Learning Arithmetic as a Learning Arithmetic as a Foundation for Learning Foundation for Learning

AlgebraAlgebraDeveloping relational Developing relational

thinkingthinking

Adapted from…Adapted from…Thomas CarpenterThomas Carpenter

University of Wisconsin-MadisonUniversity of Wisconsin-Madison

Defining AlgebraDefining Algebra Many adults equate school algebra with Many adults equate school algebra with

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

Never the twain shall meetNever the twain shall meet The artificial separation of arithmetic The artificial separation of arithmetic

and algebra deprives students of and algebra deprives students of powerful ways of thinking about powerful ways of thinking about mathematics in the early grades and mathematics in the early grades and makes it more difficult for them to makes it more difficult for them to learn algebra in the later grades. learn algebra in the later grades.

Arithmetic vs AlgebraArithmetic vs Algebra

ArithmeticArithmetic Calculating Calculating

answersanswers

= signifies the = signifies the answer is nextanswer is next

AlgebraAlgebra Transforming Transforming

expressionsexpressions

= as a relation= as a relation

Arithmetic U AlgebraArithmetic U Algebra

ArithmeticArithmetic Transforming Transforming

expressionsexpressions

= as a relation= as a relation

AlgebraAlgebra Transforming Transforming

expressionsexpressions

= as a relation= as a relation

Developing Algebraic Developing Algebraic Reasoning in Elementary Reasoning in Elementary

SchoolSchool Rather than teaching algebra Rather than teaching algebra

procedures to elementary school procedures to elementary school children, our goal is to support them children, our goal is to support them to develop ways of thinking about to develop ways of thinking about arithmetic that are more consistent arithmetic that are more consistent with the ways that students have to with the ways that students have to think to learn algebra successfully.think to learn algebra successfully.

Developing Algebraic Developing Algebraic Reasoning in Elementary Reasoning in Elementary

SchoolSchool Enhances the learning of Enhances the learning of

arithmetic in the elementary arithmetic in the elementary grades. grades.

Smoothes the transition to Smoothes the transition to learning algebra in middle school learning algebra in middle school and high school.and high school.

Relational ThinkingRelational Thinking Focusing on relations rather than only on Focusing on relations rather than only on

calculating answerscalculating answers Looking at expressions and equations in Looking at expressions and equations in

their entirety rather than as procedures to their entirety rather than as procedures to be carried out step by stepbe carried out step by step

Engaging in anticipatory thinkingEngaging in anticipatory thinking Using fundamental properties of arithmetic Using fundamental properties of arithmetic

to relate or transform quantities and to relate or transform quantities and expressionsexpressions

Recomposing numbers and expressionsRecomposing numbers and expressions Flexible use of operations and relationsFlexible use of operations and relations

6 + 2 = 6 + 2 = □ □ + 3+ 3

6 + 2 = 6 + 2 = □ □ + 3+ 3 David: It’s 5.David: It’s 5. Ms. F: How do you know it’s 5, David?Ms. F: How do you know it’s 5, David? David: It’s 6 + 2 there. There’s a 3 David: It’s 6 + 2 there. There’s a 3

there. I couldn’t decide between 5 there. I couldn’t decide between 5 and 7. Three was one more than 2, and 7. Three was one more than 2, and 5 was one less than 6. So it was 5and 5 was one less than 6. So it was 5

57 + 38 = 56 + 3957 + 38 = 56 + 39 David: I know it’s true, because it’s like the other one David: I know it’s true, because it’s like the other one

I did, 6 + 2 is the same as 5 + 3.I did, 6 + 2 is the same as 5 + 3. Ms. F. It’s the same. How is it the same?Ms. F. It’s the same. How is it the same? David: 57 is right there, and 56 is there, and 6 is 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 and 5 is there, and there is 38 there and 39 there. there.

Ms. F. I’m a little confused. You said the 57 is like the Ms. F. I’m a little confused. You said the 57 is like the 5 and the 56 is like the 6. Why?5 and the 56 is like the 6. Why?

David: Because the 5 and the 56, they both are one David: Because the 5 and the 56, they both are one number lower than the other number. The one by the number lower than the other number. The one by the higher number is lowest, and the one by the lowest higher number is lowest, and the one by the lowest number up there would be more. So it’s true.number up there would be more. So it’s true.

