MENINGITIS Laurie J Burton, MD PEM Fellows Conference December 6, 2006.
Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model...
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![Page 1: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/1.jpg)
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
![Page 2: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/2.jpg)
These ideas useALGEBRA PIECES
and the MATH IN THE MIND’S EYE curriculum developed at Portland State University
(see handout for access)
![Page 3: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/3.jpg)
What are ALGEBRA PIECES?
The first pieces are BLACK AND RED TILES which model integers:
Black Square = 1 Red Square = -1
![Page 4: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/4.jpg)
INTEGER OPERATIONS Addition
2 + 3
3 black
5 black total = 5
2 black group
![Page 5: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/5.jpg)
INTEGER OPERATIONS Addition
-2 + -3
5 red total = -5
group2 red 3 red
![Page 6: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/6.jpg)
INTEGER OPERATIONS Addition
-2 + 3
3 black
Black/Red pair: Net Value (NV) = 0
Total NV = 1
group2 red
![Page 7: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/7.jpg)
INTEGER OPERATIONSSubtraction
2 - 3
2 black
Take Away??
Still Net Value: 2
3 blackAdd R/B pairs
![Page 8: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/8.jpg)
INTEGER OPERATIONSSubtraction
2 - 3
Net Value: 2
Take away 3 black
2 - 3 = -1
![Page 9: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/9.jpg)
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.
![Page 10: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/10.jpg)
This is excellent for understanding “subtracting a negative is equivalent to
adding a positive.”
![Page 11: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/11.jpg)
INTEGER OPERATIONSMultiplication
2 x 3Edges:
NV 2 & NV 3
![Page 12: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/12.jpg)
Fill in using edge dimensions
INTEGER OPERATIONSMultiplication
2 x 3
Net Value = 62 x 3 = 6
![Page 13: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/13.jpg)
INTEGER OPERATIONSMultiplication
-2 x 3Edges:
NV -2 & NV 3
![Page 14: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/14.jpg)
Fill in with black
INTEGER OPERATIONSMultiplication
-2 x 3
![Page 15: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/15.jpg)
INTEGER OPERATIONSMultiplication
-2 x 3
Net Value = -6-2 x 3 = -6
Red edge indicates FLIP along
corresponding column or row
![Page 16: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/16.jpg)
-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
![Page 17: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/17.jpg)
BEYONDINTEGER OPERATIONS
The next important phase is understanding sequences
and patterns corresponding to a sequence of natural
numbers.
![Page 18: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/18.jpg)
TOOTHPICK PATTERNS
Students learn to abstract using
simple patterns
![Page 19: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/19.jpg)
TOOTHPICK PATTERNS
These “loop diagrams” help
the students see the pattern here is
3n + 1: n = figure #
![Page 20: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/20.jpg)
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
![Page 21: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/21.jpg)
ALGEBRA SQUARES
Black N2
Red -N2
Edge lengths match n stripsPieces rotate
![Page 22: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/22.jpg)
Patterns with Algebra Pieces
Students learn to see the abstract pattern in
sequences such as these
![Page 23: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/23.jpg)
Patterns with Algebra Pieces
N (N +1)2 -4
![Page 24: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/24.jpg)
Working with Algebra PiecesMultiplying
(N + 3)(N - 2)
First you set up the edges
N + 3 N - 2
![Page 25: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/25.jpg)
(N + 3)(N - 2)
Now you fill in according to
the edge lengths
FirstN x N = N2
![Page 26: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/26.jpg)
(N + 3)(N - 2)
Inside3 x N = 3N
OutsideN x -2 = -2N
Last 3 x -2 = -6
![Page 27: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/27.jpg)
(N + 3)(N - 2)
(N + 3)(N - 2) = N2 - 2N + 3N - 6
= N2 + N - 6
![Page 28: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/28.jpg)
(N + 3)(N - 2)
This is an excellent method for students to use to understand
algebraic partial products
![Page 29: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/29.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
=
![Page 30: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/30.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
=
Subtract 4N from both sets: same as adding
-4n
![Page 31: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/31.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
=
Subtract -8 from both
sets
![Page 32: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/32.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
= 0
NV -6 -(-8) = 2
![Page 33: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/33.jpg)
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
![Page 34: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/34.jpg)
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
![Page 35: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/35.jpg)
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
![Page 36: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/36.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
Form a rectangle that makes sense
= 0
![Page 37: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/37.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
Lay in edge pieces
= 0
![Page 38: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/38.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
Measure the edge sets
= 0
N - 1N - 2
![Page 39: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/39.jpg)
Solving Equations N2 + N - 6 = 4N - 8?
= 0
(N - 2)(N - 1) = 0 (N - 2) = 0, N = 2
or (N - 1) = 0, N = 1
![Page 40: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/40.jpg)
This last example; using natural number domain for the solutions, was clearly
contrived.
![Page 41: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/41.jpg)
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.
![Page 42: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/42.jpg)
Materials
Math in the Mind’s Eye Lesson Plans:
Math Learning Center
Burton: SabbaticalClassroom use modules
![Page 43: Northwest Two Year College Mathematics Conference 2006 Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie Burton.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d7a5503460f94a5dc8d/html5/thumbnails/43.jpg)
Packets for today:“Advanced Practice”
Integer work stands alone
Algebraic work; quality exploration provides solid
foundation