Fraction Multiplication:
Reasoning & Word Problems

Predict the size of a product WITHOUT computing — then solve real-world problems.

📚

Subject

Math · Grade 5

⏱️

Duration

50 min

🎯

Standards

5.NF.B.5 · 5.NF.B.6

📋 Standards & Objectives

📜Standards

5.NF.B.5Interpret multiplication as scaling (resizing) — compare the size of a product to the size of one factor.

5.NF.B.5aCompare the size of a product to the size of one factor when multiplying by a fraction greater than or less than 1.

5.NF.B.5bExplain why multiplying by a fraction greater than 1 gives a product greater than the given number.

5.NF.B.6Solve real-world problems involving multiplication of fractions and mixed numbers.

🎯SWBAT

Predict whether a product will be greater or less than a given factor WITHOUT computing.
Explain why multiplying by a fraction > 1 grows a number and multiplying by a fraction < 1 shrinks it.
Solve real-world fraction multiplication word problems using area models and the algorithm.
Write SBA-style responses with an answer, shown math, and a reasoning sentence.

📖 Key Vocabulary

Click any word to see examples

📝Scaling

Making something bigger or smaller by multiplying. In math, multiplying is a way of scaling a number up or down.

Multiplying 12 by 54 → scales it UP to 15.

Multiplying 12 by 34 → scales it DOWN to 9.

Multiplying 12 by 1 → no scaling, stays 12.

📝Factor

A number being multiplied. Every multiplication has two factors that work together to make a product.

In 7 × 3 = 21, the factors are 7 and 3.

In 34 × 25, the factors are 34 and 25.

📝Product

The answer to a multiplication problem. Today we'll compare the product to one of the factors.

In 7 × 3 = 21, the product is 21.

In 25 × 34 = 620, the product is 620 (or 310).

📝Reasoning

Explaining WHY your answer makes sense using math rules — not just showing the work.

"The product is less than 42 because 7/8 is less than 1, so multiplying by it shrinks the other factor." ← That's reasoning!

On SBA, "explain your reasoning" means: give your answer + show your math + tell WHY your strategy works.

🔄 Spiral Warm-Up #1

Fraction × Fraction — from yesterday

Solve:

35 × 23 = ?

Remember: multiply numerators, multiply denominators.

35 × 23 = 3 × 25 × 3 = 615 = 25

Simplify 615 by dividing top & bottom by 3.

🔄 Spiral Warm-Up #2

Mixed number subtraction — unlike denominators

Solve:

314 − 123 = ?

Find a common denominator first. You may need to borrow!

Common denominator = 12.

3312 − 1812

Borrow 1 whole from the 3 → 21512 − 1812

= 1712

🍅 The Ketchup Problem

Think-pair-share — no computing yet!

John's big sister spilled ketchup on his homework. The problem reads:

74 × 🍅 BLOB 🍅 = ___

The hidden number under the ketchup is greater than zero. What can John tell for sure about the answer WITHOUT seeing the hidden number?

A) The answer is greater than 74

B) The answer is greater than the ketchup number

C) The answer is a fraction

D) The answer is less than the ketchup number

💡 Ketchup Problem — The Answer

Let's reason through it

The answer is B — the answer is greater than the ketchup number!

Why? Look at the factor we CAN see: 74.

74 = 134 ← bigger than 1!

Since 74 is greater than 1, multiplying ANY positive number by it will give you MORE than that number — no matter what the number is!

🎯 Big Idea: You can reason about the SIZE of products WITHOUT computing.

📝 SBA Tip #4 — "Explain Your Reasoning" Has 3 Parts

Miss one → you lose points

When SBA says "explain your reasoning," they want ALL THREE parts:

① ANSWER + ② MATH shown + ③ WHY it works

❌ Weak Answer

"The answer is 815."

Only 1 part out of 3. Missing the math AND the reasoning. Loses points.

✅ Strong Answer

① Answer: 815
② Math: 23 × 45 = 2×43×5 = 815
③ Why: The answer is less than 45 because I multiplied by 23, which is less than 1, so the product shrinks.

✨ Today's Big Idea

The whole lesson in one sentence

🎯 Multiplication is scaling.

