I WANT to teach vectors!

I haven’t looked at enough math teacher blogs or talked to enough math teacher friends to know for sure, but I am assuming I’m weird for this one. I was doing some long term planning for my first go at Precalculus, and realizing that I get to teach about vectors was really REALLY exciting. I’m not particularly sure why. I do have some ideas that I think are pretty fun on how to hook students into the math, but I also know that they can be super practical when applied in physics or maybe other subjects.

I want to start off by revealing a slide that looks something like this.VectorIntro

And have them marinate in the craziness for a minute. Then explain that this can be true if we are talking about vectors, which have a magnitude and a direction, so on and so forth.

 

Also, I’m kind of hoping for a windy day, so that I can take my students outside and have a student throw a beach ball one direction, and have the wind act as another vector adding to the vector representing the velocity and direction of the throw. I guess if necessary it can be in the hallway with a big box fan providing the wind. Any thoughts on how to raise the rigor on these ideas so I don’t just jump into boring practice after that?

Story in the (Caffeinated) Data

Inspired by Dan Meyer‘s 2012 Annual Report, I decided to start tracking data on myself in February. Thanks to the Keep Track Pro android app on my phone, I have been tallying each and every coffee I drank since February 18. I’ve had a lot of fun self-analyzing through this activity and am trying to figure out a way to get students doing something similar in AP Statistics this year.

On to some graphs!

Distribution of Counts of Coffees in a Day

Distribution of Counts of Coffees in a Day

 

First, I decided to do a bar chart on the counts of coffees I drank in a day. I’d never gone above three cups in a day, but somehow survived the 18 days I had no coffee.

With this preliminary information in mind, I decided to think about the Law of Large Numbers and a moving average. The basics of the theorem suggest that over time, the average of some experimental outcome will approach the “true” value. I wanted to infer what my “true” average coffee intake per day was. Using some Microsoft Excel formula wizardry, I calculated an average after each day of the tracker, and decided to plot this cumulative average over time.

Cumulative Average of Coffees per Day

 

I noticed an interesting pattern here. As the Law of Large Numbers suggested, the cumulative average began to settle by May at around 1 coffee per day. Thinking about my routine during the school year, I would normally have a coffee at home before work, and every once in a while, maybe one or two times a week I would have another after work at home. Then I noticed that the graph became unsettled again after school ended and revealed an upward trend. At this point, I wanted to examine in a little more detail the reasons for the instability. Instead of looking at the cumulative average, more Excel wizardry enabled me to look at a running average, only looking at the prior 14 day period. This displayed a higher resolution of my caffeination habits.

Cumulative average (blue) and 14 day running average (red)

Cumulative average (blue) and 14 day running average (red)

I noticed a few regions of interest in this graph. First, around late March to early April, the running average reveals a drop in my drinking habits. I roast my own coffee, and I bet this period corresponds to a time when I had forgotten to re-order beans to roast and had to wait for them to come in. Secondly, I notice that about 2 weeks after school ends, my moving average starts to increase drastically. The red trendline begins to settle around 2 cups per day. This corresponds to what happens when I am at home without much structure. I either make several cups of coffee during the day or hang out at the coffee shop. The instability in the cumulative average (blue) is understandable based on understanding the running average (red).