More 3D Geometry


After my last post, Mike Lawler gave me all of these awesome ideas for my 3D geometry class. Considering that my class has been working on nets, I was most fascinated by the dodecahedron that folds into a cube, which came from Simon Gregg.

When I first watched the gif animation, I just couldn’t figure out what was going on. I thought, “I’ve got to show this to my students!” Thursday was that day. I tasked them with a build challenge. Of course 55 minutes wasn’t enough time to complete anything, but students had drawings (which gave us insight into the construction)

a CAD rendering (completed during a snow day)

and a previously constructed dodecahedron that had been re-purposed (completed during lunch).

So, thanks Mike, for the inspiration, and thanks #MTBoS for being there helping us to support each other.

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“How Do We Know That?”


I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?

I really felt that before we could go down the road of answering that question, my students needed a bit more knowledge and exploration around these shapes, and maybe some thinking around tiling the plane would help, too. So we spent some time trying to draw them, counting faces, edges, and vertices, visualizing what they might look like with vertices cut off, unfolding them into nets, and wondering why regular hexagons tiled a plane, but regular pentagons did not. We played around with the sides – a lot – and even talked about this thing called vertex angle defect. Then we returned to the question of why only five. Students were able to connect the need for some defect (angles totaling less than 360 degrees) and the ability to create a 3-dimensional figure. Through the investigation, they were able to see that the only combinations of regular polygons that worked (by sharing a vertex) would be 3, 4, and 5 equilateral triangles, 3 squares, and 3 regular pentagons. They could give solid reasons why 6 triangles, 4 squares, 4 pentagons, and any number of other regular polygons could not be used to create a new Platonic solid.

I had not anticipated this question, and had not included it in my plans. But, because it was asked, thankfully, by a student, it pushed us into thinking more deeply about these shapes (and their definition). And, ultimately, my students were able to answer the “why only five” question for themselves.


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Stacking Pennies


There are lots of things I love about my classes – my students are at the top, but I also am enjoying teaching a course for the 2nd or 3rd time. This activity is for a statistics course called “Designing Experiments and Studies.” It’s a course that lasts for one trimester (about 12 weeks) and we have just completed week 4. The Penny Stacking activity is an introduction to experiments. I ran this lesson last week (Jan 12 & 13).

The rules of penny stacking are simple:

  • You can only touch one penny at a time.
  • Once the penny is placed on the stack, you cannot move it.
  • Stack as many pennies as you can without having the stack fall over.
  • Each student stacks pennies once, with either their dominant hand or their non-dominant hand.
  • Students are randomly selected to stack pennies with either their dominant or non-dominant hand.

Before embarking on the experiment, we made the following predictions:

  • More pennies would be stacked with the dominant hand (although a few students disagreed and thought the results would be the same for both hands).
  • A few students thought the ratio of pennies stacked with dominant hand to non-dominant hand would be 3 to 1.
  • The range of pennies stacked with dominant hand would be 15-45 pennies, while the penny stacks from the non-dominant hand would range from 10-35 pennies.

Then it’s off to conduct the experiment. This is pretty tricky given that up to four students sit at a table and our building is so old that if someone walks across the floor upstairs, our floor will shake. Here are the results:

penny stacksHow would you interpret these results?

We calculated the means: dominant -> 30.9 pennies; non-dominant -> 26.25 pennies. It’s clear that, on average, the number of pennies stacked with the dominant hand is greater than the number of pennies stacked with the non-dominant hand. But is that difference in the means (of 4.65 pennies) significant? Is it unusually large? Is it more than what we might expect from randomizing the results?

To check this out, we randomize the results. Each pair of students received a stack of cards. Each card had a result. The partners shuffled up the cards and dealt them out in a stack of 10 (for the dominant hand) and a stack of 8 (for the non-dominant hand), calculated the means, and then subtracted (dominant – non-dominant). They each did this a couple of times and we made a histogram from the results of the randomization test.


(It’s a Google Sheets histogram – I don’t know how to get rid of the space between the bars)

If you look at our difference of 4.65 compared to these randomized results, it looks pretty common – not at all unusual – to get such a result. If you think that our randomization test was too small (with only 24 randomizations), then you can use the Randomization Distribution tool from Core Math Tools, a free suite of tools available from NCTM. And it’s the only tool that I know that runs this test effectively. Here are the results from 1000 runs, just like the card shuffling but faster.

more pennies

penny summaryAnd you can even get summary statistics that show that our result was within 1 standard deviation of the mean of the results that we got from randomizing the data. Not a very unusual result at all.

We followed this up on Thursday with an experiment inspired by an example from NCTM’s Focus in High School Mathematics: Reasoning and Sense Making – memorizing three letter “words.” Based on the experiment described in the book, I created random lists of three letter words and three letter “words.” The lists of words were meaningful, like cat, dog, act, tap, while the lists of “words” were nonsense, like nbg, rji, pxe, ghl. Students were randomly assigned to receive either a list of meaningful words or a list of nonsense words. They were then given 60 seconds to memorize as many words as possible. Like with the penny stacking, I made them predict what they thought the results might be. What would your predictions be?


