To multiply fractions:
① Multiply numerators (top × top).
② Multiply denominators (bottom × bottom).
③ Simplify if possible.
Example: 23 × 45 = 2×43×5 = 815
⚠️ Don't add! Just multiply straight across.
Fraction × Fraction
Multiplying Fractions with Area Models
📚
Subject
Math · Grade 5
⏱️
Duration
50 min
🎯
Standard
5.NF.B.4a
📋 Standards & Objectives
📜Standards
5.NF.B.4aInterpret the product (a/b) × (c/d) as parts of a partition — a of b equal parts of c/d.
5.NF.B.4bFind the area of a rectangle with fractional side lengths by tiling with unit fraction rectangles.
5.NF.B.5aCompare the size of a product to the size of one factor based on the size of the other.
🎯SWBAT
- Multiply a fraction by a fraction using an area model.
- Find the area of a rectangle with fractional side lengths.
- Explain why multiplying by a fraction less than 1 makes the product smaller.
📖 Key Vocabulary
Click any word to see examples
📝Product
The answer you get when you multiply two numbers.
In 6 × 4 = 24, the product is 24.
In 12 × 13 = 16, the product is 16.
📝Factor
A number you multiply with another number to get a product.
In 6 × 4 = 24, the factors are 6 and 4.
In 34 × 45, the factors are 34 and 45.
📝Area Model
A rectangle split into rows and columns to show fraction multiplication. The overlap shows the product.
A rectangle split into 2 rows and 4 columns makes 8 equal parts — perfect for showing area models of 12 × 14.
An area model is a picture of multiplication — it also gives the area of a rectangle with fractional sides.
📝Numerator
The top number of a fraction. It tells how many parts you have.
In 34, the numerator is 3.
When you multiply fractions, you multiply the numerators together: 23 × 45 → 2 × 4 = 8.
📝Denominator
The bottom number of a fraction. It tells how many equal parts the whole is split into.
In 34, the denominator is 4 — the whole is split into 4 equal parts.
When you multiply fractions, you multiply the denominators together: 23 × 45 → 3 × 5 = 15.
📝Scaling
Making a number bigger or smaller by multiplying. Multiplying by a fraction less than 1 shrinks the number.
34 × 45 = 35 → scaling 34 DOWN to 35.
Multiplying by 2 scales UP (doubles). Multiplying by 12 scales DOWN (halves).
🔥 Spiral Warm-Up
Let's wake up our fraction brains!
Before we learn something NEW…
We'll review whole number × fraction and fraction addition — two skills you'll use every single day.
👉 Try each problem in your notebook before I click to reveal.
🔄 Warm-Up #1
Whole number × fraction
6 × 58 = ?
Multiply the whole number by the numerator, keep the denominator:
6 × 58 = 6 × 58 = 308
= 3 68 = 3 34
🔄 Warm-Up #2
Fraction addition with unlike denominators
23 + 34 = ?
Find a common denominator. Twelfths work: 3 × 4 = 12.
23 = 812 and 34 = 912
812 + 912 = 1712 = 1 512
⚠️ We ADDED denominators only to find the common one. When we add fractions, we only add numerators once they match!
🚀 Hook: What Does This Even Mean?
Today's big question
What does 12 × 13 mean?
The word "of" is a clue. Multiplying fractions means taking a fraction of another fraction.
It means ONE HALF of ONE THIRD.
Blue = 13 (one column of 3). Green = HALF of that column = 16 of the whole.
🤯 The Weird Truth
This is going to feel strange…
⚠️ Get ready for a plot twist:
When you multiply two factors that are both less than 1…
the product is SMALLER than either factor!
12 is less than 1 • 13 is less than 1
12 × 13 = 16 ← smaller than BOTH
That feels weird, right? Multiplying is supposed to make things bigger! Today we'll see why this happens.
⭐ SBA Tip #3 — Draw a Model, Even on the Test
Test-taking strategy
💡 Area models aren't just for learning
They're a scoring strategy. On SBA constructed response, a correct model can earn you partial credit even if your computation has a small error.
📝 If you see "Show your work" on the test → DRAW THE MODEL.
A picture shows the scorer you understand what multiplication MEANS, not just the rule.
📐Example: 23 × 12
Overlap = 2 out of 6 = 13
✅ Scorer sees: rows, columns, overlap labeled → partial credit even if the final fraction has a simplification error.
👨🏫 I Do: The Area Model
Watch how I set this up
We're going to solve 14 × 12 using an area model. Here's how it works:
- Draw a rectangle (any rectangle).
- Use the first fraction's denominator to split it into rows.
- Use the second fraction's denominator to split it into columns.
