## Geometry: Which Pasta Does Mueller’s Want You to Buy?

So, I’m walking through Publix about a week or two ago, and I noticed this:

Mueller’s makes both regular length and pot-sized pasta.  Since I’m giving a presentation next week involving problem based instruction in the geometry classroom, I saw this as an opportunity excuse to develop a new geometry lesson (I really miss teaching geometry sometimes).  So, I started talking to my co-presenter and asked her to help me develop something useful for us to present with.

My co-presenter is a first year teacher and I quickly found out that she’s not very familiar or comfortable with problem based instruction.  Since she’s my mentee, I thought it would be a good idea to develop a lesson with her and then team teach in her classroom.

So we started with the question: Which would Mueller’s rather you purchase?

Her student’s sat there and started at me for a few moments.  I encouraged them to talk with each other and figure out how they could answer the question.  Two minutes later, we discussed what they wanted to compare and what questions they had for us:

They were insightful.  The only thing they didn’t ask about (that I was expecting) was a question about shipping.  My mentee and I quickly steered the conversation toward packaging costs.  So, her students wanted some measurements:

I made up the cardboard cost…maybe I can get Mueller’s to give me that info.

Given that this lesson was given during a point in the curriculum that has nothing to do with surface area, they did okay with calculations.  I was surprised to see that 3 or 4 out of the 8 groups thought that volume was the appropriate measurement to use (rather than going straight to surface area).  After some quick conversations, most were right on track.  Ultimately, the most difficult calculation was converting the cost from square inches to square centimeters.

The calculations bear out that the pot sized pasta is cheaper to package.

I’m encouraged that my mentee is interested in continuing to create lessons like these.  I’m taken back a bit that this isn’t the norm in most classrooms.  I have to find a way to keep encouraging my colleagues to keep bettering themselves and their students.

## AP Statistics: Relations Between Categorical Data

I was rehashing some of my old files and planning a lesson to introduce two way tables to my students.  I’ve always had trouble getting them to buy into the meaning/purpose/importance of two way tables.  And looking back through my old lessons, I figured out why:  I was focused solely on the two way tables…not the relationship between two categorical variables.  I decided to change that this year.

This was my students’ warm up when they walked into the room.  There was not an explanation given…just for them to take 60 seconds to fill this out.  Then I went into story mode:

We discussed the beginnings of why boys “like” blue and girls “like” pink and why the idea switched from the original boys should like pink and girls should like blue.  It’s an interesting story and my students had good insights from their history classes.

So I told my students to look at the relationship between their gender and their favorite color.  There wasn’t any other direction given.  So they decided to collect their data:

I asked them to discuss any trends that they saw.  There was a bunch of statements that didn’t say a whole lot.  “Some males like blue” or “One girl likes green.”  Nothing of consequence.  Finally, someone said that they had a better way to organize the data:

Now we were getting somewhere…unfortunately, today was PSAT/Blood Drive/Some Chorus Thing day and I was missing a lot of students.  They weren’t seeing any useful trends.

So, we looked at data I knew had some trends:

Students gathered some great insights into how society views education through their look at marginal and conditional distributions.  Some interesting discussion ensued.

Most importantly, the two way table served to enhance our look at relationships between categorical variables.  We actually talked about the nature of statistics (I don’t know if that’s a real term…if it isn’t, I’m coining it now).  I’m curious to see if this approach pays off with two way tables.

## AP Statistics: More Extrapolation

After Friday, I didn’t feel like my students had a firm grasp on the definition of extrapolation and its (possible) consequences. In case you forgot:

So, I was messing around on google, looking for some good examples of extrapolation, and I found this perplexing graph:

We started with an explanation of the stock market, the NASDAQ, and the possible implications of each of them. We also discussed what could happen with inaccurate predictions in this context.

So, I decided to cut the graph back to the original data, and have my students extrapolate and predict the NASDAQ at the end of 2003.

