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

Lesson 6.2Division Expressions with Fractions and Mixed Numbers

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can divide a fraction by a fraction and divide with mixed numbers by multiplying by the reciprocal.

Start hereWords for this lesson

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

WordWhat it meansExample
DividendSpanish: Dividendo The total number being divided into equal groups (the number inside the division bracket). In 8/9 ÷ 2/9, 8/9 is the dividend.
DivisorSpanish: Divisor The number of equal groups you are dividing into (the number outside the division bracket). In 8/9 ÷ 2/9, 2/9 is the divisor.
ReciprocalSpanish: Recíproco A fraction turned upside down. 2/9 → 9/2
QuotientSpanish: Cociente The answer when you divide. 8/9 ÷ 2/9 = 4
SimplifySpanish: Simplificar To write a fraction with smaller numbers that names the same amount, like 2/4 = 1/2. 8/12 → GCF is 4 → 8÷4 / 12÷4 = 2/3
Mixed numberSpanish: Número mixto A whole number and a fraction written together, like 2 1/2. It must be rewritten as an improper fraction before dividing. 2 1/2 = 5/2

How it worksWorked exampledividing a fraction by a fraction

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

A pitcher holds 8/9 liter of lemonade. Each glass takes 2/9 liter. How many glasses can be filled?

  1. Write the division. 8/9 ÷ 2/9
  2. Keep, Change, Flip. Keep 8/9, change ÷ to ×, flip the divisor to 9/2
  3. Multiply across. 8/9 × 9/2 = 72/18
  4. Simplify. 72/18 = 4 glasses

Answer: The pitcher fills 4 glasses that each hold 2/9 liter.

Now you trySame steps, your turn

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

A roll holds 6/7 meter of tape. Each label uses 2/7 meter. How many labels can be made?

  1. Write the division. ÷
  2. Keep, Change, Flip. 6/7 × /
  3. Multiply across. /
  4. Simplify. labels
Answer:number of labels

Say it and write itSentence starters

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

  • To divide by , I multiply by its reciprocal .
  • The quotient counts how many groups of fit inside .
  • I did not need a common denominator because .
  • After multiplying across, I simplified to .

Word bank reciprocaldividenddivisorquotientkeepchangeflipmultiply acrosssimplifygroupsfit insidetape diagram

Watch outA common mistake

A common mistake in Divide Fractions is flipping the wrong fraction: students flip the dividend (the first fraction) instead of the divisor (the second fraction). For example, solving 3/4 ÷ 5/8 by flipping 3/4 to 4/3 and keeping 5/8 gives 4/3 × 5/8 = 20/24 = 5/6 — but only the divisor gets flipped. Keep 3/4, change ÷ to ×, and flip only the divisor 5/8 to its reciprocal 8/5: 3/4 × 8/5 = 24/20 = 6/5. Before you submit, check that the fraction you kept (the dividend) is still facing the same way it started.

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

Lesson 6.2Division Expressions with Fractions and Mixed Numbers

Version ASupported practice

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

Learning target I can divide a fraction by a fraction and divide with mixed numbers by multiplying by the reciprocal.

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

    Sort each division: will the quotient be GREATER than 1 or LESS than 1?

    Groups: Quotient Greater Than 1 Quotient Less Than 1
    • 3/4 ÷ 1/4
    • 5/6 ÷ 1/3
    • 2/3 ÷ 1/6
    • 1/4 ÷ 1/2
    • 1/3 ÷ 2/3
    • 2/5 ÷ 4/5

    Hint Work out each quotient first (keep, change, flip), then compare that quotient to 1.

Part 2Practice the skillshow every step
  1. 2Circle the letter of the best answer. Show how you know.

    What is 1/2 ÷ 1/4?

    1. A1/8
    2. B1/2
    3. C2
    4. D4

    Hint This asks how many 1/4-size pieces fit inside 1/2 — the pieces are smaller than the amount.

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

    What is 3/4 ÷ 1/8?

    1. A3/32
    2. B6
    3. C8/3
    4. D24

    Hint How many 1/8-pieces fit into 3/4? Eighths are small, so expect several.

