Summer Lesson Planning: The Galton Board

Well, I’ve started planning some activities/interesting questions to add to my AP Stats curriculum.  I’ve noticed that my teaching lacks the ability to have my students see the normal distribution happen naturally.  Every year, I’m forced to have my students take my word that the normal distribution is…well…really real.

So, seeing that I’ve had some time to develop some lesson ideas (which is code for I get to watch The Price is Right)…and I remember seeing this:

Note: I’m going to come up with some binomial distribution lesson from this video too

I’ve taught workshops for teachers that involve probability, the normal distribution and the mention of galton boards (go ahead and click it…I made the hyperlink).  But the mention of the galton board was just that…a mention.  I wanted to have my students experience it!

I scoured the internet for an inexpensive (it is the summer afterall) galton board.  No such luck.  I tried to find a video on the internet that I could pull data from.  Even less luck.  So I made my own videos.

This is meant to be a ten minute #threeactmath inspired (gotta love Dan Meyer) activity.

It starts with:

Then, after some class discussion, they’ll find out there a total of 500 balls dropped.  Their job is to predict what will happen.

2.2 The Normal Distribution_3

The hope is to start making connections between the shape of the distribution and probability.  There are more ways to a middle slot than an end slot; meaning that there should be more in the middle.  The distribution should be symmetric (and we’ll discuss what would make it not symmetric).  Basically, building the idea of unimodal and symmetric for a normal distribution.

Then they’ll finally see the answer:

More than likely, my students won’t be correct in their guesses, but the conversation is really the important part.  I’m hoping to bring this back when we talk about binomial distributions/probability as well.  The more I write about this, the more I realize there’s some really rich discussion can come from this.

Please feel free to comment and discuss any thoughts you have.

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AP Statistics: How Many Penguins?

A little while ago, we began a unit on sampling and experimental design.  In the recent past, I have not done the shift from descriptive to inferential statistics any sort of justice.  Before exploring any sort of formal sampling methods, I wanted to have my students experience the idea of using a set of data to estimate a parameter.

I started the lesson with an example of this process that they’re familiar with:

5.1 Samples and Populations_1

My students were given a screenshot of this ESPN poll and I asked them to tell me anything they could about it (as well as ask any questions they wanted).  With a bit of probing, my students were able to map out the process/purpose of this poll:  ESPN wants to take a small group of fans’ opinion on who will win the NFC East and use that to generalize to the population (of which we never came to a consensus about).

Next, I wanted to put them in a situation where the process they described could be used.  I remembered that I had seen an interesting question while working with FCR-STEM facilitating a summer statistics workshop:

5.1 Samples and Populations_5

The premise is that this is an overhead shot of a section of Antarctica and each dot represents a penguin.  My students’ goal was to estimate the number of penguins.  First, I did a cheap method of getting some engagement…I had them guess how many penguins there are:

5.1 Samples and Populations_6

Then began the process of how to use the information we had quickly and efficiently. We discussed that, while it is possible to count every penguin, it was inefficient and really darn annoying to do so.  We devised a plan (or at least the class did): everybody chooses a square on the grid, we average those number of penguins and then multiply by 100.  Personally, I thought it was a good plan.

This is where the new material began for this unit.  We needed to discuss how each person would select their square.  A discussion of randomness and sampling ensued.  We decided that the calculator could be used for random digit generation and we could select rows and columns randomly:

5.1 Samples and Populations_7

Then, each student selected their square and converted that to a number of penguins:

5.1 Samples and Populations_8

We came up with a class estimate of 500 penguins.  As it turns out, that was a pretty good estimate:

5.1 Samples and Populations_9

The next few lessons are about designing proper samples of all types, and what makes a bad sample bad.  Overall, I believe my students have a direction for the next few months thanks to this lesson.  The only change I will make in the future would be to take away the grid from the first picture I should them about the penguins.  I feel like I led them too much.  I want them to come up with that idea themselves.

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.

4.3 Relations in Categorical Data_1

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:

4.3 Relations in Categorical Data_2

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:

4.3 Relations in Categorical Data_3

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:

4.3 Relations in Categorical Data_4

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:

4.3 Relations in Categorical Data_5

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.

Giving a Useful Context for Standard Deviation

Wow, it’s been a month since school started. Evidently, I’ve been busier than I realized…but it’s time to get back into regularly reflecting on my teaching.

Over the summer, I saw an interesting statistics task as an introduction to variability:

Rank the following from most fair to least fair:
1.2 Variance and Standard Deviation_2

My students’ prior knowledge only consisted of how to calculate the mean. I gave this task in order to introduce variability and create a need for a numerical measurement of how spread out values are.

After giving my students three minutes with a partner, here’s how everything sorted out:
1.2 Variance and Standard Deviation_3

I asked the group how they defined “fair” in this task (since I never defined it for them). They decided that the more uniform that the distribution is, the more fair.

The entire class agreed on allocation A being the most fair, allocation D being the next most fair, and allocation B being the least fair. The real problem came with ranking the order of allocations C and E. I opened the floor for debate and there were good arguments for both allocations. My students then turned to me for help. They wanted a way to settle the argument. So, we discussed standard deviation:

1.2 Variance and Standard Deviation_4

…and we practiced calculating…

1.2 Variance and Standard Deviation_5

My students then went back and calculated the standard deviation for each of the allocations and re-ranked them:

1.2 Variance and Standard Deviation_3

My hope was to create a need for standard deviation. My students decided that standard deviation is a measure of how spread out data is from the mean. They also concluded that the larger the standard deviation is, the more spread out the values are. This task lead to a great 45+ minute statistical conversation…hopefully, I can find some more good tasks for future concepts.