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.

First Homework Assignment

Alright.  We’re back online.  I know it’s been a while.

This year, it’s time to really revamp Algebra II.  I have to do a better job of advocating for thinking mathematically and persevering through problems.  So we’re going to start day one.

Here’s their homework assignment:

Mr. Cloud tries to be a healthy person and takes multi-vitamins daily.  He absolutely LOVES the gummy ones!

Here they are:


As he was finishing this bottle, Mr. Cloud thinks he was ripped off (he didn’t get what he paid for).  Here’s what he saw:


Circle One:

Did Mr. Cloud get ripped off?     YES         NO         Can’t Tell

Provide a well written justification for your answer:



So that’s it.  That’s the whole assignment.  And I’m going to leave this right here.  Feel free to comment and argue your point!



Algebra II Honors: Introducing Function Composition

In Algebra II, our second semester begins with function operations and inverses.  Since function composition is a very commonly used concept outside of the math classroom, I wanted to introduce the idea within a context.

So I opened the lesson with this picture:

6.2 Function Composition_4

The goal for my students was to determine how much I paid for gas the day before.  I gave them some information:

6.2 Function Composition_6

My students wanted to know how much gas was and then proceeded to spend 45 seconds commenting on how inexpensive gas is at the moment (I figured this would happen given that my a lot of my students are starting to drive).

Then I told them how many miles I had driven:

6.2 Function Composition_5

Now that they knew the price and the fact I had driven 117.2 miles, I wanted them to make a prediction.

I got anything from a $15 prediction to a $40 prediction.  I was surprised to see that when I probed them about why they predicted the price that they did, quite a few of my students used the fact that I was filling up about a half tank of gas.  Questions and comments started flying about how many gallons of gas my car’s tank would hold.  I didn’t have an answer for them, and I tried to question how that would be a useful piece of information.  They said that if they knew how many gallons the tank was in total, they could figure out how many gallons half a tank is and use that to determine how much I paid.

A conversation about how they didn’t know exactly what proportion of my gas tank was being filled.  They weren’t happy…they said it was about half of a tank…I said that using the word “about” makes that information unusable…one or two students still weren’t happy.

However, we did come to the conclusion that it would be nice to know how many gallons of gas my car needed…and I could help them with that:

6.2 Function Composition_7

Then, I sent them off in their groups to determine how much I paid.  Five minutes later, I saw a lot of work that looked like this:

6.2 Function Composition_9

This was typical of the work I saw from my students.  There were a lot of good conversations about keeping track of units and how the process worked.  After discussing with students the answer:

6.2 Function Composition_8

We had a conversation about what the process entailed.  They decided that they took a mileage and turned it into a number of gallons, then took those gallons and turned it into a price.  I then defined this as a composition of functions.  Then we decontextualized and practiced.

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: 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)

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.

Summer Lesson Building (Statistics: Graphical Representations of Center and Spread)

So, I currently have a large amount of teacher “stuff” floating around my head at the moment.

1) I don’t think I’ll be needed to teach a section of geometry this year. This didn’t hit me too hard when I first found out, but, the more I ponder it, the more upset I get that I won’t be able to reinvent that curriculum. I spent a lot of time working on that class last year, and had good EOC results…and it’s all a bunch of irrelevant complaining. It’s time to move on and teach what’s in front of me…Stats and Algebra II.

2) I’ve been reading Daniel Willingham’s book “Why Don’t Students Like School?” He’s a cognitive scientist connecting how the brain works to classroom performance. I feel like this book will change my approach in my classroom (more on that in a later post).

3) I need to do a better job of incorporating reading in my content area. I’m tired of being given the pass that it’s “hard to incorporate reading into a math classroom.” I have to create an environment that cultivates math as an everyday experience that goes beyond the classroom.

And this is where I need some help. #3.

Recently, I was reading one of my favorite blogs about the Philadelphia Eagles, Bleeding Green Nation, and I came across this article:

Lesson Ideas File_12

I love how this blog (and a few others about my favorite Philadelphia sports) will incorporate actual statistical arguments into their analysis of my favorite NFL team. I have decided that my students need to experience this article. There’s such rich conversation that will come out of my students reading this article.

I’m hoping that the conversation will lead here:

Lesson Ideas File_13

Lesson Ideas File_11

I know that I want to ask my students the following questions:
-Which team do you expect to have the most wins? How do you know this?
-What does the “height” of a teams graph represent?
-Is it possible that the Cardinals have more wins than the Seahawks? How do you know?
-Is it likely that the Cardinals have more wins than the Seahawks? How do you know?
-What is the difference between the last two questions?
-For which team is it easiest to predict the number of wins? How do you know?
-For which team is it hardest to predict the number of wins? How do you know?
-Explain how measures of center and measures of spread are shown in this graph.

I need other thoughts of teachers experienced in incorporating these type of activity in class. I’m hoping for a wonderful discussion. Any and all thoughts are welcome!

Summer Lesson Building (Geometry: Distance and Midpoint)

One of the first lessons this coming year in Geometry will be a “review” of distance and midpoint. I’ve spent the last couple of hours trying to figure out the best way to conceptually work with both of these topics and still keep the lesson in the neighborhood of 50 minutes. I’m stuck somewhere between treating this as a review of material I know my students have seen in the past and going full-on inquiry based learning. I’m leaning towards expecting them to need a reminder about from where distance and midpoint formulas come. I am using this lesson to create situations where there is a need for distance and midpoint.

I need to make sure I have my students focus on distance and midpoint on the coordinate plane. To me, the first step is create a need for the coordinate plane. I’ll open with lesson with a subway map of New York City:
1.6 Distance and Midpoint_2

Then, I’ll focus my students to a question:
1.6 Distance and Midpoint_3

I want them to guess. I want an argument based on what they think they know. I want them to say “it looks like…” and “what’s the scale?” Then, I hope someone finally asks for a way to determine the scaling (I’ll bet they won’t use the term “scale” but whatever works).
1.6 Distance and Midpoint_4
Now they can use the length of a block as the unit of measure. But the streets and avenues aren’t equally spaced! I guess they’ll have to come up with a conversion! I expect some students to remember distance, or that there’s a formula for it. That’s not the purpose of the map. I want them to see how much easier it is to estimate with a grid. Now we have a reason for the coordinate plane!

So we’ll start working on a distance formula:
1.6 Distance and Midpoint_5
We’ll work with the Pythagorean Theorem, figure out the process, add in the general case, and derive the distance formula. Then we’ll practice in a non-contextualized situation:
1.6 Distance and Midpoint_6

1.6 Distance and Midpoint_7

Then I’ll give them a “real life” situation that requires them to ask for a coordinate plane:
1.6 Distance and Midpoint_8
and once they ask:
1.6 Distance and Midpoint_9

For part two of the lesson, we’ll focus on midpoint. I’ll give my student the following four graphs:
1.6 Distance and Midpoint_10

1.6 Distance and Midpoint_11

1.6 Distance and Midpoint_12

1.6 Distance and Midpoint_13

The only direction I’ll give them is to find the midpoint of each of the line segments. Their goal is to define the midpoint for themselves, then come up with some way to find it:

1.6 Distance and Midpoint_14

We’ll come up with a class consensus for a definition and a formula for the midpoint.

Then we’ll practice:
1.6 Distance and Midpoint_15

1.6 Distance and Midpoint_16

1.6 Distance and Midpoint_17

Overall, I’m not thrilled with this lesson. I think it will do its job, but I’ll keep searching for a better approach. Also, I’m expecting this lesson to last more than one class period. I really don’t see how I can expect to finish this in less than a day and half without losing the student centered approach. Thought?