Snacking on Statistics and Variability

One of my goals this year in Algebra 2 has been to include more discrete math, statistics, and probability when I can. I’ve been convinced by all sorts of smart people that as traditional as it may be to have Calculus as the ultimate goal for math students, statistics and probability are the math that people are more likely to need to use. It compels me to include it in my courses as more than a separate unit.

As if I didn’t need another reason, we are also in a spell of reviewing properties of radicals, and it’s refreshing to get my students thinking differently after a period of simplifying, multiplying, and rationalizing.

I gave them the following scenario:

  • Imagine yourself in twenty years – you are, of course, rich and famous. You are hiring someone to fly your personal jet. your last pilot fell asleep on the job, though he was luckily parked at the gate when it happened.

Two pilots have applied for the position, both equally qualified as pilots. In order to help you make your decision (and avoid the previous situation), you have asked them to keep track of how many hours of sleep they get over a two week period before the interview.

Two weeks later, they return to you with the following data:

  • What differences do you notice about the two pilots?
  • What calculations would you make to describe any quantitative differences between them?
  • Which one would you hire? Why?

Note: This data is completely made up. My new semi-obsession is in using normal distributions to mess up clean functions and force my students (in physics and math) to deal with messy data.

The students almost immediately started calculating means – exactly what I would have expected them to do given what they have been taught to do when faced with a table of data like this. Some did so manually, others used the Geogebra file that generated the data to make their calculation.

The results were fairly consistent – everyone chose the second pilot. When asked why, they said the pilot gets more sleep on average, and so would be the better choice.

When I asked who was more consistent in their sleep, they were easily able to identify the first pilot. When asked why, many had explanations that correctly danced around how most of the data was closer to the average. No students really brought up this fact before I asked though, which leads me to believe they observed one of two things:

  • The importance of the  consistency doesn’t really matter given the difference in the means for how much sleep the pilots got.
  • They didn’t think to look at consistency at all.

Some other interesting tidbits:

  • None of the students thought to construct a histogram to look at the data. When asked, about half of the class said they knew how to construct a histogram. I didn’t dig any deeper to flesh this out. I was going to throw one together in Geogebra, but decided that might be something we should look at with more time available.
  • Half of the class that is taking AP Psychology didn’t think about finding standard deviation. Again, I didn’t dig any deeper to find if this was because they didn’t know that it might apply here, or because they thought the values of the means were more important.

There is plenty here to generate discussion, but the one thing I wonder about is if variation about a mean is a concept that comes naturally to students to consider when given a set of 1-D data. One of my professors mentioned offhand in an experimental design class that any measurement you take is a distribution, a point which I have never forgotten. Up to that moment, I had never really thought much about it either.

Sure, I had collected data in my biology, chemistry, and physics classes before and knew I had to take multiple data points. All I knew then was that doing so made my data “better”. More data makes things better. Get it? My understanding in high school science was also that you never measure the same quantity at the exact same value ten times in a row because someone in your lab group is always messing it up or doing it wrong. Averaging things together smooths that out. I don’t recall ever discussing in either math or science class that the true beauty of statistics comes from managing, communicating, and understanding variability in data that will never really go away. I have always shuddered when students write lab report conclusions that discuss how “the data are/is wrong because” rather than focusing on what the data reveals about an experiment.  We definitely want to work to minimize experimental error, but sometimes the variation in the data is an important characteristic of what is being measured.

Maybe this is something that needs to be explicitly taught in the way we present statistics to our students. It seems like something that needs to be drawn out over time, rather than in one big statistics unit of a course that focuses on other things. I think using  technology to handle the mechanics of calculating statistical quantities allows students to focus more on what the statistics say and develop their intuition about it. We risk letting the important ideas of variation and statistics collect dust and stagnate as  another box of content for students to throw in the closet of their busy, distracted brains.

Impressing the parents with Wolfram Alpha…it’s for your own good, kids.

I received a few emails from parents recently wondering how to help their children get better in math. Parents often apologize for not being strong at math themselves, and the students, in my case all teenagers, have trouble communicating with parents about, well, a lot of things, let alone math. Creating a genuine way for children and parents to communicate with each other about math has always been difficult. Thankfully, tools like Wolfram Alpha can come to the rescue.

