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.


Day 5: A Quiz, Special Right Triangles and Cups

Day five.  The first Friday.  My students were tired.  I was tired.  The first week was successful; at least, I felt that it was.

AP Statistics: The first week culminated in a quiz.  I did learn that we need to work on communication through writing.  I feel like they have a good grasp on what we’re learning, but they’re having trouble communicating it.


In precalculus, today’s lesson is to connect calculating trigonometric function values with their knowledge of special right triangles.  I was a bit concerned about their memory of special right triangles, so their warm up had them working with 45-45-90 triangles:

1.2 Trig Functions and Special Right Triangles_1

My hope was that they’d quickly realize the pattern that the length of the hypotenuse is the leg times the square root of two.  They did, and they had the “oh yeah” moment.  I regret not proving this fact to them.  I need to make sure when we’re coming up with rules and theorems that I’m more formal about it; those students who go on to do higher mathematics need that exposure.

After spending time with 30-60-90 triangles, the end of class culminated in:

1.2 Trig Functions and Special Right Triangles_6

The part my students are struggling with the most is the simplification of fractions with square roots.  I’m trying to convince them that they’ll become comfortable with them; they just need to be willing to practice.

Algebra II Honors:

I’m a bit concerned that my students don’t know what makes a linear function linear.  We took a day out to discover what linear really means.  I’ve told them that as we look at functions, we’re going to study them in three ways: a graph, a set of coordinates, and an equation.  They knew what makes a graph of a function linear, but not the other two representations.  So we explored:

1.3 Classifying Patterns_1

Each pair of students received three cups and a ruler.  From that information, they needed to make their estimation.

In case you want to play along:

1.3 Classifying Patterns_2

Most groups realized that there is a constant growth in height caused by the lip of the cup.  From that measurement (which is really 1.5ish centimeters…even though it looks like 2 centimeters in the picture) they extrapolated their guesses.

1.3 Classifying Patterns_4

A few groups got to 116 cups.  Most were within 10 cups.

From their realization that the height of the stack grows at a constant rate in relation to the number of cups, we had the discussion of what linear means.  Ideas about rate of change, y-intercepts, and functions were all discussed.  The hope was this concrete example would let them see that constant change (slope) is what makes a function linear.  This led to other discussion and playing with geogebra.  Fun was had by all (or at least me)!

Day 3: Histograms, Trigonometric Functions, and Function Transformations

Three days into the school year, my students and I are still getting used to each other.  Some classes have figured out that I prefer dialogue between teacher, student and other students.  Other classes have not started to trust that idea.  I’ll continue to encourage them to work together.

AP Statistics:

Today’s goal was to appropriately display quantitative data.  Most of the conversation focused on making histograms, stemplots and dotplots.  We did collect some interesting data though:

1.1 Histograms_3 1.1 Histograms_4

All of the data points were between 52 and 70 seconds.  Not too shabby.  Today was also the first day of AP Exam prep (even though they didn’t realize it).  Once we listed all 20 of the times from the class, I stepped back and said “now describe the data.” They had to work together to get all of the aspects they needed to describe: shape, center, spread, outliers/gaps/clusters.  They had some great conversations too!


We started class connecting what they learned about converting from degrees to radians and how to calculate arc length (with angles in both degrees and radians).  Man, the connections that were made and the speed at which they picked it up was impressive.  I’m really excited about their potential.

Speaking of their quickness, we defined three new trig functions they hadn’t worked with before: secant, cosecant and tangent.  Rather than drill methods to solving problems, I threw this with them without any hints:

1.2 Trigonometric Functions Lesson_5

The ease in which they figured out to draw a right triangle, and use the Pythagorean Theorem was great!  I figured this would challenge them at first, but I was definitely wrong.  I definitely need to step up my game.

Algebra II:

We’ve begun to go deep into the world of functions.  I’m noticing that some of my students are struggling to see the big picture.  In today’s activity the focus was supposed to be on how changes in a function affects its graph:

1.2 Transforming Functions Day 1_2 1.2 Transforming Functions Day 1_3 1.2 Transforming Functions Day 1_4 1.2 Transforming Functions Day 1_5

We ended up so bogged down in the details of the order of operations and plotting points that we lost sight of the big picture.  Hopefully their homework tonight can help re-focus them.

