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College Algebra for Teachers
Laurie BurtonWestern Oregon UniversityMAA PREP Active Learning WorkshopJuly 7, 2003“Algebra Theme” Day
My “College Algebra” Course
I teach a course, College Algebra for Teachers, where we use the black and red tiles and algebra pieces you received to explore algebra in a visual and meaningful way.
The students learn to use the pieces and then work their way through a variety of visual activities that connect to symbolic use.
Student Path Students model integers and
integer operations.
Students see & describe patterns both verbally and symbolically.
Students “see” variables and attach meaning to abstractions such as n, 2n, n2 and 2n + n2
Student Path
Students describe variables and work with variables in expressions - visually, verbally and symbolically
Students work with variables in equations- visually, verbally and symbolically
Critical Ideas
For me, it is critically important that students understand the ideas behind the algebra visually and verbally (for this the algebra piece models are useful) and that they connect these ideas, symbolically to the pencil and paper algorithms that one usually uses to perform algebraic tasks.
Materials
I use the Math in the Mind's Eye Lessons, developed at Portland State University and that are now published by the Math Learning Center.
MME has an "algebra collection" which consists of five units.
The Five MME Algebra Units
Modeling Integers (operations)
Picturing Algebra
Graphing Algebraic Relationships
Modeling Real and Complex Numbers
Sketching solutions to Algebraic Equations
Algebra Units used in Course
In my undergraduate class we only complete the first 3 units. This gets the students well into understanding how to determine solutions to factorable quadratic equations.
With the graduate students this summer (Maria and I are team teaching an MS in ED content course for MS teachers) we should complete all five units.
Goals for our Morning Activities
You have already looked at adding and subtracting integers.
I'll show you, quickly, how multiplication of integers works in the MME lessons.
Goals for our Morning Activities
We will work through Toothpick & Tile Pattern activities to give you a “flavor” for the MME materials.
You will design class activities using a MME lesson as the focus.
Arrays & Integer MultiplicationI will run briefly through the main ideas in
MME VI - 3: Counting Piece ArraysMME VI - 4: Multiplication and Division of Integers
For reference, the complete MME lessons are already in your notebook under: College Algebra-Materials
Counting Piece Array Set up
MaterialsBlack & Red Tiles
Standard matrix conventions: Rows are horizontalColumns are vertical
ARRAY NET VALUESThe net value of an array is the net value of the collection of pieces in the array
Using Counting Piece Arrays
Initially, students start with an all black array and explore flipping rows and columns to see how arrays work.
Students need to particularly note what happens when they flip first a row then a column or a column then a row and combinations of these operations.
Students quickly see the flips are hard to track on arrays.
Array Edge Pieces
Edge pieces are introduced
Edge sets have the same “values” as a regular collection. 3B edges = +3, etc.
Working with Edge Pieces
Students spend time exploring working with arrays using edge pieces.
In particular they notice that an array has net value 0 when at least one edge set has an even number of black and red pieces
This helps the students see the usefulness of using minimal edge sets.
Using Arrays to Multiply
Examples using minimal edge
set collections: 2 x 3 2 x -3 -2 x 3 -2 x -3
FIRST ACTIVITYMaterialsToothpicksToothpick Squares Activity Sheet(filed under College Algebra-Materials)
Today we’ll work in teams on the (abbreviated) worksheet version of the toothpick square lesson
For reference, the complete MME lesson is already in your notebook under: College Algebra-Materials
SECOND ACTIVITY
Tile Patterns, Part I from MME IX-2, Picturing Algebra
We’ll all work on this together
For reference, the complete MME lesson is already in your notebook under: College Algebra-Materials
Tile Patterns, Part IMaterialsComplete Algebra Piece SetsIX-Master 1
Refer to the overhead and your copy of Master 1
Use black and red tiles to determine the next figure
Numerical Computations
Circle/Mark 3 different NUMERICAL ways to count these tiles.
Extend your ideas to the 20th figure. How many tiles are there?
Use all 3 methods to compute.
Verbal Methods
Convert your generalized numerical methods to WORDS.
Check all 3 of your methods on the 20th figure.
Symbolic Methods
Convert your numerical and word methods to SYMBOLS.
Check all 3 of your methods on the 20th figure.
A Glimpse...MaterialsComplete Algebra Piece SetsActivity Sheet IX-2-B
Consider Activity Sheet IX-2-B
Let’s work through it together
What about those White Pieces?
The white pieces are “simple.”
They are called: n-frames
n-frames are used to model sequences with n an integer vs. n a whole number.
The “o” side is opposite n or ”-n,” the other side is “n”.
n-Frame ExampleMaterialsComplete Algebra Piece SetsIX-6 Master 1
Consider IX-6 Master 1
If we think of this as an extended sequence, continuing infinitely in both directions, we can index the sequence positions using n an integer.
What is the nth tile arrangement?
VIDEO VIGNETTE
COLLEGE ALGEBRAFALL 2002
Participants Design Class Activity
Focus Topic Choices MME IX-3 MME IX-4 Natural pair MME IX-5 MME IX-6 Natural pair MME IX-7
All in your notebook under: College Algebra-Materials. This completes the “Picturing Algebra” MME unit.
Activity “Design” Ideas
Work on lesson choices and jot list of class ideas.
Create formal lesson plan focused on MME lesson
Create formal worksheet or worksheet/lesson plan ideas focused on MME lesson
or … ???
Activity Share Options Write on overheads Use a computer and project work Use a computer and print overhead Scan written or printed materials (project,
print overheads or handouts) Create a poster to shareor …???
Materials found at material station
Sharing Activities
1:00 - 1:45 p.m.
Please work in 3 or 4 groups with 3 or 4 participants in each group.