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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Geometry: Which Pasta Does Mueller’s Want You to Buy?

So, I’m walking through Publix about a week or two ago, and I noticed this:

9.3 Mueller's Pasta_1

Mueller’s makes both regular length and pot-sized pasta.  Since I’m giving a presentation next week involving problem based instruction in the geometry classroom, I saw this as an opportunity excuse to develop a new geometry lesson (I really miss teaching geometry sometimes).  So, I started talking to my co-presenter and asked her to help me develop something useful for us to present with.

My co-presenter is a first year teacher and I quickly found out that she’s not very familiar or comfortable with problem based instruction.  Since she’s my mentee, I thought it would be a good idea to develop a lesson with her and then team teach in her classroom.

 

So we started with the question: Which would Mueller’s rather you purchase?

9.3 Mueller's Pasta_1

Her student’s sat there and started at me for a few moments.  I encouraged them to talk with each other and figure out how they could answer the question.  Two minutes later, we discussed what they wanted to compare and what questions they had for us:

9.3 Mueller's Pasta_2

They were insightful.  The only thing they didn’t ask about (that I was expecting) was a question about shipping.  My mentee and I quickly steered the conversation toward packaging costs.  So, her students wanted some measurements:

9.3 Mueller's Pasta_3 9.3 Mueller's Pasta_4 9.3 Mueller's Pasta_5 9.3 Mueller's Pasta_6 9.3 Mueller's Pasta_7 9.3 Mueller's Pasta_8 9.3 Mueller's Pasta_9

I made up the cardboard cost…maybe I can get Mueller’s to give me that info.

Given that this lesson was given during a point in the curriculum that has nothing to do with surface area, they did okay with calculations.  I was surprised to see that 3 or 4 out of the 8 groups thought that volume was the appropriate measurement to use (rather than going straight to surface area).  After some quick conversations, most were right on track.  Ultimately, the most difficult calculation was converting the cost from square inches to square centimeters.

9.3 Mueller's Pasta_10

The calculations bear out that the pot sized pasta is cheaper to package.

I’m encouraged that my mentee is interested in continuing to create lessons like these.  I’m taken back a bit that this isn’t the norm in most classrooms.  I have to find a way to keep encouraging my colleagues to keep bettering themselves and their students.

Summer Statistics Institute Day 8 (Exhuastion)

We have reached the point where we’re going through the motions…or at least I am. 8+ hours a day of professional development for two weeks is a lot for the participants to endure.

The hour long presentation I gave today wasn’t anything special. We discussed inference and looked at some simulated data. Most of the next two days focuses on the connection between simulation, probability and inference. I’ll give detailed updated on activities tomorrow.

With all that is being asked of the participants, I don’t find them immersing themselves in the lessons and activities. I will be presenting most of the day tomorrow; I’ll find a way to make their experience richer.

Summer Statistics Institute Day 7 (Back in the Saddle Again)

I’m back to teaching again!!…and I’m sure my participants are oh so excited.

Our new unit is about understanding probability. I want my students, my teachers and their students to understand probability as so much more than the probability of a red marble coming out of a bag. I want them to understand probability as a way to make predictions and drive decisions. So we started here:

brain

The focus of the discussion was on the idea of randomness and what should have happened. I kept emphasizing that their students will need to tackle the idea that what does happen and what should happen may not be the same thing. There’s a drastic difference the in the questions.

We started discussing numerical probability with this video:

We had a quick discussion about the “probability number line” and what different values meant.

Most of the lesson revolved around the following situation:

You and your partner have a bet. You are going to play a dice game and the winner receives $100,000,000. Each of you will roll one die. Next you will sum the two dice. If the sum is 2, 3, 4, 5, or 6, player 1 wins. If the sum is 7, 8, 9, 10, or 11, player two wins. If 12 is rolled, then the round is a tie and will be rerolled.

I asked the question of whether the game is fair. We simulated the game and calculated the theoretical probabilities. There was a lot of interesting discussion. We have a foundation laid for dealing with experimental and theoretical probabilities. We’ll see how they take the lesson in the morning!

Summer Statistics Institute Day 6 (I NEED TO TEACH)

There’s so much that I want to say, but I’m not sure how much of it I should say on a public forum. I didn’t have any lessons to teach today. It was another day of answering questions and giving input for those who wanted it. Some brief notes from the day:

1) I NEED TO TEACH. Sitting and watching someone else teach in a way that is the antithesis of everything you believe about teaching and learning is frustrating. Really really frustrating.

2) I did manage to pick up an activity to teach tomorrow on probability and its meaning. It’s not a favorite activity of mine, but it’ll give me a chance to grow.

3) I’m really proud of the group of teaching I’ve been working with. They’re building lessons about statistics and they’re asking great questions. These 25 teachers really want to grow as professionals and it makes me proud to be a teacher.

4) It still amazes me how unprepared a lot (the term “a lot” may be hyperbole but I’m sensing a lot of discomfort) of our teachers are to teach statistics the right way. I feel like we’re trying to fill a needed ocean of statistical knowledge with a leaky bucket.

Probability starts tomorrow…be prepared!

Summer Statistics Institute Day 5 (Learning Self-Control)

I did not have any teaching responsibilities today. The participants finished up analyzing one variables statistics and began their journey into bivariate data. My big focus today was restraining myself to not interject anytime I would have done something different. It was difficult to not interject. I never said my way is better. And I only interjected twice. I have found that being a presenter is boring when you’re not actually presenting.

Statistics Summer Institute Day 4 (Hijacking)

I wasn’t scheduled to teach anything on day 4 of the institute, but anyone that knows me understands that I can’t sit there and not be involved. I somehow managed to talk one of the facilitators into letting me teach the lesson on the meaning of the mean and standard deviation…although, I’m betting he let me think I talked him into it.

