Improv and Math Make Magic

In the year 2000, Chicago's famed Second City put on a new show titled "Second City 4.0". In this show was a sketch created by an amazing writer and comedian, Tami Sagher (@tambone). In the sketch Tami played a teacher proving that the square root of two is irrational. I still remember it to this day and think of it often when I am leading a proof in one of my classes. There are many lessons I take from that one scene that I reflect on. (I only wish I remembered the details of the sketch more)

First, a little background. I left a computer consulting gig in the year 1998. At this time I needed a life change and math education was going to be it. Turns out a brief 6 year stint performing improv in Chicago came along with my new teaching ambition. I trained at Second City and iO and performed at iO Chicago briefly as well as with the Improvaholics. This was an amazing time in my life where there was comedy and education but the two worlds did not cross often. I would teach by day and perform by night (sometimes well into the night). I would come to teach again bleary eyed in the morning and still do a good job. Improv made me a better observer, a better listener, and a better reactor. Certainly important attributes in any classroom. However, the beauty of math and humor of improv rarely intersected. Not until Sagher's sketch.

In this sketch, Sagher is a teacher presenting a proof that the square root of 2 is irrational. The proof is complete and correct - it would be familiar to most of you reading this. However, Sagher endures what most of us deal with on a day-to-day basis - somewhat if not wholly disinterested students. There is a lot of fun with the back and forth between students and teacher and the teacher's determination to deliver the proof with some history behind it.

I love the sketch for many reasons. For one it reminds me that I need to be on my toes with my students. Despite our students' not fully understanding the beauty of math, they should still be shown the beauty. Also, we need to roll with the punches. We shouldn't take ourselves too seriously. This is a fun business. Not many people get to do what we do - think, create, invent, interact, reflect, and impact. Which brings me to my last observation. I teach so many kids whose first passion is not math, but yet these students have talent in math. There are artists, writers, and actors who know their math. To them I say, "Look at what this person did with math. She performed it and put it out there in the medium she was most comfortable with." Math is in places we can't even imagine or in every place we imagine.

On a side note, I contacted Second City to see if I could get the script of the scene. Unfortunately, this is not available. The works at Second City are under lock and key. I get it - especially since this sketch is part of a current show (click the link for a shameless plug because I love improv that much). I wanted to use the proof/sketch in my class to share the art mixed with the math and celebrate the two together. It is my hope that at some point works like this scene can be shared when there seems to be some academic value. I promise I will not abuse the right to use. Please note that the person I contacted at Second City was more than helpful and pleasant.

Grow Dino! Skeptically looking at volumes of similar solids...

A couple quick ideas. This is my first post so hooray for me. Second, I got excited about this lesson because I love advertising claims. (If anyone has a mathura advertisement please send it my way!) The idea of someone crafting a message that will be impressive and sell an object while remaining truthful is intriguing to me. For example, below is a picture of a Comet cleaner I used in my home the other day.

See that big red banner at the top? Impressive, right? Huge letters stating 48% MORE! More than 16.6 oz. Wow - what a claim. Turns out to be true. After a quick calculation I found this 25 oz can to be 48% more than 16.8 oz. I'm not sure why this is important especially considering it still cost more than 16.8 oz. I love the fact that some group of people crafted this and probably did some mock-ups of the red banner. 

This advertising was easy to confirm. But what about another claim...Enter the Grow Dino's. 

In Geometry we have been studying similarity and proportions with area and volume. I decided to introduce the Dino activity before discussing area and volume with my students but after we have discussed similar shapes and indirect measurement.

Here is a website where I bought some Grow Dino's. The site does not sell the same ones pictured anymore, but I think these types of grow things can be purchased from many different sites. Just Google "grow Dino 600%".

The claim is that these dinos will expand 600% in 72 hours.

DAY 1:

 I showed my students the ad and gave them each a dino. My first task was to have students try to articulate what the claim meant.  I was surprised by how difficult this was for my students. Almost all of them just restated the claim that the dino would become 600% bigger in a generic way. I pushed them to describe in what way the dinos would be bigger. Their answers revolved around the length or the height of the dinosaur. One student mentioned the volume of the dino would change by 6 times. No one mentioned surface area at all. I wasn't surprised by this, but I did wonder whether or not I should have brought it up. I withheld and in the end I am glad I did because of the surprise factor we experienced.

After our discussion we then took paper and began measuring. We first took some initial measurements of the dinos' length and then drew the anticipated length to see what 6 times the length looked like. We eventually cut the paper to be a shell for the up-scaled dino. Everyone was pretty skeptical that the toy would grow this much. The students also measured the volume using a graduated cylinder. All this info was entered in a Google spreadsheet by the students. They also named their dinos for a little fun. All dinos were then thrown into a big clear bucket so they could grow.

I liked this brief intro. The paper shell was a big hit because it hinted at how much bigger the dino would have grown in terms of volume (by 216 times if the 600% referenced a change in length). In the future I think I might photocopy each of the dinos on some grid paper. Perhaps the students could then sketch the 6x longer dino on this paper and then we would have a hint as to the impact on area of an object due to the scaling.

DAY 2 & 3:
We checked in with the dinos and measured the length after each day. After day 1 the dinos had grown to about 130% of their original length. Day 2 - about to 200%. This means that the volume had already changed by over 600% but the students didn't know this yet. They now were convinced that the dinos would not grow 600%.  During the rest of class we started looking at the relationship between area and volume of similar objects. I started with the following question that I fabricated just to pique their interest.

The students get a huge kick out of the idea of this coin and whether or not to make the purchase. After this we learned more about this relationship and tried some problems.

Reflecting on this day, I wish I better documented the growth of the dinos each day. I think there are a lot of things we could do with this data. Even to see what the graph of the growth looked like. Is it logistic? Not sure. Next year I will revamp my spreadsheets to better capture this info.

DAY 4:
At this point the students have been working with the idea of similar objects ratio of area and volume is the square and cube of the ratio of linear dimensions. We made some final measurements of length and volume and found out that neither of the ratios (length or volume) fit with the ad's 600% claim. On average the length grew by a factor of 2.32 and the volume grew by a factor of 12.99. Why didn't the advertisers use this 1299% growth for their ad? That's awesome. I asked the students about this and they thought that number while true was simply not believable in an ad. We took our ratios and then did the math to see what the ratios of length, area, and volume would be with our observations. We saw the area ratio was pretty close to 6. Strong evidence that the ad's claim was true and that the claim was referring to a change in surface area. This was a big surprise to us all. Below is the spreadsheet we used to summarize our findings and an image of our dino, Bob.

Overall, we had a great time investigating our grow dinos. The experience was a good companion to our study of areas and volumes of similar objects.

Here's a time lapse video I found on YouTube.
http://youtu.be/2SQewNzwyOc