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

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# What Do Arithmetic Computation and “Real World” Math Have to Do with Algebra or Algebraic Thinking? - PowerPoint PPT Presentation

What Do Arithmetic Computation and “Real World” Math Have to Do with Algebra or Algebraic Thinking?. Johnny W. Lott jlott@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?

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

Johnny W. Lott

jlott@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 value
• Algorithms
Arithmetic Computation?
• Would your students say that

35 = 8 or

47 = 11?

• What would they say about

3 + 5 or

5 + 3?

Place value?
• Would your students say that

310 + 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?
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 Problem
• Squares on a Checkerboard Problem
• Tying the String to Get Married Problem
• 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?
• 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 hand
• Tie ends on top two at a time.
• Tie ends on bottom two at a time
• If a full loop is obtained, I can get a marriage license. How likely?
• 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.
• Can you prove it?
Checkerboard/Grains of Rice
• Substitute the “Would You Work for Me?” Problem.
• Would you?
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?
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.