1. 2008 May 29Algebra: Grades K-2: slide 1 Welcome to Grades K-2 Mathematics Content Sessions The focus is Algebraic Thinking. The goal is to help you understand this mathematics better to support your implementation of the K-8 Mathematics Standards.

2. 2008 May 29Algebra: Grades K-2: slide 2 Importance of Algebraic Thinking Algebraic thinking includes understanding and use of properties of numbers and relationships among numbers. Algebraic Thinking was chosen as the content focus because it lays the foundation for the learning of algebra in middle school and high school. It takes most students a long time to develop algebraic thinking, so it is important to begin this work in the primary grades.

3. 2008 May 29Algebra: Grades K-2: slide 3 Introduction of Facilitators

4. 2008 May 29Algebra: Grades K-2: slide 4 Introduction of Participants In a minute or two: 1. Introduce yourself. 2. Describe an important moment in your life that contributed to your becoming a mathematics educator. 3. Describe a moment in which you hit a mathematical wall and had to struggle with learning.

5. 2008 May 29Algebra: Grades K-2: slide 5 Overview Some of the problems may be appropriate for students to complete, but other problems are intended ONLY for you as teachers. As you work the problems, think about how you might adapt them for the students you teach. Also, think about what Performance Expectations these problems might exemplify.

6. 2008 May 29Algebra: Grades K-2: slide 6 Problem Set 1 The focus of Problem Set 1 is understanding equality. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

7. 2008 May 29Algebra: Grades K-2: slide 7 Problem 1.1 What is the mathematics underlying the concept of equality? That is, what would you want students to say if you asked, What does it mean for two things to be equal?

8. 2008 May 29Algebra: Grades K-2: slide 8 Problem 1.2 What do we want students to understand about the equal sign (=)?

9. 2008 May 29Algebra: Grades K-2: slide 9 Problem 1.3 Carpenter, Franke, and Levi (2003, p. 9) report data (shown on the next slide) showing students responses to the question below. What number would you put in the box to make this a true number sentence? 8 + 4 = + 5 What do you notice in the data? What conclusions can you draw?

10. 2008 May 29Algebra: Grades K-2: slide 10 Problem 1.3 What number would you put in the box to make this a true number sentence? 8 + 4 = + 5 What do you notice in the data? What conclusions can you draw?

11. 2008 May 29Algebra: Grades K-2: slide 11 Problem 1.4 Write two or three learning targets for equality and the equal sign. Be as precise as possible; that is, what do you want students to know about the equal sign? Where in the K-8 Mathematics Standards do these ideas appear?

12. 2008 May 29Algebra: Grades K-2: slide 12 Problem Set 2 The focus of Problem Set 2 is number relationships. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

13. 2008 May 29Algebra: Grades K-2: slide 13 Problem 2.1 Which of these number sentences are true, and which are false? Explain your thinking. a. 8 + 7 = 15 b. 15 = 8 + 7 c. 8 + 7 = 8 + 7 d. 8 + 7 = 7 + 8 e. 8 + 7 = 9 + 6 f. 8 + 7 = 87 g. 8 = 8

14. 2008 May 29Algebra: Grades K-2: slide 14 Problem 2.2 What number(s) could go in the box to make each number sentence true? Explain your thinking. a. 8 + 7 = b. 8 + 7 = + 7 c. + 7 = 8 + 7 d. + 8 = 8 + 7 e. 15 = f. = 7 g. 39 + 57 = + 59h. + 82 = 143 + 89

15. 2008 May 29Algebra: Grades K-2: slide 15 Problem 2.3 What number(s) could be substituted for N to make each number sentence true? Explain your thinking. a. 8 + 7 = N b. 8 + 7 = N + 7 c. N + 7 = 8 + 7 d. N + 8 = 8 + 7 e. 15 = N f. N = 7 g. 39 + 57 = N + 59 h. N + 82 = 143 + 89

16. 2008 May 29Algebra: Grades K-2: slide 16 Problem 2.4 Design a sequence of true/false and/or open number sentences that you might use to engage your students in thinking about the equal sign. Describe why you selected the problems you did.

