Wow. What a week. We had our first week of classes and our first Flex Friday.

Our schedule is an interesting mix of current tradition and pushing the envelope. The students have eight classes and there are no study halls. There are four classes every day, so the days alternate. Classes meet either Monday & Wednesday or Tuesday & Thursday. So, during this past week, I met all of my students twice. I am teaching seven classes this year. Yes, seven of eight blocks. It’s a lot. But it’s what’s necessary during this first year. So we are all doing what’s necessary. I am teaching all of the 9th graders and 2/3 of the 10th graders. That gives me about 105 students, which averages to about 15 per class. Not bad, really. I just wish I had a little more time to chat with colleagues. I’m hopeful that will come.

So what did we do this week?

## 9th grade

We had a bit of fun with Dan Meyer’s Pyramid of Pennies problem. After watching the video a couple of times, I had them write down the first question that came to mind (and then a couple more). They shared their questions with each other and with me. Many questions were common from class to class, but some were quite different. Here’s a listing:

- How tall is it?
- How much does it weigh?
- How much money is it?
- How did they keep the pennies from falling?
- How many pennies is that?
- How long did it take?
- Where did they get the pennies?
- How many pies can you buy with all those pennies?
- How many pennies are in each row?
- Why a pyramid?
- How many people did it take to build?
- How old were the creators?
- What’s the oldest penny?
- Why time lapse instead of sped up video?
- Were there any patterns to the penny placement?
- Are they real pennies?
- Who has that much time?
- Is the change between each layer a constant?
- What did they do with the pennies afterwards?

As they set out to figure out how many pennies were in the completed pyramid, I heard lots of great discussion about how the layers were formed. If the bottom layer was 40 stacks by 40 stacks, then was then next layer up 38 by 38 or 39 by 39? If there were 1600 stack in the bottom layer, then did the next layer up have 1596 because it lost four stacks from the corners? Each team made some assumptions and then proceeded with their computations from there. Several teams were able to see their plans through, making some adjustments if the numbers didn’t seem to be making sense. Other teams ran out of time. But that was okay. They had done enough to share strategies. I especially liked that when one group said, “We multiplied 40 x 40 x 13 to get the number of pennies in the bottom layer, then we did 39 x 39 x 13 to get the next layer, then 38 x 38 x 13, and so on. Then we added up all the layers to get the total,” another group said, “We did the same thing. We just multiplied by 13 at the end.” Huh? How can it be the same if you multiplied by 13 at a different time, I asked. The response: “They found the number of pennies in each layer. We found the number of stacks in the pyramid and then multiplied by 13 to get the number of pennies.” Isn’t that beautiful?

On the second day of class (Wednesday/Thursday) we played around a bit more with the patterns of pennies. Since just about every group had focused on squaring numbers to figure out how many pennies were in each layer, I asked them to take a look at these numbers: 1, 4, 9, 16, 25, … and describe any patterns they discovered. Looking for and describing patterns is key to thinking critically about mathematics (as well as lots of other things). Most groups found the difference pattern. At least one group in each class found a pattern by looking at the final digit of each square number. Interestingly, no groups attempted to represent these numbers visually. I guess we’ll have to go back to that. They were also able to tell me that the number of stack on layer *n* would be *n*^2. (Maybe someone can teach me how to show this properly using latex. I tried and tried and couldn’t get it to work.)

Then I gave them this pattern: 1, 5, 14, 30, 55, 91, 140, … and asked them to find the next few numbers in the sequence. Again, this involves looking for patterns. I also challenged them to come up with a formula, suspecting that they would not be able to. They’re 9th graders, after all. I’m happy to say that nobody gave up. They really tried to come up with a formula. They were thinking recursively, of course, but don’t yet have any language for that. Again, that’s okay. I found out a lot about these students during those two classes.

## 10th grade

Again, I turned to Dan Meyer. But this time we tried out the Penny Circle. It was really the first time I’d ever asked my students to do math through an online guided activity. I’d been through it myself, first, and it seemed pretty straight-forward and reasonable. And while the conversations during class were all good, the data that I received on the back end (teacher dashboard) was not so helpful. It’s not any fault of what Dan & Desmos put together. I love what they put together. I just forgot that I would be using it with 10th grade boys. (Yes, most of our 10th graders are boys – there are only a handful of girls in the 10th grade.) So, I got some very silly, anonymous results. Thankfully, nothing was school inappropriate!

