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Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and So

Dr. Laurie Burton presents the use of Algebra Pieces to model algebraic expressions and solve equations in the Northwest Two Year College Mathematics Conference 2006.

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Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and So

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  1. Northwest Two Year CollegeMathematics Conference 2006Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie BurtonMathematics DepartmentWestern Oregon Universitywww.wou.edu/~burtonl

  2. These ideas useALGEBRA PIECES and the MATH IN THE MIND’S EYE curriculum developed at Portland State University (see handout for access)

  3. What are ALGEBRA PIECES? The first pieces are BLACK AND RED TILES which model integers: Black Square = 1 Red Square = -1

  4. 2 black 3 black group INTEGER OPERATIONS Addition 2 + 3 5 black total = 5

  5. 2 red 3 red group INTEGER OPERATIONS Addition -2 + -3 5 red total = -5

  6. 2 red 3 black group INTEGER OPERATIONS Addition -2 + 3 Black/Red pair: Net Value (NV) = 0Total NV = 1

  7. 3 black 2 black Add R/B pairs INTEGER OPERATIONSSubtraction 2 - 3 Take Away?? Still Net Value: 2

  8. Take away 3 black INTEGER OPERATIONSSubtraction 2 - 3 Net Value: 2 2 - 3 = -1

  9. You can see that all integer subtraction models may be solved by simply added B/R--Net Value 0 pairs until you have the correct amount of black or red tiles to subtract.

  10. This is excellent for understanding “subtracting a negative is equivalent to adding a positive.”

  11. Edges: NV 2 & NV 3 INTEGER OPERATIONSMultiplication 2 x 3

  12. Fill in using edge dimensions INTEGER OPERATIONSMultiplication 2 x 3 Net Value = 62 x 3 = 6

  13. Edges: NV -2 & NV 3 INTEGER OPERATIONSMultiplication -2 x 3

  14. Fill in with black INTEGER OPERATIONSMultiplication -2 x 3

  15. Red edge indicates FLIP along corresponding column or row INTEGER OPERATIONSMultiplication -2 x 3 Net Value = -6-2 x 3 = -6

  16. -2 x -3 would result in TWO FLIPS (down the columns, across the rows) and an all black result to show -2 x -3 = 6These models can also show INTEGER DIVISION

  17. BEYONDINTEGER OPERATIONS The next important phase is understanding sequences and patterns corresponding to a sequence of natural numbers.

  18. TOOTHPICK PATTERNS Students learn to abstract using simple patterns

  19. TOOTHPICK PATTERNS These “loop diagrams” help the students see the pattern here is 3n + 1: n = figure #

  20. B / R ALGEBRA PIECES These pieces are used for sequences with Natural Number domain Black N, N ≥ 0Edge NRed -N, -N < 0Edge -N Pieces rotate

  21. ALGEBRA SQUARES Black N2Red -N2Edge lengths match n stripsPieces rotate

  22. Patterns with Algebra Pieces Students learn to see the abstract pattern in sequences such as these

  23. N (N +1)2 -4 Patterns with Algebra Pieces

  24. N + 3 N - 2 Working with Algebra PiecesMultiplying(N + 3)(N - 2) First you set up the edges

  25. (N + 3)(N - 2) Now you fill in according to the edge lengths FirstN x N = N2

  26. (N + 3)(N - 2) Inside3 x N = 3N Last 3 x -2 = -6 OutsideN x -2 = -2N

  27. (N + 3)(N - 2) (N + 3)(N - 2) = N2 - 2N + 3N - 6 = N2 + N - 6

  28. (N + 3)(N - 2) This is an excellent method for students to use to understand algebraic partial products

  29. Solving Equations N2 + N - 6 = 4N - 8? =

  30. Solving Equations N2 + N - 6 = 4N - 8? = Subtract 4N from both sets: same as adding -4n

  31. Solving Equations N2 + N - 6 = 4N - 8? Subtract -8 from both sets =

  32. NV -6 -(-8) = 2 Solving Equations N2 + N - 6 = 4N - 8? = 0

  33. Solving Equations N2 + N - 6 = 4N - 8? = 0 Students now try to factor by forming a rectangleNote the constant partial product will always be all black or all red

  34. Solving Equations N2 + N - 6 = 4N - 8? = 0 Thus, there must be 2 n strips by 1 n strip to create a 2 black square blockTake away all NV=0 Black/Red pairs

  35. Solving Equations N2 + N - 6 = 4N - 8? = 0 Thus, there must be 2 n strips by 1 n strip to create a 2 black square blockTake away all NV=0 Black/Red pairs

  36. Solving Equations N2 + N - 6 = 4N - 8? = 0 Form a rectangle that makes sense

  37. Solving Equations N2 + N - 6 = 4N - 8? = 0 Lay in edge pieces

  38. N - 1N - 2 Solving Equations N2 + N - 6 = 4N - 8? = 0 Measure the edge sets

  39. Solving Equations N2 + N - 6 = 4N - 8? = 0 (N - 2)(N - 1) = 0 (N - 2) = 0, N = 2or (N - 1) = 0, N = 1

  40. This last example; using natural number domain for the solutions, was clearly contrived.

  41. In fact, the curriculum extends to using neutral pieces (white) to represent x and -x allowing them to extend to integer domain and connect all of this work to graphing in the “usual” way.

  42. Materials Math in the Mind’s Eye Lesson Plans:Math Learning Center Burton: SabbaticalClassroom use modules

  43. Packets for today:“Advanced Practice” Integer work stands alone Algebraic work; quality exploration provides solid foundation

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