Multiplication
Fifth Grade Mathematics - Complete Guide
📊 Multiplication Patterns Over Increasing Place Values
Understanding the Pattern:
When multiplying by powers of 10 (10, 100, 1000), the digits shift to the left, making the number larger.
Place Value Pattern Formula:
Number × 10 = Move 1 place left (add 1 zero)
Number × 100 = Move 2 places left (add 2 zeros)
Number × 1000 = Move 3 places left (add 3 zeros)
Pattern Examples:
Pattern with 5:
- ( 5 × 1 = 5 )
- ( 5 × 10 = 50 ) (added 1 zero)
- ( 5 × 100 = 500 ) (added 2 zeros)
- ( 5 × 1{,}000 = 5{,}000 ) (added 3 zeros)
- ( 5 × 10{,}000 = 50{,}000 ) (added 4 zeros)
Pattern with 8:
- ( 8 × 1 = 8 )
- ( 8 × 10 = 80 )
- ( 8 × 100 = 800 )
- ( 8 × 1{,}000 = 8{,}000 )
- ( 8 × 10{,}000 = 80{,}000 )
Key Rule:
Count the zeros in the power of 10, then add that many zeros to your number
🔢 Multiply Numbers Ending in Zeros
Three-Step Method:
- Ignore the zeros temporarily
- Multiply the non-zero digits
- Add back all the zeros to the product
Formula:
Multiply non-zero parts + Add total number of zeros
Type 1: One Number Ending in Zeros:
Example: ( 4{,}000 × 6 )
Step 1: Ignore zeros: ( 4 × 6 = 24 )
Step 2: Count zeros in 4,000: 3 zeros
Step 3: Add 3 zeros to 24
Answer: 24,000
Type 2: Both Numbers Ending in Zeros:
Example: ( 50 × 8{,}000 )
Step 1: Ignore zeros: ( 5 × 8 = 40 )
Step 2: Count total zeros: 50 has 1 zero, 8,000 has 3 zeros = 4 zeros total
Step 3: Add 4 zeros to 40
Answer: 400,000
Example: ( 200 × 3{,}000 )
Step 1: ( 2 × 3 = 6 )
Step 2: 200 has 2 zeros + 3,000 has 3 zeros = 5 zeros total
Step 3: Add 5 zeros to 6
Answer: 600,000
Example: ( 400 × 50 )
Step 1: ( 4 × 5 = 20 )
Step 2: 400 has 2 zeros + 50 has 1 zero = 3 zeros total
Step 3: Add 3 zeros to 20
Answer: 20,000
⚡ Multiply a Whole Number by a Power of Ten
Power of Ten Rule:
Number × ( 10^n ) = Add ( n ) zeros to the number
Examples:
( 7 × 10^1 = 7 × 10 = 70 )
( 7 × 10^2 = 7 × 100 = 700 )
( 7 × 10^3 = 7 × 1{,}000 = 7{,}000 )
( 45 × 10^2 = 45 × 100 = 4{,}500 )
( 123 × 10^4 = 123 × 10{,}000 = 1{,}230{,}000 )
📐 Estimate Products
What is Estimation?
Estimating products means finding an approximate answer by rounding the numbers before multiplying. This helps check if exact answers are reasonable.
Estimation Steps:
- Round each factor to its highest place value
- Multiply the rounded numbers
- Compare with actual answer to check reasonableness
Estimation Examples:
Example 1: Estimate ( 48 × 72 )
Step 1: Round to nearest ten: ( 48 ⇒ 50 ), ( 72 ⇒ 70 )
Step 2: Multiply: ( 50 × 70 = 3{,}500 )
Estimate: 3,500
(Actual: ( 48 × 72 = 3{,}456 ))
Example 2: Estimate ( 387 × 52 )
Step 1: Round: ( 387 → 400 ), ( 52 → 50 )
Step 2: Multiply: ( 400 × 50 = 20{,}000 )
Estimate: 20,000
Example 3: Estimate ( 2{,}876 × 9 )
Step 1: Round: ( 2{,}876 → 3{,}000 ), ( 9 → 10 )
Step 2: Multiply: ( 3{,}000 × 10 = 30{,}000 )
Estimate: 30,000
✖️ Multiply by 1-Digit Numbers
Standard Algorithm:
- Write numbers vertically, lining up place values
- Multiply starting from the ones place
- Regroup (carry) when necessary
- Continue through all place values
Example: ( 4{,}567 × 8 )
4,567
× 8
_______
36,536
Steps:
- ( 7 × 8 = 56 ) (write 6, carry 5)
- ( 6 × 8 = 48 + 5 = 53 ) (write 3, carry 5)
- ( 5 × 8 = 40 + 5 = 45 ) (write 5, carry 4)
- ( 4 × 8 = 32 + 4 = 36 ) (write 36)
Multiplication Formula:
Factor × Factor = Product
✖️✖️ Multiply by 2-Digit Numbers
Two-Step Process:
- Multiply by the ones digit
