Multiply Whole Numbers - Sixth Grade
Complete Notes & Formulas
What is Multiplication?
Multiplication is repeated addition. It's a faster way to add the same number multiple times.
a × b = Product
Example: 5 × 3 = 15
(5 + 5 + 5 = 15)
Key Vocabulary
Factors: Numbers being multiplied (e.g., in 5 × 3, both 5 and 3 are factors)
Product: The result of multiplication (e.g., 5 × 3 = 15, product is 15)
Multiplicand: The first number (being multiplied)
Multiplier: The second number (what you multiply by)
Partial Product: Intermediate results when multiplying multi-digit numbers
1. Multiply Whole Numbers
Steps for Multiplication
Step 1: Write numbers vertically, aligning the rightmost digits
Step 2: Multiply each digit of the bottom number by the top number
Step 3: Start with the ones place, then tens, hundreds, etc.
Step 4: For each new digit, shift one place to the left (add a zero)
Step 5: Add all partial products to get the final answer
Example 1: One-Digit × Multi-Digit
Problem: Multiply 234 × 7
Step-by-step:
234
× 7
--------
• 7 × 4 (ones) = 28 → write 8, carry 2
• 7 × 3 (tens) = 21 + 2 (carry) = 23 → write 3, carry 2
• 7 × 2 (hundreds) = 14 + 2 (carry) = 16
²²
234
× 7
--------
1,638
Answer: 1,638
Example 2: Two-Digit × Two-Digit
Problem: Multiply 43 × 26
Step 1: Multiply by ones place (6)
43
× 26
------
258 ← (43 × 6)
Step 2: Multiply by tens place (20)
43
× 26
------
258
+860 ← (43 × 20)
------
1,118
Answer: 1,118
Example 3: Three-Digit × Two-Digit
Problem: Multiply 124 × 26
124
× 26
-------
744 ← (124 × 6)
+2480 ← (124 × 20)
-------
3,224
Answer: 3,224
2. Multiply Numbers Ending in Zeros
Shortcut Method
Steps to Multiply with Zeros:
1. Multiply the non-zero digits
2. Count total zeros in both numbers
3. Add those zeros to your product
Example 1: Both Numbers End in Zeros
Problem: Multiply 300 × 40
Step 1: Multiply non-zero parts: 3 × 4 = 12
Step 2: Count zeros: 300 has 2 zeros, 40 has 1 zero = 3 total zeros
Step 3: Add zeros to product: 12 + 000 = 12,000
Answer: 12,000
Example 2: One Number Ends in Zero
Problem: Multiply 250 × 8
Step 1: Multiply: 25 × 8 = 200
Step 2: Count zeros in 250: 1 zero
Step 3: Add zero: 200 + 0 = 2,000
Answer: 2,000
Example 3: Large Numbers with Zeros
Problem: Multiply 6,000 × 500
Step 1: Multiply: 6 × 5 = 30
Step 2: Count zeros: 6,000 has 3 zeros, 500 has 2 zeros = 5 total
Step 3: Add zeros: 30 + 00000 = 3,000,000
Answer: 3,000,000
Quick Tip: When multiplying by 10, 100, or 1000, just add that many zeros!
Example: 45 × 100 = 4,500 (add 2 zeros)
3. Properties of Multiplication
A. Commutative Property
The order of factors doesn't change the product
a × b = b × a
Example: 7 × 5 = 5 × 7 = 35
B. Associative Property
The grouping of factors doesn't change the product
(a × b) × c = a × (b × c)
Example: (2 × 3) × 4 = 2 × (3 × 4) = 24
C. Identity Property
Multiplicative Identity: Any number multiplied by 1 equals itself
a × 1 = 1 × a = a
Example: 25 × 1 = 25
D. Zero Property
Zero Property: Any number multiplied by 0 equals 0
a × 0 = 0 × a = 0
Example: 1,000 × 0 = 0
E. Distributive Property
a × (b + c) = (a × b) + (a × c)
Example: 3 × (4 + 5) = (3 × 4) + (3 × 5)
3 × 9 = 12 + 15 = 27
4. Estimate Products
Why Estimate?
Estimating helps you check if your answer is reasonable and allows for quick mental math.