However, it’s now difficult for me to use their data to figure out who needs some help understanding the relationship between diameter and area of a circle. Although the easy answer is: most of them. I’m not sure that this new presentation of the original problem prompted creative problem-solving and curiosity in the way that the penny pyramid problem did for the 9th graders. Maybe that wasn’t the point. But part of me wishes that I had given them some real pennies and real circles and had them collect the data that way. These kids didn’t have the opportunity to think about their own questions after watching the video. I’m just not sure their curiosity was sparked.

I’ll keep working at it. I know that I can spark some curiosity around math for this group. But they will be a bit tougher than the 9th graders. I have my work cut out for me.

## Flex Friday

One of the founding principles of Baxter Academy is that students work on projects. Big projects. Long term projects. Meaningful projects. To give kids time to work on these big, long term, meaningful projects we have Flex Friday. We have no regular classes on Friday. Instead, the time is devoted to project work (mostly). Since it was our first Friday and students do not yet have projects to work on, we teachers gave some presentations of possible projects. The ideas ranged from building a noise & dust containment system for our CNC router, to figuring out the best possible lunch program for our school, to building a greenhouse, to designing a video game, to researching the ethnomusicology of Maine. Some students have their own ideas that they are hoping to pursue this year, but the rest now have lots of good ideas to choose from.

Since we have this gift of Flex Friday time, we also thought that it would be good to get out and about into Portland. We are only a couple of blocks from the Old Port, after all. So, on Friday afternoon groups of kids with their advisors went to different locations around town. I was with the group the went to the Portland Public Library. They’ve had a recent renovation and the new building is awesome. Bright and inviting, this is a place where I would not mind spending an afternoon. We signed all the kids up for library cards – I got one, too. The little city of Portland has a great library full of wonderful resources. I can’t wait to begin exploring them all. Now, to find the time …

Hi Pamela, thanks for writing all of this up. It’s a lot of fun following your exploits. Can I ask a few follow-up questions?

1. How did you go from the list of questions to, “How many pennies are there?”

2. Did you have the students guess the answer first? How did you handle that and how did it go?

3. How did students get all the important information?

4. What kind of inappropriate responses did you get on the Penny activity? Just lousy data? Or something else?

Thanks!

Hi Dan,

To answer your questions …

1. I wanted to focus on math-y questions and not research questions. So, I suggested to my students that, if they really wanted to, they could search for information about the giant penny pyramid to find the answers to their who and why and when questions. When we took all of those questions away, it seemed pretty clear that if we could answer the “How many pennies are there?” question, then it would be easy to figure out what it was worth, how much it weighed, how tall it was, etc. So, we agreed to start there.

2. I did have them write down guess first, and write down a number that was too low and a number that was too high. I didn’t think to have them share at that point. Duh! Obviously, I should have asked them to share that information.

3. For the important information, I asked them to brainstorm with their groups (3-4) what they would need to know. I recorded a bunch of ideas that they came up with. Typically, requests for the number of pennies in each stack, the dimensions of the base, and information about single pennies (weight, thickness, diameter) came up in each class. So, I shared your Act 2 data for these requests. Additionally, I emphasized that by looking at the video, I noticed that there was one stack on top and four stacks in the layer just below the top. They also requested information like the height of the pyramid, the height of a stack, the number of layers. A couple of students attempted to count the number of layers from the paused video. I had five classes play with the Penny Pyramid. They were really into it and having fun.

4. I did the Penny Circle with two classes. I guess “inappropriate” is a bit strong. I think silly or goofy is more accurate. For example, Richard Nixon was in one of my classes. He guessed 20 pennies for the big circle. And Bob Dylan guessed 93,264 pennies.

Thanks for asking me those questions. It made me think even more about what I did during those classes. I know that it was the right way to start the year. Kids were smiling – it was clear that they were not accustomed to this kind of work in math class.