1. Region 11: Math & Science Teacher Center Solving Equations

2. Where we’ve been… Equality Ratio & Proportion Pattern Generalization

3. Patterns Review Patterns Review - Justify It! 3rd Grade MCA Practice Problem 1 4 7 10

4. Patterns Review Patterns Review - Justify It! 3rd Grade MCA Practice Problem 1 4 7 10

5. With a partner, talk about how a third grader would talk about this problem: Draw the next figure in the pattern How many dots will be in the next figure? Describe how make the pattern Patterns Review Adult Challenge: How many dots would be in the nth figure?

6. Patterns Review Share in Grade Level Groups: What strategies did you see kids use? What did the students find most challenging? What growth did you see? What surprised you?

7. Patterns Review Watch video and record: Good questions Not-so-good questions

8. Patterns Review Discuss video at your tables: What questions helped most to get your students to explain the explicit rule for different patterns? What did you learn about scaffolding questions? How do you know when you are done asking questions?

9. Solving Equations Goals: Identify a developmental sequence for solving equations Structure “math talk “about expressions & equations Emphasize equivalence and how we communicate this with students Discuss properties of equations Work with a partner to solve problems on page 1

10. Solve. Find all values that make the statement true.

11. Solve.

12. Solve.

13. Solve.

14. Solve.

15. Solve.

16. What does solve mean?

17. Break

18. Think/Pair/Share… Define expression Define equation

19. Big Idea! We solve equations because we can make them true or false. We don’t solve expressions because we can’t make them either true or false.

20. Expression vs. Equation Share and discuss in your group as you work on page 2: How would you write the directions? How would you want/expect students to show their work?…What would you write on the board?

21. Expression vs. Equation

27. Directions make a big difference! Directions depend on context and where you are in the curriculum Expressions are not equations No one “right way” to show work Most middle school textbook authors have thought carefully about what strategies to use to solve equations

28. CGI Algebra Video Clips

29. Benchmarks in Student Thinking About The Equal Sign Description 1 Children are asked to be specific about what they think the equal sign means 2 Children accept as true some number sentence that is not of the form a + b = c 3 Children recognize that the equal sign represents a relation between two equal numbers, particularly through calculation (relational understanding of the equal sign) 4 Children are able to compare the mathematical expressions using relational thinking without carrying out the calculations (relational understanding across the equal sign) Note: These benchmarks are a guide, not a firm sequence

30. CGI Algebra Video Clips Solving the Equation (4 th grader, 2 minutes, 42 seconds)

31. CGI Algebra Video Clips Solving the Equation (4 th grader, 1 minutes, 51 seconds)

32. CGI Algebra Video Clips Solving the Equation (4 th grader, 39 seconds)

33. CGI Algebra Video Clips Solving the Equation (4 th grader, 51 seconds)

34. Benchmarks in Student Thinking About The Equal Sign Description 1 Children are asked to be specific about what they think the equal sign means 2 Children accept as true some number sentence that is not of the form a + b = c 3 Children recognize that the equal sign represents a relation between two equal numbers, particularly through calculation (relational understanding of the equal sign) 4 Children are able to compare the mathematical expressions using relational thinking without carrying out the calculations (relational understanding across the equal sign) Note: These benchmarks are a guide, not a firm sequence

35. Lunch

36. Methods for solving linear equations of the form ax ± b = cx ± d Traditional Approach vs. Functions Approach

37. Traditional approach for solving linear equations of the form ax ± b = cx ± d 1)Use of number facts (solve mentally) Example: 3 + x = 7 Not-so-good for: 3x + 7 = 5x – 14

38. 2) Generate and evaluate (“guess and check” or “trial and error substitution”) Example: 2x + 3 = 4x – 7 Not-so-good for: 3x – 7 = 10 – 4x Traditional approach for solving linear equations of the form ax ± b = cx ± d

39. 3) a. Undoing (or working backwards) Example: 20 = 3x – 4 24 = 3 x 8 = x Not-so-good for: 3x + 7 = 5x – 14 Traditional approach for solving linear equations of the form ax ± b = cx ± d

40. 3) b. Undoing 17 = 3p – 1 x 3- 1 + 1 ÷ 3 p3p3p3p – 1 17186 Traditional approach for solving linear equations of the form ax ± b = cx ± d

41. 4) Cover-up Example: k + k + 13 = k + 20 k + k + 13 = k + 13 + 7 k = 7 Not-so-good for: 3x + 7 = 25 – 5x Traditional approach for solving linear equations of the form ax ± b = cx ± d

42. 5) Transposing (change side-change sign) Example: 3x = 8 5x = 15 x = 3 Not-so-good for: – 7– 2x+ 2x7 + Traditional approach for solving linear equations of the form ax ± b = cx ± d