57 + 38 = 56 + 3957 + 38 = 56 + 39

(56 + 1) + 38 = 56 + (1 + (56 + 1) + 38 = 56 + (1 + 38)38)

Recomposing numbersRecomposing numbers

8 + 7 = 8 + 7 = □□8 + (2 + 5) =8 + (2 + 5) = □□(8 + 2) + 5 =(8 + 2) + 5 = □□

Recomposing numbersRecomposing numbers

½ + ¾ = ½ + ¾ = □□½ + (½ + ¼) = ½ + (½ + ¼) = □□

Using basic propertiesUsing basic propertiesRelating arithmetic and algebraRelating arithmetic and algebra

70 + 40 = 7 X 10 + 4 X 10 70 + 40 = 7 X 10 + 4 X 10 = (7 + 4) X 10= (7 + 4) X 10 = 110= 110

7/12 + 4/12 = 7(1/12) + 4(1/12)7/12 + 4/12 = 7(1/12) + 4(1/12) = (7 + 4) X 1/12= (7 + 4) X 1/12

7a + 4a = 7(a) + 4(a)7a + 4a = 7(a) + 4(a) = (7 + 4)a = (7 + 4)a

= 11a= 11a

Using basic propertiesUsing basic properties(not)(not)

7a + 4 b = 11ab7a + 4 b = 11ab

XX22 - X - 2 = 0 - X - 2 = 0(X – 2)(X + 1) = 0(X – 2)(X + 1) = 0X + 1 = 0X + 1 = 0 X – 2 = X – 2 = 00X = -1X = -1 X = 2X = 2

XX22 - X - 2 = 0 - X - 2 = 0(X + 1)(X - 2) = 0(X + 1)(X - 2) = 0X + 1 = 0X + 1 = 0 X – 2 = X – 2 = 00X = -1X = -1 X = 2X = 2

(X + 1)(X - 2) = 6(X + 1)(X - 2) = 6

XX22 - X - 2 = 0 - X - 2 = 0(X + 1)(X - 2) = 0(X + 1)(X - 2) = 0X + 1 = 0X + 1 = 0 X – 2 = X – 2 = 00X = -1X = -1 X = 2X = 2

(X + 1)(X - 2) = 6(X + 1)(X - 2) = 6X + 1 = 6X + 1 = 6 X – 2 = X – 2 = 66X = 5X = 5 X = 8X = 8

XX22 - X - 2 = 0 - X - 2 = 0(X + 1)(X - 2) = 0(X + 1)(X - 2) = 0X + 1 = 0X + 1 = 0 X – 2 = 0X – 2 = 0X = -1X = -1 X = 2X = 2

(X + 1)(X - 2) = 6(X + 1)(X - 2) = 6X + 1 = 6X + 1 = 6 X – 2 = 6X – 2 = 6X = 5X = 5 X = 8X = 8

(5 + 1)(5 - 2) = 18(5 + 1)(5 - 2) = 18 (8 + 1)(8 -2) = 54 (8 + 1)(8 -2) = 54

Multiplication properties of Multiplication properties of zerozero

ax0 = 0ax0 = 0

axb = 0 implies a = 0 or b = axb = 0 implies a = 0 or b = 00

Equality as a relationEquality as a relation

8 + 4 = 8 + 4 = □□ + 5 + 5

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

ResponsResponse/Gradee/Grade

77 1212 1717 12 & 1712 & 17

1 and 21 and 2 ?? ?? ?? ??

3 and 43 and 4 ?? ?? ?? ??

5 and 65 and 6 ?? ?? ?? ??

