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Math problem set Grade 7 Math

Longmarsh Plant Operator Rounds: A Grade 7 Unit-Rate Set

Students find rounds-per-hour rates from fractional shift times, then compute regular pay, overtime, and the class sessions needed to finish a renewal requirement.

Objectives

  1. Students will compute unit rates from quantities divided by fractional hours.
  2. Students will calculate regular and overtime pay for a shift using mixed numbers.
  3. Students will determine how many training sessions meet a stated renewal requirement.

Vocabulary

Key words: rounds, unit rate, overtime, certification

Teacher overview

Longmarsh is a fictional water treatment plant, and every log entry, wage, and training figure below was written for this problem set. Operators walk rounds through the plant on a schedule, reading meters and checking tanks, and their logs rarely land on whole hours. Operators typically enter the field with a high school diploma, on-the-job training, and a state license earned by exam, while engineers, laboratory chemists, and utility managers at the same plant arrive through certificate programs, four-year degrees, or graduate study. The misconception this set corrects is multiplying by the fractional time rather than dividing by it, which inflates the rate whenever a shift runs longer than an hour and shrinks it whenever the shift is a fraction of an hour.

Longmarsh Plant Operator Rounds: A Grade 7 Unit-Rate Set — Grade 7 Math
Name: ____________________________ Date: ______________

Problems

Longmarsh is a fictional plant, and all of these numbers were written for this problem set. Operators log what they finish and how long it took. Show each rate with its unit.

  1. An operator finished 3 rounds of the plant in 1 1/2 hours. What is that rate in rounds per hour?
  2. A second operator read 14 meters in 2/3 of an hour. What is that rate in meters per hour?
  3. An operator filled 3/4 of a chemical day-tank in 2/5 of an hour. At that rate, how many tanks per hour is the operator filling?
  4. Longmarsh pays this operator $26.40 an hour. The operator worked 7 1/2 regular hours. What is the regular pay for the shift?
  5. Overtime at Longmarsh pays 1.5 times the regular rate. The operator worked 2 1/4 overtime hours at $26.40 an hour as the base. What is the overtime pay?
  6. Use problems 4 and 5. What is the operator's total pay for the day?
  7. Longmarsh operators must record 20 hours of training every two years to renew a license. This operator has 12.5 hours. How many more hours are needed?
  8. Each training class session is worth 1 1/4 hours. How many sessions does the operator need to reach the 7.5 hours still required?
  9. Two operators worked different shift lengths last week. One logged more total rounds, and the other had a higher rounds-per-hour rate. Explain why the rate is the better number for comparing how quickly each operator worked.

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 rounds per hour.
    Divide rounds by hours: 3 / 1.5 = 2 rounds per hour. Written as fractions, 3 / (3/2) = 3 x (2/3) = 2.
  2. Answer: 21 meters per hour.
    Divide meters by hours: 14 / (2/3) = 14 x (3/2) = 21 meters per hour.
  3. Answer: 1.875 tanks per hour.
    Divide the tank fraction by the hours: (3/4) / (2/5) = (3/4) x (5/2) = 15/8 = 1.875 tanks per hour.
  4. Answer: $198.00.
    Multiply the rate by the hours: 26.40 x 7.5 = 198.00 in regular pay.
  5. Answer: $89.10.
    The overtime rate is 26.40 x 1.5 = 39.60 an hour. Multiply by the overtime hours: 39.60 x 2.25 = 89.10.
  6. Answer: $287.10.
    Add the two amounts: 198.00 + 89.10 = 287.10 for the day.
  7. Answer: 7.5 hours.
    Subtract what is already recorded: 20 - 12.5 = 7.5 hours still required.
  8. Answer: 6 sessions.
    Divide the hours still needed by the hours in a session: 7.5 / 1.25 = 6 sessions.
  9. Answer: The rate measures speed, while the total also depends on how long the shift was.
    An operator scheduled for ten hours will log more rounds than one scheduled for six if both work at the same pace. Dividing rounds by hours puts both operators on the same one-hour basis, which is what problems 1 through 3 do.

Differentiation

Support

Give students the reciprocal step written out for problems 1 through 3 at the fictional Longmarsh plant, showing that dividing by 2/3 is the same as multiplying by 3/2. For problems 4 through 6, provide a three-line pay stub template labeled regular pay, overtime pay, and total.

Extension

Ask students to find the hourly rate the fictional plant would have to pay so that the same 7 1/2 regular hours and 2 1/4 overtime hours produced a total of exactly $310.00, then to explain why scaling the $26.40 rate by the ratio of the two totals (310 / 287.10) gives the same answer.

ELL supports

Preteach "rounds," "day-tank," and "overtime" with a simple plant map of the fictional Longmarsh plant showing the stops an operator makes in order. Provide the sentence frame for problem 9: "The rate is better because it compares ____ for the same amount of ____."

Extensions

Students can interview a local utility about how operator shifts are scheduled and how license renewal hours are earned in their state, then compare what they learn with the fictional Longmarsh requirements. A second follow-on asks students to build a table of rounds per hour for five made-up shift logs and to rank the shifts, explaining which comparison would change if only total rounds were used.