43. 6) Equivalent equations (performing the same operation on both sides) Example: 17 = 3x – 7 17 + 7 = 3x – 7 + 7 24 = 3x 24/3 = 3x/3 8 = x Traditional approach for solving linear equations of the form ax ± b = cx ± d

44. Group Task #1 Break into six table groups Write one or two good problems for each method on the table When the bell rings, move to the next table Traditional approach for solving linear equations of the form ax ± b = cx ± d

45. Group Task #2 Move to the table where you started Work the problems using that particular method Star any problems that can’t be solved easily with the method Determine the minimum benchmark level needed to solve these problems Be ready to report out Traditional approach for solving linear equations of the form ax ± b = cx ± d

46. Functions approach for solving linear equations of the form ax ± b = cx ± d 1) Table using graphing calculator (similar to guess and check) Example: 3x – 4 = x + 6 x = 5 Y1Y1 Y2Y2

47. Functions approach for solving linear equations of the form ax ± b = cx ± d 2) Graphing Example: 3x – 4 = x + 6 x = 5 Y1Y1 Y2Y2

48. Why standards? “By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more-sophisticated work in algebra in the middle grades and high school.” - NCTM, 2000, p.37

49. True or False Your textbook determines the algebra concepts and skills that you should cover at a particular grade level. False : In a standards-based system, the focus is shifted from what is TAUGHT to what is LEARNED. The standards tell us what students should know and be able to do.  TRUE FALSE   don’t know

50. True or False Algebra content has been shifted down and now starts in the middle grades. Algebra content has been shifted down and now starts in the middle grades. False: Algebra and algebraic thinking are integrated across K-11 in the state standards. Every teacher has to do his/her part to give students the opportunity to learn the grade-level content.  TRUE FALSE   don’t know

51. Sort the Standards Involving Solving Equations In your groups, Look through individual standards Classify for grades 1 - 8 Ask for an answer key when finished Take time to reflect on standards with your group members

52. BIG IDEA TRADITIONAL ALGEBRA I

53. 8th GRADE STANDARDS BIG IDEA

54. Break

55. Equivalence - Expressions On page 7: State directions for each problem Simplify each expression or equation one operation at a time, leaving a trail down of equivalent expressions or equivalent equations.

56. Equivalence - Expressions Directions:

57. Equivalence - Expressions Directions:

58. Equivalence - Expressions Directions: Check:

59. Equivalence - Expressions Expressions are equivalent… …if every line has the same value for the same value of x

60. Equivalence - Expressions Equations are equivalent… …if they have the same solution set (but each line has a different value for a given value of x).

61. Equivalence - Expressions Can I add 5? … 10?

62. STRETCH

63. Balance metaphor for equations Same weight on both sides page 8 of the handout

64. Balance metaphor for equations Build a balance to solve: 4 x + 4 = 12

65. Balance metaphor for equations Build a balance to solve: x + y = 7

66. Balance metaphor for equations Build a balance to solve: 2 x + 4 = 2 x + 9

67. Problems with balance metaphor Negatives

68. Problems with balance metaphors Subtraction

69. Metaphors for equations – Balance (same weight on both sides) 1. Open Number sentences (CGI)

70. Metaphors for equations – Balance (same weight on both sides) ii.Algebra Tiles (McDougal Littell) or Bags of Gold (CMP)

71. Metaphors for equations – Balance (same weight on both sides) 3.Equation Mat (CPM – Algebra Connections)

72. STRETCH

73. Baseline Assessment

74. 1. Find the value of m that makes the number sentence below true. 12 = 4 m

75. Baseline Assessment 2. Find the value of b that makes the number sentence below true. 15 + 3 b = 42

76. Baseline Assessment 3. Find the value of x that makes the number sentence below true. 12 x - 10 = 6 x + 32

77. Baseline Assessment 4. Find the value of n that makes the number sentence below true. Show your steps to demonstrate how you solved the problem. 4 + n - 2 + 5 = 11 + 3 + 5

78. Baseline Assessment 5. Balance A is balanced. The amount on the left side of Balance A is tripled for Balance B. Draw in what should appear on the right side of Balance B to be balanced. Explain why your answer works.

79. Baseline Assessment 6. Determine which of the equations below are equivalent to: 3 b – 4 = b + 6 Circle yes, no or do not know for each part.

80. Properties of Equations Addition Property of Equality Words Adding the same number to each side of an equation produces an equivalent equation Algebra from McDougal Littell Math Course 2

81. Properties of Equations What other properties maintain equivalence? Words Algebra

82. Big Ideas We want to: Learn the different methods that children use to solve linear equations Learn how to select examples to push kids from less sophisticated methods (i.e. guess and check) to more algebraic methods Learn different metaphors for helping students solve equations

83. Evaluation On an index card, please record: P: one positive from today’s work M: one ‘minus’ or concern from today’s work I: something that you found interesting or intriguing from today’s work. Thanks for your feedback.