Properties of Equality, Identity, and Operations September 11, 2014 Essential Question: Can I justify solving an equation using mathematical properties?

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)

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)

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)

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 = = 5

Zero Property of Multiplication n · 0 = 0 Simply stated, any number times zero equals zero.

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 = ½ n = 7 ½ n (2) = 7(2) n = 14

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 = or 6 + (-3) = 6 + (- 3).

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

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.

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?

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?