Understand Multiplication with Mixed Numbers | Fifth Grade
Complete Notes & Formulas
1. Multiply a Mixed Number by a Whole Number Using a Model I (Basic Visual Models)
Definition: Use visual models (circles, rectangles, or bars) to represent multiplying a mixed number by a whole number through repeated addition.
🔑 Key Concept:
Mixed Number × Whole Number = Repeated Addition
Example: 2 1/3 × 4 means "four groups of 2 1/3"
📝 Steps Using Visual Models:
- Step 1: Draw the mixed number using circles or rectangles
- Step 2: Repeat the model as many times as the whole number
- Step 3: Count all whole shapes and fractional parts
- Step 4: Combine and simplify to get the final answer
✏️ Example 1: 1 1/2 × 3 (Using Circle Model)
Step 1: Draw 1 1/2 (1 whole circle + 1/2 circle)
Step 2: Repeat this model 3 times
Step 3: Count: 3 whole circles + 3 halves
Step 4: 3 + 3/2 = 3 + 1 1/2 = 4 1/2
Answer: 4 1/2
✏️ Example 2: 2 1/4 × 2
Visual: Draw 2 1/4 twice
First group: 2 wholes + 1/4
Second group: 2 wholes + 1/4
Total: 4 wholes + 2/4 = 4 + 1/2 = 4 1/2
Answer: 4 1/2
2. Multiply a Mixed Number by a Whole Number Using a Model II (Converting Method)
Definition: Convert the mixed number to an improper fraction first, then multiply by the whole number, and use models to verify.
📐 Formula Method:
Step 1: Convert W a/b to improper fraction = (W × b + a)/b
Step 2: Multiply improper fraction by whole number n
Step 3: (Improper fraction × n)/b = Result
📝 Detailed Steps:
- Convert mixed number to improper fraction
- Write whole number as a fraction (n = n/1)
- Multiply numerators and denominators
- Simplify the result
- Convert back to mixed number if needed
✏️ Example 1: 2 1/3 × 4
Step 1: Convert 2 1/3 to improper fraction
2 1/3 = (2 × 3 + 1)/3 = 7/3
Step 2: Multiply by 4
7/3 × 4 = 7/3 × 4/1 = (7 × 4)/(3 × 1) = 28/3
Step 3: Convert to mixed number
28/3 = 9 1/3
Answer: 9 1/3
✏️ Example 2: 1 3/4 × 3
Convert: 1 3/4 = (1 × 4 + 3)/4 = 7/4
Multiply: 7/4 × 3 = 21/4
Convert: 21/4 = 5 1/4
Answer: 5 1/4
3. Multiply with Mixed Numbers Using Area Models
Definition: Use area models (rectangular grids) to multiply two mixed numbers by breaking each into whole numbers and fractions, finding partial products, then adding them together.
📐 Area Model Method:
Break: (W₁ + a/b) × (W₂ + c/d)
Create 4 rectangular sections and find each area
📝 Four Partial Products:
Area 1 (Top Left)
Whole × Whole
W₁ × W₂
Area 2 (Top Right)
Whole × Fraction
W₁ × c/d
Area 3 (Bottom Left)
Fraction × Whole
a/b × W₂
Area 4 (Bottom Right)
Fraction × Fraction
a/b × c/d
Total Area = Area 1 + Area 2 + Area 3 + Area 4
✏️ Example 1: 1 1/2 × 2 1/3 (Using Area Model)
Break apart: (1 + 1/2) × (2 + 1/3)
Find 4 partial products:
Area 1: 1 × 2 = 2
Area 2: 1 × 1/3 = 1/3
Area 3: 1/2 × 2 = 2/2 = 1
Area 4: 1/2 × 1/3 = 1/6
Add all areas:
2 + 1/3 + 1 + 1/6 = 3 + 1/3 + 1/6
Convert to common denominator (6): 3 + 2/6 + 1/6 = 3 3/6 = 3 1/2
Answer: 3 1/2
✏️ Example 2: 2 1/4 × 1 1/2
Break apart: (2 + 1/4) × (1 + 1/2)
Four partial products:
• 2 × 1 = 2
• 2 × 1/2 = 2/2 = 1
• 1/4 × 1 = 1/4
• 1/4 × 1/2 = 1/8
Add: 2 + 1 + 1/4 + 1/8 = 3 + 2/8 + 1/8 = 3 3/8
Answer: 3 3/8
✏️ Example 3: Area Problem - 3 1/2 × 2 1/4 feet
Find the area of a rectangle with length 3 1/2 feet and width 2 1/4 feet.
Break apart: (3 + 1/2) × (2 + 1/4)
Four areas:
• 3 × 2 = 6
• 3 × 1/4 = 3/4
• 1/2 × 2 = 1
• 1/2 × 1/4 = 1/8
Total: 6 + 3/4 + 1 + 1/8 = 7 + 6/8 + 1/8 = 7 7/8
Answer: 7 7/8 square feet
Comparison of Multiplication Methods
| Method | Best For | Advantage |
|---|---|---|
| Visual Model | Small numbers, beginners | Easy to understand visually |
| Converting Method | Mixed × Whole number | Quick and accurate |
| Area Model | Mixed × Mixed number | Shows all partial products clearly |
Quick Reference Chart
| Model Type | Formula/Method | Example |
|---|---|---|
| Visual Repeated Addition | Draw mixed number n times | 1 1/2 × 3 = draw 1 1/2 three times |
| Convert to Improper | W a/b → (W×b+a)/b × n | 2 1/3 × 4 = 7/3 × 4 = 28/3 |
| Area Model (4 parts) | W×W + W×F + F×W + F×F | 1 1/2 × 2 1/3 = (1×2)+(1×1/3)+(1/2×2)+(1/2×1/3) |
💡 Key Formulas:
Mixed to Improper
W a/b = (W × b + a)/b
Improper to Mixed
Divide numerator by denominator
Area Model
Sum of 4 partial products
Area Formula
Length × Width
🔑 Key Tips for Success:
- Use visual models to understand what multiplication means with mixed numbers
- Convert mixed numbers to improper fractions for easier calculation
- In area models, break mixed numbers into whole and fractional parts
- Find all four partial products in area models (W×W, W×F, F×W, F×F)
- Add all partial products together to get final answer
- Always simplify and convert back to mixed numbers when appropriate
- Check your answer by estimating (round mixed numbers to nearest whole)
- Remember: Multiplying makes the answer larger when multiplying by numbers > 1
📚 Fifth Grade Understand Multiplication with Mixed Numbers - Complete Study Guide
Master these concepts for math excellence! ✨