Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.11Fraction Division Problem Solving

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can solve real-world problems by writing and solving fraction division equations.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
ModelSpanish: Modelo A picture or math way to show a problem so you can solve it. A bar model showing 3/4 split into 1/8-size pieces, or the equation 3/4 ÷ 1/8 = 6
EquationSpanish: Ecuación A math sentence with an equal sign showing both sides are the same. 3/4 ÷ 1/8 = 6 — the left side (division) equals the right side (answer)
SolutionSpanish: Solución A number that makes the equation or inequality true. In 3/4 ÷ 1/8 = 6, the solution is 6 portions
ReasonablenessSpanish: Razonabilidad Checking if your answer makes sense. 3 ÷ 1/4 = 12. Is 12 reasonable? Yes, because 1/4 is small so many pieces fit into 3.
Inverse operationsSpanish: Operaciones inversas Two math actions that undo each other, like × and ÷. If 3/4 ÷ 1/8 = 6, then 6 × 1/8 = 6/8 = 3/4. Multiplication checks division.

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Agent Torres has 2/3 gallon of dust and each test uses 1/6 gallon. I ask: how many tests fit?

  1. The total 2/3 is the dividend. The size of each test 1/6 is the divisor. My equation is 2/3 ÷ 1/6.
  2. Keep 2/3, change to ×, flip 1/6 to 6/1: 2/3 × 6/1 = 12/3 = 4.
  3. So she can run 4 tests. I check by multiplying back: 4 × 1/6 = 4/6 = 2/3. It matches the total, so the answer is reasonable.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

A ribbon is 3/4 yard and each bow uses 1/4 yard. The total 3/4 is the dividend; 1/4 is the divisor.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • The total amount is the because it is being split into groups. The size of each group is the because we are dividing by that amount.

Word bank ModelEquationSolutionReasonablenessInverse operationsdividenddivisortotalgroupsplitequation

Watch outA common mistake

A common mistake in Fraction Division Problem Solving is reversing which fraction is the dividend and which is the divisor when setting up the equation from a word problem. For example: a pitcher holds 3/4 cup of punch, and each serving is 1/8 cup — how many servings? The correct setup is total ÷ group size = 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6 servings. A student who reverses the setup calculates 1/8 ÷ 3/4 = 1/8 × 4/3 = 4/24 = 1/6, an answer smaller than 1 that can't possibly be a serving count. Before you submit, check: does my answer make sense as a NUMBER OF GROUPS? If the setup gives a fraction less than 1 for a 'how many' question, you likely swapped the dividend and divisor.

Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.11Fraction Division Problem Solving

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can solve real-world problems by writing and solving fraction division equations.

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Sort each word problem: should you DIVIDE or MULTIPLY to solve it?

    Groups: Division Problem Multiplication Problem
    • How many 1/3-cup scoops fit into 2 cups? (2 ÷ 1/3)
    • A ribbon is 3/4 yard. Each piece is 1/8 yard. How many pieces? (3/4 ÷ 1/8)
    • You run 1/2 mile, 3 times. How far total? (1/2 × 3)
    • A recipe uses 2/3 cup. You make half a recipe. How much? (2/3 × 1/2)

    Hint Re-read each story: does it ask 'how many fit?' or 'how much is a part of the amount?'

Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    A ribbon is 4/5 of a yard long. Each bow uses 1/10 of a yard. Which equation finds how many bows can be made?

    1. A1/10 ÷ 4/5
    2. B4/5 ÷ 1/10
    3. C4/5 × 1/10
    4. D4/5 + 1/10

    Hint Name the roles: 4/5 yard is ALL the ribbon; 1/10 yard is what one bow uses.

    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    A detective has 1/2 gallon of solution. Each test uses 1/8 gallon. How many tests can be run?

    1. A1/16
    2. B1/4
    3. C4
    4. D16

    Hint The total is 1/2 gallon; each test drains 1/8 gallon — you're counting tests.

    Rewrite each fraction
    Work it out
    Simplify
    1. 1Same size piecesCommon denominator first.
    2. 2OperateOnly then add or subtract.
    3. 3SimplifyDivide out common factors.
  3. 4Circle the letter of the best answer. Show how you know.

    A trail is 3/4 mile long. A jogger runs laps of 1/4 mile each. How many laps does the jogger complete?

    1. A1/3 lap
    2. B3 laps
    3. C3/16 lap
    4. D4 laps

    Hint The trail is 3/4 mile total, and every lap covers exactly 1/4 mile.

    Rewrite each fraction
    Work it out
    Simplify
    1. 1Same size piecesCommon denominator first.
    2. 2OperateOnly then add or subtract.
    3. 3SimplifyDivide out common factors.
  4. 5Circle the letter of the best answer. Show how you know.

    A board is 5/6 foot long. Each shelf bracket needs 1/6 foot. How many brackets fit?