2424

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

ResponsResponse/Gradee/Grade

77 1212 1717 12 & 1712 & 17

1 and 21 and 2 55 5858 1313 88

3 and 43 and 4 99 4949 2525 1010

5 and 65 and 6 22 7676 2121 22

Challenge--Try this!Challenge--Try this! What are the different responses What are the different responses

that students may give to the that students may give to the following open number sentence:following open number sentence:

9 + 7 = 9 + 7 = + 8+ 8

2626

Challenging students’ Challenging students’ conceptions of equalityconceptions of equality

9 + 5 = 149 + 5 = 14 9 + 5 = 14 + 09 + 5 = 14 + 0 9 + 5 = 0 + 149 + 5 = 0 + 14 9 + 5 = 13 + 19 + 5 = 13 + 1

Challenging students’ Challenging students’ conceptions of equalityconceptions of equality

7 + 4 = 117 + 4 = 11 11 = 7 + 411 = 7 + 4 11 = 1111 = 11 7 + 4 = 7 + 47 + 4 = 7 + 4 7 + 4 = 4 + 77 + 4 = 4 + 7

Correct solutions to 8 + 4 = Correct solutions to 8 + 4 = + + 5 before and after instruction5 before and after instruction

GradeGrade Before Inst.Before Inst. After Inst.After Inst. 1 and 21 and 2 55 6666

3 and 43 and 4 99 7272

5 and 65 and 6 22 8484

Learning to think relationally, Learning to think relationally, thinking relationally to learnthinking relationally to learn

Using true/false and open number Using true/false and open number sentences (equations) to engage sentences (equations) to engage students in thinking more flexibly students in thinking more flexibly and more deeply about arithmeticand more deeply about arithmetic

Learning to think relationally, Learning to think relationally, thinking relationally to learnthinking relationally to learn

Using true/false and open number Using true/false and open number sentences (equations) to engage sentences (equations) to engage students in thinking more flexibly students in thinking more flexibly and more deeply about arithmeticand more deeply about arithmetic

in ways that are consistent with the in ways that are consistent with the ways that they need to think about ways that they need to think about algebra.algebra.

True and false number True and false number sentencessentences

7 + 5 = 127 + 5 = 12 5 + 6 = 135 + 6 = 13 457 + 356 = 543457 + 356 = 543 7 7 13/1613/16 – 2 – 2 17/1817/18 = 4 = 4 11/1511/15 12÷0 = 012÷0 = 0

Challenge--Try this!Challenge--Try this! Construct a series of true/false Construct a series of true/false

sentences that might be used to sentences that might be used to elicit elicit oneone of the conjectures in of the conjectures in Table 4.1Table 4.1

(on p. 54-55).(on p. 54-55).

3434

Learning to think Learning to think relationallyrelationally

26 + 18 - 18 = 26 + 18 - 18 = 17 - 9 + 8 = 17 - 9 + 8 =

Learning to think Learning to think relationallyrelationally

750 + 387 +250 = 750 + 387 +250 =

7 + 9 + 8 + 3 + 1 = 7 + 9 + 8 + 3 + 1 =

More challenging problemsMore challenging problems

A. 98 + 62 = 93 + 63 + A. 98 + 62 = 93 + 63 +

B. 82 – 39 = 85 – 37 - B. 82 – 39 = 85 – 37 -

C. 45 – 28 = C. 45 – 28 = - 24 - 24

True or FalseTrue or False 35 + 47 = 37 + 4535 + 47 = 37 + 45

35 × 47 = 37 × 4535 × 47 = 37 × 45

35 + 47 = 37 + 4535 + 47 = 37 + 45

TrueTrue 30 + 5 + 40 + 7 = 30 + 7 + 40 + 530 + 5 + 40 + 7 = 30 + 7 + 40 + 5

35 × 47 = 37 × 4535 × 47 = 37 × 45

FalseFalse

35 × 47 = 37 × 4535 × 47 = 37 × 45FalseFalse

35 × 47 =35 × 47 =(30 + 5) × (40 + 7) =(30 + 5) × (40 + 7) =(30 + 5) ×40 + (30 + 5) × 7 =(30 + 5) ×40 + (30 + 5) × 7 =(30×40 + 5×40) + (30×7 + 5×7)(30×40 + 5×40) + (30×7 + 5×7)

37 × 45 = 37 × 45 = (30 + 7) × (40 + 5) =(30 + 7) × (40 + 5) =(30 + 7) ×40 + (30 + 7) × 5 =(30 + 7) ×40 + (30 + 7) × 5 =(30×40 + 7×40) + (30×5 + 7×5)(30×40 + 7×40) + (30×5 + 7×5)

