## Shoulda Woulda Coulda

Posted in algebra, geometry, lessons on November 15th, 2011 38 Comments »

Two things I'd do if I were still doing the job instead of just talking about it:

**Set Up The Expected Value Spinner**

I don't think people who understand expected value understand how hard it is for other people to understand expected value.

Let's say I roll a die. I ask if you want to bet on an even number coming up or a five. You're bright. You pick the even number. It has a 3/6 shot versus a 1/6 shot for the five. But what if I said I'd pay you $150 if the even number comes up and $600 for the five. What if I said I'd keep on giving you that same bet every day for the rest of your life? This is where expected value steps in and puts a number on the *value* of each bet, not its probability. The expected value of the even number bet is (3/6) * $150 or $75. The expected value of the five bet is (1/6) * $600 or $100. The five bet will score you more money over time.

This is tricky to fathom in gambling where superstition rules the day. ("Tails never fails," betting your anniversary on the pick six, blowing on the dice, etc.) So one month before our formal discussion of expected value, I'd print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I'd seal my own bet in an envelope.

I'd ask a new student to spin it every day for a month. We'd tally up the cash at the end of the month as the introduction to our discussion of expected value.

So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?

**BTW**: You could make an argument that a computer simulation of the spinner would be better since you could run it millions of times and all on the same day. My guess is that your simulation would be less convincing and less fun for your students than the daily spin, but you could definitely make that argument.

**Host A Steepest / Shallowest Stairs Competition**

*Tonight's homework*: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.

Your students should then determine whose stairs were the steepest and the shallowest and you'll post those photos at the front of the classroom. You'll make a big fuss over them. Then you'll post a bounty for stairs that will knock them off their perch.

One interesting thing about slope is that it doesn't have a unit, so you don't need a measuring tape or a ruler to calculate it. Anything your students have on hand will work, including their hands.

Be prepared for a contentious discussion about the difference between the *tallest* steps and the *steepest* steps. It's possible to design steps that are extremely shallow but too tall for anyone to climb up. Wrap your students' heads around *that* one.

Be prepared also for students who can't shake the sense that *math is here* every time they climb up a new set of stairs.

What a cool job the rest of y'all have.

[photo credits: moyogo, vulcho]

**2012 Jan 17**: Useful description and modifications from James Cleveland.