17. 2008 May 29Algebra: Grades K-2: slide 17 Reflection What did you learn (or re-learn) from solving these problems? Where in the K-8 Mathematics Standards do these ideas appear? The Standards uses the word, equation, instead of the phrase, number sentence. What is the difference in these two terms?

18. 2008 May 29Algebra: Grades K-2: slide 18 Problem Set 3 The focus of Problem Set 3 is making number sentences true. You may work alone or with colleagues to solve the problems in set 3.1. When you are done, share your solutions with others.

19. 2008 May 29Algebra: Grades K-2: slide 19 Problem 3.1 In each number sentence, what number(s) could be substituted for the variable to make that number sentence true? Explain your thinking. a. 6 + 9 = 8 + 10 + d b. 9 - 6 = 8 - 4 + g c. 10 - 6 = 8 - 4 + a d. 5 + 8 + d = 6 + 9 + d

20. 2008 May 29Algebra: Grades K-2: slide 20 Problem 3.1 In each number sentence, what number(s) could be substituted for the variable to make that number sentence true? Explain your thinking. e. 5 + 8 - d = 6 + 9 - d f. 5 + 8 + d + d = 6 + 9 + d g. 5 + 8 - d - d = 6 + 9 - d h. 5 + 8 - d = 6 + 9 - d - d

21. 2008 May 29Algebra: Grades K-2: slide 21 Problem 3.2 a. Solve: d + d + d - 20 = 16 b. Look at video 5.1 (Carpenter, et al., 2003). Focus your attention on the students strategy. Does the students strategy illustrate algebraic thinking?

22. 2008 May 29Algebra: Grades K-2: slide 22 Problem 3.3 a. Solve: k + k + 13 = k + 20 b. Look at video 5.2 (Carpenter, et al., 2003). Focus your attention on the students strategy. Does the students strategy illustrate algebraic thinking?

23. 2008 May 29Algebra: Grades K-2: slide 23 Problem 3.4 How are the strategies used in videos 5.1 and 5.2 alike? How are they different?

24. 2008 May 29Algebra: Grades K-2: slide 24 Reflection How might your students solve these or similar problems? What strategies might they use? How could you adapt these problems for your teaching? Where in the K-8 Mathematics Standards do these ideas appear?

25. 2008 May 29Algebra: Grades K-2: slide 25 Problem Set 4 The focus of Problem Set 4 is relational thinking. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

26. 2008 May 29Algebra: Grades K-2: slide 26 Relational Thinking Make an argument that this equation is true, WITHOUT computing each sum. 25 + 17 = 24 + 18 Make an argument that this equation is false, WITHOUT computing each difference. 25 - 17 = 24 - 18

27. 2008 May 29Algebra: Grades K-2: slide 27 Problem 4.1 Which number sentences are true and which are false? Justify your answers. a. 3,765 + 2,987 = 3,565 + 3,187 b. 4,013 – 2,333 = 4,043 – 2,363 c. 8,041 – 3,762 = 8,051 – 3,752 d. 5,328 + 3,933 = 8,328 + 933 e. 6,789 – 6,345 = 789 - 345

28. 2008 May 29Algebra: Grades K-2: slide 28 Problem 4.2 Rank the following problem from easiest to most difficult (for students). Justify your choices. a. 73 + 56 = 71 + d b. 92 – 57 = g - 56 c. 68 + b = 57 + 69 d. 56 – 23 = f - 25 e. 96 + 67 = 67 + p f. 87 + 45 = y + 46 g. 74 – 37 = 75 - q

29. 2008 May 29Algebra: Grades K-2: slide 29 Problem 4.3 Decide whether each number sentence below is true or false. Justify your choices. How do you think students would justify the choices? a. 56 = 50 + 6 b. 87 = 7 + 80 c. 93 = 9 + 30 d. 94 = 80 + 14 e. 94 = 70 + 24 f. 246 = 24 x 10 + 6 g. 47 + 38 = 40 + 30 + 7 + 8 h. 78 + 24 = 98 + 4 i. 63 – 28 = 60 – 20 – 3 – 8 j. 63 – 28 = 60 – 20 + 3 – 8

30. 2008 May 29Algebra: Grades K-2: slide 30 Problem 4.4 Create a set of problems that might encourage students to use relational thinking. Be ready to explain the grade-level of your problems and justify your choices of numbers.

