Learning Arithmetic as a Foundation for Learning Algebra

1. Learning Arithmetic as a Foundation for Learning Algebra Developing relational thinking Adapted from… Thomas Carpenter University of Wisconsin-Madison

2. Defining Algebra Many adults equate school algebra with symbol manipulation– solving complicated equations and simplifying algebraic expressions. Indeed, the algebraic symbols and the procedures for working with them are a towering , historic mathematical accomplishment and are critical in mathematical work. But algebra is more than moving symbols around. Students need to understand the concepts of algebra, the structures and principles that govern the manipulation of the symbols, and how the symbols themselves can be used for recording ideas and gaining insights into situations. (NCTM, 2000, p. 37)

3. Never the twain shall meet • The artificial separation of arithmetic and algebra deprives students of powerful ways of thinking about mathematics in the early grades and makes it more difficult for them to learn algebra in the later grades.

4. Arithmetic Calculating answers = signifies the answer is next Algebra Transforming expressions = as a relation Arithmetic vs Algebra

5. Arithmetic Transforming expressions = as a relation Algebra Transforming expressions = as a relation Arithmetic U Algebra

6. Developing Algebraic Reasoning in Elementary School • Rather than teaching algebra procedures to elementary school children, our goal is to support them to develop ways of thinking about arithmetic that are more consistent with the ways that students have to think to learn algebra successfully.

7. Developing Algebraic Reasoning in Elementary School • Enhances the learning of arithmetic in the elementary grades. • Smoothes the transition to learning algebra in middle school and high school.

8. Relational Thinking • Focusing on relations rather than only on calculating answers • Looking at expressions and equations in their entirety rather than as procedures to be carried out step by step • Engaging in anticipatory thinking • Using fundamental properties of arithmetic to relate or transform quantities and expressions • Recomposing numbers and expressions • Flexible use of operations and relations

9. 6 + 2 = □ + 3

11. 6 + 2 = □ + 3 • David: It’s 5. • Ms. F: How do you know it’s 5, David? • David: It’s 6 + 2 there. There’s a 3 there. I couldn’t decide between 5 and 7. Three was one more than 2, and 5 was one less than 6. So it was 5

12. 57 + 38 = 56 + 39 • David: I know it’s true, because it’s like the other one I did, 6 + 2 is the same as 5 + 3. • Ms. F. It’s the same. How is it the same? • David: 57 is right there, and 56 is there, and 6 is there and 5 is there, and there is 38 there and 39 there. • Ms. F. I’m a little confused. You said the 57 is like the 5 and the 56 is like the 6. Why? • David: Because the 5 and the 56, they both are one number lower than the other number. The one by the higher number is lowest, and the one by the lowest number up there would be more. So it’s true.

13. 57 + 38 = 56 + 39 (56 + 1) + 38 = 56 + (1 + 38)

14. Recomposing numbers 8 + 7 = □ 8 + (2 + 5) =□ (8 + 2) + 5 =□

15. Recomposing numbers ½ + ¾ = □ ½ + (½ + ¼) = □

16. Using basic propertiesRelating arithmetic and algebra 70 + 40 = 7 X 10 + 4 X 10 = (7 + 4) X 10 = 110 7/12 + 4/12 = 7(1/12) + 4(1/12) = (7 + 4) X 1/12 7a + 4a = 7(a) + 4(a) = (7 + 4)a = 11a

17. Using basic properties(not) 7a + 4 b = 11ab

18. X2 - X - 2 = 0(X – 2)(X + 1) = 0 X + 1 = 0 X – 2 = 0X = -1 X = 2

19. X2 - X - 2 = 0(X + 1)(X - 2) = 0 X + 1 = 0 X – 2 = 0X = -1 X = 2(X + 1)(X - 2) = 6

20. X2 - X - 2 = 0(X + 1)(X - 2) = 0 X + 1 = 0 X – 2 = 0X = -1 X = 2(X + 1)(X - 2) = 6 X + 1 = 6 X – 2 = 6X = 5 X = 8

21. X2 - X - 2 = 0(X + 1)(X - 2) = 0 X + 1 = 0 X – 2 = 0X = -1 X = 2(X + 1)(X - 2) = 6 X + 1 = 6 X – 2 = 6X = 5 X = 8(5 + 1)(5 - 2) = 18 (8 + 1)(8 -2) = 54

