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

Lesson 6.1Division Expressions with Fractions and Whole Numbers

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can explain what dividing with a whole number and a unit fraction means — how many groups fit, or how much is in one group — find the quotient with a model, and check it by multiplying.

Why This Matters Agent Reyes finds a 3-foot strip of tape at the crime scene. The lab needs the tape cut into 1/4-foot sections for analysis. How many sections can be made? The answer cracks the code to the locked evidence vault.

Start hereWords for this lesson

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

WordWhat it meansExample
ReciprocalSpanish: Recíproco A fraction turned upside down. 1/4 → 4/1
QuotientSpanish: Cociente The answer when you divide. 8 ÷ 1/2 = 16 is the quotient.
DividendSpanish: Dividendo The total number being divided into equal groups (the number inside the division bracket). In 8 ÷ 1/2, 8 is the dividend.
DivisorSpanish: Divisor The number of equal groups you are dividing into (the number outside the division bracket). In 8 ÷ 1/2, 1/2 is the divisor.
Unit fractionSpanish: Fracción unitaria A fraction with 1 on top, like 1/4. 1/3, 1/8, 1/10
Multiplicative inverseSpanish: Inverso multiplicativo Two numbers whose product is 1; each is the reciprocal of the other. 4 and 1/4 · 2/3 and 3/2

How it worksWorked example

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

A cook has 4 cups of broth. How many 1/3-cup servings can she pour?

  1. Write the division: 4 ÷ 1/3.
  2. Keep 4, change ÷ to ×, flip 1/3 to 3: 4 × 3 = 12.
  3. Check: 12 × 1/3 = 4 cups.

Answer: 12 servings

Now you trySame steps, your turn

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

Let's try 2 ÷ 1/3. The bar model counted 6 thirds inside 2 wholes — now let's see why.

Answer:

Say it and write itSentence starters

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

  • To divide by a fraction, I multiply by its .
  • I wrote over 1 so both numbers are fractions.
  • The quotient is because that many groups fit inside.
  • Dividing by a fraction smaller than one whole makes the quotient .

Word bank reciprocalquotientdividenddivisorunit fractionkeepchangeflipmultiplygroupsfit insidelarger

Watch out!A common mistake

A common mistake in Interpret Division of Fractions is assuming that dividing always makes the answer smaller. For 4 ÷ 1/3, students reason 'dividing shrinks the number' and compute 4 ÷ 3 ≈ 1.3 instead of asking how many 1/3-foot pieces fit into 4 feet (each whole foot holds 3 thirds, so 4 × 3 = 12). Before you submit, check: if the divisor is a fraction smaller than 1, the quotient must be BIGGER than the dividend, not smaller.

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

Lesson 6.1Division Expressions with Fractions and Whole Numbers

Version ASupported practice

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

Learning target I can explain what dividing with a whole number and a unit fraction means — how many groups fit, or how much is in one group — find the quotient with a model, and check it by multiplying.

Why This Matters Agent Reyes finds a 3-foot strip of tape at the crime scene. The lab needs the tape cut into 1/4-foot sections for analysis. How many sections can be made? The answer cracks the code to the locked evidence vault.

Part 1Understand the ideamodels and meaning
  1. 1Circle the letter of the best answer. Show how you know.

    What does 3 ÷ 1/4 mean?

    1. AHow many 1/4-size pieces fit into 3
    2. B1/4 of 3
    3. C3 groups of 1/4
    4. D3 minus 1/4

    Hint Re-read the expression: 3 is the total amount and 1/4 is the size of one piece. What question could that setup be asking?

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

    Which expression means 'how many 1/3-size pieces are in 2'?

    1. A1/3 ÷ 2
    2. B2 ÷ 1/3
    3. C2 × 1/3
    4. D2 + 1/3

    Hint Find the two jobs: which number is the total being split, and which is the size of each piece?

    Show your work
Part 2Practice the skillshow every step
  1. 3Circle the letter of the best answer. Show how you know.

    What is 2/3 ÷ 4?

    1. A1/6
    2. B2/7
    3. C3/8
    4. D8/3

    Hint The whole number is second, so write the 4 as 4/1.

    Show your work
Part 3Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    A rope is 4 feet long. How many 1/2-foot pieces can be cut from it?

    1. A1/2 piece
    2. B2 pieces
    3. C4 pieces
    4. D8 pieces

    Hint Note the numbers: the rope is 4 feet long, and every piece is 1/2 foot — smaller than a whole foot.

    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.
  2. 5Circle the letter of the best answer. Show how you know.

    A half-pan of brownies is shared equally among 4 people. How much of the whole pan does each get?

    1. A1/8
    2. B1/2
    3. C2
    4. D8

    Hint Draw half a pan, then split that half into 4 equal parts.

    Show your work
  3. 6Circle the letter of the best answer. Show how you know.

    Agent Cole has 3/4 of a pound of fingerprint powder and splits it equally into 2 kits. How much powder goes in each kit?

    1. A1 1/2 pounds
    2. B1/8 pound
    3. C3/8 pound
    4. D3/4 pound

    Hint Draw 3/4 of a bar, then cut each of those three fourths in half.