35 × 47 = 37 × 45 35 × 47 = 37 × 45 FalseFalse

35 × 47 =35 × 47 =(30 + 5) × (40 + 7) =(30 + 5) × (40 + 7) =(30 + 5) ×40 + (30 + 5) × 7 =(30 + 5) ×40 + (30 + 5) × 7 =((30×4030×40 + + 5×405×40) + () + (30×730×7 + + 5×75×7))

37 × 45 = 37 × 45 = (30 + 7) × (40 + 5) =(30 + 7) × (40 + 5) =(30 + 7) ×40 + (30 + 7) × 5 =(30 + 7) ×40 + (30 + 7) × 5 =((30×4030×40 + + 7×407×40) + () + (30×530×5 + + 7×57×5))

Parallels with multiplying Parallels with multiplying binomialsbinomials

(X + 7)(X + 5) = (X + 7)(X + 5) = (X + 7)X + (X + 7) 5 =(X + 7)X + (X + 7) 5 = XX22 + 7X +5X + 35 = + 7X +5X + 35 = XX22 +(7 +5)X + 35 = +(7 +5)X + 35 = XX22 +12X + 35 +12X + 35

Thinking relationally to Thinking relationally to learnlearn

Learning number facts with Learning number facts with understandingunderstanding

Constructing algorithms and Constructing algorithms and procedures for operating on whole procedures for operating on whole numbers and fractionsnumbers and fractions

Number sentences to Number sentences to develop Relational Thinkingdevelop Relational Thinking

(Large numbers are used to discourage calculation)(Large numbers are used to discourage calculation) Rank from easiest to most difficultRank from easiest to most difficult a) 73 + 56 = 71 + da) 73 + 56 = 71 + d b) 92 – 57 = g – 56b) 92 – 57 = g – 56 c) 68 + b = 57 + 69c) 68 + b = 57 + 69 d) 56 – 23 = f – 25d) 56 – 23 = f – 25 e) 96 + 67 = 67 + pe) 96 + 67 = 67 + p f) 87 + 45 = y + 46f) 87 + 45 = y + 46 g) 74 – 37 = 75 - qg) 74 – 37 = 75 - q

Learning Multiplication facts Learning Multiplication facts usingusing

relational thinkingrelational thinking

Julie Koehler Zeringue Julie Koehler Zeringue

A learning trajectory for A learning trajectory for thinking relationally thinking relationally

Starting to think relationallyStarting to think relationally The equal sign as a relational symbolThe equal sign as a relational symbol Using relational thinking to learn multiplicationUsing relational thinking to learn multiplication

Multiplication as repeated additionMultiplication as repeated addition Beginning to use the distributive propertyBeginning to use the distributive property Recognizing relations involving doubles, fives, and Recognizing relations involving doubles, fives, and

tenstens Appropriating relational strategies to derive Appropriating relational strategies to derive

number factsnumber facts

Multiplication as repeated Multiplication as repeated additionaddition

  3 3 7 = 7 + 7 + 7 7 = 7 + 7 + 74 4 7 = 7 + 7 + 7 + b 7 = 7 + 7 + 7 + b6 + 6 = 2 6 + 6 = 2 6 6 2 2 9 = h + h 9 = h + h

Beginning to use the distributive Beginning to use the distributive propertyproperty

336 + 6 = 46 + 6 = 466 336 + 3 = 46 + 3 = 466 554 = 24 = 24 + 4 + 84 + 4 + 8 556 = 36 = 36 + g6 + g   667 = a7 = a7 + b7 + b77 667 = h7 = h7 + h7 + h77

Multiplication factsMultiplication facts

xx 11 22 33 44 55 66 77 88 9911 11 22 33 44 55 66 77 88 9922 22 44 66 88 1010 1212 1414 1616 181833 33 66 99 1212 1515 1818 2121 2424 272744 44 88 11

