Johnny W. Lott jlott@mso.umt

What Do Arithmetic Computation and “Real World” Math Have to Do with Algebra or Algebraic

Thinking?

Johnny W. Lottjlott@mso.umt.edu

What ties if any does arithmetic have to

algebra?A different way to put this is the

following:Is everything that we teach in algebra

new?What should we think about if we talk

about algebraic thinking?

Arithmetic Computation?

What do you need to know?Place valueAlgorithms

Arithmetic Computation?

Would your students say that35 = 8 or47 = 11?

What would they say about 3 + 5 or 5 + 3?

Place value?

Would your students say that310 + 5 = 8

What would they say about 3 + 5 or 5 + 3?

How are place value and algebraic symbolism related?

Look at worksheet 1.

Do the arithmetic as directed.

What happens when you look at decimals?

What is the meaning of 431.25?

What would this look like in expanded form?

Do worksheet 2.How are decimals related to algebra?

Look at growing patterns.

Use Exploring Houses.Use Building with Toothpicks.Use Tile Patterns.

Considering Patterns

Will more than one pattern work?How many does it take to decide a

pattern?Can you prove your answer?

How Tall Are the Cups?

2 inches

7 inches

How tall is a stack of 100 cups?

What are your favorite problems to solve?

Locker ProblemSquares on a Checkerboard

ProblemTying the String to Get Married

Problem

Lockers all open.

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Second student goes through1 C 3 C 5 C 7 C 9 C

11

C 13

C 15

C 17

C 19

C

21

C 23

C 25

C 27

C 29

C

31

C 33

C 35

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C

41

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C 47

C 49

C

51

C 53

C 55

C 57

C 59

C

61

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C

71

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C

81

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91

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C 99

C

Third student goes through1 C C C 5 6 7 C C C

11

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C C C 17

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C

C C 23

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C C C 29

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C C C 35

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C C C

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C C C 47

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C

C C 53

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C C C 65

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C C C

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C C C 77

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C

C C 83

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C C C 95

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C C C

Fourth student goes through

1 C C 4 5 6 7 8 C C

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C 13

C C 16

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C C 23

C 25

C C 28

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C C 35

C 37

C C 40

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C C 47

C 49

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C 52

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C C

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C 97

C C 100

Questions to ask

If 1000 students go through the school and change the state of doors, how many times is door 72 touched?

What is the final state of door 432?Who touched door 46 last?What is the relation of the door

number and the number of factors?

Squares on a Checkerboard Problem

Give me one grain of wheat for the first square.

Give me two grains for the second square.

Give me four grains for the third square and continue.

How many grains in all when the board is filled?

Questions to ask

Would you take only the grains on the 64th square or would you take all the grains on the first 63 squares if given the option?

How many grains are on the 15th square?

Tying the String to Get Married Problem

Six strings in my handTie ends on top two at a time.Tie ends on bottom two at a

timeIf a full loop is obtained, I can

get a marriage license. How likely?

Questions to ask

Is a person’s chance of getting a license more than 50% in the first year?

Does the probability of getting the marriage license change in a second year if the license is not obtained in the first year?

Suppose there are only five strings. Is the probability more or less? Four strings? Three strings? Two strings? One string?

Twist old problems

Locker problem gave perfect squares.

Try the pig problem--even with young kids.

Pig Problem

A farmer sold n cows for n dollars each. With the proceeds, she bought an odd number of sheep at $10 each, and a pig for less than $10. How much did the pig cost?

Think perfect squares.

Why?Think of the ones digit of the

proceeds.Think of the tens digit of the

proceeds.Look at a table. What is your answer?Can you prove it?

Checkerboard/Grains of Rice

Substitute the “Would You Work for Me?” Problem.

Would you?

Day My Scheme G's Scheme1 $1.00 $1.002 $0.50 $0.503

Algebra

Algebra is a civil right. Robert Moses What types of formulas are used in

spreadsheets? Teachers, what types of formulas are used in

your retirement packages? Students, how can you tell how long

medication stays in your blood stream? How do you decide on pricing for concert

tickets?

Algebra Continued

How do you learn multiplication facts?

Why do you learn multiplication tables?

6, 12, 18, 24, 30,What is the pattern here that 3rdgraders know?

Number of term with term below1 2 3 4 5 6… n6 12 18 24 30 36…7 13 19 25 31 37…

Yet More Algebra!

Consider addition and all the pairs that add to 12; now that add to 18; now that add to 0. What do they have in common?

Try the same with multiplication.