Water for Schools
Menu

Lesson plan Grade 6 Math

Vayle Plant's Feed Jugs: Dividing Fractions of a Gallon

A grade 6 lesson in a treatment plant's chemical room: students divide fractions and mixed numbers to find how long a jug of solution lasts at a set feed rate.

Objectives

  1. Students will interpret a quotient of fractions as the number of equal portions in a larger amount.
  2. Students will divide mixed numbers by fractions and check the result with a drawn fraction model.
  3. Students will explain what the leftover amount means when a jug does not divide evenly.

Vocabulary

Key words: quotient, reciprocal, mixed number, feed rate, solution

Teacher overview

The chemical room at a small treatment plant holds jugs of solution and a small pump that draws a set amount every hour. An operator planning a shift asks the same question of every jug: how many hours will it last at that rate? Answering it means dividing one fraction by another. The misconception this lesson addresses is the belief that dividing always makes a number smaller. Students who hold it will reject an answer such as ten rinses from five sixths of a gallon, so the lesson pours a physical example before any symbolic work and keeps returning to the question of how many portions fit.

Procedure

Hook (5 min): Fill a clear pitcher to three quarters of a gallon and pour it into small identical cups that each hold one eighth of a gallon, counting aloud as you go. Stop at six and write on the board: three quarters divided by one eighth equals 6. Ask students why the answer is larger than either number in the problem.

Direct instruction (12 min): Work two examples set in the chemical room of the fictional Vayle plant. First, model three quarters divided by one eighth with a drawn bar cut into eighths, then with the reciprocal rule, showing that both give 6. Second, model two thirds divided by one sixth the same way for an answer of 4. Name the question the division answers each time: how many portions of the smaller size fit in the amount on hand.

Guided practice (10 min): Pairs work two problems that are not on the handout: how many one sixth gallon doses come from one half gallon, which is 3, and how many hours a 3 gallon jug lasts at three quarters of a gallon per hour, which is 4. Require a drawn model beside each answer, and check both models before pairs continue.

Independent practice (13 min): Students complete handout questions 1 through 8. Circulate during questions 7 and 8, where the jug does not divide evenly, and ask students to point to the leftover portion in their drawing before writing an answer.

Closure (5 min): Ask every student to show their answer to handout question 5 on their fingers, ten fingers and then two. Look for that answer of 12 from at least four of every five students. If fewer students have it, rework question 5 on the board with a bar model before the period ends.

Vayle Plant's Feed Jugs: Dividing Fractions of a Gallon — Grade 6 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Vayle plant keeps its treatment chemicals as a mixed solution in jugs. A small pump draws a set amount each hour. Solve each problem and draw a fraction model for at least three of them.

  1. A jug holds three quarters of a gallon. The pump draws one eighth of a gallon each hour. How many hours does the jug last?
  2. A jug holds 2 and one quarter gallons. The pump draws three eighths of a gallon each hour. How many hours does the jug last?
  3. An operator has five sixths of a gallon of rinse solution and uses one twelfth of a gallon for each rinse. How many rinses does the operator get?
  4. A drum holds 4 and one half gallons. It is poured into jugs that each hold one half gallon. How many jugs does it fill?
  5. A tank holds 8 gallons of solution. It is poured into jugs that each hold two thirds of a gallon. How many jugs does it fill?
  6. Three fifths of a gallon of solution is shared equally among 4 feed lines. How many gallons does each line get?
  7. A drum holds 5 gallons. It is poured into jugs that each hold three quarters of a gallon. How many jugs can be filled completely?
  8. Start again with the 5 gallon drum and the three quarter gallon jugs from question 7. After every full jug is filled, how many gallons are left in the drum?

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: 6 hours.
    Three quarters divided by one eighth is three quarters times eight, which is 24 divided by 4, or 6.
  2. Answer: 6 hours.
    Write 2 and one quarter as nine quarters. Nine quarters divided by three eighths is nine quarters times eight thirds, which is 72 divided by 12, or 6.
  3. Answer: 10 rinses.
    Five sixths divided by one twelfth is five sixths times twelve, which is 60 divided by 6, or 10.
  4. Answer: 9 jugs.
    Write 4 and one half as nine halves. Nine halves divided by one half is nine halves times two, which is 18 divided by 2, or 9.
  5. Answer: 12 jugs.
    Eight divided by two thirds is eight times three halves, which is 24 divided by 2, or 12.
  6. Answer: 3/20 gallon.
    Three fifths divided by 4 is three fifths times one quarter, which is three twentieths of a gallon. As a decimal that is 0.15 gallon.
  7. Answer: 6 jugs.
    Five divided by three quarters is five times four thirds, which is 20 divided by 3, or 6 and two thirds. Only 6 jugs can be filled completely.
  8. Answer: 1/2 gallon.
    The 6 full jugs hold 6 times three quarters, which is 4 and one half gallons. Five gallons minus 4 and one half gallons leaves one half gallon in the drum.

Differentiation

Support

Give these students a printed bar divided into twenty-fourths for each problem, so the drawing step at the fictional Vayle plant is shading rather than partitioning. Restrict their required set to questions 1, 3, 4, and 5, all of which divide evenly, and add question 7 only after those four are correct.

Extension

Ask these students to write their own problem for the fictional Vayle plant in which a 3 and one third gallon drum is poured into jugs of an equal size they choose, so the answer comes out a whole number, and to prove their choice works. Then have them find a jug size for the same drum that leaves exactly one third of a gallon over.

ELL supports

Pre-teach jug, feed rate, and leftover with the pitcher and small cups from the hook still on the desk, pouring once for each word. Provide this sentence frame for questions 7 and 8: "Each jug holds ___ gallon, so ___ full jugs can be filled and ___ gallon is left."

Extensions

Have students plan a full 24 hour day for the pump at the fictional Vayle plant, choosing a jug size and a feed rate so that no jug is left partly full at the end of a shift. Asking them to justify the pair of numbers in writing turns the division into a scheduling decision, which is the form the question takes in a real chemical room.