College Algebra for Teachers
Laurie Burton
Western Oregon University
MAA PREP
Active Learning Workshop
July 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
Equations
solutions
to
Algebraic
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 Multiplication
I will run briefly through the main ideas
in
MME VI - 3: Counting Piece Arrays
MME 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
Materials
Black & Red Tiles
Standard matrix conventions:
Rows are horizontal
Columns are vertical
ARRAY NET VALUES
The 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:

2x 3

2 x -3
 -2 x 3
 -2 x -3
FIRST ACTIVITY
Materials
Toothpicks
Toothpick 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 I
Materials
Complete Algebra Piece Sets
IX-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...
Materials
 Complete Algebra Piece Sets
 Activity 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 Example
Materials
 Complete Algebra Piece Sets
 IX-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 ALGEBRA
FALL 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 share
or
…???
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.