## 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.

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.

## How Far to the Horizon?

In geometry, the focus is on circles. We have finished a unit consisting of parts of circles, arc length, angles in a circle and segments in a circle. I decided to give them the following problem as a review:

I’ve been to Chicago a few times, and have gone up to the observation deck in the John Hancock building each time.

I originally played with this question in Algebra II and used this to set up a need for solving square root equations (which google was able to give me for the distance to the horizon…and a formula which one of my students found quickly on their phone). This time, in Geometry, I wanted to calculate the distance without any formulas. My students’ first question was how far off of the ground was that picture taken:

We discussed which of those heights was most appropriate and decided to use the height of the observation deck.

This is where I had to help them out a bit. I had to give my students a nudge in the right direction.

They figured they needed to use circles, so they asked for the radius of the Earth.

As we drew on the Earth, my students noticed that my line of sight from the observation deck created a tangent line. We were able to create a right triangle, and use what we know about right triangle trigonometry to calculate the central angle created between the building and my line of sight. Once we had the central angle, we used arc length and calculated the distance to be approximately 39.3 miles.

And here was the answer based on what I was able to find online:

## Mr. Cloud Likes to Run

In geometry, we started a unit on angles and segments in circles. Since we are at the end of the softball season, I decided to use a situation that I’ve encountered to help them figure out how to calculate arc length.

I like to work out and run during the hour I have after school and before needing to be at softball to start pre-game warm ups. In order to save some time, I usually run around the school or on the softball field. So the question arises:

I define a pole as from the left field foul pole to the right field foul pole and back. My students immediately asked for some measurements and some sort of “proof” that the field was circular.

My students quickly figured out that the 90 degree angle made by the foul lines was a quarter of the circle and tied that to the idea that they were using a percentage of the circumference. As a group, we came up with a general formula and then worked some practice problems dealing with arc length.

During the past few years of teaching Algebra II, I have found that teaching students to recognize and understand the different types of discontinuity in a function is quite difficult. I’ve found it even more difficult to have the students discover the different discontinuitites for themselves. This year, I decided that a Wizard of Oz analogy was the way to go. Here’s what I want my students to be able to work with:

And here’s what I set up in the front of my classroom:

The idea is that the path is not continuous (my students determine this for themselves based on their definition of continuous that they develop for the warm-up to this lesson) and I have booby-trapped the path with each type of discontinuity for them to discover and define.

Removable Discontinuity:

Notice that brick 3 is conveniently missing. My students were quick to notice a discontinuity. I asked them to describe what happened and to name the discontinuity. With some probing, they got to the idea that the brick has been removed and they coined the term “removable discontinuity.”

Jump Discontinuity:

In each of these cases, none of the bricks are removed, they’re just not part of the same path. My students were quick to notice this and decided that these aren’t removable discontinuities. Since they had to jump to get from brick 6 to brick 7, they named this discontinuity “jump discontinuity.”

Asymptotic Discontinuity:

My students thought they were at the end of the road when they got to brick number 12.

However, I told them that there’s a brick 14 in the next room.

Since we had spent some time recently discussing asymptotes and how they apply to exponential functions, my students quickly figured out that the wall represented an asymptote and that we had an asymptotic discontinuity.

This path will stay on the floor of my classroom for me to refer back to when we discuss any discontinuities of rational functions. This way, I hope that they can have a meaningful way to connect something abstract in an equation or graph to their definition of each discontinuity.

## Mr. Cloud Makes Dinner Part 2

So yesterday’s and today’s lesson in geometry started with this picture:

The question asked of them was “Will the pot of sauce overflow when the meatballs are added?” My students were quick to break that question down to “how many meatballs will make the pot overflow?” They’ve gotten to the point that they ask for measurements very quickly:

They knew that we’ve been working with volume, but we had not discussed cylinders yet (they were at least calling the pot a cylinder). They asked about how to do the volume of a cylinder and that turned into a 30 minute lesson involing some basic calculus ideas. The bell rang and they didn’t get their answer.

Today’s lesson started with a review of the homework from the previous night. Once they felt grounded with cylinders, we brought back up the measurements. They calculated their volume of the pot. They quickly realized that the meatballs are spheres. On went a lesson about the volume of a sphere. Once they were okay with that, we got some measurments:

I didn’t give them every meatball’s diameter, but they decided that 4.5 cm was a “good enough” estimate (they actually found a use for an average). Then came their work:

The general consensus was that 11 would be okay but meatball #12 would be too many. Tomorrow we get to see if they were right!

## Is Our School’s Kicker Better Than A Professional?

I have the luxury, nay, the honor, of teaching the high school varsity kicker in my statistics class. He’s been talking about how good he is, and I wanted to give him the chance to show off. We are learning how to run significance tests for proportions and I decided to take a different approach in how we addressed the topic. I didn’t tell my students about what new significance test we were learning, rather, I just gave them this:

The kicker’s picture has been removed for obvious reasons. The slide did not have any of the writing on it; we added that as needed during the lesson. I asked my students how we could answer who was a better kicker. The first thing they decided was that they needed to see our kicker in action.

I asked him to kick ten field goals from 40 yards. 9 out of 10 isn’t bad. Definitely better than I would ever do. Then my students wanted to compare what our kicker did to the professional. I was ready for them:

We spent some time dissecting the statistics and decided that the only important value was that the professional was a 0.729 career kicker from 40-49 yards. A lot of great conversation came from whether this was a fair comparison, what stats were relevant, could we treat this as a population value, etc.

We started running the significance test, and my students led me through what should happen. They made the hypotheses, decided they needed a proportions test, figured out that a binomial distribution applies, and ran the test.