31. 2008 May 29Algebra: Grades K-2: slide 31 Reflection What is relational thinking? Why is relational thinking important for students to be able to do? Where in the K-8 Mathematics Standards do these ideas appear?

32. 2008 May 29Algebra: Grades K-2: slide 32 Problem Set 5 The focus of Problem Set 5 is properties of operations. You may work alone or with colleagues to solve the problems in set 5.1. When you are done, share your solutions with others.

33. 2008 May 29Algebra: Grades K-2: slide 33 Problem 5.1 Make four groups with each group exploring one operation. a. Explore the properties of addition. b. Explore the properties of subtraction. c. Explore the properties of multiplication. d. Explore the properties of division.

34. 2008 May 29Algebra: Grades K-2: slide 34 Problem 5.2 Describe the commutative and associative properties for addition. Represent these properties using symbols. Do the same for multiplication. Can you do the same for subtraction and division? Why or why not?

35. 2008 May 29Algebra: Grades K-2: slide 35 Problem 5.3 Read this equation aloud using words rather than symbols: a + b = (a + 1) + (b - 1) What mathematical idea does this equation represent? Is the equation true or false? Explain your answer.

36. 2008 May 29Algebra: Grades K-2: slide 36 Problem 5.4 Look at video 3.3 (Carpenter, et al., 2003). Focus your attention on the strategies the student uses. What, if anything, do you think the student learned during this interview? What problem would you pose to check your hypothesis?

37. 2008 May 29Algebra: Grades K-2: slide 37 Reflection What do students need to know about the properties of operations? How can you help students learn those properties? Where in the K-8 Mathematics Standards do these ideas appear?

38. 2008 May 29Algebra: Grades K-2: slide 38 Welcome to Grades K-2 Mathematics Content Sessions The focus is Algebraic Thinking. The goal is to help you understand this mathematics better to support your implementation of the K-8 Mathematics Standards.

39. 2008 May 29Algebra: Grades K-2: slide 39 Problem Set 6 The focus of Problem Set 6 is justification and proof. You may work alone or with colleagues to solve the problems in set 6.1. When you are done, share your solutions with others.

40. 2008 May 29Algebra: Grades K-2: slide 40 Problem 6.1 True or false: a - b - c = a - (b + c). Justify your answer.

41. 2008 May 29Algebra: Grades K-2: slide 41 Problem 6.2 If you have 5 sodas and each person gets half a soda, how many people will get to drink soda? True or false: N ÷ 1/2 = 2 x N Justify your answer. True or false: N ÷ 1/3 = 3 x N Justify your answer.

42. 2008 May 29Algebra: Grades K-2: slide 42 Problem 6.3 Look at video 7.2. Focus your attention on the students explanations. What do you think this student understands about proof? How do the interviewers questions help reveal what the student knows?

43. 2008 May 29Algebra: Grades K-2: slide 43 Reflection What do you look for in a childs argument about whether something is true or false? How sophisticated can you expect childrens arguments to be? Where in the K-8 Mathematics Standards do these ideas appear?

44. 2008 May 29Algebra: Grades K-2: slide 44 Problem Set 7 The focus of Problem Set 7 is what happens and why. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

45. 2008 May 29Algebra: Grades K-2: slide 45 Problem 7.1 True or false? Explain your answers. a. 87 ÷ 5 and 7 ÷ 5 have the same remainder b. 876 ÷ 5 and 6 ÷ 5 have the same remainder c. 895 ÷ 5 and 5 ÷ 5 have the same remainder d. If abc represents a 3-digit number, abc ÷ 5 and c ÷ 5 have the same remainder e. What rule can you state for divisibility by 5? Justify the rule.

46. 2008 May 29Algebra: Grades K-2: slide 46 Problem 7.2 True or false? Explain your answers. a. 65 ÷ 3 and (6 + 5) ÷ 3 have the same remainder b. 652 ÷ 3 and (6 + 5 + 2) ÷ 3 have the same remainder c. 651 ÷ 3 and (6 + 5 + 1) ÷ 3 have the same remainder d. If abc represents a 3-digit number, abc ÷ 3 and (a + b + c) ÷ 3 have the same remainder e. What rule can you state for divisibility by 3? Justify the rule.

