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Understanding Properties of Addition and Multiplication for Easier Computation

In this lesson, we explore the essential properties of addition and multiplication that simplify computations. Highlighting the Commutative Property, we learn that rearranging addends or factors does not affect the sum or product, respectively. The Associative Property teaches us that changing the grouping of numbers also yields the same results. Additionally, the Distributive Property illustrates how to distribute multiplication over addition or subtraction. We will work through examples and practice exercises to reinforce these key mathematical concepts.

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Understanding Properties of Addition and Multiplication for Easier Computation

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  1. TCAP Coach Lesson Lesson 3: Properties of Addition and Multiplication Pages: 21-25

  2. Getting the Idea • The properties of addition and multiplication can make computation easier and faster. • The Commutative Property of Addition states that changing the order of the addends does not change the sum. • The Commutative Property of Multiplication states that changing the order of the factors does not change the product.

  3. Example 1 • What is the value of x? 9.8+4.7 = X +9.8 • Strategy: Use the Commutative Property of Addition. Changing the order of the addends does not change the sum. 9.8+4.7 = 4.7+9.8 14.5 = 14.5 Solution: The value of x is 4.7

  4. Example 2 • What is the value of y? 3.95 X y = 2.3 X 3.95 • Strategy: Use the Commutative Property of Multiplication. Changing the order of the factors does not changes the product. 3.95 X 2.3 = 2.3 X 3.95 9.085 = 9.085 • Solution: The value of y is 2.3

  5. Associative Properties • The Associative Property of Addition states that changing the grouping of the addends does not change the sum. • The Associative property of Multiplication states that changing the grouping of the factors does not change the product. • When you use associative properties, try to find a grouping of numbers that makes the computation easier to do mentally.

  6. Example 3 • 14.3+(12.7+13.27)=[ ] • Strategy: Use the Associative Property of Addition Step 1: Use the Associative Property to regroup the addends. 14.3+(12.7+13.47) = (14.3+12.7) +13.47 Step 2: Add inside the parentheses first. (14.3+12.7)+13.47 27 +13.47 Step 3: Find the final sum. 27+13.47=40.47 • Solution: 14.3+(12.7+13.47)=40.47

  7. Example 4 • (32x6)x15= [ ] • Strategy: Use the Associative Property of Multiplication Step 1: Use the associative property of multiplication to regroup the factors. (32x6)x15 = 32x(6x15) Step 2: Multiply inside the parentheses first. 32x(6x15) 32x90 Step 3: Find the final product. 32x90=2,880 • Solution: (32x6)x15=2,880

  8. Distributive Property • The Distributive Property of Multiplication over Addition states that to multiply a sum by a number, you can multiply each addend by the number and add the products. • For any numbers a,b,and c, a(b+c)= (axb)+(axc)

  9. Example 5 • 76x47= [ ] • Strategy: Use the Distributive Property of Multiplication over Addition Step 1: Write one factor as the sum of two numbers. 47=40+7 76x47=76x(40x7)+(76x7) Step 2: Multiply each set of factors. (76x40)+(76x7) 3,040 + 532 Step 3: Add the products. 3,040+532=3,572 • Solution: 76x47=3,572

  10. Distributive Property • The Distributive Property of Multiplication over Subtraction states that to multiply a difference by a number, you can multiply each of the two numbers by that number and then find the difference in the products. • For any numbers a, b, and c • a(b-c) = ab-ac

  11. Example 6 • 83X58= [ ] • Strategy: Use the Distributive Property of Multiplication over Subtraction. Step1: Write one factor as the difference of two numbers. 58=60-2 83 X 58 = 83 x (60-2) - (83x2) Step 2: Multiply each set of factors (83x60) – (83x2) 4,980 - 166 Step 3: Subtract the products. 4,980-166= 4,814 • Solution: 83x58= 4,814

  12. Other Properties to help you Multiply • The Identity Property of Addition states that when one addend is 0, the sum is equal to the other addend. Ex. 3.82+0=3.82 • The Identity Property of Multiplication states that when one factor is 1, the product is equal to the other factor. Ex. 4.25x1=4.25 • The Zero Property of Multiplication states that when one factor is 0, the product is equal to 0. Ex. 5.73x0=0

  13. Lesson Practice • You are now to complete the Lesson Practice. • Questions 1-8 • Page 25

  14. Exit Ticket • Define: • Commutative Property of Addition • Associative Property of Multiplication • Distributive Property of Multiplication over Addition

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