To introduce the mean, I asked the participants define it in their groups. Most of the participants defined the mean as a process (sum the data and divide by the number of data points). Our goal was to get to a definition of the mean that wasn’t exclusively a process. I had some volunteers come to the front of the room and gave each of them different amounts of blocks. Next, the volunteers had to create a “fair” allocation of those blocks. With two volunteers, they just shared blocks from one person to the next. Even when there were three people, the volunteers just shared blocks until they all had the same amount. Once there were six volunteers, it got a little harder. Eventually someone simplified the situation by putting all of the blocks into one group and dividing them evenly amongst the volunteers. There was the a-ha moment of what the mean really is.

Next came a conversation about the standard deviation. Most of the participants have some knowledge of the standard deviation calculation, but not really of its meaning/use. We started here:

fair allocation

I asked them to rank the allocations from most fair to least fair. The entire group didn’t have a problem with allocation A being the most fair, allocation D being the second most fair and allocation B being the least fair. However, there was a relatively heated argument about whether C or E was fairer. This created a need for the standard deviation. The participants walked me through a way to measure fairness (spread) and we did some calculations:

standard deviation outcomes

The task is made easier with the values for the standard deviation.

I still had to completely change the prescribed lesson given to me. I think I’ve just decided I’m going to take liberties wherever I feel the need. The participants haven’t complained that what we’re doing doesn’t match their binder; at least they haven’t complained yet.

Statistics Summer Institute Day 3 (The Loss of Freedom)

Day 3 of the statistics institute was my day. All day. Every Lesson. We explored every possible graphical representation to help describe a set of data that we could. What made day 3 special was the emphasis on quantitative data.

The first activity we spent a couple of hours on is similar to the second day of AP statistics. Typically, I start with a fun size package of M&Ms and ask the class how many M&Ms are in the package. Eventually they get to the fact that they have to open the package and count. Then, I pull out a second package and ask the same question. The pattern continues for a couple more packages until the class decides that there’s variability involved and they would feel more confident with a larger sample. We collect a large(r) sample and I have groups surmise ways to organize the data. Fun with dotplots! This lesson was similar; I was required to use raisins for this one though (ick).

The other lessons typically followed the same way: collect data and find a good way to organize the data. The vast majority of the conversation revolved around what types of graphs fit what types of data and how to convert from one to the next.

I got two things out of today:
1. Teaching for 7 hours without repeating material is exhausting.

2. The lack of freedom involved with a big professional development institutes really frustrates me. I’m not saying that the activities that I’m being asked to use are bad or that I’m not allowed to put my spin on them, but I feel constrained. I’m spoiled with the ability to create my own curriculum when I teach and I want to do the same here. And working with powerpoint here reminds me of how inflexible it is for a math teacher to use.

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!

Statistics Summer Institute Day 1 (Mostly Nerves and Fear)

I have the wonderful opportunity to facilitate a two week long summer institute on using statistics in an Algebra classroom. I am working with two other facilitators to help 25 classroom teachers learn everything from what a statistical question is to the basics of probability and statistical inference. Our goals as facilitators is to give these teachers the confidence and ability to teach statistics at a meaningful level to their students next year and beyond.

Day 1: The goals of day 1 include familiarizing the teachers with the common core…er…I mean mathematics florida standards, and have them start tackling the idea of what statistics is. One of the other facilitators is tasked with starting the conversation about what statistics is and how its used.

My task on day 1 is to familiarize the teachers with the standards for mathematical practice. Preparing for this hour long presentation I figured I had two options. I could do every presentation about the math practices that I’ve seen and dryly lecture about what the are and what they should look like…or I could throw them into a lesson that I’ve done in the past and force them to live the math practices. I decided to use the car crash problem. Here’s the link if you missed it: https://corycloud.wordpress.com/2014/03/01/who-is-at-fault-for-this-car-accident/

We started with:
5.3 Intro to Radical Graphs Car Accident Problem_1 - Copy

The teachers decided they couldn’t answer who was at fault for the accident. So, I gave them:

5.3 Intro to Radical Graphs Car Accident Problem_2 - Copy

5.3 Intro to Radical Graphs Car Accident Problem_3 - Copy

They decided to check to see if the relationship between stopping distance and speed was proportional. After a few minutes they decided it wasn’t. Eventually, with some probing, they decided the relationship wasn’t linear:

skid mark scatterplot

skid mark scatterplot linear

skid mark scatterplot square root

They wanted to know the stopping distance of the white car:

5.3 Intro to Radical Graphs Car Accident Problem_5 - Copy

They surmised that the speed of the white car was about 68mph and that that car was at fault for the accident.

That part of the lesson went as planned. Some people were very comfortable with using the math practices in the classroom, some people weren’t comfortable at all, and some people decided not to participate too much in the lesson. It was your typical professional development.

There was one conversation that interested me. A few people “liked the idea” of what we did but concluded that it wasn’t realistic in a middle/high school classroom. Their argument was based on the fact that they had an end of course exam to prepare for. They needed to teach to the test. There’s no way they could do this style of lesson and get anything accomplished in time for the EOC. I tried to assure them that what we did is possible. Coincidentally enough, my geometry EOC scores came in yesterday. Geometry is the class that I’ve spent the most time with creating these styles of lessons and 88% of my class passed the EOC. I really want these teachers to buy into using these math practices for the sake of their students’ learning and loving of math. I’m going to spend every opportunity that I get to present to show them how this can look in a classroom. Hopefully, for the sake of my profession, they decide that this is how a math classroom should look.