My biggest mistake was giving them the data with the graph. Most of my students wanted to use their calculator and come up with the best prediction model using logarithms. I’m not upset that they decided to go there…I just wanted them to give a quick prediction based off of the trend they could see. Eventually, they got here:

…and we looked at the predictions of the entire class (in which all of them followed the same basic trends):

Finally, we looked at the actual value:

In this case, they were wrong. By a lot. And we discussed why. I’m must hoping that this helps emphasize what extrapolation is and its implications…

## AP Statistics: Introducing Extrapolation

Okay, this was the last slide that my classes saw. I wanted you to have a working knowledge of what extrapolation means in statistics.

After we finished our quiz on non-linear regression modeling, I wanted a 15 minute lesson introducing extrapolation. I didn’t want my students to hear me Peanuts teacher a definition and give them several examples that don’t mean anything to them. I’ve tried this…they forget the term approximately 4.87 seconds after they leave my class.

So Coach Helms isn’t the happiest or most photogenic person to use here, but he’s conveniently next door to my room and the kids love him (not to mention he’s a pretty good math teacher).

To answer the question, my students wanted to build a model that relates the age of a person and their height. It was about this time that some people started to have some problems; they started trying to argue that type of model wouldn’t help us. I muttered some quick response (pretending that I didn’t believe them), and we trudged on through some data:

So, the data wasn’t really that creepy in class. I went around our elementary school measuring some students. Real data from real people. I just can’t show you their likeness on a public forum like this; hence why you have creepy smiley faced kids.

I asked my students to use this data to create a prediction model for age vs height. Then I asked them to use that model to predict the heights of a 16, 30 and 50 year old person. More students started to argue that we couldn’t use our model for the 30 and 50 year old…I made them do it any way:

They decided that their model was a good predictor of values (they made the argument using the correlation coefficient), but they didn’t believe their predictions for the 30 and 50 year old.

Then we looked at the answers:

As it turns out, their prediction for the 16 year old was pretty accurate, but, as they suspected, it wasn’t for the 30 and 50 year old.

Now is when we defined extrapolation:

We had a nice five minute discussion of what extrapolation is, when it’s used, and whether it’s good or bad (or both). We’ll find out Monday how well it sticks.

## AP Statistics: Introduction to Non-Linear Regression

I’ve been struggling with an interesting way to introduce non-linear regression in AP Statistics. I wanted a quick situation to put my students in to give a reason that non-linear regression is necessary to learn. Then, over the summer, I had an idea while playing video games.

So I nerded out today in class:

Skyrim is a role playing game (and I love good RPGs set in medieval times). Here’s a brief rundown if you need it:

http://en.wikipedia.org/wiki/The_Elder_Scrolls_V:_Skyrim

While playing Skyrim, I noticed that smithing was not leveling up linearly. So I had the brilliant idea to gather some data for my students to explore. I gave my students four pieces of data and asked them how many iron daggers I would need to smith to reach level 100 in smithing.

Here’s what I mean by smithing: http://www.uesp.net/wiki/Skyrim:Smithing

I didn’t inform my students that the data isn’t linear. I didn’t tell them how to approach the problem. I just gave them data and six minutes to make a prediction:

The conversations I heard from my students gave me great insight into their understanding of linear regression. 99% of my students were able to build a least squares regression model and predict that it would take me 516 daggers to reach level 100. There were, however, a few students in each class that looked at the scatter plot and residual plot and decided that their prediction wasn’t accurate. Even with a correlation of r = 0.99, they knew there had a be a better fit.

After a brief class discussion, I informed them that it would take in the neighborhood of 550-560 daggers to reach a level 100 and then showed them a better, non-linear model that gives a more accurate prediction.

The purpose of this activity was to get conversation started about non-linear modeling, and give a context where it could be useful. This activity was meant only as an introduction…now we get to spend three days with logarithms to actually find those models that best fit the data!! Woo! (I’m excited)