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 vs Algebra

• Arithmetic
• Calculating answers
• signifies the answer is next

• Algebra
• Transforming expressions
• as a relation

5
Arithmetic U Algebra

• Arithmetic
• Transforming expressions
• as a relation

• Algebra
• Transforming expressions
• as a relation

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
10
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6 2 ? 3

• David Its 5.
• Ms. F How do you know its 5, David?
• David Its 6 2 there. Theres a 3 there. I
  couldnt 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 its true, because its like the
  other one I did, 6 2 is the same as 5 3.
• Ms. F. Its 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. Im 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 its
  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
Response/Grade 7 12 17 12 17
1 and 2 ? ? ? ?
3 and 4 ? ? ? ?
5 and 6 ? ? ? ?
25
Percent of Students Offering Various Solutions to
8 4 ? 5
Response/Grade 7 12 17 12 17
1 and 2 5 58 13 8
3 and 4 9 49 25 10
5 and 6 2 76 21 2
26
Challenge--Try this!

• What are the different responses that students
  may give to the following open number sentence
• 9 7 ?? 8

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Challenging students conceptions of equality

• 9 5 14
• 9 5 14 0
• 9 5 0 14
• 9 5 13 1

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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
Grade Before Inst. After Inst.
1 and 2 5 66
3 and 4 9 72
5 and 6 2 84
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
• 120 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

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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
• (3040 540) (307 57)
• 37 45
• (30 7) (40 5)
• (30 7) 40 (30 7) 5
• (3040 740) (305 75)

42
35 47 37 45 False

• 35 47
• (30 5) (40 7)
• (30 5) 40 (30 5) 7
• (3040 540) (307 57)
• 37 45
• (30 7) (40 5)
• (30 7) 40 (30 7) 5
• (3040 740) (305 75)

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
x 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
51
Generating number facts based on doubles

• 3 8 2 8 8
• 3 8 16 8
• 3 8 8 3
• 4 9 2 9 2 9

52
Multiplication facts
x 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
53
Generating number facts for nines and fives

• 9 7 10 7 7
• 9 7 10 7 9
• 9 7 10 7 - ?
• 7 5 10 10 10 5

54
Multiplication facts
x 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
55
Appropriating relational strategies to derive
number facts

• 6 6 6 5 6
• 7 8 7 9 7

56
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

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3?7 7 7 7

• Ms L Could you read that number sentence for me
  and tell me if it is true or false?
• Kelly Three times 7 is the same as 7 plus 7
  plus 7. Thats true, because times means groups
  of and there are 3 groups of 7, 3 times 7 just
  says it in a shorter way.
• Ms L Ok, nice explanation.

59
3?7 14 7

• Ms L How about this, 3?7 14 7, is that true
  or false?
• Kelly Its true.
• Ms L Wow, that was quick, how do you know that
  is true?
• Kelly Can we go back up here pointing to 3?7
  7 7 7?
• Ms L Sure.
• Kelly Seven and 7 is 14, that is right here
  drawing a line connecting two 7s in the first
  number sentence and writing 14 under them.
  Fourteen went right into here pointing to the 14
  in the second number sentence. Then there is one
  7 left pointing to the third 7 in the first
  number sentence, and that went right here
  pointing to the last 7 in the second number
  sentence.

60
4?6 12 12

• Ms L Ok, I have another one for you 4?6 12
  12, true or false?
• Kelly That is true.
• Ms L Ok, how did you get that one so quickly?
• Kelly Six plus 6 is 12, in this case, there are
  4 groups of 6, so it is like this writing 6 6
  6 6. Six and 6 is 12, that leaves another 6
  and 6, and that equals 12. So one 12 is here and
  one 12 went here indicating the two 12s in the
  problem. What Im trying to say is there are
  four 6s and you broke them in half and made them
  into two 12s.

61
4?6 12 12 Continued

• Ms L Nice! Kelly, do you know right away what 4
  times 6 is?
• Kelly Yes.
• Ms L What is it?
• Kelly Its pause thirty- long pause two.
• Ms L Ok, do you know what 12 plus 12 is?
• Kelly Yeah. That is the same thing, 32.
• Ms L Do you have a way of doing 12 plus 12, to
  check it?
• Kelly Well, there are two 10s, 20- oh wait, I
  was thinking of a different one!
• Ms L You were thinking of a different
  multiplication problem?
• Kelly Yes. 4 times 6 is 24, because 10 and 10
  is 20, and 2 and 2 is 4, put those together and
  its 24.

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4?7 ?

• Ms L Ok, here is another one. Four times 7
  equals box. I want you tell me what you would
  put in the box to make this a true number
  sentence.
• Kelly That would be short pause 28.
• Ms L Ok, how did you get 28?
• Kelly Well, I kinda had other problems that
  went into this problem. If you go up here
  pointing to 3?7 7 7 7 3 times 7 is the
  same as 7 plus 7 plus 7. That problem helped me
  and I used it with this problem, pointing to 3?7
  14 7 3 sevens is the same as 14 and 7 You
  add one more seven and that goes right to here.
  Then she points to 4?6 12 12. This problem
  also helped me because 4?7 is like My mind went
  back up to here pointing to 3?7 14 7, and I
  said, there is another 7 so I could put those two
  7s together, thats 14, and there are two 14s, 10
  and 10 is 20, 4 and 4 is 8, 28.

