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What Algebra Is Traditional and What Is Reality?

What Algebra Is Traditional and What Is Reality?. Johnny W. Lott Past President, NCTM jlott@mso.umt.edu. What does “algebra” mean to you?. Why the switch to algebraic thinking versus algebra?. What is algebraic thinking? Do I ever engage in it? Do my students ever engage in it?

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What Algebra Is Traditional and What Is Reality?

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  1. What Algebra Is Traditional and What Is Reality? Johnny W. Lott Past President, NCTM jlott@mso.umt.edu

  2. What does “algebra” mean to you?

  3. Why the switch to algebraic thinking versus algebra? • What is algebraic thinking? • Do I ever engage in it? • Do my students ever engage in it? • Does anyone think that it is important? • What do parents think about algebra for the very young?

  4. What are NCTM’s Thoughts? • All students need algebraic thinking to learn analytical and reasoning skills. • With algebraic thinking as a strand, students see interconnections in math.

  5. Algebraic thinking as a strand requires re-evaluating your entire mathematics curriculum and re-thinking teaching methods used.

  6. What are some guiding themes for an algebraic thinking strand? • Mathematical Language • Patterns • Relations and Functions • Multiple Representations • Modeling • Structure

  7. Contributing Themes • Problem Solving • Communication • Connections • Reasoning

  8. Principles • All students need algebraic thinking. • Algebraic reasoning is as fundamental as algebraic manipulation. • Appropriate tools should be used to teach algebraic thinking. Do you accept these principles?

  9. Now, I get to pose problems; you get to work. • Would you work for me?

  10. On the first day, I will pay you $1000 early in the morning., At the end of the day, you must pay me a commission of $100. At the end of the day we will both determine your next days’ salary and my commission. I will double what you have at the end of the day, but you must double the amount that you pay me. Will you work for me for a month?

  11. What is a variable?

  12. Mathematical Language • Pick any number; keep it a secret. • Add 15. • Multiply the sum by 4. • Subtract 8 • Divide the difference by 4. • Subtract 12 • Tell me your answer and I know the original number.

  13. I can find your number. Let n be the number. n + 15 4(n + 15) 4(n + 15) – 8 [4(n + 15) – 8]/4 [4(n + 15) – 8]/4 – 12 What is the result?

  14. Patterns, Relations, Functions • Use pattern blocks. • Use Illuminations from www.nctm.org/illuminations/ • They can be fun! • O, T, T, F, F, S, S, E, N, T, … • 61, 52, 63, ____, 46, 18, 001 • They can drive you nuts!

  15. Let’s try common patterns and push them. • Think of all the sum families that add to 12. • Think of all the sum families that add to 15. • Think of all the sum families that add to 20. • And so on. • Graph the pairs in the sum families of each one. • What do you see?

  16. Explore “over 1 and down 2” on a hundreds chart. • 1 2 3 4 5 6 7 8 9 10 • 11 12 13 14 15 16 17 18 19 20 • 21 22 23 24 25 26 27 28 29 30 • 31 32 33 34 35 36 37 38 39 40 • 41 42 43 44 45 46 47 48 49 50 • 51 52 53 54 55 56 57 58 59 60 • 61 62 63 64 65 66 67 68 69 70 • 71 72 73 74 75 76 77 78 79 80 • 81 82 83 84 85 86 87 88 89 90

  17. Patterns, Relations and Functions • Sally is hosting a party. Every time the doorbell rings, three more people come to join the party. Identify the pattern and write a mathematical sentence for this pattern.

  18. Consider the pattern that follows: • 4, 7, 10, 13, 16, 19, … • Let’s use a spreadsheet.

  19. Use your imagination and the pseudo-spreadsheet on the overhead.

  20. Where did we move from lower grades to high school? • Can the same type of problem be used?

  21. What does any linear function look like? • f(n) = an + b • What does almost every linear pattern (sequence) look like? • Retreat to the overhead!

  22. Patterns, Relations, and Functions • With technology, use a pendulum to measure a 5-second interval accurately. • How? • Use a motion detector to collect data that measures the motion of a pendulum. Examine the graph; determine the function.

  23. Patterns, Relations, and Functions--for Most • Spreadsheet yields recursive formula • Algebra yields explicit formula • Which is better? • What are we learning?

  24. How do you prove that the pattern is what you say it is? • At what level? • How would you do this in high school?

  25. Revisit a Twisted Old Problem • Consider all the pairs whose product is 12 • Consider all the pairs whose product is 24. • Consider all the pairs whose product is 36. • Consider all the pairs whose product is 72. • What do these graphs look like?

  26. Back to Early GradesTeddy Bear Mathematics • A set of teddy bears are on a shelf in a child’s classroom. Count the number. Represent a relation between the number of bears and the total number of ears. • Try eyes. • Try paws. • Try noses

  27. Move to Figure This! • Examples abound where algebraic thinking exists and equations can be solved.

  28. With the prices on the coming slide, which has the cheapest prices?

  29. Old BUT Good Modeling Problems • How do you model operations with integers? • Postman Stories • Elevator • Number Lines • Can this be extended to algebraic thinking?

  30. For many • A “wave” is often seen at sporting events. How long would it take the wave to travel around this room? • How many times would you have to simulate this to be reasonably sure that you could write a formula for the process?

  31. Modeling for Algebra • According to The Missoulian, the trees in a certain land area are being cut at a rate of 15% per year. The lumber company claims that it replants 2000 trees every year in this area. Discuss the future tree production of this land area if this plan continues.

  32. Old BUT Good Modeling Problems • How do you model operations with integers? • Postman Stories • Elevator • Number Lines • Can this be extended to algebraic thinking?

  33. For many • A “wave” is often seen at sporting events. How long would it take the wave to travel around this room? • How many times would you have to simulate this to be reasonably sure that you could write a formula for the process?

  34. Modeling for 9-12 • According to The Missoulian, the trees in a certain land area are being cut at a rate of 15% per year. The lumber company claims that it replants 2000 trees every year in this area. Discuss the future tree production of this land area if this plan continues.

  35. Structure • The stained glass window consists of two regular hexagons. How many ways can you represent the green as a part of the whole?

  36. Two examples--more? What is the structure that is being presented here? What are other ways to show this structure?

  37. More Structure • The set of all square matrices • Operations of addition and multiplication What properties of real numbers still hold?

  38. How does an algebraic thinking strand fit with the remainder of your mathematics program?

  39. Selected References • Montana Council of Teachers of Mathematics. Algebra: A Strand PreK-12. Missoula, MT: MCTM, 1998. • www.nctm.org • www.figurethis.org • www.nctm.org and search for illuminations

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