Algebra 1 SC Standards:

1. Field Properties and Axioms of Equality Mrs. Hacker Algebra 1 Grade: 8th Algebra 1 SC Standards: A.4 Carry out a procedure using the properties of real numbers (including commutative, associative, and distributive) to simplify expressions. O: SWBAT recognize and use the properties of identity, equality, distributive, commutative and associative properties to simplify expressions.

2. For any number a, a + 0 = 0 + a. “Identity à gets itself back" General: x + 0 = x 0 + z = z For any number x, x * 1 = 1 * x = x “Identity à gets itself back" General: a *1 = a 1 * q = q For an number r, r*0 = 0*r = 0 “Always returns (or is = to) a 0.” General: a * 0 = 0 0 * x = 0 0 = c*0 Additive Identity Property Ex: 1 3 + 0 = 3 Ex: 2 –7 + 0 = –7 Ex: 3 ½ + 0 = ½ Multiplicative Identity Property Ex.1 -1/3 * 1 = -1/3 Ex.2 7 = 1 * 7 Ex.3 -14 * 1 = -14 Multiplicative Property of Zero Ex.1 1 * 0 = 0 Ex.2 -8 * 0 = 0 Ex.3 ¼ * 0 = 0

3. Your Turn! Ex.1 Using the additive identity property, solve each equation. a. a + 0 = b. 3 + 0 = c. 0 + x = a 3 x Ex.2 Using the multiplicative property of zero, solve each equation. a. x * 0= b. 3(0) 0 0 Ex.3 Name the multiplicative inverse of each number. c. a d. -1/x a. 5 b. -1/3 -x 1/5 1/a -3

4. Multiplicative Inverse Property For every non-zero number a/b, where b does not equal 0, there is exactly one number, b/a such that a/b * b/a =1. "Flip the fraction and multiply to get 1" General: x/1 * 1/x = 1 1/a * a/1 = 1 r/s * s/r = 1 For any number a, a = a. General: x = x, -r = -r “Any number is equal to itself” For any numbers, a and b, if a = b, then b = a "Mirror Image" General: if a + b = c, then c = a + b Ex. 1 ½ * 2/1 = 1 Ex. 2 -4/1 * -1/4 = 1 Ex. 3 7/9 * 9/7 = 1 Reflexive Property Ex.1 3 = 3 Ex.2 - ½ = - ½ Ex. 3 -75 = -75 Symmetric Property Ex.1 if 3 + 5 =8, then 8 = 3 + 5 Ex.2 if ½ + ¼ = ¾ , then ¾ = ½ + ¼

5. Your Turn! Ex.4 Name the multiplicative inverse of each number or variable. Assume that no variable equals zero. a. 5 b. x c. 2/3 1/5 1/x 3/2 Ex.5 Using the reflexive property, solve each equation. a. a = b. 3 = c. xy = a 3 xy Ex.6 Using the symmetric property, solve each equation. a. x + y = b. if 7 + 1 = 8, then y + x 8 = 7 + 1

6. Transitive Property For any numbers x, y, and z, if x = y and y = z, then x = z "If 1st = 2nd, and 2nd = 3rd, then 1st = 3rd” General: If a = b and b = c then a = c If x = q + r and q + r = y, then x = y If a = b, then a may be replaced by b in any expression "Think replacement" General: x = y, x + 7 = y + 7, x/3 = y/3 Ex.1 if 2 + 4 = 6 and 6 = 3 + 3, then 2 + 4 = 3 + 3 Ex.2 if -7 + 4 = -3 and -3 = 1 – 4, then -7 + 4 = 1 – 4 Ex.3 if 10 = 2(5) and 2(5) = 1(10), then 10 = (1)(10) Substitution Property Ex.1 (1+2)(3) = 3(3) Ex.2 9(4+2) = 9(6) Ex.3 14 + 2 = 16

7. Your Turn! Ex.7 Using the transitive property, solve each equation. a. If 4 * 2 = 8 and 8 = 6 + 2, then 4 * 2 = 6 + 2 b. 12 = (-3)(-4) and (-3)(-4) = (6)(2), then 0 12 = (6)(2) Ex.8 Using the substitution property, solve equation. c. If a = 7, then 5a = a. 22 b. 10 ÷ 5 4 35 -3

8. term - a number, a variable, or a product or quotient of numbers and variables. coefficient - the number in front of the term. like terms - terms that have the same variables raised to the same exponent (or power), but could have different coefficients. 3x2y term

9. Examples of terms: Are they like terms or unlike terms? Pull Pull Pull Pull Pull Pull Pull ·3x and -7x like unlike ·6w and 8y ·8y2, 2y2 and -3y2 like like ·5, 89 and 100 unlike ·4x and 4x2 unlike ·5 and 8x ·x3y4 and ½x3y4 like

10. 3y x 2x2y 4 3x2y 7x x2y 5 7y2 6x 7y 10x 6x2 7x 10x2 Identify the like terms in the following expressions: Ex.93y + 2x2y - x + 4 + 3x2y - 5 + 7x - x2y Ex.10 6x + 6x2 + 7x + 7y + 7y2+ 10x + 10x2

11. To combine like terms: ·Figure out which terms are like terms. ·Add or subtract the coefficients (# in front.) ·Attach the variable or variables at the end.

12. Simplify each expression: Ex.11 7x + 9x + 8 + 13 Ex.12 19x2 + 7y2 + 21x2 16x + 21 40x2 + 7y2 Ex.13 9a + 60a -10 Ex.14 -7n - 21 - 8 - 64n 69a - 10 -71n - 29

13. For any numbers a, b, and c: a(b+c) = ab + ac and (b+c)a = ba + ca a(b-c) = ab – ac and (b-c)a = ba – ca "Distribute one number to the rest" For any numbers a and b, a + b = b + a and ab = ba General: a(b + c) = a(c + b) and xyz = yxz "Change order" For any numbers x, y, and z, (x+y) + z = x +(y+z) and (xy)z = x(yz). "Same order, but the parenthesis go in a different spot" General: (qr)s=q(rs) 7x + (3y - 2z) = (7x + 3y) - 2z Distributive Property Ex.1 x(8+3) = 8x + 3x Ex.2 2(4+t) = 8 + 2t Ex.3 8(3m + 6) = 24m + 48 Commutative Property DOES NOT work for division or subtraction! Ex.1 7 + 6 = 6 + 7 Ex.2 3(5) = 5(3) Ex.3 4(2*7) = 4(7*2) Associative Property Ex.1 (4 + 6) + 8 = 4 + (6 + 8) Ex.2 (2c + 6) + 10 = 2c + (6 + 10) Ex.3 7(11*3) = (7*11)3

14. Your Turn! Ex.15 Using the distributive property, solve each equation. a. 2(2 + x) b. y(a - 3) c. x(1 + y) 4 + 2x ay -3y x + xy Ex.16 Using all the properties, solve each equation. b. 4a + 2b + a a. 3(4x + y) + 2x 5a + 2b 14x + 3y d. 4y4 + 3y2 + y4 c. 4 + 6(ac + 2b) + 2ab 4 + 8ac + 12b 5y4 + 3y2

15. Know your properties or else!

16. Math Hoops

17. Match your Properties again!

18. Credits: Games: http://www.quia.com/hm/26073.html http://www.math-play.com/math-basketball-properties-of-multiplication/math-basketball-properties-of-multiplication.html Images: http://math.phillipmartin.info/math_properties.htm http://www.wordle.net/