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Welcome to Grades K-2 Mathematics Content Sessions

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 . Importance of Algebraic Thinking.

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Welcome to Grades K-2 Mathematics Content Sessions

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  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. Algebra: Grades K-2: slide 1

  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. Algebra: Grades K-2: slide 2

  3. Introduction of Facilitators <Insert Facilitator Names Here> Algebra: Grades K-2: slide 3

  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. Algebra: Grades K-2: slide 4

  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. Algebra: Grades K-2: slide 5

  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. Algebra: Grades K-2: slide 6

  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?” Algebra: Grades K-2: slide 7

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

  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? Algebra: Grades K-2: slide 9

  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? Algebra: Grades K-2: slide 10

  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? Algebra: Grades K-2: slide 11

  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. Algebra: Grades K-2: slide 12

  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 Algebra: Grades K-2: slide 13

  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 =  + 59 h.  + 82 = 143 + 89 Algebra: Grades K-2: slide 14

  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 Algebra: Grades K-2: slide 15

  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. Algebra: Grades K-2: slide 16

  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? Algebra: Grades K-2: slide 17

  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. Algebra: Grades K-2: slide 18

  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 Algebra: Grades K-2: slide 19

  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 Algebra: Grades K-2: slide 20

  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 student’s strategy. Does the student’s strategy illustrate algebraic thinking? Algebra: Grades K-2: slide 21

  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 student’s strategy. Does the student’s strategy illustrate algebraic thinking? Algebra: Grades K-2: slide 22

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

  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? Algebra: Grades K-2: slide 24

  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. Algebra: Grades K-2: slide 25

  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 Algebra: Grades K-2: slide 26

  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 Algebra: Grades K-2: slide 27

  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 Algebra: Grades K-2: slide 28

  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 Algebra: Grades K-2: slide 29

  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. Algebra: Grades K-2: slide 30

  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? Algebra: Grades K-2: slide 31

  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. Algebra: Grades K-2: slide 32

  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. Algebra: Grades K-2: slide 33

  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? Algebra: Grades K-2: slide 34

  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. Algebra: Grades K-2: slide 35

  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? Algebra: Grades K-2: slide 36

  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? Algebra: Grades K-2: slide 37

  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. Algebra: Grades K-2: slide 38

  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. Algebra: Grades K-2: slide 39

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

  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. Algebra: Grades K-2: slide 41

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

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

  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. Algebra: Grades K-2: slide 44

  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. Algebra: Grades K-2: slide 45

  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. Algebra: Grades K-2: slide 46

  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? Algebra: Grades K-2: slide 47

  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? Algebra: Grades K-2: slide 48

  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? Algebra: Grades K-2: slide 49

  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. Algebra: Grades K-2: slide 50

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