For the bulk of today’s lesson students will be taking notes on matrices. Then, students will complete several practice problems. My goal for today is for students to review basic matrix operations with an emphasis on multiplying matrices (N.VM.11) and reviewing the process for matrix operations on the calculator.
I will begin my presentation with the formal definitions on Page 3 of the Flipchart - Basic matric operations. At some point during this time, I will make a show of chanting to students:
ROW BY COLUMN, ROW BY COLUMN, ROW BY COLUMN
I want to make it obvious to students that this is something I expect them to internalize. I say, “If you learn nothing else today, remember ROW BY COLUMN! ROW BY COLUMN!! ROW BY COLUMN!!!”
On pages 4-6 of the Flipchart, there are built in Clicker Questions. I include these to allow students to demonstrate their ability to apply the definitions given on page 3. I will carefully monitor the classroom while students work on writing the identity matrices on page 4 to be sure that students were able to correctly interpret the definition.
Next, I will have my students practice adding matrices. Since I am pretty confident that they will remember how to perform addition of matrices, I let my students go at this without any formal introducton. I believe that adding matrices is very intuitive even for students that have never seen it before. So, by this time in their mathematical careers this should be pretty easy.
I expect that my students will quickly realize that A + B the second example is impossible. When this realization begins to spread, I will asked the class if anyone can state WHY we cannot add Matrix A and Matrix B. I will probably pose the question like this, “How would you convince someone else that we can’t add these?” I will express it in this way to give my students the chance to produce a mathematical explanation (MP3).
On Slide 8 I model how to multiply A*B. With multiplication, I am always concerned about students getting it wrong from the beginning. Matrix multiplication is never as intuitive as matrix addition. It is, of course, a very common misconception for students to want to multiply matrices by following the same methods for addition (multiplying the entry in row 1 column 1 by the entry in the same position on the other matrix). Once students go down that road, it is hard to get them back on track.
During our Brain Dump at the start of class, my students typically mention things like, "I remember that multiplying matrices had ‘crazy rules’" Or, multiplying matrices is harder than it should be.’ So, I advise my class that some of their peers feel this way as a motivation for tuning into my demonstration of multiplying matrices.
My plan is to write out the multiplication work for A*B on the white board. I plan to use a variety of colors to illustrate what rows and columns I was multiplying. I will also talk about the need to identify if two matrices are of suitable dimensions for multiplication, and, how to determine the dimensions of the product matrix.
Once I complete my demonstration, I will ask the students do a quick 3-minute review with their table mates. My students generally benefit when I give them time to review a demo with their peers before we move on to practicing the procedure. Then, I have my students complete the remaining examples in their teams.
The Questions on Page 9 are designed to spark a class discussion that I will lead. I hope that my students will make the observation that when they multiplied by matrix C, the product was the other matrix. I want my students to make a connections to the Identity Property of Multiplication and realize the existence of the Identity Matrix.
At the end of this segment of the class, I will remind my students about how to perform matrix calculations using their Ti calculators. My students are very confident using their calculators, so this is no a major segment in my lesson.
Differentiating the Lesson: If any of my students struggle with multiply matrices, I will encourage them to use different colored pencils to circle the rows and columns. I will also focus their attention on the fact that when we multiply row x (in the first matrix) by column y (in the second matrix), the result goes in the (x,y) position in the answer matrix. Of course, I don’t use x and y. I refer to the actual positions for the rows and columns they are working on. Also, I find it helps to require students to write out all the work, rather than skip intermediate results when multiplying matrices.
To close out today’s lesson students are asked to complete some Clicker Questions (pages 13-15 of the Flipchart) where I ask them to use their calculator to find the inverse. Using the Clickers for this brief assessment will provide me with a quick check of who is able to use their calculator successfully.
For homework this evening I will assign Matrices Homework 1.