- Multiply by the tens digit (remember to add a zero placeholder)
- Add the two partial products together
Example: ( 34 × 25 )
34
× 25
_____
170 (34 × 5)
+680 (34 × 20)
_____
850
Steps:
- Step 1: ( 34 × 5 = 170 ) (first partial product)
- Step 2: ( 34 × 2 = 68 ), add 0 placeholder = 680 (second partial product)
- Step 3: ( 170 + 680 = 850 )
Example: ( 146 × 32 )
146
× 32
______
292 (146 × 2)
+4380 (146 × 30)
______
4,672
⚡ Properties of Multiplication
Five Properties of Multiplication:
Property 1: Commutative Property
( a × b = b × a )
Meaning: Changing the order of factors doesn't change the product
Examples: ( 4 × 7 = 7 × 4 = 28 )
Property 2: Associative Property
( (a × b) × c = a × (b × c) )
Meaning: Grouping of factors doesn't change the product
Examples: ( (2 × 3) × 4 = 2 × (3 × 4) = 24 )
Property 3: Distributive Property
( a × (b + c) = (a × b) + (a × c) )
Meaning: Multiply can be distributed over addition
Examples: ( 5 × (3 + 4) = (5 × 3) + (5 × 4) = 15 + 20 = 35 )
Property 4: Identity Property
( a × 1 = a )
Meaning: Any number multiplied by 1 equals itself
Examples: ( 67 × 1 = 67 ), ( 1 × 250 = 250 )
Property 5: Zero Property
( a × 0 = 0 )
Meaning: Any number multiplied by zero equals zero
Examples: ( 45 × 0 = 0 ), ( 0 × 892 = 0 )
🎯 Multiply Using Properties
Strategies Using Properties:
Strategy 1: Break Apart Numbers (Distributive Property)
Example: ( 7 × 18 )
Think: ( 18 = 10 + 8 )
Calculate: ( 7 × (10 + 8) = (7 × 10) + (7 × 8) )
( = 70 + 56 = 126 )
Strategy 2: Rearrange Factors (Commutative Property)
Example: ( 25 × 7 × 4 )
Rearrange: ( 25 × 4 × 7 )
Calculate: ( 100 × 7 = 700 )
Strategy 3: Group for Easier Multiplication (Associative Property)
Example: ( 2 × 17 × 5 )
Group: ( (2 × 5) × 17 )
Calculate: ( 10 × 17 = 170 )
Strategy 4: Double and Half
Example: ( 16 × 5 )
Think: ( 16 × 5 = (8 × 2) × 5 = 8 × 10 = 80 )
⚖️ Compare Products of Whole Numbers
Comparison Strategies:
- Estimate first to compare quickly
- Look at place values - more digits usually means larger product
- Compare factors - larger factors generally give larger products
- Calculate exactly if estimation isn't clear
Comparison Symbols:
( > ) means "greater than"
( < ) means "less than"
( = ) means "equal to"
Examples:
Compare: ( 45 × 12 ) __ ( 38 × 15 )
( 45 × 12 = 540 )
( 38 × 15 = 570 )
Answer: ( 45 × 12 < 38 × 15 )
Compare: ( 60 × 40 ) __ ( 50 × 50 )
( 60 × 40 = 2{,}400 )
( 50 × 50 = 2{,}500 )
Answer: ( 60 × 40 < 50 × 50 )
📋 Quick Reference: Multiplication Formulas
| Concept | Formula/Rule |
|---|---|
| Basic Multiplication | Factor × Factor = Product |
| Multiply by Power of 10 | Number × ( 10^n ) = Add ( n ) zeros |
| Numbers Ending in Zeros | Multiply non-zeros + Add all zeros |
| Commutative Property | ( a × b = b × a ) |
| Associative Property | ( (a × b) × c = a × (b × c) ) |
| Distributive Property | ( a × (b + c) = ab + ac ) |
| Identity Property | ( a × 1 = a ) |
| Zero Property | ( a × 0 = 0 ) |
| Estimation | Round → Multiply → Compare |
🌟 Key Takeaways
- Multiplying by powers of 10 adds zeros to numbers
- When multiplying numbers ending in zeros, multiply non-zero parts first, then add all zeros
- Estimate products by rounding factors first
- Use standard algorithm for multi-digit multiplication
- Two-digit multiplication requires two partial products
- Five properties: Commutative, Associative, Distributive, Identity, and Zero
- Properties make multiplication easier and faster