When you multiply, you're either growing a number, keeping it the same, or shrinking it.

Which one happens depends on the factor you multiply by — and we can figure it out without doing any computing.

📏 The Scaling Rule

Our anchor chart for today

🔽SHRINKS

× a factor LESS than 1

The product is smaller than the other factor.

12 × 34 = 9

🟰STAYS SAME

× a factor EQUAL to 1

The product equals the other factor.

12 × 1 = 12

🔼GROWS

× a factor GREATER than 1

The product is bigger than the other factor.

12 × 54 = 15

📊 See It on a Number Line

Three versions of 12 × something

Watch what happens to 12 when we scale it three different ways:

🔑 Same starting number (12) — three different products depending on the factor.

📓 Write This Down

Anchor chart for your notebook

Key Terms

Scaling

Factor

Product

The Scaling Rule

Multiplying is scaling.

• × a factor < 1 → product is SMALLER than the other factor (shrinks)
• × a factor = 1 → product EQUALS the other factor (stays the same)
• × a factor > 1 → product is BIGGER than the other factor (grows)

You can use this rule to predict answers WITHOUT computing!

👨‍🏫 Let's Use the Rule: 7/4 × Hidden

I'll think out loud — watch my reasoning

74 × (hidden #) = ?

🧠Step 1: Look at the factor I can see

The factor I CAN see is 74.

Is 74 greater than, less than, or equal to 1?

74 = 134 → greater than 1!

🧠Step 2: Apply the scaling rule

Factor > 1 → the product GROWS.

So 74 × (hidden #) will be BIGGER than the hidden number.

It doesn't matter what the hidden number is — the answer HAS to be bigger than it.

👨‍🏫 Now Let's Try: 1/3 × Hidden

Same thinking, different factor

13 × (hidden #) = ?

🧠Step 1: Look at the factor I can see

The factor I CAN see is 13.

Is 13 greater than, less than, or equal to 1?

13 is less than 1 (just 1 out of 3 parts).

🧠Step 2: Apply the scaling rule

Factor < 1 → the product SHRINKS.

So 13 × (hidden #) will be SMALLER than the hidden number.

Whatever the hidden number is, the answer must be less than it.

⚖️ Marcus vs. Tina — Who's Right?

Wrong-answer analysis

The claim: "Multiplying always makes numbers BIGGER."

Marcus says: TRUE ❌

"Yeah, of course! 3 × 5 = 15 and 15 is bigger than 3 AND bigger than 5. Multiplying makes things bigger. That's the whole point!"

Tina says: FALSE ✅

"Not always! 12 × 34 = 9. The answer is SMALLER than 12. If the factor is less than 1, the product shrinks."

Tina is right. The old elementary-school rule "multiplying always makes things bigger" breaks once fractions enter the picture. × a factor < 1 shrinks. × 1 stays. × a factor > 1 grows.

🚲 Real-World: Rose's Bike Ride

Now we'll actually solve one

Rose lives 34 of a mile from her cousin. She rode her bike 25 of the way before it broke. What fraction of a MILE did she ride?

🧠First: Predict with the rule

She rode 25 of 34 mile → 25 × 34.

Is 25 > 1 or < 1? Less than 1!

So the answer must be less than 34 mile.

✏️Then: Compute

25 × 34 = 620 = 310

310 of a mile

✓ 310 IS less than 34. Our prediction was right!

▦ Rose's Ride — Area Model

Why 2/5 × 3/4 = 6/20

Start with a whole mile as a square. Split it into 4 columns (for 34) and 5 rows (for 25).

25 × 34 = 620 = 310 of a mile — the green overlap.

➖ Rose's Ride — Number Line

The same answer, different model

Think of 34 mile as a line from 0 to 34. She rode 25 of that.

✓ Notice: 310 IS smaller than 34 — exactly like the scaling rule predicted, because 25 is less than 1.

✅ Quick Check

Thumbs up or thumbs down — no computing!

Is 79 × 8 greater than or less than 8?

👍 greater • 👎 less

LESS than 8.

79 is less than 1 → the product shrinks 8.

👥 We Do #1 — Without Computing

Is 5/8 × 14 greater or less than 14?

58 × 14 = ?

