Multiplication as Scaling: Predicting Products

5.NF.B.5

In this lesson:

• Predict whether a product is greater, less, or equal — without computing
• Explain why the scale factor controls the direction
• Connect multiplying by to equivalent fractions

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Predict whether a product is greater, less, or equal
2. Explain why factor > 1 stretches
3. Explain why factor < 1 shrinks
4. Connect to equivalent fractions
5. Apply scaling to real-world contexts

Do Not Solve — Just Predict

What happens to the number 9?

• — is the answer bigger or smaller than 9?
• — bigger or smaller than 9?

is bigger — 36 > 9. But . Smaller!

Scaling on the Number Line

The scale factor controls the direction: stretch, hold, or shrink

The Scale Factor Controls the Direction

The scale factor is the number you multiply by:

• Factor > 1 → product is greater (stretches)
• Factor = 1 → product is equal (holds)
• Factor < 1 → product is less (shrinks)

You don't need to compute — just check the factor against 1

Quick Check: Predict the Direction

Without computing, answer this:

Is greater than, less than, or equal to 24?

• What is the scale factor? →
• Is greater than, less than, or equal to 1?
• Since , the product is less than 24

A Sentence Frame for Predictions

Use this pattern to explain your reasoning:

"The product is ___ than ___ because the scale factor is ___ than 1."

Example: compared to

"The product is greater than because the scale factor 3 is greater than 1."

Example: Scale Factor Greater Than One

compared to

• Scale factor: 3
• Since , the product is greater than
• Verify: , and ✓

Example: Scale Factor Less Than One

compared to 7

• Scale factor:
• Since , the product is less than 7
• Verify: , and ✓

Important: is less than 7 but NOT less than 1

Compare to Either Factor — Both Work

The same expression can be compared two ways:

• Compared to 7: scale factor → product < 7
• Compared to : scale factor → product >

Both are true! The product lands between the two factors.

Your Turn: Classify These Three Expressions

For each, predict: greater than, less than, or equal?

1. 13 — scale factor is ___ 1
2. 4 — scale factor is ___ 1
3. — scale factor is ___ 1

Predict first, then check on the next slide

Answers: Classifying the Three Expressions

1. : factor → product = 13 ✓
2. : factor → product > 4 ✓
3. : factor → product < ✓

Watch Out for Improper Fractions

Is greater or less than 10?

• Is a fraction? Yes
• Is less than 1? No! — , so
• The product is greater than 10

Quick check: numerator > denominator → fraction > 1 → product stretches

Case One: Factor Greater Than One Stretches

• — the product exceeds 6
• Taking copies means 1.5 copies — more than 1 whole
• Any factor > 1 means more than one full copy

Case Two: Factor Equals One Preserves Value

Multiplying by 1 changes nothing — and :

The equivalent-fraction connection:

Multiplying by is how we make equivalent fractions!

Equivalent Fractions Through Scaling by One

Start with . Generate equivalent fractions:

• → same value ✓
• → same value ✓
• → same value ✓

Rule:

Case Three: Factor Less Than One Shrinks

• — the product is less than 12
• Taking of 12 means only 2 of 3 equal parts
• 2 out of 3 parts is less than the whole

Verify: , and ✓

The Three Cases at a Glance

This diagram covers every positive scale factor

Quick Check: Explain Your Reasoning

Without computing, explain why

Your explanation should include:

• Which number is the scale factor?
• Is it greater than, less than, or equal to 1?
• Why does that make the product less than 20?

Scaling Appears in Real-World Contexts

The principle works in every context:

• Photos: scale factor controls size (enlarge or reduce)
• Recipes: scale factor adjusts amounts (more or less)
• Maps: scale factor relates distances (bigger or smaller)

Same question every time: Is the scale factor >, =, or < 1?

Example: Resizing a Photo Three Ways

An 8-inch photo printed at three different scales:

• scale → shorter than 8 inches (shrink)
• scale → exactly 8 inches (hold)
• scale → taller than 8 inches (stretch)

Real-World Predictions: Recipe, Factory, Map

Recipe: Scale 2 cups by

• Factor → less flour needed

Factory: 500 widgets/day at capacity

• Factor → more than 500

Map: Road is inch; 1 inch = 10 miles

• Factor → shorter than 10 miles

Practice: Predict These Mixed Scaling Problems

Predict without computing, then explain:

1. Is greater, less, or equal to 18?
2. Is greater, less, or equal to 9?
3. 12-ounce drink scaled by — more, less, or same?

Write predictions before advancing

Answers: Mixed Scaling Practice Problems Revealed

1. : factor → less than 18 (verify: 15) ✓
2. : factor → greater than 9 (verify: 12) ✓
3. : factor → equal to 12 ✓

Respond to This Common Multiplication Claim

"Multiplying always gives a bigger answer."

How would you respond? Think about:

• When is this true? (factor > 1)
• When is this false? (factor < 1 or factor = 1)
• Give a specific counterexample

Key Takeaways for Multiplication as Scaling

✓ Factor > 1 → product stretches
✓ Factor = 1 → product holds ( = equivalent fractions)
✓ Factor < 1 → product shrinks

Fractions can exceed 1 (numerator > denominator)
"Less than the factor" ≠ "less than 1"
Bigger digits don't mean bigger value

What Comes Next in Fraction Multiplication

Next lesson — 5.NF.B.6:
Solving real-world problems with fraction multiplication

You'll use today's scaling reasoning to:

• Estimate answers before computing
• Check whether your computed answers are reasonable
• Solve multi-step word problems involving fractions