47. 2008 May 29Algebra: Grades K-2: slide 47 Problem 7.3 Take any 3 digits (not all the same!!) and make the greatest and least 3-digit numbers. Subtract the lesser from the greater to make the high-low difference. Repeat this process for that difference. Keep on repeating the process. What happens? Why?

48. 2008 May 29Algebra: Grades K-2: slide 48 Problem 7.4 Take a three-digit number (with digits not all the same), reverse its digits, and subtract the lesser from the greater. Reverse the digits of the result and add these two numbers. 132 becomes 231, and 231 - 132 = 99 = 099 099 becomes 990, and 099 + 990 = 1089 Try this process for several numbers. What happens? Why?

49. 2008 May 29Algebra: Grades K-2: slide 49 Reflection Why is knowledge of divisibility important for students to know? In K-2, the ideas of odd and even are important for children to learn. How are those ideas related to divisibility rules? Where in the K-8 Mathematics Standards do these ideas appear?

50. 2008 May 29Algebra: Grades K-2: slide 50 Problem Set 8 The focus of Problem Set 8 is representations. You may work alone or with colleagues to solve the problems in set 8.1. When you are done, share your solutions with others.

51. 2008 May 29Algebra: Grades K-2: slide 51 Problem 8.1 Look at video 7.1 and video 8.1. Focus your attention on students representations. How are the two representations of odd numbers alike? How are they different? Which representation is more convincing? What assumptions is each student making?

52. 2008 May 29Algebra: Grades K-2: slide 52 Problem 8.2 How could you use variables to represent an even number? An odd number? A multiple of 5?

53. 2008 May 29Algebra: Grades K-2: slide 53 Problem 8.3 Suppose that when N is divided by 3 the remainder is 1, and that when P is divided by 3 the remainder is 2. What is the remainder of N + P when you divide it by 3? What is the remainder of N - P when you divide it by 3?

54. 2008 May 29Algebra: Grades K-2: slide 54 Reflection How can representations help students reason? How can we help students learn to use representations to clarify their thinking? Where in the K-8 Mathematics Standards do these ideas appear?

55. 2008 May 29Algebra: Grades K-2: slide 55 Problem Set 9 The focus of Problem Set 1 is patterns and conjectures. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

56. 2008 May 29Algebra: Grades K-2: slide 56 Problem 9.1 What do you notice about each pair of products below. What happens? Why? 6 x 6 and 5 x 7 30 x 30 and 29 x 31 500 x 500 and 499 x 501 N x N and (N - 1) x (N + 1)

57. 2008 May 29Algebra: Grades K-2: slide 57 Problem 9.2 Guess these products. Then check your guesses. 300 x 300 and 298 x 302 300 x 300 and 295 x 305 N x N and (N - a) x (N + a)

58. 2008 May 29Algebra: Grades K-2: slide 58 Reflection What kinds of patterns might you ask K-2 students to explore? Where in the K-8 Mathematics Standards do these ideas appear?

59. 2008 May 29Algebra: Grades K-2: slide 59 Problem Set 10 The focus of Problem Set 1 is reflection on your thinking. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

60. 2008 May 29Algebra: Grades K-2: slide 60 Problem 10.1 What did you learn (or re-learn) from working on these problems? How did the videos help you understand the mathematics ideas?

61. 2008 May 29Algebra: Grades K-2: slide 61 Problem 10.2 Look back over the problems. Which ones could you use directly with children? Which ones could you adapt for use with children?

62. 2008 May 29Algebra: Grades K-2: slide 62 Problem 10.3 Where do these problems fit in the K-8 Mathematics Standards? Which problems might you share with other teachers in your school? Why would those problems be important ones to share?

63. 2008 May 29Algebra: Grades K-2: slide 63 Closing Comments Implementing the K-8 Mathematics Standards will require a deep focus of mathematics ideas at each grade. Personal understanding of these ideas will make the implementation process easier.