My Favorite Lesson

Welcome back to school!

I hope everyone is off to a great start. I had a wonderful first week back and it included my favorite lesson! I was a little disappointed that my favorite lesson fell within the first week but I hope that it is only the spring board for a fantastic year.

The first time I used my favorite lesson it wasn't even a real lesson. It was a warm-up that the students loved and it grew on its own and before I knew it class was over and my students had made an unexpected discovery.

The Set-Up:
I had heard Dan Meyer speak at the Southern California Mathematics Conference and loved the idea of having the students complete problems that allowed for creativity and deeper thinking. One of his suggestions on how to do this was to give them the answer and have them work backwards. This lesson comes the day after my students learn how to calculate determinants.

The Lesson:
On the board I write the first question:
Write a 2x2 matrix with a determinant of 11.
The students work on their own for a bit and then I release them to work together.
Then I put my answer on the board and I ask them questions:
"Did anyone come up with my answer?"
"How many people got a different answer?"
"How many different matrices do you think we could write?"
"Ok now write this one: 2x2 matrix with a determinant of -42"

At this point most students are feeling confident. I ask the class to let me know if anyone is getting stuck and I make way around helping people out and having students show me their answers. They were very excited to share. We had a great time.

"Ok next question! Write me a 3x3 matrix with a determinant of 81"

They get a little nervous but I tell them to work together and they are all over it. After a bit I encourage them to come up with a group answer and then send one person to write the group answer on the board.

I have them look over the answers on the board. Usually there are a few repeats but it is fun to see the variety. The following conversation happens spontaneously, I hear it happen as I walk around the room:
"oh they got what I got"
"I didn't think about that one"
"Um I think you forgot a number, let me help you fix it"

I look over the answers looking for something in particular. I am looking for ones and zeros. I know that if I see ones and zeros we are on track. This week I saw it with just one go at a 3x3 matrix but if I hadn't I would've gone again asking for another 3x3 with maybe a harder number to obtain, like a prime number.

At this point I tell them to get out their TI-83/84 calculators. I walk them through how to calculate the determinant using their calculators. This is normally the first time I am showing them how to calculate matrices on their calculators and that is another reason why I like this lesson. There is usually time to work through the mishaps of figuring out the sequence of correct buttons.

Now that they are comfortable with the calculator. I ask them to write a 5x5 matrix with a determinant of 43. We go through the group process again where I ask for one group member to write their answer on the board. Again I am looking for ones and zeros. Are they there? If yes, I'm good to go for the final push.

The last problem I ask for is an 11x11 with a determinant of -17.9

Their reactions are awesome. If I have timed it right, like this week, they are laughing. Some told me they felt like they were cheating. Meanwhile I have the biggest smile on my face.
This time I ask for one person in the class to write their answer on the board. I ask followup questions to the class making sure we are all on the same page. "Did you guys get something like this?"

Here it is:

-17.9 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 1

Do you see it? I can't tell you how excited I was the first time this happened. I had NO IDEA that having the class work backwards to find determinants of various sizes would lead them to figure out the Identity Matrix.

I wrap up the lesson by telling the class they are not cheating. That the pattern they found is an actual thing and that the reason it feels so easy is because it creates something that is really helpful. This pattern is called the Identity Matrix. The Identity Matrix is made up only of ones and zeros with the ones in a diagonal. The diagonal of ones is called the main diagonal. Next week we will continue seeing how useful the Identity Matrix can be.

They leave this lesson with ownership over the Identity Matrix and confidence with using their calculator for matrices. I love how the discovery process does this. It reminds me of the Interactive Math Program (IMP). I am sad that this lesson has already come to pass but I am determined to include more lessons like this throughout the year. I know that it is moments like this that make math fun and more approachable.