AP Statistics: The Law of Large Numbers

I’ve been trying to come up with a way to immerse my students in the law of large numbers without simply giving them the law and reading example after example.  To jog your memory, the law of large numbers states: as the sample size increases, the experimental probability will approach the theoretical probability…or as one of my students stated, as the sample size gets larger, the prediction (experimental probability) gets more accurate.
6.3  Laws of Large and Small Numbers_1
I’ve been playing Pass the Pigs for a long time, and I’ve always wondered about the probability of having the pigs land in the different piggy positions. (If you don’t know what I’m talking about, go here:
So, I asked them this question:
6.3  Laws of Large and Small Numbers_2
I explained to my students that we were going to modify the game to just look at rolling one pig.  The conversation started with my students deciding that the higher the point value, the harder that type of roll is to get.  someone eventually made the connection that the difficulty of the roll is inversely related to its probability.  I then asked them what the probabilities for the different rolls were.  They didn’t know,but were willing to roll some pigs to make some predictions . After some rolls, we gathered the class’s data and calculated experimental probabilities:
6.3  Laws of Large and Small Numbers_3
Tons of great conversation followed.  They argued about fairness and attempted to come up with a point system that was “fair.” After a few minutes, I brought them back to see what they were doing.  I pointed out that they were using those experimental probabilities as if they were the absolute truth.  A student quickly quipped: “but we did a lot of rolls, it should be pretty accurate.”
Then, and only then, we discussed the law of large numbers.

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.

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:

4.2 Cautions about Correlation and Regression1_4

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

4.2 Cautions about Correlation and Regression1_8

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.

4.2 Cautions about Correlation and Regression1_6

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):

4.2 Cautions about Correlation and Regression1_7

Finally, we looked at the actual value:

4.2 Cautions about Correlation and Regression1_10

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

4.2 Cautions about Correlation and Regression1_4

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, instead, we started here:

Age vs Height Data_1

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:

Age vs Height Data_3

Age vs Height Data_4

Age vs Height Data_5

Age vs Height Data_6

Age vs Height Data_7

Age vs Height Data_8

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:

Age vs Height Data_9

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:

Age vs Height Data_10

Age vs Height Data_11

Age vs Height Data_12

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:

4.2 Cautions about Correlation and Regression1_4

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:

4.1 Intro Skyrim Data_1

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

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:

4.1 Intro Skyrim Data_2

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:

4.1 Intro Skyrim Data_3

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)

Statistics Summer Institute Day 2 (The day I realized my AP Stat students are amazing!)

For all of the stress my AP Statistics students go through and put me through, I realized today that they truly are an amazing group.

Day 2 of the institute had the participants looking at types of statistical data; they dealt with statistical questions, sampling methods, and types of variables. Today, my presentation focused on they different types of variables: categorical vs quantitative.

I started the participants off with the first thing my AP Statistics students do on the first day of school. They have to answer the following questions:

1) How many cats does Mr. Cloud have?
2) What is your favorite color?
3) How many pockets are there on your clothes?
4) How many stars are there on the American flag?
5) What is your favorite pizza topping?
6) How long can you hold your breath?

The answers are interesting and trivial at the same time. After 2 minutes trying to answer the questions, the participants spent a few minutes sorting the questions in meaningful ways. Once the participants decided that Questions #2 and #5 were special because they have categorical (non-measureable numerical) responses, we spent the next two hours discussing what makes a variable categorical and the best way to summarize and represent data that’s categorical (bar graph, pie chart, frequency tables, etc.). It was a relatively uneventful two hours. There was good discussion and I emphasized when and where the standards for mathematical practice were used.

The most interesting part of the day occurred during a break. Some of the participants were discussing how they were frustrated with the amount of statistics we have done and plan on doing. They weren’t complaining, but feeling overwhelmed with the amount of material we have covered and will cover. Upon reflection, the pace and depth of the material we are working with is slower and less than (in most cases) than what we do in AP Statistics. I got a lot of perspective today on the struggles that my own students go though in AP Statistics having less statistical experience than these participants. The amount of knowledge my own students gain is pretty astonishing. I’ll definitely have a different perspective on their accomplishments from now on!