221616 2020 2424 2828 3232 3636

55 55 1010 1155

2020 2525 3030 3535 4040 4545

66 66 1212 1188

2424 3030 3636 4242 4848 5454

77 77 1414 2211

2828 3535 4242 4949 5656 6363

88 88 1616 2244

3232 4040 4848 5656 6464 7272

99 99 1818 2277

3636 4545 5454 6363 7272 8181

Generating number facts Generating number facts based on doublesbased on doubles

3 × 8 = 2 × 8 + 83 × 8 = 2 × 8 + 8 3 × 8 = 16 + 83 × 8 = 16 + 8

3 × 8 = 8 × 33 × 8 = 8 × 3

4 × 9 = 2 × 9 + 2 × 94 × 9 = 2 × 9 + 2 × 9

Multiplication factsMultiplication facts

xx 11 22 33 44 55 66 77 88 9911 11 22 33 44 55 66 77 88 9922 22 44 66 88 1010 1212 1414 1616 181833 33 66 99 1212 1515 1818 2121 2424 272744 44 88 11

221616 2020 2424 2828 3232 3636

55 55 1010 1155

2020 2525 3030 3535 4040 4545

66 66 1212 1188

2424 3030 3636 4242 4848 5454

77 77 1414 2211

2828 3535 4242 4949 5656 6363

88 88 1616 2244

3232 4040 4848 5656 6464 7272

99 99 1818 2277

3636 4545 5454 6363 7272 8181

Generating number facts for Generating number facts for nines and fivesnines and fives

9 × 7 = 10 × 7 – 79 × 7 = 10 × 7 – 7 9 × 7 = 10 × 7 – 99 × 7 = 10 × 7 – 9

9 × 7 = 10 × 7 - 9 × 7 = 10 × 7 - □□ 7 × 5 = 10 + 10 + 10 + 57 × 5 = 10 + 10 + 10 + 5

Multiplication factsMultiplication facts

xx 11 22 33 44 55 66 77 88 9911 11 22 33 44 55 66 77 88 9922 22 44 66 88 1010 1212 1414 1616 181833 33 66 99 1212 1515 1818 2121 2424 272744 44 88 11

221616 2020 2424 2828 3232 3636

55 55 1010 1155

2020 2525 3030 3535 4040 4545

66 66 1212 1188

2424 3030 3636 4242 4848 5454

77 77 1414 2211

2828 3535 4242 4949 5656 6363

88 88 1616 2244

3232 4040 4848 5656 6464 7272

99 99 1818 2277

3636 4545 5454 6363 7272 8181

Appropriating relational Appropriating relational strategies to derive number strategies to derive number

factsfacts 6 × 6 = 6 × 5 + 66 × 6 = 6 × 5 + 6 7 × 8 = 7 × 9 – 77 × 8 = 7 × 9 – 7

A learning trajectory for A learning trajectory for thinking relationallythinking relationally

Starting to think relationallyStarting to think relationally The equal sign as a relational symbolThe equal sign as a relational symbol Using relational thinking to learn Using relational thinking to learn

multiplicationmultiplication Multiplication as repeated additionMultiplication as repeated addition Beginning to use the distributive propertyBeginning to use the distributive property Recognizing relations involving doubles, Recognizing relations involving doubles,

fives, and tensfives, and tens Appropriating relational strategies to Appropriating relational strategies to

derive number factsderive number facts

337 = 7 + 7 + 77 = 7 + 7 + 7Ms LMs L: “Could you read that number : “Could you read that number

sentence for me and tell me if it is true or sentence for me and tell me if it is true or false”?false”?

KellyKelly: “Three times 7 is the same as 7 plus 7 : “Three times 7 is the same as 7 plus 7 plus 7. That’s true, because times means plus 7. That’s true, because times means groups of and there are 3 groups of 7, 3 groups of and there are 3 groups of 7, 3 times 7 just says it in a shorter way”.times 7 just says it in a shorter way”.

Ms LMs L: “Ok, nice explanation”.: “Ok, nice explanation”.

