Dividing Fractions by Fractions

Lesson 1 of 1: The Number System

In this lesson:

• Interpret fraction division two ways
• Use number lines and area models
• Apply the invert-and-multiply algorithm

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Explain what fraction division means using measurement and partitive interpretations
2. Use number line and area models to represent fraction division
3. Compute quotients of fractions using the invert-and-multiply algorithm
4. Justify why invert-and-multiply works using multiplication and division as inverse operations
5. Solve and interpret word problems involving division of fractions by fractions

What Does Division Actually Mean?

— but there are two different stories:

• Measurement: "12 cookies, bags of 4. How many bags?" → 3 bags
• Partitive: "12 cookies, 4 friends equally. How many each?" → 3 each

Same expression. Different meaning. Both apply to fractions too.

Two Ways to Interpret Fraction Division

Measurement: "How many groups of size [divisor] fit in [dividend]?"

• Divisor = size of each group — you count groups

Partitive: "Split [dividend] into [divisor] equal parts — how large is each?"

• Divisor = number of parts — you find one share

Measurement Story: Rice and Servings

Problem: You have cup of rice. Each serving is cup.

How many servings do you have?

→ "How many -cups fit in cup?" — this is measurement

Count: three -cups fit → answer: 3

Partitive Story: Pizza and Friends

of a pizza shared equally among 3 friends. How much each?

→ "Split into 3 equal parts" — partitive

Check-In: Write a Story for

Write one measurement story for this expression.

Then reason out the answer — without any algorithm.

Think: how many -sized groups fit in ?

Advance to the next slide to see the answer.

Seeing Division: Number Line Models

Visual models let us see the quotient before computing it.

Two key ideas:

• Mark the dividend on the number line
• Count how many divisor-sized segments fit inside it

We will build intuition through three examples — starting with whole-number results, then a fractional result.

Number Line Examples: Two Whole-Number Quotients

Example 1:

Count three -segments in → quotient = 3

Example 2:

Subdivide into sixths; count four -segments in → quotient = 4

Number Line: When the Quotient Is a Fraction

How many -unit segments fit in ?

• , so less than one full segment fits
• Mark within the first -unit interval
• Ask: what fraction of one -unit is ?

Since , the answer is .

Area Model: The Land Strip Problem

Problem: A rectangular strip has area sq mi and length mi. How wide is it?

Area = length × width, so width = area ÷ length:

The area model shows: width = mi

Verify: ✓

Check-In: Greater or Less Than 1?

Before computing: is the quotient greater than 1 or less than 1?

Think: how does compare to ?

, so more than one group of fits → quotient > 1

Why the Invert-and-Multiply Rule Works

If , then

Multiply both sides by (the reciprocal of ):

Check: ✓

Dividing = multiplying by the reciprocal of the divisor.

Invert and Multiply: The Four-Step Algorithm

Rule:

Four steps:

1. Identify the divisor (the fraction after ÷)
2. Write its reciprocal (flip numerator and denominator)
3. Multiply numerators together, denominators together
4. Simplify the result

Worked Examples 1 and 2

Example 1:

Divisor: → reciprocal:

Example 2:

Divisor: → reciprocal:

Worked Examples 3 and 4

Example 3: — result greater than 1

Divisor: → reciprocal:

Example 4: — simplify before multiplying

Divisor: → reciprocal:

Your Turn: Compute

Follow the four steps:

1. Identify the divisor: ____
2. Write its reciprocal: ____
3. Multiply: ____ = ____
4. Simplify: ____

Pause and work through all four steps before advancing.

Answer:

Divisor → reciprocal

Verify: ✓

Reasonableness: → quotient , and ✓

Word Problem Framework: Five Steps

1. Identify the two quantities: total, and group size or number of groups
2. Classify: measurement ("how many groups?") or partitive ("how much per group?")
3. Write the expression: total ÷ group size (or total ÷ number of parts)
4. Compute using invert-and-multiply
5. Interpret the quotient — check reasonableness

Word Problem 1: Chocolate Sharing

"3 people share lb equally. How much each?"

• Partitive: split into 3 parts
• Expression: lb per person
• Reasonable: each share total ✓

Word Problem 2: Yogurt Servings

"How many -cup servings in cup of yogurt?"

• Measurement: how many -groups fit in ?
• Expression: of a serving
• Reasonable: → less than 1 full serving, ✓

Word Problem 3: Land Strip

Strip with area sq mi, length mi. How wide?

• Area ÷ length = width → mi
• Verify: ✓

Your Turn: Create a Story Context

Create two story contexts for :

• Measurement: "How many -sized groups fit in ?"
• Partitive: " split into of a group — how much is each?"

Both answers should equal .

Practice: Two More Word Problems

A: Recipe needs cup per batch. You have cup. How many batches?

B: Hiker goes mile in hour. What speed in mph?

Write the division expression first, then compute.

Practice: Worked Solutions and Checks

A: batch

→ less than one batch ✓

B: mph

→ speed above 1 mph ✓

Key Takeaways for This Lesson

✓ Division has two interpretations: measurement (how many groups?) and partitive (how much per group?)

✓ Visual models — number lines and area models — reveal the quotient before computing

✓ Algorithm: — multiply by the reciprocal of the divisor

✓ Justify: if is the quotient, then divisor × = dividend — always verify

✓ For word problems: identify the interpretation, write the expression, compute, check reasonableness

Watch out: — do NOT divide numerators and denominators

Watch out: Flip the divisor (second fraction), NOT the dividend

Watch out: Flip first, then multiply — not the other way around

Watch out: For "how many 3/4-cup servings in 2/3 cup?" → total (2/3) ÷ group size (3/4)

What Comes Next in Grade 6

Coming up in 6.NS: multi-digit decimals, GCF, LCM

Fraction division reappears in:

• 6.EE.B.7 — solving
• 7.NS.A — dividing negative rational numbers
• Ratios, rates, and proportional reasoning