    1. A1/36
    2. B5
    3. C5/36
    4. D6

    Hint Board = 5/6 foot, each bracket = 1/6 foot — both are already in sixths.

    Write any whole number over 1
    Keep · Change · Flip
    Multiply across
    Simplify · label the units
    1. 1Whole over 13 becomes 3/1.
    2. 2KeepThe first fraction does not move.
    3. 3Change · Flip÷ becomes ×; flip the second fraction.
    4. 4CheckDividing by a piece smaller than 1 gives a BIGGER answer.
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1ProblemA detective has 4/5 gallon of cleaning fluid. Each spray bottle holds 1/5 gallon. How many bottles can she fill?
    2. 2Student's equation1/5 ÷ 4/5
    3. 3Solve1/5 × 5/4 = 5/20 = 1/4
    4. 4Answer1/4 of a bottle

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Hint Which value is the total being split up — 4/5 or 1/5?

    Correct work
    Correct answer:

Explain your thinkingHow do you decide which value is the dividend and which is the divisor in a word problem?

Sentence starter The total amount is the ___ because it is being split into groups. The size of each group is the ___ because we are dividing by that amount.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.11Fraction Division Problem Solving

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can solve real-world problems by writing and solving fraction division equations.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Solve each word problem by writing and solving an equation.

    ProblemEquationSolution
    A board is 7/8 foot long. Each bracket needs 1/4 foot. How many FULL brackets can be cut?
    A jug holds 5/6 liter. Each cup holds 1/3 liter. How many FULL cups can be filled?
    A path is 3/4 mile long. A runner does laps of 3/8 mile. How many laps fit?
    Show your work
  2. 2Write the name of the correct group on each line.

    Sort: is the equation and solution CORRECT, or is there an error?

    Groups: Correct Equation and Solution Incorrect Equation or Solution
    • A tank has 5/6 gallon. Each bucket holds 1/3 gallon. → 5/6 ÷ 1/3 = 2 1/2 buckets
    • A pipe is 2/3 foot long. Each piece is 1/6 foot. → 2/3 ÷ 1/6 = 4 pieces
    • A bag has 3/4 lb. Each scoop is 1/8 lb. → 3/4 ÷ 1/8 = 6 scoops
    • A rope is 1/2 yard. Each section is 1/4 yard. → 1/4 ÷ 1/2 = 1/2 section
    • A jug has 4/5 liter. Each cup is 2/5 liter. → 4/5 × 2/5 = 8/25 cups
    • A shelf is 7/8 foot. Each book is 1/4 foot. → 7/8 ÷ 1/4 = 7/32 books
  3. 3Use the model to solve. Show your work.
    3/4 = 6/8 → six 1/8-size pieces fit1/81/81/81/81/81/8

    How many 1/8-pound molds can be made from 3/4 pound?

    Show your work
    Answer:
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    Marcus solved 2/3 ÷ 1/4 and got 8/3 = 2 2/3. He says 'That can't be right because I started with less than 1.' Is Marcus's math correct? Is his reasoning correct?

    1. AHis math and reasoning are both correct — the answer should be less than 1
    2. BHis math is wrong — the answer should be 1/6
    3. CHis math is correct (2 2/3), but his reasoning is wrong — dividing by a small fraction gives a larger result
    4. DHis math is wrong — the answer should be 8/12
    Show your work
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1ProblemA chemist has 3/4 liter of solution. Each test tube holds 1/8 liter. How many test tubes can be filled?
    2. 2Student's equation3/4 ÷ 1/8 = 3/4 × 1/8
    3. 3Solve3/4 × 1/8 = 3/32
    4. 4Answer3/32 of a test tube

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingHow do you decide which value is the dividend and which is the divisor in a word problem?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.11Fraction Division Problem Solving

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can solve real-world problems by writing and solving fraction division equations.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Solve each multi-step problem. Show the equation and verify with multiplication.

    ProblemEquationSolutionCheck (quotient × divisor = dividend)
    A chef has 2 1/2 cups of flour. Each batch of cookies needs 1/4 cup. How many batches?
    A detective has 1 1/3 gallons of solution. Each test uses 2/3 gallon. How many tests?
    A board is 3 3/4 feet long. Each piece must be 5/8 foot. How many pieces?
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Problem-Solving Error

    1. 1ProblemA tank has 3/4 gallon of water. Each bucket holds 1/6 gallon. How many buckets can be filled?
    2. 2Student's equation1/6 ÷ 3/4
    3. 3Solve1/6 × 4/3 = 4/18 = 2/9
    4. 4Answer2/9 buckets

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 3Answer in complete sentences.

    Write your own fraction division word problem where the answer is 8. Then show the equation, solve it step by step, and explain how you know your answer is reasonable.

    Write your answer

Explain your thinkingHow do you decide which value is the dividend and which is the divisor in a word problem?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help