Expressions & Equations

1. Expressions & Equations www.njctl.org 2013-01-23

2. Setting the PowerPoint View • Use Normal View for the Interactive Elements • To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.  • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. • On the View tab, click Slide Master | Page Setup. Select On-screen Show (4:3) under Slide sized for and click Close Master View. • On the Slide Show menu, confirm that Resolution is set to 1024x768. • Use Slide Show View to Administer Assessment Items • To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 10 for an example.)

3. Table of Contents Commutative and Associative Properties Combining Like Terms Click on a topic to go to that section. The Distributive Property and Factoring Simplifying Algebraic Expressions Inverse Operations One Step Equations Two Step Equations Multi-Step Equations Distributing Fractions in Equations Translating Between Words and Equations Using Numerical and Algebraic Expressions and Equations Graphing & Writing Inequalities with One Variable Simple Inequalities involving Addition & Subtraction Simple Inequalities involving Multiplication & Division CommonCoreStandards: 7.EE.1, 7.EE.3, 7.EE.4

4. Commutative and Associative Properties Return to table of contents

5. Commutative Property of Addition: The order in which the terms of a sum are added does not change the sum. a + b = b + a 5 + 7 = 7 + 5 12= 12 Commutative Property of Multiplication: The order in which the terms of a product are multiplied does not change the product. ab = ba 4(5) = 5(4)

6. Associative Property of Addition: The order in which the terms of a sum are grouped does not change the sum. (a + b) + c = a + (b + c) (2 + 3) + 4 = 2 + (3 + 4) 5 + 4 = 2 + 7 9 = 9

7. The Commutative Property is particularly useful when you are combining integers. Example: -15 + 9 + (-4)= -15 + (-4) + 9= Changing it this way allows for the -19 + 9 = negatives to be added together first. -10

8. Associative Property of Multiplication: The order in which the terms of a product are grouped does not change the product.

9. 1 Identify the property of -5 + 3 = 3 + (-5) A Commutative Property of Addition Commutative Property of Multiplication B C Associative Property of Addition D Associative Property of Multiplication

10. 2 Identify the property of a + (b + c) = (a + c) + b A Commutative Property of Addition Commutative Property of Multiplication B C Associative Property of Addition D Associative Property of Multiplication

11. Identify the property of (3 * (-4)) *8 = 3 * ((-4) *8) 3 A Commutative Property of Addition B Commutative Property of Multiplication C Associative Property of Addition D Asociative Property of Multiplication

12. Discuss why using the Commutative Property would be useful with the following problems: 1. 4 + 3 + (-4) 2. -9 x 3 x 0 3. -5 x 7 x -2 4. -8 + 1 + (-6)

13. Combining Like Terms Return to table of contents

14. An Algebraic Expression - contains numbers, variables and at least one operation.

15. Like terms: terms in an expression that have the same variable raised to the same power Examples: LIKE TERMS NOT LIKE TERMS 6x and 2x 6x2 and 2x 5y and 8y 5x and 8y 4x2 and 7x2 4x2yand 7xy2

16. 4 Identify all of the terms like 2x A 5x B 3x2 C 5y D 12y E 2

17. 5 Identify all of the terms like 8y A 9y B 4y2 C 7y D 8 E -18x

18. 6 Identify all of the terms like 8xy A 8x B 3x2y C 39xy D 4y E -8xy

19. 7 Identify all of the terms like 2y A 51w B 2x C 3y D 2w E -10y

20. 8 Identify all of the terms like 14x2 A -5x B 8x2 C 13y2 D x E -x2

21. If two or more like terms are being added or subtracted, they can be combined. To combine like terms add/subtract the coefficient but leave the variable alone. 7x +8x =15x 9v-2v = 7v

22. Sometimes there are constant terms that can be combined. 9 + 2f + 6 = 9 + 2f + 6 = 2f + 15 Sometimes there will be both coeffients and constants to be combined. 3g+ 7+ 8g - 2 11g + 5 Notice that the sign before a given term goes with the number.

