Teaching Geometry - Expected Value

I am teaching a new prep this year - Accelerated Geometry, along with Acc Alg 1 and Acc Alg 2.  It is a lot.  I have taught a lower level Geometry course, but a while ago.  I have a colleague who also teaches it, so I have the quizzes (I make different versions). I have the homework.  (I still do each problem).  I have the Do Nows, but have to do them.  And, then of course there is the presentation - the lessons, those are mine and how I run the class.  I have the what and when, but I get to create the how and I enjoy the creation part.

This is where expected value comes in.  I have not taught it before.  I'm not even sure I learned it before.  So, I immersed myself in it. I went down the EV rabbit hole, reaching, researching, reading definitions, real life uses for it, formulas, worksheets, examples.  We have a target example in the book I knew I was going to use but I thought it might not be a good starting point. 



It has the whole area of a circle and the probability of hitting each ring.  I was teaching probability so it should be okay, but I didn't want them to enter at that point and get hung up.  I am all for starting with a challenging problem, but today's lesson was not it.

I also am doing a lot of VNPS - having the kids work at the whiteboards to discover Geometry.  I knew I would be teaching this lesson on the first day back from winter break, so I wanted to have them up and working and discussing, but I just couldn't do it.  I didn't want to force it.  It just didn't fit with that kind of lesson and that is okay.  I felt this topic had to be a whole class, step by step procedure.  That doesn't mean it had to be boring.  I just wanted to slow it down a little and take it one piece at a time.  I also decided to start at a carnival with a card game.  A standard deck of cards because we were familiar with the probability in a deck of cards.

I started like this:



I got them interested.  Some had questions - what do we win?  (I was going for 3Act style, not giving away all the information.)

I gave them some more information and polled the class.  We had some gamblers ready to play.



It went pretty smoothly and seemed to make sense.  I built up to the target.  Now they have seen the idea of EV.  I created a table to organize my work.  I had to consider all possibilities - winning and losing.  And, then it is about multiplying the "prize" by the probability.  Not so bad.  We ended up with an EV of -0.73 and we talked about what that meant.  Did they see how the carnival could rig their games?  Yes, they did.

When I introduced the target problem, I started by saying we couldn't just jump straight to the EV chart.  We had to figure out the probabilities first and that took some time, but again,  they were okay with it. We left things in terms of pi and conveniently pi/pi simplified to 1.  We walked through the problem step by step and they found success.  Then we did a couple more on their own for practice.  I liked the way the lesson went and was happy with it. 

Here is the slideshow if you are interested.