Here is the advice I gave one parent this week whose child is learning to factor quadratic trinomials:

Think of four numbers, keep them between 1 and 8. For example, 2, 1, 5, 7.

You can then write them like this: (2x+1)(5x+7) or make some negative: (2x – 1)(5x+7).

Go to Wolfram Alpha, and in the main input bar, type what you wrote, as shown below:

A page will load with a part that looks like this, a bit of the way down the page:

Give the top one (10x^2+9x-7) to him, and say to factor it. A groan at this point is natural. And then he will remember how to do it. The final answer should be the same as what you entered into the website. You can come up with new numbers and do this as much as you want – it will only make him stronger. If he has trouble, make the 1st and 3rd numbers you pick be 1, and it will simplify it a bit.

Yes, it will result in at least some expression of teenage ‘come-on-mom/dad-ery’. But that’s probably going to happen anyway, right?

How good is your model? Angry Birds edition

With Algebra 2 this week, I decided it was time to get on the Angry Birds wagon. I didn’t even mention exactly what we were going to do with it – the day before, the students found the above image in the class directory on the school server, and were immediately intrigued. This was short lived when they learned they weren’t going to find out what it would be used for until the day after.

To maximize the time spent actually mathematical modeling, I used the video Frank Noschese posted on his blog for all students. They could pick any of the three birds and do the following:

Part A:
Birds are launched at 6, 13, and 22 seconds in the video. Let’s call each one Bird A, Bird B, and Bird C.
• Take a screenshot of any of the complete paths of birds A, B, or C.
• Import the picture into Geogebra. Create the most accurate model you can for the bird you selected. What is the equation that models the path? Does it match that of your neighbors?

Part B:
• Go back to the video and the part in the video for the bird that you picked. Move forward to a frame shortly after the bird is launched, take a screenshot, and put it again into Geogebra. Can you create a model that hits the landing point you found before using only the white dots that show only the beginning of the path?

If not, find the earliest possible time at which you can do this. Post a screenshot of your model and the equations for the models you came up with for both Part A and Part B.

My hope is not to just use the excitement of using Angry Birds in class to motivate knowing how to model using quadratic functions. That seems a bit too much like a gimmick. The most interesting and realistic use (and ultimately the most powerful capability of any model) of this source of data is to come up with as accurate of a prediction of the behavior of the trajectory as is possible using minimal information. It’s easy to come up with a quadratic model that matches the entire path after the fact. Could they do this only twenty frames after launch? Ten?

The students quickly started seeing how wildly the parabola changes shape when the points being used to model the parabola are all close together. This made obvious the importance of collecting data over a range of values in creating a model – the students caught on pretty quickly to this fact.

I think Angry Birds served as a cool “something different” for the class and has a lot of potential in a math class, as it does in physics. I am hoping to use this as a springboard to have students understand the power of models and ultimately choose something to model that allows them to predict a phenomenon that is of some importance to their own adolescent worlds. I don’t exactly know what this might be, and I have some suggestions for students to make if they are unable to come up with anything, but this tends to be one of those ideas that eventually results in a few students doing some very original work. Given my interest in ultimately getting students to participate in the Google Science Fair, I think this is just the thing to push them in the right direction of making their own investigation.

A smattering of updates – the good with the bad.

I want to record a few things about the last couple of days of class here – cool stuff, some successes, some not as good, but all useful in terms of moving forward.


I have been working incredibly hard to get this class talking about their work. I have stood on chairs. I’ve given pep talks, and gotten merely nods of agreement from students, but there is this amazing resistance to sharing their work or answering questions when it is a teacher-centric moment. There are a couple students that are very willing to present, but I almost think that their willingness overshadows many others who need to get feedback from peers but don’t know how to go about it. What do I do?

We turn it into a workshop. If a student is done, great. I grab the notebook and throw it under the document camera, and we talk about it. (In my opinion, the number one reason to have a document camera in the classroom, aside from demonstrating lab procedures in science, is to make it easy and quick for students get feedback from many people at once. Want to make this even better and less confrontational? Throw up student work and use Today’s Meet to collect comments from everyone.