Step 1: Find the factor you're scaling BY → 58.
Step 2: Ask — is 58 more than, less than, or equal to 1?
Step 3: Apply the scaling rule.

LESS than 14. 58 < 1 (only 5 out of 8 parts), so multiplying 14 by it shrinks it.

SBA reasoning sentence: "The product must be less than 14 because 58 is less than 1, and multiplying by a factor less than 1 makes the product smaller than the other factor."

👥 We Do #2 — Without Computing

Is 9/5 × 7/8 greater or less than 7/8?

95 × 78 = ?

Step 1: We're comparing the product to 78, so look at the OTHER factor: 95.
Step 2: Is 95 more or less than 1? 95 = 145 → more than 1!
Step 3: Factor > 1 → GROWS.

GREATER than 78. 95 is greater than 1, so multiplying 78 by it grows it.

SBA reasoning sentence: "The product must be greater than 78 because 95 is greater than 1, and multiplying by a factor greater than 1 makes the product bigger than the other factor."

⚽ We Do #3 — Kelly's Playground

Build an area model together

At Kelly's school, 23 of the playground is grass, and 35 of the grassy area is a baseball field. What fraction of the WHOLE playground is the baseball field?

▦Area Model

✏️The Math

23 × 35 = 615 = 25

25 of the whole playground is a baseball field.

Scaling check: 23 < 1, so the answer < 35. ✓ (25 < 35)

🤝 Turn & Talk

30 seconds with your partner

🤔Discuss

Someone says: "Multiplying always makes things bigger."

Is that true? When is it true, and when is it false? Give an example of each.

Sentence starters:

• "It's true when ___ because ___"

• "It's false when ___ because ___"

📓 Before You Practice

Add this to your notebook

Remember

< 1 shrinks

= 1 stays

> 1 grows

Your Strategy

Before you compute ANY fraction multiplication problem:

1. Find the factor you're scaling by.
2. Ask: is it < 1, = 1, or > 1?
3. Predict whether the product will be smaller, the same, or bigger than the OTHER factor.
4. Compute — and check that your answer matches your prediction!

🔍 You Try #1 — Predict the Product

Reason without computing

Without Computing

Is 310 × 456 greater than or less than 456?
Explain your reasoning.

🔍 You Try #2 — Predict the Product

Reason without computing

Without Computing

Is 118 × 23 greater than or less than 23?
Explain your reasoning.

🔍 You Try #3 — Jake's Trail Mix 🥜

Word problem — compute the product

Word Problem

Jake has 34 of a pound of trail mix. He eats 23 of it.

How much trail mix did he eat? How much is left?

🔍 You Try #4 — The Swimming Pool 🏊

Word problem — compute the product

Word Problem

A swimming pool is 56 full. Then 14 of the water evaporates.

What fraction of the pool still has water?

⭐ SBA Spotlight — Multiple Choice

Test-format question

SBA-Style Question

Which statement about 78 × 42 is TRUE?

A

The product is greater than 42.

B

The product equals 42.

C ⭐

The product is less than 42.

D

The product equals 0.

⭐ SBA Spotlight — Constructed Response

Answer + math + why

SBA-Style Question

Lucas says: "If I multiply 95 by any number, my answer will always be bigger than the number I started with."

Is Lucas correct? Use the 3 parts of an explanation: answer, math, and why.

⭐ Today's Big Takeaway

Multiplication is scaling

Remember

Multiplication is scaling.
The size of the factor tells you what happens to the product.

Factor < 1

Product shrinks ⬇

Factor = 1

Product stays same ↔

Factor > 1

Product grows ⬆

And when you explain: answer + math + why.

🎫 Exit Ticket — Show What You Know

4 questions

1 — Reasoning

Without computing: Is 47 × 15 greater or less than 15? Explain your reasoning.

2 — Reasoning

Without computing: Is 83 × 56 greater or less than 56? Explain.

3 — SBA Constructed Response

A rectangular garden is 56 of a yard long and 23 of a yard wide. What is the area? Show your work and explain.

4 — Spiral Review

A bag has 36 candies. Layla takes 14 of them. How many candies does Layla take?

Fraction Multiplication:Reasoning & Word Problems

Fraction Multiplication:
Reasoning & Word Problems