Lesson 6-1: Division Expressions with Fractions and Whole Numbers

Warm-Up 1

What is 3 ÷ ½?

Vocabulary: The dividend is the number being divided (3). The divisor is the number you divide by (½).
Hint: Think: "How many halves fit into 3 wholes?"

✓Correct! There are 6 halves in 3 wholes. 3 ÷ ½ = 3 × 2 = 6.

✗Not quite. Think about how many halves fit into 3. Each whole has 2 halves, so 3 × 2 = 6. The answer is C.

Warm-Up 2

True or False: ¼ ÷ 2 = ½

Hint: Dividing a fraction by a whole number makes it smaller. Does ¼ split into 2 equal parts give you ½?

✓Correct! ¼ ÷ 2 = ⅛, not ½. Dividing a fraction by a whole number gives a smaller result.

✗¼ ÷ 2 means splitting one-fourth into 2 equal pieces. Each piece is ⅛, not ½. The statement is false.

Practice 1

Which expression shows how to divide ⅓ ÷ 4 using the reciprocal?

Vocabulary: The reciprocal of a number is its "flip." The reciprocal of 4 is ¼.
Rule: To divide by a number, multiply by its reciprocal.

Explain your reasoning:

Sentence frame: "To divide ⅓ by 4, I multiply ⅓ by the reciprocal of 4, which is ___. So the expression is ___."

✓Correct! ⅓ ÷ 4 = ⅓ × ¼ = 1/12. Multiply by the reciprocal.

✗To divide by 4, multiply by its reciprocal (¼). The answer is B: ⅓ × ¼.

Practice 2

Find each quotient.

Strategy: For a whole number ÷ fraction, multiply the whole number by the reciprocal of the fraction. For a fraction ÷ whole number, multiply the fraction by 1 over the whole number.

a. 5 ÷ ⅕ =

b. ½ ÷ 3 =

c. 4 ÷ ⅓ =

Sentence frame: "I found the quotient by multiplying ___ by the reciprocal ___. The answer is ___."

✓All correct! 5 ÷ ⅕ = 25, ½ ÷ 3 = 1/6, and 4 ÷ ⅓ = 12.

✗Some answers need another look. Remember: multiply by the reciprocal. 5 × 5 = 25, ½ × ⅓ = 1/6, and 4 × 3 = 12.

Practice 3

Sort each expression by whether the quotient is greater than 1 or less than 1.

Remember: When you divide a whole number by a fraction less than 1, the quotient is greater than the dividend. When you divide a fraction by a whole number greater than 1, the quotient is less than the original fraction.

2 ÷ ⅓

½ ÷ 5

4 ÷ ½

¼ ÷ 2

6 ÷ ¼

⅓ ÷ 4

Quotient > 1

Quotient < 1

✓All correct! Dividing a whole number by a fraction gives a quotient greater than 1. Dividing a fraction by a whole number gives a quotient less than 1.

✗Some items are in the wrong group. Whole number ÷ fraction → greater than 1. Fraction ÷ whole number → less than 1. Try again!

Practice 4

True or False: A baker has 6 cups of sugar and uses ½ cup per batch of cookies. She can make 12 batches.

Think about it: This is a grouping problem. How many groups of ½ fit into 6? Set up: 6 ÷ ½ = ?

Explain your reasoning:

Sentence frame: "6 ÷ ½ = ___ because there are ___ halves in 6 wholes. So the baker can make ___ batches."

✓Correct! 6 ÷ ½ = 12. There are 12 half-cups in 6 cups, so she can make 12 batches.

✗6 ÷ ½ = 6 × 2 = 12. There are 12 half-cups in 6 cups. The answer is True.

Practice 5

Marcus has ¾ of a pizza. He shares it equally among 3 friends. How much pizza does each friend get?

Set up: ¾ ÷ 3. Multiply by the reciprocal: ¾ × ⅓ = ?

Show your work:

Sentence frame: "¾ ÷ 3 = ¾ × ___ = ___. Each friend gets ___ of the pizza."

✓Correct! ¾ ÷ 3 = ¾ × ⅓ = 3/12 = ¼. Each friend gets one-fourth of the pizza.

✗¾ ÷ 3 = ¾ × ⅓ = 3/12 = ¼. The answer is A.

Challenge

A ribbon is 5 yards long. You need pieces that are each ⅔ of a yard. How many pieces can you cut? Will there be any ribbon left over? Explain your reasoning.

Think about it: Set up the division: 5 ÷ ⅔. Multiply by the reciprocal: 5 × &frac32; = ?
If the answer is not a whole number, the whole-number part tells you how many full pieces. The remainder tells you how much is left.

How many full pieces?

Explain your work and whether there is leftover ribbon:

Sentence frame: "5 ÷ ⅔ = 5 × ___ = ___. I can cut ___ full pieces. There is ___ yard left over because ___."

🟠 Level 0 — Extra Support

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