Multiplication
Grade 5 Math – Patterns, Properties & Strategies
Multiplication Patterns over Increasing Place Values
- Each step left on a place value chart = Multiply by 10
- Each step right = Divide by 10
- Pattern: Multiplying a number by 10, 100, 1,000... adds zeros!
Formula:
If \( a \times 10^n \), move decimal \( n \) places right or add \( n \) zeros.
E.g., \( 32 \times 1,000 = 32,000 \)
If \( a \times 10^n \), move decimal \( n \) places right or add \( n \) zeros.
E.g., \( 32 \times 1,000 = 32,000 \)
Multiply Numbers Ending in Zeros
- Multiply the non-zero digits, then add the zeros from both numbers!
- Pattern: \( 400 \times 50 = (4 \times 5) \) and two zeros from 400 + one zero from 50 = 20,000
Formula:
Let a0...0 × b0...0 = (a × b) × 10^{m+n}
Where m, n = number of zeros in each factor
Where m, n = number of zeros in each factor
Multiply a Whole Number by a Power of Ten
- Move the decimal point right by the exponent's value.
- Multiply 56 by 1000: \( 56 \times 10^3 = 56,000 \)
Formula:
\( a \times 10^n \): add \( n \) zeros to a.
Estimate Products
- Round each factor to a "friendly" number (10s, 100s, ...)
- Multiply the rounded numbers (ignoring their original digits)
- Example: \( 389 \times 72 \approx 400 \times 70 = 28,000 \)
Multiplying by 1-digit and 2-digit Numbers
- Line up numbers by rightmost digit.
- Multiply each digit, carry over, then sum partial products.
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Example:
\( 24 \times 36 = (24 \times 6) + (24 \times 30) = 144 + 720 = 864 \)
Properties of Multiplication
| Property Name | Formula | Example |
|---|---|---|
| Commutative | \( a \times b = b \times a \) | \( 3 \times 4 = 4 \times 3 = 12 \) |
| Associative | \( (a \times b) \times c = a \times (b \times c) \) | \( (2 \times 3) \times 7 = 2 \times (3 \times 7) = 42 \) |
| Distributive | \( a \times (b+c) = a\times b + a\times c \) | \( 5 \times (8+2) = 5\times 8 + 5\times 2 = 50 \) |
| Identity | \( a \times 1 = a \) | \( 8 \times 1 = 8 \) |
| Zero | \( a \times 0 = 0 \) | \( 6 \times 0 = 0 \) |
Multiply Using Properties
- Use distributive property to break numbers into easier parts.
- \( 28\times 16 = 28 \times (10+6) = (28\times 10) + (28\times 6) = 280 + 168 = 448 \)
- Use commutative/associative to rearrange and group for easier calculation.
Compare Products of Whole Numbers
- Estimate products or use properties to predict which product is larger.
- Example: Is \( 250 \times 79 \) larger than \( 220 \times 92 \)? Estimate by rounding to \( 300 \times 80 = 24,000 \) vs. \( 200 \times 100 = 20,000 \), so first is larger.
Quick Reference Summary
- When multiplying by powers of ten, count the zeros!
- Use rounding for fast estimation.
- Commutative: Order doesn't matter.
Associative: Grouping doesn't matter.
Distributive: Break numbers apart for easier calculation. - Always check you line up your digits for multi-digit multiplication!
Study Tip: Practice estimation and properties to improve mental math speed!