22. Multiplication properties of zero • ax0 = 0 • axb = 0 implies a = 0 or b = 0

23. Equality as a relation 8 + 4 = □ + 5

24. Percent of Students Offering Various Solutions to 8 + 4 =  + 5

25. Percent of Students Offering Various Solutions to 8 + 4 =  + 5

26. Challenge--Try this! • What are the different responses that students may give to the following open number sentence: 9 + 7 = + 8

28. Challenging students’ conceptions of equality • 9 + 5 = 14 • 9 + 5 = 14 + 0 • 9 + 5 = 0 + 14 • 9 + 5 = 13 + 1

29. Challenging students’ conceptions of equality • 7 + 4 = 11 • 11 = 7 + 4 • 11 = 11 • 7 + 4 = 7 + 4 • 7 + 4 = 4 + 7

30. Correct solutions to 8 + 4 =  + 5 before and after instruction

31. Learning to think relationally, thinking relationally to learn • Using true/false and open number sentences (equations) to engage students in thinking more flexibly and more deeply about arithmetic

32. Learning to think relationally, thinking relationally to learn • Using true/false and open number sentences (equations) to engage students in thinking more flexibly and more deeply about arithmetic • in ways that are consistent with the ways that they need to think about algebra.

33. True and false number sentences • 7 + 5 = 12 • 5 + 6 = 13 • 457 + 356 = 543 • 7 13/16 – 2 17/18 = 4 11/15 • 12÷0 = 0

34. Challenge--Try this! • Construct a series of true/false sentences that might be used to elicit one of the conjectures in Table 4.1 (on p. 54-55).

35. Learning to think relationally • 26 + 18 - 18 =  • 17 - 9 + 8 = 

36. Learning to think relationally • 750 + 387 +250 =  • 7 + 9 + 8 + 3 + 1 = 

37. More challenging problems A. 98 + 62 = 93 + 63 +  B. 82 – 39 = 85 – 37 -  C. 45 – 28 =  - 24

38. True or False • 35 + 47 = 37 + 45 • 35 × 47 = 37 × 45

39. 35 + 47 = 37 + 45 True 30 + 5 + 40 + 7 = 30 + 7 + 40 + 5

40. 35 × 47 = 37 × 45 False

41. 35 × 47 = 37 × 45False 35 × 47 = (30 + 5) × (40 + 7) = (30 + 5) ×40 + (30 + 5) × 7 = (30×40 + 5×40) + (30×7 + 5×7) 37 × 45 = (30 + 7) × (40 + 5) = (30 + 7) ×40 + (30 + 7) × 5 = (30×40 + 7×40) + (30×5 + 7×5)

42. 35 × 47 = 37 × 45 False 35 × 47 = (30 + 5) × (40 + 7) = (30 + 5) ×40 + (30 + 5) × 7 = (30×40 + 5×40) + (30×7 + 5×7) 37 × 45 = (30 + 7) × (40 + 5) = (30 + 7) ×40 + (30 + 7) × 5 = (30×40 + 7×40) + (30×5 + 7×5)

43. Parallels with multiplying binomials • (X + 7)(X + 5) = (X + 7)X + (X + 7) 5 = X2 + 7X +5X + 35 = X2 +(7 +5)X + 35 = X2 +12X + 35

44. Thinking relationally to learn • Learning number facts with understanding • Constructing algorithms and procedures for operating on whole numbers and fractions

45. Number sentences to develop Relational Thinking • (Large numbers are used to discourage calculation) • Rank from easiest to most difficult • 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

46. Learning Multiplication facts usingrelational thinking • Julie Koehler Zeringue

47. A learning trajectory for thinking relationally • Starting to think relationally • The equal sign as a relational symbol • Using relational thinking to learn multiplication • Multiplication as repeated addition • Beginning to use the distributive property • Recognizing relations involving doubles, fives, and tens • Appropriating relational strategies to derive number facts

48. Multiplication as repeated addition 3  7 = 7 + 7 + 7 4  7 = 7 + 7 + 7 + b 6 + 6 = 2  6 2  9 = h + h

49. Beginning to use the distributive property • 36 + 6 = 46 • 36 + 3 = 46 • 54 = 24 + 4 + 8 • 56 = 36 + g • 67 = a7 + b7 • 67 = h7 + h7

50. Multiplication facts