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P.2 Use multiplication properties to complete equations

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What are multiplication properties?

Multiplication properties are special rules that are always true. They help us understand how numbers work together when we multiply and make solving problems easier and faster.

Examples:
  • Changing the order of factors does not change the product: 3 × 5 = 5 × 3
  • Grouping factors differently does not change the product: (2 × 4) × 3 = 2 × (4 × 3)
Note

Think of properties as reliable shortcuts. Once you learn them, you can use them every time you multiply.

The commutative property of multiplication

The commutative property states that changing the order of the factors does not change the product. You can multiply numbers in any sequence.

Examples:
  • 8 × 7 = 56 and 7 × 8 = 56
  • 4 × 9 = 9 × 4
  • If 5 × 3 = 15, then 3 × 5 must also equal 15.
Note

This property only works for addition and multiplication. It does not work for subtraction or division.

The associative property of multiplication

The associative property states that when multiplying three or more numbers, the way you group them does not change the product. You change which numbers you multiply first.

Examples:
  • (3 × 2) × 5 = 3 × (2 × 5)
  • First group: (3 × 2) = 6, then 6 × 5 = 30
  • Second group: (2 × 5) = 10, then 3 × 10 = 30
Note

The parentheses ( ) tell you which numbers to multiply first. Even when the grouping changes, the final product stays the same.

The distributive property

The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.

Examples:
  • 4 × (3 + 2) = (4 × 3) + (4 × 2)
  • 4 × 5 = 12 + 8
  • 20 = 20
Note

This property helps break down harder multiplication facts into smaller, easier ones you already know.

The identity property of multiplication

The identity property states that when you multiply any number by 1, the product is that number. One is called the multiplicative identity.

Examples:
  • 9 × 1 = 9
  • 1 × 47 = 47
  • 125 × 1 = 125
Note

Any number multiplied by 1 keeps its identity. It remains the same number.

The zero property of multiplication

The zero property states that when you multiply any number by 0, the product is always 0.

Examples:
  • 6 × 0 = 0
  • 0 × 15 = 0
  • 328 × 0 = 0
Note

Zero acts like an "eraser" in multiplication. No matter how large the other factor is, the product will be zero.

Using properties to complete equations

You can use your knowledge of multiplication properties to find missing numbers in equations. Look for clues about which property is being used.

Examples:
  • Commutative: 7 × ▢ = 5 × 7 → The missing number is 5.
  • Associative: (4 × 2) × 3 = 4 × (▢ × 3) → The missing number is 2.
  • Distributive: 3 × (5 + 4) = (3 × 5) + (3 × ▢) → The missing number is 4.
  • Identity: ▢ × 9 = 9 → The missing number is 1.
Note

First, identify the property. Then, use what you know about that property to figure out the missing value.