## Testing expected values using Geogebra

I was intrigued last night looking at Dan Meyer’s blog post about the power of video to clearly define a problem in a way that a static image does not. I loved the simple idea that his video provoked in me – when does one switch from betting on blue vs. purple? This gets at the idea of expected value in a really nice and elegant way. When the discussion turned to interactivity, Geogebra was the clear choice.

I created this simple sketch (downloadable here)as a demonstration that this could easily be turned into an interactive task with some cool opportunities for collecting data from classes. I found myself explaining the task in a slightly different way to the first couple students I showed this to, so I decided to just show Dan’s video to everyone and take my own variable out of the experiment. After doing this with the Algebra 2 (10th grade) group, I did it again later with Geometry (9th) and a Calculus student that happened to be around before lunch.

The results were staggering.

Each colored point represents a single student’s choice for when they would no longer choose blue. Why they chose these was initially beyond me. The general ability level of these groups is pretty strong. After a while of thinking and chatting with students, I realized the following:

• Since the math level of the groups were fairly strong, there had to be something about the way the question was posed that was throwing them off. I got it, but something was off for them.
• The questions the students were asking were all about winning or losing. For example, if they chose purple, but the spinner landed on blue, what would happen? The assumption they had in their heads was that they would either get \$200 or nothing. Of course they would choose to wait until there was a better than 50:50 chance before switching to purple. The part about maximizing the winnings wasn’t what they understood from the task.
• When I modified the language in the sketch to say when do you ‘choose’ purple instead of ‘bet’ on the \$200  between the Algebra 2 group and the Geometry group, there wasn’t a significant change in the results. They still tended to choose percentages that were close to the 50:50 range.

I made an updated sketch that allowed students to do just that, available here in my Geogebra repository. It lets the user choose the moment for switching, simulates 500 spins, and shows the amount earned if the person stuck to either color. I tried it out on an unsuspecting student that stayed after school for some help, one of the ones that had done the task earlier.

Over the course of working with the sketch, the thing he started looking for was not when the best point to switch was, but when the switch point resulted in no difference in the amount of money earned in the long run by spinning 500 times. This, after all, was why when both winning amounts were \$100, there was no difference in choosing blue or purple. This is the idea of expected value – when are the two expected values equal? When posed this way, the student was quickly able to make a fairly good guess, even when I changed the amount of the winnings for each color using the sketch.

I’m thinking of doing this again as a quick quiz with colleagues tomorrow to see what the difference is between adults and the students given the same choice. The thing is, probably because I am a math teacher, I knew exactly what Dan was getting at when I watched the video myself – this is why I was so jazzed by the problem. I saw this as an expected value problem though.

The students had no such biases – in fact, they had more realistic ones that reflect their life experiences. This is the challenge we all face designing learning activities for the classroom. We can try our best to come up with engaging, interesting activities (and engagement was not the issue – they were into the idea) but we never know exactly how they will respond. That’s part of the excitement of the job, no?

## From projectile motion to orbits using Geogebra

I was inspired last night while watching the launch of the Mars Science Laboratory that instead of doing banked curve problems (which are cool, but take a considerable investment of algebra to get into) we would move on to investigating gravity.

The thing that took me a long time to wrap my head around when I first studied physics in high school was how a projectile really could end up orbiting the Earth. The famous Newton drawing of the cannon with successively higher launch velocities made sense. I just couldn’t picture what the transition looked like. Parabolas and circles (and ellipses for that matter) are fundamentally different shapes, and at the time the fact that they were all conic sections was too abstract of a concept for me. Eventually I just accepted that if you shoot a projectile fast enough tangentially to the surface of the Earth, it would never land, but I wanted to see it.

Fast forward to this afternoon and my old friend Geogebra. There had to be a way to give my physics students a chance to play with this and perhaps discover the concept of orbits without my telling them about it first.