337 = 14 + 77 = 14 + 7Ms LMs L: “How about this, 3: “How about this, 37 = 14 + 7, is that true or false”?7 = 14 + 7, is that true or false”?KellyKelly: “It’s true”.: “It’s true”.Ms LMs L: “Wow, that was quick, how do you know that is : “Wow, that was quick, how do you know that is

true”?true”?Kelly: “Can we go back up here [pointing to 3Kelly: “Can we go back up here [pointing to 37 = 7 + 7 + 7 = 7 + 7 +

7]”?7]”?Ms LMs L: “Sure”.: “Sure”.KellyKelly: “Seven and 7 is 14, that is right here [drawing a line : “Seven and 7 is 14, that is right here [drawing a line

connecting two 7s in the first number sentence and connecting two 7s in the first number sentence and writing 14 under them]. Fourteen went right into here writing 14 under them]. Fourteen went right into here [pointing to the 14 in the second number sentence]. [pointing to the 14 in the second number sentence]. Then there is one 7 left pointing to the third 7 in the first Then there is one 7 left pointing to the third 7 in the first number sentence], and that went right here [pointing to number sentence], and that went right here [pointing to the last 7 in the second number sentence]”.the last 7 in the second number sentence]”.

446 = 12 + 126 = 12 + 12Ms L: “Ok, I have another one for you 4Ms L: “Ok, I have another one for you 46 = 12 + 12, 6 = 12 + 12,

true or false”?true or false”?Kelly: “That is true”.Kelly: “That is true”.Ms L: “Ok, how did you get that one so quickly”?Ms L: “Ok, how did you get that one so quickly”?Kelly: “Six plus 6 is 12, in this case, there are 4 groups 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 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 is 12, that leaves another 6 and 6, and that equals 12. So one 12 is here and one 12 went here 12. So one 12 is here and one 12 went here [indicating the two 12s in the problem]. What I’m [indicating the two 12s in the problem]. What I’m trying to say is there are four 6s and you broke them trying to say is there are four 6s and you broke them in half and made them into two 12s”.in half and made them into two 12s”.

446 = 12 + 12 Continued6 = 12 + 12 ContinuedMs L: “Nice! Kelly, do you know right away what 4 times 6 is”?Ms L: “Nice! Kelly, do you know right away what 4 times 6 is”?Kelly: “Yes”.Kelly: “Yes”.Ms L: “What is it?”Ms L: “What is it?”Kelly: “It’s [pause] thirty- [long pause] two.”Kelly: “It’s [pause] thirty- [long pause] two.”Ms L: “Ok, do you know what 12 plus 12 is”?Ms L: “Ok, do you know what 12 plus 12 is”?Kelly: “Yeah. That is the same thing, 32”.Kelly: “Yeah. That is the same thing, 32”.Ms L:” Do you have a way of doing 12 plus 12, to check it”?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 Kelly: “Well, there are two 10s, 20- oh wait, I was thinking of a

different one”!different one”!Ms L: “You were thinking of a different multiplication problem”?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 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”.2 is 4, put those together and its 24”.

447 = 7 = □□Ms L: “Ok, here is another one. Four times 7 equals box. I want 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 you tell me what you would put in the box to make this a true number sentence”.number sentence”.

Kelly: “That would be [short pause] 28”.Kelly: “That would be [short pause] 28”.Ms L: “Ok, how did you get 28”?Ms L: “Ok, how did you get 28”?Kelly: “Well, I kinda had other problems… that went into this Kelly: “Well, I kinda had other problems… that went into this

problem. If you go up here [pointing to 3problem. If you go up here [pointing to 37 = 7 + 7 + 7] 3 7 = 7 + 7 + 7] 3 times 7 is the same as 7 plus 7 plus 7. That problem helped me times 7 is the same as 7 plus 7 plus 7. That problem helped me and I used it with this problem, [pointing to 3and I used it with this problem, [pointing to 37 = 14 + 7] 3 7 = 14 + 7] 3 sevens is the same as 14 and 7… You add one more seven and sevens is the same as 14 and 7… You add one more seven and that goes right to here. [Then she points to 4that goes right to here. [Then she points to 46 = 12 + 12.] This 6 = 12 + 12.] This problem also helped me because 4problem also helped me because 47 is like… My mind went 7 is like… My mind went back up to here [pointing to 3back up to here [pointing to 37 = 14 + 7], and I said, there is 7 = 14 + 7], and I said, there is another 7 so I could put those two 7s together, that’s 14, and another 7 so I could put those two 7s together, that’s 14, and there are two 14s, 10 and 10 is 20, 4 and 4 is 8, 28”.there are two 14s, 10 and 10 is 20, 4 and 4 is 8, 28”.

