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Understanding Properties of Addition: Identity, Commutative, and Associative Laws

This educational overview covers fundamental properties of addition, including the Identity Property (a + 0 = a), Commutative Property (a + b = b + a), and Associative Property ((a + b) + c = a + (b + c)). These properties are essential for performing arithmetic operations efficiently. Examples illustrate each property, emphasizing their application in mental math. Additionally, the Compensation method for simplifying addition of decimal numbers is discussed, showing its practical usefulness when adding numbers close to whole values.

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Understanding Properties of Addition: Identity, Commutative, and Associative Laws

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  1. Properties of Addition

  2. Identity Property of Addition • a + 0 = a • 17 + 0 = ? • 17 • -123 + 0 = ? • -123

  3. Commutative Property of Addition • Changing the order of the addends does not change their sum. • a + b = b + a • 23 + 17 = 17 + 23 • 50 + 2 = 2 + 50

  4. NO SUCH THING! Commutative Property of Subtraction • a – b = b – a • 12 – 5 = 5 – 12

  5. Associative Property of Addition • Changing the grouping of the addends does not change their sum. • (a + b) + c = a + (b + c) • This property can be extremely useful. For example: • What is 23 + 125 + 7? • 23 + 7 = 30 … 30 + 125 = 155. • What is 99.5 + 17 + 0.5? • 117.

  6. Quick Check • Use mental math to find: 4.4 + 5.3 + 0.6 • 4.4 + 0.6 + 5.3 = 5 + 5.3 = 10.3 • Use mental math to find: 2.85 + 62.89 + 7.15 • 2.85 + 7.15 + 62.89 = 10 + 62.89 = 72.89

  7. Extension: Using Compensation • The sum of two numbers remains the same if you add a number to one addend and subtract the same number from the other addend. • 5 + 7 = 12 • (5 – 3) + (7 + 3) = 2 + 10 = 12 • (5 + 6) + (7 – 6) = 11 + 1 = 12 • Why is this useful?

  8. Compensation is useful when adding decimals • $4.96 + $3.79 • ($4.96 + $0.04) + ($3.79 - $0.04) • = $5.00 + $3.75 • = $8.75

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