What does it take to be a good problem solver?

That is our essential question for this unit in Algebra 2. I am still new to essential questions but I like this one. It helps students focus on the problem solving strategies we learn in math class but also gives them a chance to thing about the problems they face outside of class. This question reminds me of one of my favorite mathematical principles, "Make sense of problems and persevere in solving them". I know not everyone likes math or wants to get a degree in math but life is full of problems and hardships. In math class we can practice persevering through problems and making sense of the problem before us without giving up.

The problem solving in this unit is focusing on solving quadratic equations. We are learning several different methods and have started having conversations about why we have different methods, which one is easiest, and are there times when one method is better than another. We have already reviewed solving by factoring and with the quadratic formula. Today we learned solve by completing the square. I have a seven step box strategy that helps students stay organized. For some students this is one of the longest and sometimes ugliest problem they have every face. My goal is that they persevere through the length and feel confident. That definitely happened today! Here is a copy of the notes:

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.

Spring Break is not as exciting as teaching math

It's the day before spring break and I have been counting down to this moment for awhile now.
ITS FINALLY HERE. Spring break is a chance to rest and find new energy.
But then today as I was planning for my lessons for when we come back from spring break I started to get really excited! I found myself thinking..."I wish I was teaching tomorrow". WOW! I've been looking forward to spring break for new energy when all I needed to do was excited about my lesson plans :) Here they are:

AP Calculus: We are are in full prep mode. We are switching everyday between multiple choice and free response. The other day we had "The Worst Day Ever" with is my take on the surprise tests done by Mr. Calculus over at Diamond Bar High School. 20 questions multiple choice pop quiz style. I told them that it was "The Worst Day Ever" and our goal until after the AP Test is that every day will be better. So after spring break we are going to have another surprise test but it is going to be BETTER for several reasons. One, I am actually letting them study for it :) and 2) they know what to expect. Already WAY better than "The Worst Day Ever". The goal of course is that the AP Test will be "The Best Day Ever".

Precalculus: My students definitely needed a break from trig so they've had three weeks of probability but now we are back to wrap up the trig. We are going to be working with Law of Sines and Law of Cosines. From reading other fantastic blogs I am preparing myself for a tough lesson. However, when I was at the CUE conference I was inspired by Joe Corippo to make sure that I am teaching more than what students can google. Students will be given a list of  "googable" things and then as a class we will figure out what to DO with them. (Use appropriate tools strategically #mathpractices)

Algebra 2: Two words. Function Notation. I have only been teaching for four years but so far I have been really disappointed with the level of understanding most students have for function notation. Using the hashtag MTBoS I was able to hunt down some expert advice via Mrs. E's Classroom Blog. We are going to notice and wonder as a class for several examples and create a class definition of function notation BEFORE we start to use it. Again the approach is sure they could google the definition but can they google how to discover, wonder, and notice function notation?

Needless to say I am looking forward to coming back after spring break!
Are you? What are you doing in your class that you are excited about?