- Shade the rows for the first fraction, then the columns for the second.
- The overlap = the product.
👨🏫 14 × 12 — Step 1
Divide into halves, shade 1 row
1 of 2 rows shaded
The second factor is 12.
Split the rectangle into 2 equal rows (halves).
Shade 1 of the 2 rows. That's 12.
👨🏫 14 × 12 — Step 2
Now divide into fourths
1 column (14) overlaps the shaded half
The first factor is 14.
Split the same rectangle into 4 equal columns.
Shade 1 of the 4 columns. That's 14.
The GREEN square is where they overlap.
💡 14 × 12 = ?
Count the overlap
Total pieces in the rectangle: 8 (2 rows × 4 columns).
Green overlap: 1 piece.
14 × 12 = 18
Shortcut: 1 × 14 × 2 = 18
👨🏫 13 × 14
Same process — your brain is building the pattern
4 rows × 3 cols = 12 pieces
4 rows (fourths), 3 columns (thirds). Shade 1 column and 1 row.
Overlap: 1 piece out of 12.
13 × 14 = 112
👨🏫 34 × 45 — Setup
Now the numerators aren't 1 — watch what happens
5 rows × 4 cols = 20 pieces
5 rows (fifths) and 4 columns (fourths).
Shade 4 of the 5 rows (45).
Shade 3 of the 4 columns (34).
Now count the green overlap…
💡 34 × 45 = ?
Count the overlap and simplify
Green pieces: 12 • Total pieces: 20
34 × 45 = 1220
Now simplify — both 12 and 20 are divisible by 4:
12 ÷ 420 ÷ 4 = 35
Multiply across:
3 × 44 × 5 = 1220 = 35 ✓
The shortcut matches the area model. Always.
📜 THE ALGORITHM
The shortcut — now that you UNDERSTAND why it works
Multiplying Fractions: 3 Steps
- Multiply numerator × numerator
- Multiply denominator × denominator
- Simplify if you can
ab × cd = a × cb × d
⚠️ The algorithm is the SHORTCUT, not the UNDERSTANDING. The area model is why it works.
📓 Write This Down
Open your notebook!
Key Terms
Multiply Across
Area Model
Simplify
Notes
📏 The Shrinking Rule
Why is the product smaller?
Multiplying by a factor less than 1 makes the answer SMALLER.
Look: we start with 34. When we multiply by 45 (which is less than 1), the answer 35 is smaller than 34.
34 × 45 = 35
35 = 0.6 < 34 = 0.75 ✓ smaller!
Taking four-fifths OF something means taking most of it, but not all. So you end up with less.
📐 Number Line Check
35 < 34
Starting at 34 and multiplying by 45 pulls us LEFT to 35.
🔭 What If a Factor Is GREATER Than 1?
Sneak peek at Thursday's lesson
54 × 23 = ?
54 is GREATER than 1 • 23 is LESS than 1
5 × 24 × 3 = 1012 = 56
Notice: 56 is greater than 23 (0.83 > 0.67) but less than 54 (0.83 < 1.25).
When one factor is bigger than 1 and the other is smaller than 1, the product lands BETWEEN them. 🤯 We'll explore this Thursday.
⚠️ Marcus vs. Tina
Who solved it correctly?
Problem: 12 × 13 = ?
❌ Marcus says 25
"I added the tops (1+1=2) and added the bottoms (2+3=5)."
Marcus added when he should have multiplied. That's the rule for adding fractions (and even then, you need common denominators first). Wrong operation entirely.
✅ Tina says 16
"I multiplied across: 1×12×3 = 16."
Tina used the multiplication rule. The area model confirms: 1 green square out of 6 total = 16. Tina is correct.
The rule: When you MULTIPLY fractions, you do NOT add — you multiply across. Adding fractions needs common denominators; multiplying does not.
✅ Quick Check
Thumbs up or thumbs down
TRUE or FALSE?
"When I multiply two fractions, I need a common denominator first."
👍 Thumbs up if TRUE • 👎 Thumbs down if FALSE
👎 FALSE
Common denominators are for adding. For multiplying, just multiply across!
🔄 Bridge to We Do
What we just learned → what's next
✅What we just learned
Use an area model OR the "multiply across" algorithm to multiply fractions.
👥Next: We Do
I'll solve 3 problems WITH you. You'll shade the area models on your paper as I go.
👥 We Do #1: 12 × 23
Shade the array together
2 rows (halves), 3 columns (thirds).
Shade 1 row, then 2 columns.
1 × 22 × 3 = 26 = 13
Overlap: 2 green squares out of 6 = 26 → simplify to 13.
