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Properties of Equality, Identity, and Operations September 11, 2014 Essential Question: Can I justify solving an equation using mathematical properties?
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Commutative Property a + b = b + a (a)(b) = (b)(a) The Commutative Property states that the order of the numbers may change and the sum/product will remain the same. This property applies to only addition and multiplication; NOT subtraction and division. 2 + 3 = 3 + 2 (2)(3) = (3)(2)
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Associative Property (a + b) + c = a + (b + c) (a · b) · c = a · (b · c) The Associative Property states that the grouping of numbers can change and the sum/product will remain the same. This property also applies to both addition and multiplication. (2 + 4) + 5 = 2 + (4 + 5) (2 · 4) · 5 = 2 · (4 · 5)
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Distributive Property of Multiplication a (b + c) = a(b) + a(c) a (b – c) = a(b) – a(c) The Distributive Property takes a number and multiplies it by everything inside the parentheses. This property works over addition and subtraction. 2(3 + 4) = 2(3) + 2(4) 2 (5 – 2) = 2(5) – 2(2)
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Identity Properties n · 1 = n n + 0 = n This property shows how a given number is itself when multiplied by 1 or added to 0. The one and zero act like mirrors. 4 · 1 = 4 5 + 0 = 5
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Zero Property of Multiplication n · 0 = 0 Simply stated, any number times zero equals zero.
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Multiplicative Inverse Property ½ (2) = 1 This property is helpful when solving equations where there is a fraction “attached” to a variable by multiplication. The normal inverse operation for multiplication is division, but in this case, you will multiply both sides of the equation by the reciprocal of the fraction. ½ n – 3 = 4 ½ n -3 + 3 = 4 + 3 ½ n = 7 ½ n (2) = 7(2) n = 14
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Addition Property of Equality If a = b, then a + c = b + c or a + (-c) = b + (-c) The addition property of equality says that if you may add equal quantities to each side of the equation & still have equal quantities Example In if-then form: If 6 = 6 ; then 6 + 3 = 6 + 3 or 6 + (-3) = 6 + (- 3).
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Subtraction Property of Equality If a = b, then a – c = b – c. The subtraction property of equality says that if you may subtract equal quantities to each side of the equation & still have equal quantities Example In if-then form: If 6 = 6 ; then 6 - 3 = 6 - 3
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Multiplication Property of Equality If a = b, then ac = bc The multiplication property of equality says that if you may multiply equal quantities to each side of the equation & still have equal quantities. In if-then form: If 6 = 6 ; then 6 * 3 = 6 * 3.
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Division Property of Equality If a = b and c ≠ 0, then a ÷ c = b ÷ c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation. In if-then form: If 6 = 6 ; then 6 ÷ 3 = 6 ÷ 3 Why can’t C be 0?
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Properties of Equality Turn and Talk Notice, after using any of the properties of equality, the numbers are still equal. Why do you think it is important to learn these properties?
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