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Johnny W. Lott jlott@mso.umt

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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Johnny W. Lott jlott@mso.umt

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

  2. 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?

  3. Arithmetic Computation? • What do you need to know? • Place value • Algorithms

  4. Arithmetic Computation? • Would your students say that 35 = 8 or 47 = 11? • What would they say about 3 + 5 or 5 + 3?

  5. 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?

  6. Look at worksheet 1. • Do the arithmetic as directed.

  7. 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?

  8. Look at growing patterns. • Use Exploring Houses. • Use Building with Toothpicks. • Use Tile Patterns.

  9. Considering Patterns • Will more than one pattern work? • How many does it take to decide a pattern? • Can you prove your answer?

  10. How Tall Are the Cups? 2 inches 7 inches How tall is a stack of 100 cups?

  11. What are your favorite problems to solve? • Locker Problem • Squares on a Checkerboard Problem • Tying the String to Get Married Problem

  12. Lockers all open.

  13. Second student goes through

  14. Third student goes through

  15. Fourth student goes through

  16. 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?

  17. 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?

  18. 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?

  19. 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?

  20. 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?

  21. Twist old problems • Locker problem gave perfect squares. • Try the pig problem--even with young kids.

  22. 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?

  23. 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?

  24. Checkerboard/Grains of Rice • Substitute the “Would You Work for Me?” Problem. • Would you?

  25. 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?

  26. Algebra Continued • How do you learn multiplication facts? • Why do you learn multiplication tables?

  27. 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.

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