    Show your work

Explain your thinkingWhy does dividing by a unit fraction always give a number larger than the dividend?

Sentence starter Dividing by 1/3 means finding how many ___ fit into ___, so the quotient is ___ than the dividend.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
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 6.1Division Expressions with Fractions and Whole Numbers

Version BCore practice

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

Learning target I can explain what dividing with a whole number and a unit fraction means — how many groups fit, or how much is in one group — find the quotient with a model, and check it by multiplying.

Why This Matters Agent Reyes finds a 3-foot strip of tape at the crime scene. The lab needs the tape cut into 1/4-foot sections for analysis. How many sections can be made? The answer cracks the code to the locked evidence vault.

Part 1Understand the ideamodels and meaning
  1. 1Write the letter of the matching item on each line.

    Match each division expression to its quotient.

    1. 5 ÷ 1/2
    2. 6 ÷ 1/3
    3. 2 ÷ 1/6
    4. 4 ÷ 1/4
    5. 3 ÷ 1/5
    6. 7 ÷ 1/2
    • A10
    • B18
    • C12
    • D16
    • E15
    • F14
  2. 2Use the model to solve. Show your work.
    3 wholes split into sixths — count all the 1/6 pieces1/61/61/61/61/61/61/61/61/61/61/61/61/61/61/61/61/61/6

    How many 1/6-size pieces fit into 3 wholes?

    Show your work
    Answer:
  3. 3Mark and label each value on the number line.

    Place each result on the number line

    024681012141618202224
    Show your thinking
  4. 4Circle the letter of the best answer. Show how you know.

    A shelf is 5 feet long. Books are each 1/4 foot wide. How many books fit on the shelf? Which expression represents this?

    1. A1/4 ÷ 5 = 1/20 book
    2. B5 ÷ 1/4 = 20 books
    3. C5 × 1/4 = 1 1/4 books
    4. D5 + 1/4 = 5 1/4 books
    Show your work
  5. 5Write the letter of the matching item on each line.

    Match each fraction ÷ whole number expression to its quotient.

    1. 3/4 ÷ 2
    2. 2/3 ÷ 4
    3. 5/6 ÷ 2
    4. 3/5 ÷ 3
    5. 4/9 ÷ 2
    6. 5/8 ÷ 5
    • A3/8
    • B1/6
    • C5/12
    • D1/5
    • E2/9
    • F1/8
Part 2Apply itreal-world problems
  1. 6Circle the letter of the best answer. Show how you know.

    A 5/6-mile stretch of the evidence trail is split equally among 3 search teams. How much of a mile does each team walk?

    1. A1/18 mile
    2. B5/2 miles
    3. C5/6 mile
    4. D5/18 mile
    Write any whole number over 1
    Keep · Change · Flip
    Multiply across
    Simplify · label the units

Explain your thinkingWhy does dividing by a unit fraction always give a number larger than the dividend?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
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 6.1Division Expressions with Fractions and Whole Numbers

ChallengeExtension

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

Learning target I can explain what dividing with a whole number and a unit fraction means — how many groups fit, or how much is in one group — find the quotient with a model, and check it by multiplying.

Why This Matters Agent Reyes finds a 3-foot strip of tape at the crime scene. The lab needs the tape cut into 1/4-foot sections for analysis. How many sections can be made? The answer cracks the code to the locked evidence vault.

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

    Complete the pattern: as the unit fraction divisor gets smaller, what happens to the quotient?

    ExpressionQuotientPattern
    2 ÷ 1/2
    2 ÷ 1/3
    2 ÷ 1/4
    2 ÷ 1/6
    2 ÷ 1/10
    Show your work
  2. 2Complete the table. Show how you found each value.

    Complete the pattern: the same 3/4 shared among more and more people.

    ExpressionQuotientPattern
    3/4 ÷ 2
    3/4 ÷ 3
    3/4 ÷ 4
    3/4 ÷ 6
    Show your work
Part 2Explain and correctreasoning
  1. 3Find the mistake. Explain it, then show the correct work.

    Find the Interpretation Error

    1. 1ProblemInterpret 4 ÷ 1/5
    2. 2Student says4 ÷ 1/5 means 1/5 of 4
    3. 3Student's answer4/5

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

    Correct work
    Correct answer:
  2. 4Answer in complete sentences.

    Dividing 6 by 1/2 gives 12, but dividing 6 by 2 gives 3. How can dividing by a smaller number (1/2) give a bigger answer than dividing by a larger number (2)? Explain using a real-world example.

    Write your answer
  3. 5Find the mistake. Explain it, then show the correct work.

    Spot the Mistake — When the Numerator Is Not 1

    1. 1ProblemA 3/4-pound bag of powder is split equally into 2 kits. Find 3/4 ÷ 2.
    2. 2Student writesThe numerator is not a 1, so I will divide straight across: 3 ÷ 2 on top and 4 ÷ 2 on the bottom.
    3. 3Student's answer1.5/2 of a pound

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

    Correct work
    Correct answer:

Explain your thinkingWhy does dividing by a unit fraction always give a number larger than the dividend?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help