    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    What is 2/3 ÷ 1/3?

    1. A1/2
    2. B2
    3. C2/9
    4. D6

    Hint Both fractions are thirds — you're counting how many 1/3-pieces sit inside 2/3.

    Show your work
Part 3Apply itreal-world problems
  1. 5Circle the letter of the best answer. Show how you know.

    What is the reciprocal of 3/5?

    1. A1/3
    2. B3/5
    3. C5/3
    4. D5/5

    Hint Re-read 'reciprocal' — it's about the two numbers of the fraction trading places.

    Rewrite each fraction
    Work it out
    Simplify
    1. 1Same size piecesCommon denominator first.
    2. 2OperateOnly then add or subtract.
    3. 3SimplifyDivide out common factors.
  2. 6Circle the letter of the best answer. Show how you know.

    What is the first step in evaluating 2 1/2 ÷ 1/4?

    1. ARewrite 2 1/2 as the improper fraction 5/2
    2. BFlip 2 1/2 to 1/2 2
    3. CDivide 2 by 1/4 and ignore the half
    4. DAdd 2 + 1/2 + 1/4

    Hint Which number in this problem is a mixed number?

    Show your work

Explain your thinkingWhy does 5/6 ÷ 1/12 = 10? How does the bar model show this?

Sentence starter 5/6 ÷ 1/12 = 10 because 5/6 is the same as ___ twelfths, and ___ twelfths divided by 1 twelfth = ___.

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 6.2Division Expressions with Fractions and Mixed Numbers

Version BCore practice

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

Learning target I can divide a fraction by a fraction and divide with mixed numbers by multiplying by the reciprocal.

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

    Divide each pair of fractions using Keep, Change, Flip.

    ProblemKeep, Change, FlipQuotient
    1/2 ÷ 1/6
    3/4 ÷ 1/4
    2/3 ÷ 1/3
    5/8 ÷ 1/4
    Show your work
  2. 2Write the letter of the matching item on each line.

    Match each fraction division to its quotient.

    1. 4/5 ÷ 2/5
    2. 3/4 ÷ 3/8
    3. 5/6 ÷ 1/3
    4. 2/3 ÷ 1/6
    5. 7/8 ÷ 1/4
    6. 1/2 ÷ 3/4
    • A2
    • B2
    • C2 1/2
    • D4
    • E3 1/2
    • F2/3
  3. 3Mark and label each value on the number line.

    Place each quotient on the number line.

    0123456
    Show your thinking
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    A recipe needs 2/3 cup of sugar. You only have a 1/6-cup scoop. How many scoops do you need?

    1. A2 scoops
    2. B3 scoops
    3. C4 scoops
    4. D6 scoops
    Rewrite each fraction
    Work it out
    Simplify
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1Problem2/3 ÷ 1/2
    2. 2Change ÷ to × (but forgot to flip)2/3 × 1/2
    3. 3Multiply2/6
    4. 4Simplify1/3

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

    Correct work
    Correct answer:

Explain your thinkingWhy does 5/6 ÷ 1/12 = 10? How does the bar model show this?

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 6.2Division Expressions with Fractions and Mixed Numbers

ChallengeExtension

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

Learning target I can divide a fraction by a fraction and divide with mixed numbers by multiplying by the reciprocal.

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

    Divide, then check your answer by multiplying: quotient × divisor should equal the dividend.

    ProblemQuotientCheck: Quotient × Divisor
    5/6 ÷ 1/3
    4/9 ÷ 2/3
    7/8 ÷ 1/2
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Division Error

    1. 1Problem2/5 ÷ 3/4
    2. 2Keep, Change, Flip2/5 × 3/4
    3. 3Multiply6/20
    4. 4Simplify3/10

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

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

    Is 3/4 ÷ 1/2 the same as 1/2 ÷ 3/4? Solve both and explain why or why not. What does this tell you about the order in division?

    Write your answer

Explain your thinkingWhy does 5/6 ÷ 1/12 = 10? How does the bar model show this?

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