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How the ideas and language of algebra K-5 set the stage for algebra 6–12

How the ideas and language of algebra K-5 set the stage for algebra 6–12. E. Paul Goldenberg. 2008. With downloadable PowerPoint Ideas and approaches drawn from Think Math! a comprehensive K-5 program from Houghton Mifflin Harcourt School Publishers. http://thinkmath.edc.org.

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How the ideas and language of algebra K-5 set the stage for algebra 6–12

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  1. How the ideas and language of algebra K-5set the stage for algebra 6–12 E. Paul Goldenberg 2008

  2. With downloadable PowerPoint Ideas and approaches drawn fromThink Math!a comprehensive K-5 program fromHoughton Mifflin HarcourtSchool Publishers http://thinkmath.edc.org Before you scramble to take notes Go to intersections Go to marble bag trick Go to Kindergarten sorting, CNPs Go to 3rd grade detectives Go to “Guess my number” (mental buffer) Go to multiplication onions

  3. Algebraic language & algebraic thinking

  4. Math could be spark curiosity! Algebraic thinking Is there anything interesting about addition and subtraction sentences? 2nd grade

  5. 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 How does this work?

  6. Math could be fascinating! Algebraic language Is there anything less sexy than memorizing multiplication facts? What helps people memorize?Something memorable! 4th grade Go to “Mommy, give me…” Go to visual way to understand Go to index

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

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

  9. 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 ? ? ? ? … …

  10. 100 4 5 6 7 8 9 10 11 12 13 14 15 Take it even further What about three steps out? What about four? What about five? 75

  11. 29 30 31 32 33 34 35 36 37 38 39 40 Take it even further What about three steps out? What about four? What about five? 1200 1225

  12. 29 30 31 32 33 34 35 36 37 38 39 40 Take it even further What about two steps out? 1221 1225

  13. 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

  14. But nobody cares if kids can multiply 47  53 mentally!

  15. Algebraic thinking Science Algebraic language What do we care about, then? • 50  50 (well, 5  5 and place value) • Keeping 2500 in mind while thinking 3  3 • Subtracting 2500 – 9 • Finding the pattern • Describing the pattern

  16. n n – 1 n + 1 (7 – 1)  (7 + 1) = 7  7–1 (n– 1)  (n+ 1) = nn– 1 (n– 1)  (n+ 1)

  17. n n – 3 n + 3 (7 – 3)  (7 + 3) = 7  7–9 (n– 3) (n– 3)  (n+ 3) (n– 3)  (n+ 3) = nn– 9

  18. Distance away What to subtract 1 1 2 4 3 9 4 16 5 25 ddd Make a table

  19. n n – d n + d (7 –d)  (7 +d) = 7  7–dd (n– d)  (n+ d) = nn– (n– d)  (n+ d) = nn–dd (n– d)  (n+ d) (n– d)

  20. We also care about thinking! • Kids feel smart!Why silent teaching? • 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!

  21. One way to look at it 5  5

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

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

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

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

  26. An important propaganda break…

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

  28. It could be surprising! What could mathematics be like? Surprise! You’re good at algebra! 5th grade Go to index

  29. 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!

  30. 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!

  31. 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!

  32. 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!

  33. 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!

  34. 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!

  35. 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!

  36. 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!

  37. 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! Go to index

  38. Kids need to do it themselves…

  39. 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

  40. 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.

  41. 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

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

  43. 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 Go to index

  44. 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.

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

  46. Representing multiplication, itself

  47. 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? Go to index

  48. 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

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

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

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