Problem sequenceProblem sequence

337 = 7 + 7 + 77 = 7 + 7 + 7337 = 14 + 77 = 14 + 7446 = 12 + 126 = 12 + 12447 = 7 = □□

Inventing algorithmsInventing algorithms

612612300300 -457-457 -299-299 200200 1 1

-40-40 -5-5 155155

292 292 8787 -549-549 -49.02-49.02 -300-300 40.0040.00 5050 -2.00 -2.00

-7-7 -.02 -.02 -257-257 37.9837.98

300300-294 -294 5/85/8 66

-5/8-5/8 5 5 3/83/8

5 ½ ÷ 5 ½ ÷ ____ = =

Number choices ½, ¼, ¾, Number choices ½, ¼, ¾, 3/83/8

Choose one of the numbers Choose one of the numbers depending on the level of your depending on the level of your students.students.

Change the problem to add Change the problem to add additional challenge as needed.additional challenge as needed.

Allow students to choose one to vary Allow students to choose one to vary the problem and allow choice.the problem and allow choice.

5 ½ ÷ 5 ½ ÷ ____ = =

Number choices ½, ¼, ¾, Number choices ½, ¼, ¾, 3/83/8 Put the problem into a context.Put the problem into a context.

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

Solve for 3/8 cup of sugar for a batch.Solve for 3/8 cup of sugar for a batch.

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

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

5 ½ ÷ 3/8 = 5 ½ ÷ 3/8 =

× 3/8 = 5 ½× 3/8 = 5 ½

× × 3/83/8 = 5 ½ = 5 ½ 8 × 3/8 = 38 × 3/8 = 3 4 × 3/8 would be ½ of 3 or 1 ½4 × 3/8 would be ½ of 3 or 1 ½ 12 × 3/8 = 4 ½ => 4 ½ cups makes 12 batches12 × 3/8 = 4 ½ => 4 ½ cups makes 12 batches Need to use 1 more cup of sugarNeed to use 1 more cup of sugar Because 8 × 3/8 = 3, a third as much would be 1 Because 8 × 3/8 = 3, a third as much would be 1 i.e. 1/3 × (8 × 3/8) = 1i.e. 1/3 × (8 × 3/8) = 1 So you need 1/3 of 8, which is 8/3 So you need 1/3 of 8, which is 8/3 i.e. 1 cup makes 8/3 batches.i.e. 1 cup makes 8/3 batches. So altogether you get a total of 12 + 8/3 or 14 2/3 batches So altogether you get a total of 12 + 8/3 or 14 2/3 batches

× × 3/83/8 = 5 ½ = 5 ½

8 × 3/8 = 38 × 3/8 = 3

½(8 × 3/8) = ½ × 3½(8 × 3/8) = ½ × 3

(½×8)×3/8 = 1 ½ (½×8)×3/8 = 1 ½

4 ×3/8 = 1 ½4 ×3/8 = 1 ½

Next subgoal:Next subgoal:How many 3/8 cups to use the remaining How many 3/8 cups to use the remaining

cup?cup? 8 × 3/8 = 38 × 3/8 = 3

1/3 × (8 × 3/8) = 1/3 × 31/3 × (8 × 3/8) = 1/3 × 3

(1/3×8)×3/8 = 1(1/3×8)×3/8 = 1

8/3 ×3/8 = 18/3 ×3/8 = 1

Putting the parts togetherPutting the parts together (8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) = (8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) =

3 + 1 ½ + 1 = 5 ½3 + 1 ½ + 1 = 5 ½

AndAnd

(8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) =(8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) = (8 + 4 + 8/3) ×3/8 = 14 2/3 ×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 So 5 ½ cups of sugar makes 14 2/3 batches of 3/8 cups of sugar sugar

Solving equationsSolving equations k + k + 13 = k + 20k + k + 13 = k + 20

From arithmetic to algebraic From arithmetic to algebraic reasoningreasoning

Attend to relations rather than Attend to relations rather than teaching only step by step proceduresteaching only step by step procedures

Align the teaching of arithmetic with Align the teaching of arithmetic with the concepts and skills students need the concepts and skills students need to learn algebra to learn algebra