The most crucial thing that seems to loosen everyone up for this conversation is that we start out with a compliment. Not “you got the right answer”. Usually I tolerate a couple “the handwriting is really neat” and “I like that you can draw a straight line” comments before I say let’s have some comments that focus on the mathematics here. I also give effusive and public thanks to the person whose work is up there (often not fully with their permission, but this is because I am trying to break them of the habit of only wanting to share work that is perfect.) This praise often includes how Student X (who may be not on task but is refocused by being called out) is appreciative that he/she is seeing how a peer was thinking, whether it was incorrect or not. I also noticed that after starting to do this, all students are now doing a better job of writing out their work rather than saying “I’ll do it right on the test, right now I just want to get a quick answer.”

Algebra 2

We had a few students absent yesterday (which, based on our class size, knocks out a significant portion of the group) so I decided to bite the bullet and do some Python programming with them. We used the Introduction to Python activity made by Google. We are a 1:1 Mac school, and I had everyone install the Python 3 package for OS 10.6 and above. This worked well in the activities up through exercise 8. After this, students were then supposed to write programs using a new window in IDLE. I did not do my research well enough, unfortunately, as I read shortly afterward that IDLE is a bit unstable on Macs due to issues with the GUI module. At this point, however, we were at the end of the period, so it wasn’t the end of the world. I will be able to do more with them now that they have at least seen it.

How would I gauge the student response? Much less resistance than I thought. They seemed to really enjoy figuring out what they were doing, especially with the % operator. That took a long time. Then one student asked if the word was ‘remainder’ in English, and the rest slapped their heads as they simultaneously figured it out. Everyone enjoyed the change of pace.

For homework, in addition to doing some review problems for the unit exam this week, I had them look at the programs here at the class wiki page.


I had great success giving students immediate feedback on the physics test they took last week by giving them the solutions to look at before handing it in. I had them write feedback for themselves in colored pencils to distinguish their feedback from their original writing. In most cases, students caught their own mistakes and saw the errors in their reasoning right away. I liked many of the notes that students left for themselves.

This was after reading about Frank Noschese’s experience doing this with his students after a quiz. I realize that this is something powerful that should be done during the learning cycle rather than with a summative assessment – but it also satisfied a lot of their needs to know when they left how they did. Even getting a test back a couple days later, the sense of urgency is lost. I had them walking out of the room talking about the physics rather than talking about how great it was not to be taking a test anymore.

Today we started figuring out circular motion. We played broom ball in the hallway with a simple task – get good at making the medicine ball go around in a circle using only the broom as the source of force.

We then came in and tried to figure out what was going on. I took pictures of all of their diagrams showing velocity and the applied force to the ball.

It was really interesting to see how they talked to each other about their diagrams. I think they were pretty close to reality too, particularly since the 4 kilogram medicine ball really didn’t have enough momentum to make it very far (even on a smooth marble floor) without needing a bit of a tangential force to keep its speed constant. They were pretty much agreed on the fact that velocity was tangent and net force was at least pointed into the circle. To what extent it was pointed in, there wasn’t a consensus. So Weinberg thinks he’s all smart, and throws up the Geogebra sketch he put together for this very purpose:

All I did was put together the same diagram that is generally in textbooks for deriving the characteristics of centripetal acceleration. We weren’t going to go through the steps – I just wanted them to see a quick little demo of how as point C was brought closer to B, that the change in velocity approached the radial direction. Just to see it. Suddenly the students were all messed up. Direction of change of velocity? Why is there a direction for change in velocity? We eventually settled on doing some vector diagrams to show why this is, but it certainly took me down a notch. If these students had trouble with this diagram, what were the students who I showed this diagram and did the full derivation in previous years thinking?

Patience and trust – I appreciate that they didn’t jump out the windows to escape the madness.


All in all, some good things happening in the math tower. Definitely enjoying the experimentation and movement AWAY from lecturing and using the I do, we do, you do model, but there are going to be days when you try something and it bombs. Pick up the pieces, remind the students you appreciate their patience, and be ready to try again the next day.