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, n 2 and 2n + n 2

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 Multiplication I will run briefly through the main ideas in *MME VI - 3: Counting Piece Arrays *MME VI - 4: Multiplication and Division of Integers 4For reference, the complete MME lessons are already in your notebook under: College Algebra-Materials

Counting Piece Array Set up Materials äBlack & 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:  2 x 3  2 x -3  -2 x 3  -2 x -3

FIRST ACTIVITY Materials äToothpicks äToothpick Squares Activity Sheet (filed under College Algebra-Materials) Today we’ll work in teams on the (abbreviated) worksheet version of the toothpick square lesson 4For 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 4For reference, the complete MME lesson is already in your notebook under: College Algebra-Materials

Tile Patterns, Part I Materials äComplete Algebra Piece Sets äIX-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 äComplete Algebra Piece Sets äActivity 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 äComplete Algebra Piece Sets äIX-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 4All 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  …??? 4Materials 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.