They figured out that the sample size that our kicker kicked wasn’t large enough for us to draw any usable conclusions (and I set that up on purpose). My students were so into it that they asked how large of a sample size was needed to draw valid conclusions.

I wasn’t expecting them to ask that questions, but loved that they did. They took ownership in the problem and drove the conversation themselves. All I did was guide the conversation and point out what they already knew!

## How Long Until Mr. Cloud’s Out of Water?

Today in geometry, I had an empty 2.5 gallon container of water. I raised the question “How long did it take for me to drain the container?”

The first step was for the students to decide what was a reasonable guess:

Once it was time to get to mathematical business, they quickly asked for some measurements. We’ve just started with calculating volume, so they decided for length, width, and height:

There was a quick conversation about the water level at the start of the draining, and a quick conversation about the accuracy of our calculations since the container isn’t a perfect rectangular prism.

Finally, someone asked how fast the water drains out of the container. I couldn’t answer that question for them, but I could give them this:

It took me 13 seconds to fill up the smaller container. My students were content at this time and went about their calculations. Most of the calculations were along these lines:

Seemed reasonable. Most people were in this ballpark. So we got our answer:

This confounded them. They weren’t right. They looked to me for answers. No calculation errors were found. They concluded that the flow rate wasn’t constant and that the amount of water in the container helped determine how fast the water comes out. Huge insight!! Now if only we offered physics!

## Mr. Cloud Cooks Dinner

My wife was going to be late getting home from class and I was on my own for dinner. I was going through the fridge and noticed that it was time for a cleaning. We had some veggies and such that weren’t quite in a condition to eat…so I thought that it was a perfect opportunity to make a geometry lesson on cross sections!

The idea is for students to have to give their best interpretation of what they think the cross section of each cut would be. This lesson isn’t anything too special or ground breaking. Just a nice twist to give my students a real-world look at what cross sections are and how they’re already familiar with them.

## Rumors and The Pentagon

Two short problems from class today:

First, in Algebra II, we started discussing logarithms. Rather than just jumping straight into the definition, I gave my students an example (with a lot of pitfalls) that made a logarithm a necessary tool to use. Here is the situation:

Josh has decided that it would be great not to have to attend school next Wednesday (April 9th). Therefore, he starts a rumor that schools will be closed that day. So, today, he told two of his friends that school would be closed. On the next day, each of these students tells 2 students and on consecutive days, each of the new students tells 2 more students and so on. If there are 1,730 students at Florida High, was started early enough for everyone to have heard it?

A lot of interesting conversation ensued. After a few minutes working with the problem, my students ended up at the following consensus:

They decided that, at most (which took them a while to realize that some students may tell the same friends), 1023 students would hear the rumor, which is not enough. One student pointed out that if Josh had started the rumor one day sooner, everyone would have found at (at least by the numbers). There was great conversation about the pattern for the number of new people that heard the rumor and the pattern for the total number of people that heard the rumor were exponential. Some formulas were derived. Then this happened:

The first question turned into a simple plug and chug in the formula issue. The second question led to some good discussion. My students decided that they needed something new…something they didn’t have…something that would allow them to find a missing exponent…a logarithm!!

Secondly, in geometry, we derived the formula for the area of a regular polygon. The interesting conversation came from an application problem.

I gave them some interesting information about The Pentagon. I gave them the area of the building and the apothem (without telling them that’s what they had) and asked them for the perimeter.

I know, it’s a really straight forward textbook style of question. I was excited, however, to see how interested some of my typically unengaged students were with this question. The really exciting part was when I asked them how long it would take Bethany to walk the perimeter of the E-ring. There wasn’t guessing, there wasn’t assuming…they went straight to: get her up in the front of the room and let’s do some measuring. They decided to use proportions and they decided that they needed to convert their answer to minutes. It’s really wonderful to see how they’re taking ownership in the problems and are driving the conversations. I did very little probing. I was truly facilitating and only offering suggestions. I can’t wait until this model can be fully implemented!

**Notes: I didn’t think up either situation. I found the rumor problem in a textbook and the pentagon idea in another blog. I just took them and made them fit my classroom.**

## Two Area Problems

Geometric solids week is coming! Prior to starting a unit on surface area and volume, I wanted to spend a couple days working on their concept of area. I pillaged a couple of text books for some ideas.

Problem One:

We ended up having a great conversation about the best way to answer the question. They immediately wanted to tell me what they knew. I showed them the results from the poll here on my blog (thanks to all of you who voted), and asked them to vote themselves:

Then I asked them to justify their votes. Some students started to go into defining what larger meant (a really important idea). They wanted to give me that they “knew” that the soccer field is larger, but did not have a mathematical justification. Finally they asked for some measurements:

Some calculations ensued:

They eventually decided the area of the soccer field was larger. A student made a great point that the football field is longer and that we had to make sure we were clear about how we defined “larger.”

Problem Two:

These deals were taken straight from Papa John’s website. They immediately wanted measurements.

I wanted my students to explore which deal was better, and sent this problem home with them to work on. The work you see on the slide did not come easily. They came back with thoughts that I wasn’t expecting. I felt like this question was tailor made for area, yet a few of my students didn’t go there at all. They wanted to make the argument that they get “more inches of pizza” with the larger pizza deal, so it was worth it. We took some time to explore if an extra inch of pizza is the same whether you start with a one inch pizza or a fifty inch pizza:

They decided that that an extra inch isn’t always the same in a circle. Eventually, with more probing than I expected, we got to the idea of a unit rate. I know they’ve seen unit rates and they’ve seen area, but I don’t know that they’d seen them together before. To everyone’s surprise (including mine) the unit rates for both deals works out to cost about \$0.06 per square inch of pizza. Therefore, there isn’t a better deal…only how much pizza you actually want!