Division Expressions with Fractions and Mixed Numbers

Divide fractions and mixed numbers using models, improper fractions, and the reciprocal strategy.

6.NOS.A.1 · Fractions & Mixed Numbers
Level
Guided practice with vocabulary support

🟠 Level 0 — Extra Support

Sentence starters: “First, I…” · “The answer is… because…” · “I know this because…”
W

Warm-Up

2 questions
Warm-Up 1
What is 2½ written as an improper fraction?
Vocabulary: A mixed number has a whole part and a fraction part. An improper fraction has a numerator larger than or equal to the denominator.
How to convert: Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.
✓Correct! 2 × 2 + 1 = 5. So 2½ = 5/2.
✗Multiply: 2 × 2 = 4, then add 1: 4 + 1 = 5. Keep the denominator 2. The answer is B: 5/2.
Warm-Up 2
True or False: To divide by a fraction, you multiply by its reciprocal.
Vocabulary: The reciprocal of a fraction is the "flip." For example, the reciprocal of ⅔ is 3/2.
✓Correct! Dividing by a fraction is the same as multiplying by its reciprocal. This is the key rule for fraction division.
✗This is actually true. The "Keep, Change, Flip" rule means you keep the first fraction, change ÷ to ×, and flip the second fraction.
P

Practice

5 questions
Practice 1
What is 2½ ÷ ¼?
Steps:
1. Convert the mixed number: 2½ = 5/2
2. Multiply by the reciprocal of ¼: 5/2 × 4/1
3. Simplify: 20/2 = ?
Show your steps:

Sentence frame: "I converted 2½ to ___. Then I multiplied by the reciprocal of ¼, which is ___. The answer is ___."

✓Correct! 5/2 × 4/1 = 20/2 = 10. There are 10 quarter-pieces in 2½.
✗Convert: 2½ = 5/2. Multiply by reciprocal: 5/2 × 4 = 20/2 = 10. The answer is C.
Practice 2
Convert each mixed number to an improper fraction, then find the quotient.
Strategy: Convert all mixed numbers to improper fractions first. Then multiply the first fraction by the reciprocal of the second.
a. 1⅓ ÷ ⅔ =
b. 3½ ÷ ½ =
c. ¾ ÷ 1½ =

Sentence frame: "I converted ___ to ___. Then I multiplied by the reciprocal ___ and got ___."

✓All correct! 4/3 × 3/2 = 2, 7/2 × 2 = 7, and 3/4 × 2/3 = 1/2.
✗Check your work: 1⅓ = 4/3, so 4/3 × 3/2 = 2. 3½ = 7/2, so 7/2 × 2 = 7. ¾ ÷ 3/2 = 3/4 × 2/3 = 1/2.
Practice 3
Match each division expression to its equivalent multiplication expression.
Remember: To divide by a fraction, multiply by its reciprocal. Flip the divisor and change ÷ to ×.
5/3 × 2/1
7/4 × 3/1
3/2 × 4/3
1⅔ ÷ ½
1¾ ÷ ⅓
1½ ÷ ¾
✓All matched correctly! Remember: convert the mixed number, then multiply by the reciprocal of the divisor.
✗Some matches are incorrect. Convert each mixed number to an improper fraction, then flip the divisor and multiply. Try again!
Practice 4
A recipe calls for ⅔ cup of flour. You have 3⅓ cups. True or False: You can make 5 batches of the recipe.
Set up: 3⅓ ÷ ⅔. Convert: 3⅓ = 10/3. Multiply by reciprocal: 10/3 × 3/2 = ?
Show your reasoning:

Sentence frame: "3⅓ ÷ ⅔ = ___ × ___ = ___. Since the quotient is ___, you can make ___ batches."

✓Correct! 10/3 × 3/2 = 30/6 = 5. You have exactly enough flour for 5 batches.
✗10/3 × 3/2 = 30/6 = 5. The answer is True — you can make exactly 5 batches.
Practice 5
What is ⅚ ÷ 1¼?
Steps:
1. Convert 1¼ to an improper fraction: 5/4
2. Multiply ⅚ by the reciprocal of 5/4: ⅚ × 4/5
3. Simplify
Explain your steps:

Sentence frame: "I converted 1¼ to ___. Then I multiplied ⅚ × ___ = ___."

✓Correct! ⅚ × 4/5 = 20/30 = 2/3.
✗Convert 1¼ = 5/4. Then: ⅚ × 4/5 = 20/30 = 2/3. The answer is B.
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Challenge

1 question
Challenge
A carpenter has a board that is 4½ feet long. She needs to cut it into pieces that are each ¾ of a foot long. How many full pieces can she cut? How much board is left over? Show all your work.
Think about it: Set up: 4½ ÷ ¾. Convert 4½ to 9/2. Multiply by the reciprocal: 9/2 × 4/3 = ?
If the answer is not a whole number, the whole-number part = full pieces. The fractional part × ¾ = leftover length.
How many full pieces?
Show your work and explain the leftover:

Sentence frame: "4½ ÷ ¾ = ___ × ___ = ___. I can cut ___ full pieces with ___ foot left over."