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Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton Mathematics Department Western Oregon University www.wou.edu/~burtonl
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These ideas use ALGEBRA PIECES and the MATH IN THE MIND’S EYE curriculum developed at Portland State University (see handout for access)
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What are ALGEBRA PIECES? The first pieces are BLACK AND RED TILES which model integers: Black Square = 1Red Square = -1
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INTEGER OPERATIONS Addition 2 + 3 3 black 5 black total = 5 2 black group
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INTEGER OPERATIONS Addition -2 + -3 5 red total = -5 group 2 red 3 red
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INTEGER OPERATIONS Addition -2 + 3 3 black Black/Red pair: Net Value (NV) = 0 Total NV = 1 group 2 red
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INTEGER OPERATIONS Subtraction 2 - 3 2 black Take Away?? Still Net Value: 2 3 black Add R/B pairs
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INTEGER OPERATIONS Subtraction 2 - 3 Net Value: 2 Take away 3 black 2 - 3 = -1
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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.
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This is excellent for understanding “subtracting a negative is equivalent to adding a positive.”
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INTEGER OPERATIONS Multiplication 2 x 3 Edges: NV 2 & NV 3
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Fill in using edge dimensions INTEGER OPERATIONS Multiplication 2 x 3 Net Value = 6 2 x 3 = 6
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INTEGER OPERATIONS Multiplication -2 x 3 Edges: NV -2 & NV 3
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Fill in with black INTEGER OPERATIONS Multiplication -2 x 3
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INTEGER OPERATIONS Multiplication -2 x 3 Net Value = -6 -2 x 3 = -6 Red edge indicates FLIP along corresponding column or row
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-2 x -3 would result in TWO FLIPS (down the columns, across the rows) and an all black result to show -2 x -3 = 6 These models can also show INTEGER DIVISION
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BEYOND INTEGER OPERATIONS The next important phase is understanding sequences and patterns corresponding to a sequence of natural numbers.
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TOOTHPICK PATTERNS Students learn to abstract using simple patterns
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TOOTHPICK PATTERNS These “loop diagrams” help the students see the pattern here is 3n + 1: n = figure #
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B / R ALGEBRA PIECES These pieces are used for sequences with Natural Number domain Black N, N ≥ 0 Edge N Red -N, -N < 0 Edge -N Pieces rotate
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ALGEBRA SQUARES Black N 2 Red -N 2 Edge lengths match n strips Pieces rotate
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Patterns with Algebra Pieces Students learn to see the abstract pattern in sequences such as these
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Patterns with Algebra Pieces N (N +1) 2 -4
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Working with Algebra Pieces Multiplying (N + 3)(N - 2) First you set up the edges N + 3 N - 2
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(N + 3)(N - 2) Now you fill in according to the edge lengths First N x N = N 2
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(N + 3)(N - 2) Inside 3 x N = 3N Outside N x -2 = -2N Last 3 x -2 = -6
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(N + 3)(N - 2) (N + 3)(N - 2) = N 2 - 2N + 3N - 6 = N 2 + N - 6
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(N + 3)(N - 2) This is an excellent method for students to use to understand algebraic partial products
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Solving Equations N 2 + N - 6 = 4N - 8? =
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= Subtract 4N from both sets: same as adding -4n
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Solving Equations N 2 + N - 6 = 4N - 8? = Subtract -8 from both sets
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Solving Equations N 2 + N - 6 = 4N - 8? = 0 NV -6 -(-8) = 2
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Solving Equations N 2 + N - 6 = 4N - 8? Students now try to factor by forming a rectangle Note the constant partial product will always be all black or all red = 0
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Solving Equations N 2 + N - 6 = 4N - 8? Thus, there must be 2 n strips by 1 n strip to create a 2 black square block Take away all NV=0 Black/Red pairs = 0
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Solving Equations N 2 + N - 6 = 4N - 8? Thus, there must be 2 n strips by 1 n strip to create a 2 black square block Take away all NV=0 Black/Red pairs = 0
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Solving Equations N 2 + N - 6 = 4N - 8? Form a rectangle that makes sense = 0
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Solving Equations N 2 + N - 6 = 4N - 8? Lay in edge pieces = 0
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Solving Equations N 2 + N - 6 = 4N - 8? Measure the edge sets = 0 N - 1 N - 2
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Solving Equations N 2 + N - 6 = 4N - 8? = 0 (N - 2)(N - 1) = 0 (N - 2) = 0, N = 2 or (N - 1) = 0, N = 1
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This last example; using natural number domain for the solutions, was clearly contrived.
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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.
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Materials Math in the Mind’s Eye Lesson Plans: Math Learning Center Burton: Sabbatical Classroom use modules
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Packets for today: “Advanced Practice” Integer work stands alone Algebraic work; quality exploration provides solid foundation
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