Deriving and Applying Fraction Division

In this lesson:

• Derive the fraction-division rule from multiplication
• Compute and verify with the rule
• Solve the standard's three word problems
• Confirm: divisor under 1 makes the quotient bigger

What You Will Be Able to Do

By the end, you will:

1. Derive from
2. Apply the rule and verify by multiplying back
3. Solve fraction-division word problems
4. Recognize when the quotient exceeds the dividend

Recap: The Yogurt Visual From Yesterday

Yesterday's number line gave us:

Today: how do we COMPUTE this without the picture?

Division Is the Inverse of Multiplication

For any numbers, means .

So means .

Find the that makes this multiplication true.

Set Up the Equation for q

We want the where:

How do we isolate ? Multiply both sides by something that cancels the 3/4.

Multiply Both Sides by 4/3

The reciprocal of the divisor undoes the divisor.

See the Derivation Step by Step

Three moves: setup → multiply both sides → cancel.

The Cancellation Leaves q Alone

So , leaving:

Verify: 3/4 of 8/9 is 2/3

The standard's own framing: "(2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3."

The General Fraction Division Rule

For any fractions with :

Multiply by the reciprocal of the DIVISOR — only the divisor inverts.

Apply the Rule: Four-Fifths Divided by Two-Thirds

Multiply by reciprocal of divisor:

Verify:

Apply Again: Three-Eighths Divided by Three-Quarters

Verify:

Now Finish the Standard's Three Word Problems

Yesterday we set them up and used pictures. Today we compute and verify.

• Chocolate (partitive)
• Yogurt (measurement)
• Strip-of-land (missing-factor)

Same workflow: identify → set up → compute → verify.

Chocolate: Each Person Gets 1/6 lb

Setup:

Verify:

Yogurt: 8/9 of One Serving

Setup:

Verify:

Strip of Land: 2/3 mi Wide

Setup:

Verify:

Surprise: Division Can Make Things BIGGER

When the divisor is between 0 and 1, the quotient is larger than the dividend.

"How many 1/4s fit in 1/2?" — each 1/4 is small, so several fit.

See It: 1/2 ÷ 1/4 = 2

Two 1/4s fit in 1/2. And .

Try It: 3/4 lb Bag of Nuts

A bag holds 3/4 lb of nuts.

(a) Share equally among 4 people — how much each?

(b) Bagged into 1/8-lb portions — how many portions?

Compute both. Verify both.

Answers and Verifications for the Bag

(a) lb each. Check:

(b) portions. Check:

Key Takeaways for Fraction Division

• Rule:
• Verify every quotient: quotient × divisor = dividend
• Only the divisor inverts (the dividend stays put)
• Divisor between 0 and 1 → quotient is BIGGER than dividend

Coming Up: Fraction Division With Negatives

In Grade 7:

• Fraction division extends to negative numbers (7.NS.A)
• Complex-fraction unit rates become computable (7.RP.A)
• The rule stays the same