23. Try These: 1.) 2b +6g(3) + 4f + 9f 2.) 9j + 3 + 24h + 6 + 7h + 3 3.) 7a + 4 + 2a -1 9 + 8c -12 + 5c 4.) 8x + 56xy + 5y

24. 9 8x + 3x = 11x A True B False

25. 10 7x + 7y = 14xy A True B False

26. 11 2x + 3x = 5x A True B False

27. 12 9x + 5y = 14xy A True B False

28. 13 6x + 2x = 8x2 A True B False

29. 14 -15y + 7y = -8y A True B False

30. 15 -6 + y + 8 = 2y A True B False

31. 16 -7y + 9y = 2y A True B False

32. 17 9x + 4 + 2x = A 15x B 11x + 4 C 13x + 2x D 9x + 6x

33. 18 12x + 3x + 7 - 5 A 15x + 7 - 5 B 13x C 17x D 15x + 2

34. 19 -4x - 6 + 2x - 14 A -22x B -2x - 20 C -6x +20 D 22x

35. The Distributive Property and Factoring Return to table of contents

36. An Area Model Imagine that you have two rooms next to each other. Both are 4 feet long. One is 7 feet wide and the other is 3 feet wide . How could you express the area of those two rooms together? 4 3 7

37. 4 4 4 7 + 3 7 3 You could multiply 4 by 7, then 4 by 3 and add them 4(7) + 4(3) = 28 + 12 = 40 You could add 7 + 3 and then multiply by 4 4(7+3)= 4(10)= 40 OR Either way, the area is 40 feet2:

38. An Area Model Imagine that you have two rooms next to each other. Both are 4 yards long. One is 3 yards wide and you don't know how wide the other is. How could you express the area of those two rooms together? 4 x 3

39. You cannot add x and 3 because they aren't like terms, so you can only do it by multiplying 4 by x and 4 by 3 and adding 4(x) + 4(3)= 4x + 12 The area of the two rooms is 4x + 12 (Note: 4x cannot be combined with 12) 4 x + 3

40. The Distributive Property Finding the area of the rectangles demonstrates the distributive property. Use the distributive property when expressions are written like so: a(b + c) 4(x + 2) 4(x) + 4(2) 4x + 8 The 4 is distributed to each term of the sum (x + 2)

41. Write an expression equivalent to: 5(y + 4) 5(y) + 5(4) 5y + 20 6(x + 2) 3(x + 4) 4(x - 5) 7(x - 1) Remember to distribute the 5 to the y and the 4

42. The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions. EXAMPLE: -2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6 3(4x - 6) = 3(4x) - 3(6) = 12x - 18 -2 (x - 3) = -2(x) - (-2)(3) = -2x + 6 TRY THESE: 3(4x + 2) = -1(6m + 4) = -3(2x - 5) = Be careful with your signs!

43. Keep in mind that when there is a negative sign on the outside of the parenthesis it really is a -1. For example: -(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7 What do you notice about the original problem and its answer? Remove to see answer. The numbers are turned to their opposites. Try these: -(9x + 3) = -(-5x + 1) = -(2x - 4) = -(-x - 6) =

44. 20 4(2 + 5) = 4(2) + 5 A True B False

45. 21 8(x + 9) = 8(x) + 8(9) A True B False

46. 22 -4(x + 6) = -4 + 4(6) A True B False

47. 23 3(x - 4) = 3(x) - 3(4) A True B False

48. 24 Use the distributive property to rewrite the expression without parentheses 3(x + 4) A 3x + 4 B 3x + 12 C x + 12 D 7x

49. 25 Use the distributive property to rewrite the expression without parentheses 5(x + 7) A x + 35 B 5x + 7 5x + 35 C D 40x

50. 26 Use the distributive property to rewrite the expression without parentheses (x + 5)2 A 2x + 5 B 2x + 10 C x + 10 D 12x