Steps to Estimate Products
Step 1: Round each factor to its highest place value
Step 2: Multiply the rounded numbers
Step 3: Use mental math for the calculation
Rounding Rules
• If digit is 0-4: Round DOWN
• If digit is 5-9: Round UP
• Round to nearest 10, 100, or 1000
Example 1: Estimate Two-Digit Multiplication
Problem: Estimate 47 × 32
Step 1: Round to nearest 10
47 rounds to 50
32 rounds to 30
Step 2: Multiply rounded numbers
50 × 30 = 1,500
Estimated Product: 1,500
(Actual: 47 × 32 = 1,504)
Example 2: Estimate Three-Digit Multiplication
Problem: Estimate 678 × 849
Step 1: Round to nearest 100
678 rounds to 700
849 rounds to 800
Step 2: Multiply
700 × 800 = 560,000
Estimated Product: 560,000
(Actual: 678 × 849 = 575,622)
Example 3: Range Estimation
Problem: Estimate a range for 45 × 67
Lower Estimate (round down):
40 × 60 = 2,400
Upper Estimate (round up):
50 × 70 = 3,500
Range: Between 2,400 and 3,500
(Actual: 45 × 67 = 3,015)
5. Multiplication Word Problems
Key Multiplication Keywords
• Times
• Product
• Each
• Per
• Total (when groups are equal)
• At this rate
• Twice, triple, quadruple
Example 1: Simple Word Problem
Problem: A movie theater has 24 rows of seats. Each row has 18 seats. How many seats are there in total?
Given:
Number of rows = 24
Seats per row = 18
Operation: Multiplication (keyword: "each")
Solution:
Total seats = 24 × 18 = 432
Answer: 432 seats
Example 2: Numbers Ending in Zeros
Problem: A warehouse stores 500 boxes on each floor. If there are 30 floors, how many boxes can the warehouse store?
Solution using shortcut:
500 × 30
= 5 × 3 = 15
Add 3 zeros (2 from 500, 1 from 30)
= 15,000
Answer: 15,000 boxes
Example 3: Multi-Step Problem
Problem: A school has 15 classrooms. Each classroom has 8 rows of desks with 4 desks in each row. How many desks are in the school?
Step 1: Find desks per classroom
8 rows × 4 desks = 32 desks per classroom
Step 2: Find total desks
15 classrooms × 32 desks = 480 desks
Answer: 480 desks
Example 4: Estimation Word Problem
Problem: A bookstore sells about 89 books per day. Estimate how many books they sell in 31 days.
Round the numbers:
89 rounds to 90
31 rounds to 30
Estimate:
90 × 30 = 2,700
Estimated Answer: About 2,700 books
(Actual: 89 × 31 = 2,759)
Quick Reference: Multiplication Facts
| Property | Formula | Example |
|---|---|---|
| Commutative | a × b = b × a | 4 × 5 = 5 × 4 |
| Associative | (a × b) × c = a × (b × c) | (2 × 3) × 4 = 2 × (3 × 4) |
| Identity | a × 1 = a | 25 × 1 = 25 |
| Zero | a × 0 = 0 | 100 × 0 = 0 |
💡 Important Tips to Remember
✓ Align digits properly when multiplying vertically
✓ Start multiplying from the rightmost digit
✓ Carry when products are 10 or more
✓ Shift left (add zero) for each new digit in multiplier
✓ Multiplying by 10, 100, 1000: just add zeros!
✓ Order doesn't matter: 3 × 5 = 5 × 3 (commutative)
✓ Estimate first to check if your answer makes sense
✓ Any number × 1 = itself
✓ Any number × 0 = 0
✓ Check your work with estimation or division
🧠 Memory Tricks & Strategies
Multiplication Steps:
"Right to Left, Bottom to Top!" (multiply from right, work through each digit)
Numbers Ending in Zeros:
"Multiply what's not zero, then add the zeros!"
300 × 40 = (3 × 4) + (000) = 12,000
Commutative Property:
"Flip the factors, same result!"
Estimating:
"Round 'em up, Round 'em down, Multiply and you're homeward bound!"
Zero Property:
"Anything times zero = zero, no exceptions, my hero!"
Check Your Work:
Estimate first, calculate, then check: Does it make sense?
Master Multiplication! × ✖️ 🔢
Practice daily to build speed and accuracy!