# Raising the Bar: Modeling Part One

A great feature of following and reading the work of great math teachers is you don't have to reinvent the wheel. Though:

Nowak retweeted it, so I think I'm on to something.

Last year I had an idea: we have a quadratics unit in Pre Cal, what can we do to spice it up a little? In a fit of 11th hour inspiration (a tactic I got really good at last year), I developed an activity.  A year later, I thought about some problems:

• Kids struggled with the math component, many were horribly confused by a simple Algebra 1 task of solving y = a (x - h)^2 + k when y, x, h, and k are known
• If they understood the math, they may not have understood the meaning of what they did
• Technology was useful and yet limiting: making their own tennis ball toss videos was neat, but 6 iPads for 30 kids sucked, most never had a hand in analyzing what they made

I developed some answers to this on my own, and a big help came down on the wings of angels when I first peeped Penny Circle at TMC13. Conveniently, the class deployable version was ready a couple weeks before I needed it. I used a combination of things to zip up and down the abstraction ladder to help make tennis ball tossing a more in-depth activity.

### Curiosity

In the morning I taped paper circles of random sizes to the tables. Towards the end of class I handed out real pennies and had them fill in their circle. I wrote down the data they collected and then proposed a question: alright, how many do you think fit in a 12" circle?

Each table had to submit a guess. I gave them a few minutes to discuss. It was fascinating to listen to. One girl pulled pennies out of her purse. Others reached for calculators and started to guess at how the problem scaled. Some eye balled it from the picture. Several asked for the diameter of a single penny (which I withheld) to try and scale from there.  An unknown side effect, most were looking for linear ways because the circles I used were so close in size. Guesses ranged from the 60s to the 200s. There seemed to be different prevailing strategies in each class. Interesting observation was the number who were willing to trust the arbitrary scaling method they determined and verified with the calculator over just looking at the picture for a minute. Calculators are smart, don't you know?

Then the torture part: no answers until tomorrow.

### Data Collection

The next day, groups were given the opportunity to revise their guesses if they wanted. One student said he fiddled with the problem at work, using a 12" pizza pan and change from the register.

Still not giving any answers, I had them fire up the Penny Circle. I used 20 iPads per class and a unique code for each class. Incredibly minor technology issues (one errant iPad needed its WiFi reset), and no chokes on the Desmos end.

Beautiful silence during this part. The bar was low, the data we needed easy to get. I had to nudge a few when it came to the modeling portion, but none found the challenge scary at all. At the conclusion, some awards were handed out for best initial guess and best model.

### OMG Future

The best part is how well the back end works. I had the teacher view of Penny Circle up and running and it kept pace with the whole room all day. To answer a question about teacher dashboards: no, I never had the time to check it while the students worked. It only became useful AFTER the legwork was done and we started our discussion.

These views let me generate a few questions: what's a risk of modeling? what if you aren't careful about the parameters of the model you choose? how does this resemble weather reports about hurricanes?

And then the biggie: now wait a second, we built a model for a 22" circle, right? Can this help us solve our 12" circle problem? Many pointed to the presence of 12 along the x-axis to tell me "there, it should be right there!" I pulled the equation of the class' best model and said, well can this tell me an exact prediction? "Yeah, just find f(12)!" (some really did say it that way, I was so excited) The student generated models gave answers in the 196-205 range. While we were talking about this, a group was instructed to fill the 12" circle for real:

The result? A number between 205 - 230. Imagine that! This math stuff might actually work. Remember the guesses I took the day before? As we were having the discussion and I revealed the model, you could see the expressions of sadness as various tables realized how inaccurately they guessed.

### Going Further

What do all these pennies have to do with tennis balls? Penny Circle helped introduce a vertex form quadratic. The day after Penny Circle we analyzed the basketball videos, introduced the vocabulary of vertex, minimum, and maximum. At this point I introduced the math workload. We calculated the min/max, how to determine whether a point is a min or max, how a standard form quadratic can become vertex form and finally, how to analyze a random quadratic graph to create a vertex form equation.  And then, how to analyze a basketball video to do the same.

Next time in Part Two: Build a Better Tennis Ball Video

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