Enhance the learning of arithmetic Enhance the learning of arithmetic Provide a foundation for and smooth Provide a foundation for and smooth

the transition to learning algebrathe transition to learning algebra

Learning arithmetic and algebra with Learning arithmetic and algebra with understandingunderstanding

Algebra for allAlgebra for all Not watering down algebra to teach Not watering down algebra to teach

isolated proceduresisolated procedures Develop algebraic reasoning rather than Develop algebraic reasoning rather than

teaching meaningless algebraic proceduresteaching meaningless algebraic procedures Learning arithmetic and algebra grounded Learning arithmetic and algebra grounded

in fundamental properties of number and in fundamental properties of number and number operationsnumber operations

AssignmentAssignmentWhat you will do before the next meetingWhat you will do before the next meeting

You will be writing a series of problems that You will be writing a series of problems that you might use with your students to you might use with your students to encourage them to begin to look for relations.encourage them to begin to look for relations.Write a problem to assess student thinkingWrite a problem to assess student thinkingPredict student responsesPredict student responsesWrite a series of problems to address your Write a series of problems to address your students…back up…extend???students…back up…extend???Try these with your students Try these with your students

Assignment cont’dAssignment cont’d You will be planning a lesson with your You will be planning a lesson with your

colleagues for a lesson study cyclecolleagues for a lesson study cycle As a team choose a topic to be taught As a team choose a topic to be taught

on December 6on December 6thth

Choose a topic that is typically difficult Choose a topic that is typically difficult for studentsfor students

Bring planning materials to the October Bring planning materials to the October sessionsession

ChallengeChallenge Addition is associative, but Addition is associative, but

subtraction is not. How about the subtraction is not. How about the following:following:

a)a) Is (a + b) - c = a + (b - c) true for all Is (a + b) - c = a + (b - c) true for all numbers?numbers?

b)b) Is (a - b) + c = a - (b + c) true for all Is (a - b) + c = a - (b + c) true for all numbers?numbers?

Thinking MathematicallyThinking Mathematically p. 120 #4 p. 120 #4

8181

ChallengeChallenge What kind of number do you get What kind of number do you get

when you add three odd when you add three odd numbers?numbers?

Can you justify your response?Can you justify your response?

Thinking MathematicallyThinking Mathematically p.103 #1 p.103 #1

8282

Challenge--Try this!Challenge--Try this! Design a sequence of true/false Design a sequence of true/false

and/or open sentences that you and/or open sentences that you might use to engage your might use to engage your students in thinking about the students in thinking about the equal sign.equal sign.

Thinking MathematicallyThinking Mathematically p. 24 #4 p. 24 #4

8383

ReferencesReferences Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Carpenter, T.P., Franke, M.L., & Levi, L. (2003).

Thinking mathematically: Thinking mathematically: IntegratingIntegrating arithmetic arithmetic and algebra in the elementary school.and algebra in the elementary school. Portsmouth, Portsmouth, NH: Heinemann.NH: Heinemann.

Carpenter, T. P., Franke, M.L., & Levi, L. (2005). Carpenter, T. P., Franke, M.L., & Levi, L. (2005). Algebra in Elementary School. ZDM. 37(1), 1-7.Algebra in Elementary School. ZDM. 37(1), 1-7.

Carpenter, T. P., Levi, L., Berman, P., & Pligge, M. Carpenter, T. P., Levi, L., Berman, P., & Pligge, M. (2005). Developing algebraic reasoning in the (2005). Developing algebraic reasoning in the elementary school. In T. A. Romberg , T. P. elementary school. In T. A. Romberg , T. P. Carpenter, & F. Dremock (Eds). Carpenter, & F. Dremock (Eds). Understanding Understanding mathematics and science matters.mathematics and science matters. Mahwah, NJ: Mahwah, NJ: Erlbaum.Erlbaum.

Jacobs, V.J., Franke, M.L., Carpenter, T. P., Levi, L., & Jacobs, V.J., Franke, M.L., Carpenter, T. P., Levi, L., & Battey, D. (2007) A large-scale study of professional Battey, D. (2007) A large-scale study of professional development focused on children’s algebraic development focused on children’s algebraic reasoning in elementary school. reasoning in elementary school. Journal for Research Journal for Research in Mathematics Education, 38,in Mathematics Education, 38, 258-288 258-288..