👥 We Do #2: 23 × 56
Bigger grid — stay careful
3 rows × 6 cols = 18 total
Shade 2 rows (23) and 5 columns (56).
2 × 53 × 6 = 1018 = 59
10 green out of 18 → divide by 2 → 59.
👥 We Do #3: 34 × 45
Algorithm first — then verify with the model
Step 1 — Use the algorithm:
34 × 45 = 3 × 44 × 5 = 1220 = 35
Yes! We did this one in the I Do (slide 16). The area model showed 12 green out of 20 total = 35. Same answer both ways. ✓
This is the whole point: the algorithm is just a faster version of what the model shows.
🤝 Turn & Talk
Reason without computing
🤔Discuss with a partner — 60 seconds
Is 23 × 34 greater than 23, or less than 23?
No computing! Use the shrinking rule to explain your answer.
Sentence starter: "I think it's ___ than 23 because ___."
LESS than 23. Because 34 is less than 1, multiplying 23 by it will shrink the answer. (If you do the math: 612 = 12, which is less than 23. ✓)
✏️ You Try — Directions
Your turn to fly solo
Before You Start
- Open your notebook to today's page.
- Solve each problem — use the area model OR the algorithm.
- Always simplify your final answer.
- Try before I click to reveal the answer.
🔍 You Try #1
Shade the blank array
3 rows × 5 cols — shade it!
13 × 15
1 × 13 × 5 = 115
1 overlap out of 15 total.
🔍 You Try #2
Algorithm + simplify
25 × 34 = ?
2 × 35 × 4 = 620
Simplify — both divisible by 2:
6 ÷ 220 ÷ 2 = 310
🔍 You Try #3
Multiply across + simplify
56 × 23 = ?
5 × 26 × 3 = 1018
Simplify — both divisible by 2:
10 ÷ 218 ÷ 2 = 59
🌱 You Try #4 — Word Problem
Real-world fraction multiplication
A garden is 34 of a yard long and 25 of a yard wide. What is the area of the garden?
💡 Remember: Area = length × width. And draw a model if you want partial credit on a test!
Area = 34 × 25
3 × 24 × 5 = 620 = 310
310 square yard
⚠️ Don't forget the UNIT on a word problem: square yard (sq yd).
⭐ SBA Spotlight #1 — Distractor Analysis
Learning the traps
What is 34 × 25?
A59
❌ Added BOTH: 3+2=5, 4+5=9. Treating multiply like add.
B69
❌ Multiplied numerators (3×2=6) but ADDED denominators (4+5=9). Mixed rules.
C✅ 620 = 310
✅ Multiplied across: 3×24×5 = 620 → simplifies to 310. Both forms can be correct on the test — read carefully!
D520
❌ Added numerators (3+2=5) but multiplied denominators (4×5=20). Half-right is still wrong.
💡 The SBA ALWAYS includes "add both" traps. Train your eye: multiply means MULTIPLY across.
⭐ SBA Spotlight #2 — Constructed Response
Modeling a strong written answer
Prompt: "Explain why 23 × 45 is LESS than 45. Use words, a model, or both. Then compute 23 × 45 and check that your answer matches your explanation."
1️⃣Answer
23 × 45 = 815
2️⃣Explanation
23 is less than 1. Multiplying by any factor less than 1 shrinks the other factor — so the product has to be less than 45.
3️⃣Check
815 ≈ 0.53 45 = 0.8 ✓ 0.53 < 0.8 — my answer matches my explanation.
💡 All 3 parts — answer, explanation, check — earn points on SBA.
📓 Summary Note
Write 1 sentence in the bottom of your notes
Write 1 Sentence
In the bottom strip of your Cornell notes page, write ONE sentence explaining what you learned today about multiplying fractions by fractions.
Sentence starter: "To multiply a fraction by a fraction, I ___ and the product is always ___ when both factors are less than 1."
🎫 Exit Ticket
Show what you know — 4 questions
1️⃣Compute + Model
23 × 35 = ?
Show work — area model OR algorithm
2×33×5 = 615 = 25
2️⃣Compute
34 × 27 = ?
Simplify your answer
3×24×7 = 628 = 314
3️⃣⭐ SBA — MC + Explain
Which is TRUE about 12 × 58?
A) Product > 58 B) Product < 58
C) Product = 58 D) Product > 1
Circle the letter AND explain.
B — Product < 58
Explanation: 12 is less than 1, so multiplying shrinks 58. Check: 516 < 58. ✓
4️⃣🔄 Spiral Review
42 students are in the lunchroom. 37 are 5th graders. How many 5th graders?
37 of 42 = 42 × 37 = 1267 = 18 students