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From Buttons to Algebra:

From Buttons to Algebra:. Paul Goldenberg http://thinkmath.edc.org Some ideas from the newest NSF program, Think Math!. Learning the ideas and language of algebra, K-12. from and Harcourt School Publishers Rice University, Houston, Sept 2007. http://thinkmath.edc.org

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From Buttons to Algebra:

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  1. From Buttons to Algebra: Paul Goldenberg http://thinkmath.edc.org Some ideas from the newest NSF program, Think Math! Learning the ideas and language of algebra, K-12 from and Harcourt School Publishers Rice University, Houston, Sept 2007

  2. http://thinkmath.edc.org With downloadable PowerPoint at http://www.edc.org/thinkmath/ Before you scramble to take notes

  3. It could be spark curiosity! What could mathematics be like? Is there anything interesting about addition and subtraction sentences?

  4. Write two number sentences… To 2nd graders: see if you can find some that don’t work! 4 + 2 = 6 3 + 1 = 4 + = 3 10 7

  5. It could be fascinating! What could mathematics be like? Is there anything less sexy than memorizing multiplication facts? What helps people memorize?Something memorable!

  6. 2 3 4 5 6 7 8 9 10 11 12 13 38 18 39 19 40 20 21 41 42 22 Teaching without talking Shhh… Students thinking! Wow! Will it always work? Big numbers? 35 80 15 36 81 16 ? ? 1600 ? … …

  7. Take it a step further What about two steps out?

  8. 16 2 3 4 5 6 7 8 58 28 9 59 29 10 30 60 11 12 31 61 13 32 62 Teaching without talking Shhh… Students thinking! Again?! Always? Find some bigger examples. 12 60 64 ? ? ? ? … …

  9. Take it even further What about three steps out? What about four? What about five?

  10. 47 48 49 50 51 52 53 “Mommy!Give me a 2-digit number!” about 50 2500 • “OK, um, 53” • “Hmm, well… • …OK, I’ll pick 47, and I can multiply those numbers faster than you can!” To do… 5347 I think… 5050(well, 5  5 and …)… 2500 Minus 3  3– 9 2491

  11. Why bother? • Kids feel smart! • Teachers feel smart! • Practice.Gives practice. Helps me memorize, because it’s memorable! • Something new. Foreshadows algebra. In fact, kids record it with algebraic language! • And something to wonder about: How does it work? It matters!

  12. One way to look at it 5  5

  13. One way to look at it Removing a column leaves 5  4

  14. One way to look at it Replacing as a row leaves 6  4 with one left over.

  15. One way to look at it Removing the leftover leaves 6  4 showing that it is one less than 5 5.

  16. How does it work? 47 50 53 50  50 – 3 3 3 = 53 47 47 3

  17. An important propaganda break…

  18. “Math talent” is made, not found • We all “know” that some people have… musical ears, mathematical minds, a natural aptitude for languages…. • We gotta stop believing it’s all in the genes! • And we are equally endowed with much of it

  19. A number trick • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  20. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  21. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  22. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  23. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  24. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  25. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  26. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  27. How did it work? • Think of a number. • Add 3. • Double the result. • Subtract 4. • Divide the result by 2. • Subtract the number you first thought of. • Your answer is 1!

  28. Kids need to do it themselves…

  29. Using notation: following steps Words Pictures Dana Cory Sandy Chris 5 Think of a number. 10 Double it. 16 Add 6. Divide by 2.What did you get? 8 7 3 20

  30. Pictures Using notation: undoing steps Words Dana Cory Sandy Chris 4 5 Think of a number. 8 10 Double it. 14 16 Add 6. Divide by 2.What did you get? 8 7 3 20 Hard to undo using the words. Much easier to undo using the notation.

  31. Using notation: simplifying steps Words Pictures Dana Cory Sandy Chris 4 5 Think of a number. 10 Double it. 16 Add 6. Divide by 2.What did you get? 8 7 3 20

  32. Why a number trick? Why bags? • Computational practice, but much more • Notation helps them understand the trick. • Notation helps them invent new tricks. • Notation helps them undo the trick. • But most important, the idea that notation/representation is powerful!

  33. Children are language learners… • They are pattern-finders, abstracters… • …natural sponges for language in context. n 10 8 28 18 17 58 57 n – 8 2 0 20 3 4

  34. A game in grade 3 ones digit < 5 tens digit is 7, 8, or 9 hundreds digit > 6 the number is even the number is a multiple of 5 tens digit < ones digit the tens digit is greater than the hundreds digit the number is divisible by 3 the ones digit is twice the tens digit

  35. 3rd grade detectives! I. I am even. II. All of my digits < 5 h t u III. h + t + u = 9 1 4 4 432 342 234 324 144 414 IV. I am less than 400. 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 V. Exactly two of my digits are the same.

  36. Is it all puzzles and tricks? • No. (And that’s too bad, by the way!) • Curiosity. How to start what we can’t finish. • Cats play/practice pouncing; sharpen claws. • We play/practice, too. We’ve evolved fancy brains.

  37. Representing processes • Bags and letters can represent numbers. • We need also to represent… • ideas — multiplication • processes — the multiplication algorithm

  38. Representing multiplication, itself

  39. Naming intersections, first grade Put a red house at the intersection of A street and N avenue. Where is the green house?How do we go fromthe green house tothe school?

  40. Combinatorics, beginning of 2nd • How many two-letter words can you make, starting with a red letter and ending with a purple letter? a i s n t

  41. Multiplication, coordinates, phonics? a i s n t in as at

  42. Multiplication, coordinates, phonics? w s ill it ink b p st ick ack ing br tr

  43. Similar questions, similar image Four skirts and three shirts: how many outfits? Five flavors of ice cream and four toppings: how many sundaes? (one scoop, one topping) How many 2-block towers can you make from four differently-colored Lego blocks?

  44. Representing 22  17 22 17

  45. Representing the algorithm 20 2 10 7

  46. Representing the algorithm 20 2 20 200 10 7 14 140

  47. Representing the algorithm 20 2 20 220 200 10 7 154 14 140 34 340 374

  48. 1 22 x 17 154 220 374 Representing the algorithm 20 2 20 220 200 10 7 154 14 140 34 340 374

  49. 1 17 x 22 34 340 374 Representing the algorithm 20 2 20 220 200 10 7 154 14 140 34 340 374

  50. 22 17 374 22  17 = 374

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