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

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 MINDS 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
INTEGER OPERATIONS Addition
2 3
5 black total 5
5
INTEGER OPERATIONS Addition
-2 -3
5 red total -5
6
INTEGER OPERATIONS Addition
-2 3
Black/Red pair Net Value (NV) 0Total NV 1
7
INTEGER OPERATIONSSubtraction
2 - 3
Take Away??
Still Net Value 2
8
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
INTEGER OPERATIONSMultiplication
2 x 3
12
INTEGER OPERATIONSMultiplication
2 x 3
Net Value 62 x 3 6
13
INTEGER OPERATIONSMultiplication
-2 x 3
14
INTEGER OPERATIONSMultiplication
-2 x 3
15
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 lt 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
Patterns with Algebra Pieces
24
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
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 NV0
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 NV0
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
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 Minds Eye Lesson PlansMath
Learning Center
Burton SabbaticalClassroom use modules
43
Packets for todayAdvanced Practice
Integer work stands alone
Algebraic work quality exploration provides
solid foundation