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 4: More Function Transformations, Percentiles, and a Golf Match

I know this is many days late, but this post refers to last Thursday (eep).  I’m going to do my best to get caught up this week.  Day 4 was a bit less eventful than most.  We spent a lot of time defining and transforming.

Precalculus:  This class was definitely the least eventful of the day.  I had to leave school early to go to a golf match.  So, we debriefed homework, and then I went on my way (while they did some conversion practice).  Unfortunately, this will happen more than I’d hope for this nine weeks.

AP Statistics:  We started the class with a warm-up that I modified from FSU’s Learning Systems Institute MFAS project (I think…I mean…I’m pretty sure).

1.1 Frequencies, Percentiles, and Ogives_1

We used this set of questions to emphasize details and clarity in their writing.  Defining the variable as clearly as possible, how they calculated certain values, etc.

The rest of class discuss percentiles and their usage.  We started the conversation with the least imaginative example I could think of:

1.1 Frequencies, Percentiles, and Ogives_6

Then came the hard-hitting question: “What does the national percentile represent?”

We spent the rest of the class calculating frequencies, relative frequencies (percents), and cumulative relative frequencies (percentiles).  Then we learned how to create and interpret ogives (they show percentiles versus variable values).  Nothing too spectacular, but necessary.

Algebra II:

We continued our conversation about transformations of functions.

1.2 Transforming Functions Day 2_2

This is the style of question I want them to be able to answer.  General trends for general functions.  Not memorizing rules, but knowing the 3 affects the input and the 2 affects the output.  Understanding that changing inputs affects the x-direction and changing the outputs affects the y-direction.  We’re really going to develop function ideas throughout the course of the year.

The rest of the activity was:

1.2 Transforming Functions Day 2_4 1.2 Transforming Functions Day 2_5 1.2 Transforming Functions Day 2_6 1.2 Transforming Functions Day 2_7

I could hear them starting to hypothesize what would happen (and justify their reasoning).  They’re still a bit afraid to be wrong.  I’m trying to convince them that they learn more from being wrong than being right all of the time, but they’re still a bit apprehensive.  It’s getting better though; we’ll stick with it.

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.

Day 2: Displaying Categorical Data, Functions and Radians

It’s Day 2 and I’m tired.  I need to work on my conditioning.  My lessons were a bit dryer than I’d prefer today.  Here are some highlights from today.

AP Statistics:

The goal today was to be able to summarize univariate categorical data (and start the conversation about summarizing quantitative data).  The lesson started with having my students collect some categorical data and telling them organize/graph the outcomes.  Every student chose to make a bar graph.  We discussed features of a bar graph and its advantages over a pie chart.  The most interesting part of the lesson, however, happened with this slide:

1.1 Bar Graphs, Dot Plots, Stem Plots_6

We used remote responders so I could get instant feedback from the class and found that 80% of them missed this question (the answer is E by the way).  After discussion, we came to the consensus that there was a reading issue.  Whether they read too quickly, or not carefully enough, I need to keep an eye on this and help with their critical reading skills.

Algebra II Honors:

In class today, we had a crash course on everything they should know about functions.  Discussions included domain and range, with proper notation.  Interestingly enough, they did struggle with the domain and range of the triangle here:

1.1 Intro to Functions lesson_4

They wanted to tell me that the triangle had three points.  After quelling that misconception, we realized that we can write domain and range using inequalities.  I’m glad we came across that gap in their knowledge.

After domain and range, we discussed the input/output idea behind a function…and we got to watch one of my favorite educational videos.

The conversation we have during this video is really rich.  The nuggetizer really gets to the input/output idea without an equation.  good stuff.  and it’s entertaining.


Today we explored radians.  The warm-up allowed them to review circumference and arc length.

1.1 Radians and Arc Length_1

I’m finding they’re a little rusty/apprehensive about fractions.  I need to make sure we get better quickly.

My hope today was that they would figure out the conversion to go from degrees to radians.  I found an intriguing activity in the textbook we currently use.  I modified to fit my style, but it has the same bones:

1.1 Radians and Arc Length_2

I was thrilled with how quickly they found that the s/r ratio is constant (in this case pi/3).  We defined that ratio as the number of radians, and my students decided that ratio measured the angle.  pi/3 is equivalent to 60 degrees.  Then we derived how to convert from one measurement to the other.

Day 1: Variables, Solving Equations and DMS

This year I’ll be teaching AP Statistics, Algebra II Honors and Precalculus.  I want transparency and an open dialogue about how to transform my curriculum in these classes.  Two of the classes, I’ve taught for a few years.  One of the classes is brand new to me.  I won’t tip you off to which is which, but feel free to comment freely.

My hope is to post something about each class every day…so here we go…

AP Statistics:

Today’s focus was on types of variables.  Mainly the difference between quantitative and categorical variables.

We started with this warm up:

Chapter 1 Introduction_1

The hope was that they would talk to each other about the questions, and I was curious who would ask me about question #1 (I have three cats…just so you know).

Without answering any of the questions, I showed this slide:

Chapter 1 Introduction_2

I loved that most of the students grouped the questions in two important ways. 1) They grouped the questions based on whether there was one distinct answer versus the answers having possible variability (which showed that they knew there was a difference with a statistical question versus a non-statistical question).  2) They grouped the questions based on whether the possible answers were numerical or non-numerical.

At this point they’ve stated that they know of quantitative versus categorical variables.  So, I wanted to dive into the grey areas and get a really good definition of quantitative and categorical.  I asked them to classify the following situations:

Chapter 1 Introduction_5

Through working on classifying these variables, my students came to some interesting conclusions.  First, they were able to better state that quantitative variables require measurement, not just a numerical response.  Second, and most interestingly, there was an in depth conversation (student started) that quantitative variables could be treated categorically; they concluded that someone has to be clear in stating their expectations for measurement.

Algebra II Honors:

To ease into the new year, I wanted to make sure my algebra II students were able to solve one variable linear equations.  Sounds easy enough (and they thought it was too), but I wanted to give a different spin on it.

Unit 1 Intro (Solving Equations)_1

This question required that they keep everything in balance.  I wanted to encourage my students to work together and be self-sufficient in checking whether their solution is correct.  After having a consensus on a solution, they were asked to find the value of the missing shape in each of the following mobiles:

Unit 1 Intro (Solving Equations)_2 Unit 1 Intro (Solving Equations)_3

The first one was to build confidence; no problems…the triangle is worth 10.  In the second one, there were a bunch of methods used.  Some guessed and checked.  Others set up an equation.  A few noticed that a triangle and square cancelled each other out on the two sides and they were really solving 2 squares equals 1 triangle…which someone quickly noticed that they did the exact same thing if they set up an equation.  Amazing conversations.  Then we solved some equations.


We started the period with practice using dimensional analysis to convert value.

1.1 Measuring in Degrees_1

The purpose of this was to put a seed in their mind for figuring out how to convert angle measurements from degrees into degree-minutes-seconds form.  After defining how many minutes are in a degree (60) and how many seconds are in a minute (60) (then explaining how the Sumerians worked in a base 60 number system…which is why we measure time the way we do), I had them complete the following conversion:

1.1 Measuring in Degrees_5

I question whether I made this too leading, but everyone came up with a method that worked for them.  I challenged them for homework to come up with a method to convert an angle measured in DMS back to degrees.  We’ll see how they did tomorrow.


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: 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.

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!