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Problem sequence

• 3?7 7 7 7
• 3?7 14 7
• 4?6 12 12
• 4?7 ?

64
Inventing algorithms

• 612 300
• -457 -299
• 200 1
• -40
• -5
• 155

65

• 292 87
• -549 -49.02
• -300 40.00
• 50 -2.00
• -7 -.02
• -257 37.98

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• 300
• -294 5/8
• 6
• -5/8
• 5 3/8

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5 ½ __ ? Number choices ½, ¼, ¾, 3/8

• Choose one of the numbers depending on the level
  of your students.
• Change the problem to add additional challenge as
  needed.
• Allow students to choose one to vary the problem
  and allow choice.

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5 ½ __ ? Number choices ½, ¼, ¾, 3/8

• Put the problem into a context.
• It takes __ of a cup of sugar to make a batch of
  cookies. I have 5 ½ cups of sugar. How many
  batches of cookies can I make?
• Solve for 3/8 cup of sugar for a batch.

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It takes 3/8 of a cup of sugar to make a batch of
cookies. I have 5 ½ cups of sugar. How many
batches of cookies can I make?
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It takes 3/8 of a cup of sugar to make a batch of
cookies. I have 5 ½ cups of sugar. How many
batches of cookies can I make?

• 5 ½ 3/8 ?
• ? 3/8 5 ½

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? 3/8 5 ½

• 8 3/8 3
• 4 3/8 would be ½ of 3 or 1 ½
• 12 3/8 4 ½ gt 4 ½ cups makes 12 batches
• Need to use 1 more cup of sugar
• Because 8 3/8 3, a third as much would be 1
  
• i.e. 1/3 (8 3/8) 1
• So you need 1/3 of 8, which is 8/3
• i.e. 1 cup makes 8/3 batches.
• So altogether you get a total of 12 8/3 or 14
  2/3 batches

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? 3/8 5 ½

• 8 3/8 3
• ½(8 3/8) ½ 3
• (½8)3/8 1 ½
• 4 3/8 1 ½

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Next subgoalHow many 3/8 cups to use the
remaining cup?

• 8 3/8 3
• 1/3 (8 3/8) 1/3 3
• (1/38)3/8 1
• 8/3 3/8 1

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Putting the parts together

• (8 3/8) (4 3/8) (8/3 3/8)
• 3 1 ½ 1 5 ½
• And
• (8 3/8) (4 3/8) (8/3 3/8)
• (8 4 8/3) 3/8 14 2/3 3/8
• So 5 ½ cups of sugar makes 14 2/3 batches of 3/8
  cups of sugar

75
Solving equations

• k k 13 k 20

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From arithmetic to algebraic reasoning

• Attend to relations rather than teaching only
  step by step procedures
• Align the teaching of arithmetic with the
  concepts and skills students need to learn
  algebra
• Enhance the learning of arithmetic
• Provide a foundation for and smooth the
  transition to learning algebra

78
Learning arithmetic and algebra with understanding

• Algebra for all
• Not watering down algebra to teach isolated
  procedures
• Develop algebraic reasoning rather than teaching
  meaningless algebraic procedures
• Learning arithmetic and algebra grounded in
  fundamental properties of number and number
  operations

79
AssignmentWhat you will do before the next
meeting

• You will be writing a series of problems that
  you might use with your students to encourage
  them to begin to look for relations.
• Write a problem to assess student thinking
• Predict student responses
• Write a series of problems to address your
  studentsback upextend???
• Try these with your students

80
Assignment contd

• You will be planning a lesson with your
  colleagues for a lesson study cycle
• As a team choose a topic to be taught on December
  6th
• Choose a topic that is typically difficult for
  students
• Bring planning materials to the October session

81
Challenge

• Addition is associative, but subtraction is not.
  How about the following
• Is (a b) - c a (b - c) true for all
  numbers?
• Is (a - b) c a - (b c) true for all
  numbers?
• Thinking
  Mathematically p. 120 4

82
Challenge

• What kind of number do you get when you add three
  odd numbers?
• Can you justify your response?
• Thinking Mathematically p.103 1

83
Challenge--Try this!

• Design a sequence of true/false and/or open
  sentences that you might use to engage your
  students in thinking about the equal sign.
• Thinking Mathematically p. 24 4

84
References

• Carpenter, T.P., Franke, M.L., Levi, L. (2003).
  Thinking mathematically Integrating arithmetic
  and algebra in the elementary school. Portsmouth,
  NH Heinemann.
• Carpenter, T. P., Franke, M.L., Levi, L.
  (2005). Algebra in Elementary School. ZDM. 37(1),
  1-7.
• Carpenter, T. P., Levi, L., Berman, P., Pligge,
  M. (2005). Developing algebraic reasoning in the
  elementary school. In T. A. Romberg , T. P.
  Carpenter, F. Dremock (Eds). Understanding
  mathematics and science matters. Mahwah, NJ
  Erlbaum.
• Jacobs, V.J., Franke, M.L., Carpenter, T. P.,
  Levi, L., Battey, D. (2007) A large-scale study
  of professional development focused on childrens
  algebraic reasoning in elementary school. Journal
  for Research in Mathematics Education, 38,
  258-288.