the table. How much runoff came from the 1.0-inch storm?
  2. Use the table. What is the ratio of runoff to rainfall for the 1.5-inch storm, in million gallons per inch?
  3. Check the ratio for every row of the table. Is the relationship between rainfall and runoff proportional? Explain how you decided.
  4. What is the constant of proportionality for Redmoss Basin, in million gallons per inch?
  5. Let n stand for rainfall in inches and r stand for runoff in million gallons. Write an equation that gives r from n.
  6. Use your equation from item 5. How much runoff comes from a 3.2-inch storm?
  7. The Amber Fork floods when runoff passes 90 million gallons. Write an inequality using your equation, then solve it to find the rainfall that causes a flood.
  8. A neighboring fictional basin, Moss Gable, produced 18 million gallons from 1.0 inch of rain. How many more million gallons per inch does Redmoss Basin produce than Moss Gable?
  9. Moss Gable received 2.5 inches of rain. Using its own constant of 18 million gallons per inch, how much runoff did it produce?
  10. Moss Gable is mostly forest, and Redmoss Basin has more streets and parking lots. Explain why the paved basin has the larger constant of proportionality.

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: 24 million gallons.
    The table row for 1.0 inch lists 24 million gallons of runoff.
  2. Answer: 24 million gallons per inch.
    Divide runoff by rainfall: 36 / 1.5 = 24 million gallons per inch.
  3. Answer: Yes, the relationship is proportional.
    Every row gives the same quotient: 12 / 0.5 = 24, 24 / 1.0 = 24, 36 / 1.5 = 24, 48 / 2.0 = 24, and 60 / 2.5 = 24. Because the ratio of runoff to rainfall is constant, the relationship is proportional.
  4. Answer: 24 million gallons per inch.
    The constant of proportionality is the shared quotient found in item 3.
  5. Answer: r = 24n.
    Runoff equals the constant of proportionality times the rainfall.
  6. Answer: 76.8 million gallons.
    Substitute 3.2 for n: r = 24 x 3.2 = 76.8 million gallons.
  7. Answer: 3.75 inches.
    Write 24n greater than 90, then divide both sides by 24 to get n greater than 3.75. Rainfall above 3.75 inches sends the Amber Fork over its flood level.
  8. Answer: 6 million gallons per inch.
    Subtract the two constants: 24 - 18 = 6 million gallons per inch.
  9. Answer: 45 million gallons.
    Multiply the constant by the rainfall: 18 x 2.5 = 45 million gallons.
  10. Answer: Pavement sends a larger share of each inch of rain to the stream.
    Rain that lands on forest soil soaks in, so less of it reaches the gauge. Streets and parking lots cannot absorb rain, so nearly all of it runs off. A larger share reaching the stream from the same inch of rain is exactly what a larger constant of proportionality means.

Differentiation

Support

Give students a third column in the fictional Redmoss table labeled "runoff divided by rainfall" and have them fill it in before item 3, so the constant ratio is visible rather than remembered. For item 7, provide the inequality 24n greater than 90 already written, and ask only for the solution step.

Extension

Have students graph both fictional basins on one coordinate grid, with rainfall on the horizontal axis, and explain what the point (1, 24) means for Redmoss Basin and what the steeper line tells them about the pavement. Then ask them to predict the rainfall at which Moss Gable would produce 90 million gallons.

ELL supports

Preteach "runoff," "gauge," and "basin" with a simple cross-section sketch of the fictional Redmoss Basin showing rain falling, water flowing across the ground, and a gauge at the stream. Provide the sentence frame for item 3: "The relationship is proportional because every row gives the same ____."

Extensions

Students can look up a real stream gauge in their own county and compare several storms, then explain why real gauge data rarely gives the perfectly constant ratio that the fictional Redmoss table gives. A second follow-on has students rewrite the table for a basin that is half the size, keeping the same rainfall column, and identify the new constant.