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

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

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

INTEGER OPERATIONS Addition black 5 black total = 5 2 black group

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

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

INTEGER OPERATIONS Subtraction black Take Away?? Still Net Value: 2 3 black Add R/B pairs

INTEGER OPERATIONS Subtraction Net Value: 2 Take away 3 black = -1

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.

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

INTEGER OPERATIONS Multiplication 2 x 3 Edges: NV 2 & NV 3

Fill in using edge dimensions INTEGER OPERATIONS Multiplication 2 x 3 Net Value = 6 2 x 3 = 6

INTEGER OPERATIONS Multiplication -2 x 3 Edges: NV -2 & NV 3

Fill in with black INTEGER OPERATIONS Multiplication -2 x 3

INTEGER OPERATIONS Multiplication -2 x 3 Net Value = x 3 = -6 Red edge indicates FLIP along corresponding column or row

-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

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

TOOTHPICK PATTERNS Students learn to abstract using simple patterns

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

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

ALGEBRA SQUARES Black N 2 Red -N 2 Edge lengths match n strips Pieces rotate

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

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

Working with Algebra Pieces Multiplying (N + 3)(N - 2) First you set up the edges N + 3 N - 2

(N + 3)(N - 2) Now you fill in according to the edge lengths First N x N = N 2

(N + 3)(N - 2) Inside 3 x N = 3N Outside N x -2 = -2N Last 3 x -2 = -6

(N + 3)(N - 2) (N + 3)(N - 2) = N 2 - 2N + 3N - 6 = N 2 + N - 6

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

Solving Equations N 2 + N - 6 = 4N - 8? =

= Subtract 4N from both sets: same as adding -4n

Solving Equations N 2 + N - 6 = 4N - 8? = Subtract -8 from both sets

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

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

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

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

Solving Equations N 2 + N - 6 = 4N - 8? Form a rectangle that makes sense = 0

Solving Equations N 2 + N - 6 = 4N - 8? Lay in edge pieces = 0

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

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

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

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.

Materials Math in the Mind’s Eye Lesson Plans: Math Learning Center Burton: Sabbatical Classroom use modules

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