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 These ideas use ALGEBRA PIECES and the MATH IN THE MINDS 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 = 1

Red Square = -1 INTEGER OPERATIONS Addition 2+3 2 black group

3 black 5 black total = 5 INTEGER OPERATIONS Addition -2 + -3 2 red

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

2 red group 3 black Black/Red pair: Net Value (NV) = 0 Total NV = 1

INTEGER OPERATIONS Subtraction 2-3 Take Away?? 2 black 3 black

Add R/B pairs Still Net Value: 2 INTEGER OPERATIONS Subtraction 2-3 Take away 3 black Net Value: 2 2 - 3 = -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 2x3 Edges:

NV 2 & NV 3 INTEGER OPERATIONS Multiplication 2x3 Fill in using edge dimensions Net Value = 6

2x3=6 INTEGER OPERATIONS Multiplication -2 x 3 Edges: NV -2 & NV 3 INTEGER OPERATIONS

Multiplication -2 x 3 Fill in with black INTEGER OPERATIONS Multiplication -2 x 3 Red edge indicates

FLIP along corresponding column or row Net Value = -6 -2 x 3 = -6 -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 Red -N2 Edge lengths match n strips Pieces rotate 2

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) N+3

N-2 First you set up the edges (N + 3)(N - 2) First NxN 2 =N

Now you fill in according to the edge lengths (N + 3)(N - 2) Inside 3xN

= 3N Outside N x -2 = -2N Last 3 x -2 = -6

(N + 3)(N - 2) (N + 3)(N - 2) = 2 N - 2N + 3N - 6 = N2 + N - 6 (N + 3)(N - 2) This is an excellent

method for students to use to understand algebraic partial products Solving Equations 2 N + N - 6 = 4N - 8?

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

as adding -4n Solving Equations 2 N + N - 6 = 4N - 8? = Subtract -8

from both sets Solving Equations 2 N + N - 6 = 4N - 8? =0 NV

-6 -(-8) = 2 Solving Equations 2 N + N - 6 = 4N - 8? =0 Students now try to factor by forming a rectangle

Note the constant partial product will always be all black or all red Solving Equations 2 N + N - 6 = 4N - 8? =0

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 Solving Equations 2

N + N - 6 = 4N - 8? =0 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

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

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

edge sets N-1 N-2 Solving Equations 2 N + 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 Minds 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