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Worksheet Grade 7 Math

Ebbstone's Cast-Iron Mains: A Grade 7 Unit-Rate Worksheet

Students compute crew unit rates from ratios of fractions, convert miles of old pipe to feet, and track signed changes in the pipe still left to replace.

Objectives

  1. Students will compute a unit rate from a ratio of fractional quantities.
  2. Students will convert miles of water main into feet and back.
  3. Students will combine positive and negative changes to find a remaining pipe total.

Vocabulary

Water main
A big underground pipe that carries drinking water beneath the streets to a neighborhood. Smaller service lines branch off it to reach each building.

Also: unit rate, cast iron, complex fraction

Teacher overview

Ebbstone is a fictional town whose oldest water mains are cast iron, and every figure below was written for this worksheet. Two crews are replacing that pipe street by street, and their daily logs give the raw material for unit-rate work. The misconception to watch for is the one that shows up whenever the time is a fraction: students divide by the numerator, or they multiply by the fraction instead of dividing, and a crew that laid 450 feet in three-quarters of an hour comes out slower rather than faster. Ask students to check every rate against the question "would a full hour give more or less than what the log shows?"

Ebbstone's Cast-Iron Mains: A Grade 7 Unit-Rate Worksheet — Grade 7 Math
Name: ____________________________ Date: ______________

Student Worksheet

Ebbstone is a fictional town, and these numbers were written for this worksheet. Ebbstone is replacing its old cast-iron water mains. Two crews keep daily logs. Use one mile equals 5,280 feet where you need it.

  1. Crew A laid 450 feet of new main in 3/4 of an hour. What is Crew A's rate in feet per hour?
  2. Crew B laid 210 feet of new main in 1/2 of an hour. What is Crew B's rate in feet per hour?
  3. Use items 1 and 2. How many more feet per hour does Crew A lay than Crew B?
  4. Crew A works at 600 feet per hour. How many hours does Crew A need for a 4,200-foot street?
  5. Ebbstone still has 5 1/4 miles of cast-iron main. How many feet is that?
  6. Crew A lays 600 feet per hour and works 6 hours a day. How many feet does Crew A lay in one workday?
  7. Use item 6. How many workdays does Crew A need for 27,720 feet? Round to the nearest tenth of a day.
  8. Ebbstone's records started the year at 5.25 miles of cast iron. The crews replaced 1.75 miles in spring and 2.5 miles in summer. Then a survey found 0.6 mile that had never been recorded. How many miles of cast iron are left?
  9. A leaking joint on an old main lost 5/8 gallon of water in 1/4 of a minute. What is the leak rate in gallons per minute?
  10. Crew A laid more total feet last week than Crew B, but Crew B worked more hours. Explain why feet per hour is a fairer way to compare the two crews than total feet.

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: 600 feet per hour.
    Divide the feet by the hours: 450 / (3/4) = 450 x (4/3) = 600 feet per hour.
  2. Answer: 420 feet per hour.
    Divide the feet by the hours: 210 / (1/2) = 210 x 2 = 420 feet per hour.
  3. Answer: 180 feet per hour.
    Subtract the two rates: 600 - 420 = 180 feet per hour.
  4. Answer: 7 hours.
    Divide the street length by the rate: 4,200 / 600 = 7 hours.
  5. Answer: 27,720 feet.
    Multiply the miles by the feet in a mile: 5.25 x 5,280 = 27,720 feet.
  6. Answer: 3,600 feet.
    Multiply the hourly rate by the hours worked: 600 x 6 = 3,600 feet in a workday.
  7. Answer: 7.7 workdays.
    Divide the total feet by the daily feet: 27,720 / 3,600 = 7.7 workdays. Seven full days cover 25,200 feet, leaving 2,520 feet, and 2,520 / 3,600 = 0.7 of a day.
  8. Answer: 1.6 miles.
    Combine the changes in order: 5.25 - 1.75 = 3.5, then 3.5 - 2.5 = 1.0, then 1.0 + 0.6 = 1.6 miles. The survey correction is added because it found pipe that was missing from the record.
  9. Answer: 2.5 gallons per minute.
    Divide the gallons by the minutes: (5/8) / (1/4) = (5/8) x 4 = 20/8 = 2.5 gallons per minute.
  10. Answer: Feet per hour compares the crews on the same amount of time.
    Total feet rewards whichever crew was scheduled for more hours, which says nothing about how fast either crew works. A unit rate divides the work by the time, so both crews are measured against one hour. Item 3 shows the gap that total feet would hide.

Differentiation

Support

Give students the reciprocal step written out for items 1, 2, and 9 in the fictional Ebbstone logs, showing that dividing by 3/4 is the same as multiplying by 4/3. A number line marked in quarter-hours also helps students see why a crew's hourly rate must be larger than the feet it laid in three-quarters of an hour.

Extension

Ask students to find the number of full workdays the fictional Ebbstone crews would need if a third crew joined at 540 feet per hour and all three worked the same 6-hour day, then to explain why the answer is not simply one-third of the item 7 result.

ELL supports

Provide a labeled sketch of a street cross-section for the town of Ebbstone showing an old cast-iron main and a new main, with the words "replace," "crew," and "main" glossed. Offer the sentence frame for item 10: "Feet per hour is fairer because it compares ____ over the same ____."

Extensions

Students can collect the length of their own local utility's water main network from a public utility report and, keeping the fictional Ebbstone crew rates, estimate how many workdays a full replacement would take. A second follow-on asks students to rewrite item 8 as an expression with variables so that any set of spring, summer, and survey values can be substituted.