Northwest Two Year CollegeMathematics Conference 2006

Using Visual Algebra Pieces to Model Algebraic Expressions

and Solve Equations

Dr. Laurie BurtonMathematics Department

Western Oregon Universitywww.wou.edu/~burtonl

These ideas useALGEBRA 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 = 1 Red Square = -1

INTEGER OPERATIONS Addition

2 + 3

3 black

5 black total = 5

2 black group

INTEGER OPERATIONS Addition

-2 + -3

5 red total = -5

group2 red 3 red

INTEGER OPERATIONS Addition

-2 + 3

3 black

Black/Red pair: Net Value (NV) = 0

Total NV = 1

group2 red

INTEGER OPERATIONSSubtraction

2 - 3

2 black

Take Away??

Still Net Value: 2

3 blackAdd R/B pairs

INTEGER OPERATIONSSubtraction

2 - 3

Net Value: 2

Take away 3 black

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 OPERATIONSMultiplication

2 x 3Edges:

NV 2 & NV 3

Fill in using edge dimensions

INTEGER OPERATIONSMultiplication

2 x 3

Net Value = 62 x 3 = 6

INTEGER OPERATIONSMultiplication

-2 x 3Edges:

NV -2 & NV 3

Fill in with black

INTEGER OPERATIONSMultiplication

-2 x 3

INTEGER OPERATIONSMultiplication

-2 x 3

Net Value = -6-2 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

BEYONDINTEGER 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 ≥ 0Edge N

Red -N, -N < 0Edge -N

Pieces rotate

ALGEBRA SQUARES

Black N2

Red -N2

Edge lengths match n stripsPieces 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 PiecesMultiplying

(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

FirstN x N = N2

(N + 3)(N - 2)

Inside3 x N = 3N

OutsideN x -2 = -2N

Last 3 x -2 = -6

(N + 3)(N - 2)

(N + 3)(N - 2) = N2 - 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 N2 + N - 6 = 4N - 8?

=

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

=

Subtract 4N from both sets: same as adding

-4n

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

=

Subtract -8 from both

sets

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

= 0

NV -6 -(-8) = 2

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

Students now try to factor by forming a

rectangleNote the constant partial product will always be all black

or all red

= 0

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

Form a rectangle that makes sense

= 0

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

Lay in edge pieces

= 0

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

Measure the edge sets

= 0

N - 1N - 2

Solving Equations N2 + 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: SabbaticalClassroom use modules

Packets for today:“Advanced Practice”

Integer work stands alone

Algebraic work; quality exploration provides solid

foundation