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Working Together


Like all schools, my school has committees. And, like all schools, my school has a student government. But, unlike other schools, our student government is a single body: the Student Senate, comprised of one representative from each advisory. This year, the Student Senate split into subcommittees to align themselves with the faculty standing committees. I’m on the Academics Committee – the group charged with looking at curriculum & standards and how they are aligned. On Monday, we met with the Senate subcommittee during lunch to discuss different ways that students can demonstrate what they’ve learned. We talked about the possibility of students creating portfolios and presenting their portfolios to teachers in a particular learning area to have them assess which standards the student has met, and to what level. So, while the teachers keep working on tightening up our standards & curriculum (we’re halfway through year three), our students are developing a portfolio review process.


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About Me

It would appear that I can’t resist a challenge from the #MTBoS. And so, mtbos blogthis January, I’ve been challenged to update my “About Me” page. Feel free to check it out. I’ve updated it with a bit about the school where I teach and what happens when I talk about teaching at a public charter high school.

By the way, we will be graduating our first class this year, and that’s pretty exciting. Even more exciting is the fact that they’re actually getting accepted to college! (Of course they are. But as the first group, this is still pretty exciting for us.)

I’m hoping to blog a bit more this year. A bit more than once a month (or less). But I’m not promising anything. I’m still building brand new classes – which takes a ton of time. I have a couple of drafts still in the pipeline that need some attention. Maybe I’ll get to them soon.

In the meantime. Thanks for the push #MTBoS (and Tina C). And Happy New Year!

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Pentathlon Day!

I’ve written about Flex Friday before. It’s the day of the week when we don’t have classes. Instead, students work on their own long-term projects. It’s our attempt to teach them those valuable, but intangible skills around autonomy: time management, self-direction, interpersonal relationships, problem solving, resilience, and so on. It’s been going fairly well, but we felt that we could improve the experience. So this year we designed an experience to orient our 9th graders to what Flex Friday is all about. It was called the Flex Friday Pentathlon.

How it Worked

Our 9th graders were randomly assigned to 11 teams of 8-10 students. Each team was presented with five challenges:

  • Tug of War robot Challenge
  • Autonomous Robot Maze Challenge
  • Scientific Inquiry of Mysterious Liquids
  • Media Madness
  • Art of the Pitch

Each team was to decide how many of the challenges to tackle and who was to work on which challenge. Teams were allowed to work on as many of the five challenges as they wanted, and they were encouraged to tackle at least two of them.

What Happened Next

Teams were assigned on our the first Friday of school, September 11, and were introduced to the Pentathlon concept and the specific challenges. They spent the next couple of Fridays deciding which challenges to work on and creating their Gantt charts to help with organization and planning.

Robot-based teams chomped at the bit for their parts so they could start building – something. Young scientists began researching the types of tests they could perform to distinguish one liquid from another. Media types brainstormed the type of message they could produce with 1 minute of video. No one wanted to pitch anything.

Teams worked together – or they didn’t. Each team had a project manager and struggled with leadership, to some extent. Some of the upperclass students observed difficulties and stepped in to mentor the leaders. We (the adults) created afternoon workshops about how to research, edit video, design and build stuff, at the request of the 9th graders, to help them out.

The Final Friday

Their last day to work together as a team was November 20, two weeks before Pentathlon Day. Some teams were on track, while others were falling apart. My Media Madness team finally filmed their story. Now they had two weeks to edit it into a final cut. My Mysterious Liquids team had done all the research they could – they tested liquids using probeware, they researched characteristics, they asked for test liquids and analyzed them. They were ready. I also had two robot teams – one Tug of War and one Maze Runner. The tug of war team was falling apart, and there wasn’t anything that I could do to help them. There was plenty of frustration and blame to go around. One member of the team was absent. The other two were frantically trying to make a working robot.

On the Maze Runner team, it was all about programming. In all, there were three Maze Runners across all the teams. They were struggling with the programming. Two teams joined forces to try to learn together and help each other out. They shared code. They taught each other how to program an arduino, the processor that ran the robot. Just before lunch I heard, “We quit!” It wasn’t working. They had poured 10 weeks into the thing and couldn’t get it to do what they wanted. They were spent. I told them to take the afternoon off (there were other activities available). They went to lunch, and when they returned they asked if they could keep working on the robot – they’d had an idea!

Pentathlon Day, December 4

The Pentathlon ran from 10:30 until 1:30 on Friday, December 4. We all (9th graders and their project advisors) arrived at the Maine Irish Heritage Center (which feels like our partner campus – we can’t fit the school in our own space anymore) to witness the results of our 12 weeks of hard work. There were robots tugging war (eventually), robots running a maze (eventually), minutes of media madness, scientists investigating mysterious liquids, and, eventually, one pitch. It was a day where I worried that students would not have enough to keep them occupied – and they did run out of steam as the day wore on – but, overall, they rose to the challenge.

So, What Happened?

My Maze Runner team was up first, with no other events happening, so I was able to catch them on video. It was about what they had expected: They were stumped at how to program the robot to make a decision about turning left or turning right. As a result, they programmed it to always turn right. Here’s what happened:

I was on duty at the Tug of War Robot station. There were 7 robots at the beginning of the day. The first two robots to battle didn’t do much once the power switches were thrown. That caused the other teams to rethink about what they had created. We had a series of non-battles. In the meantime, that Maze Runner team from the video asked if they could enter their robot in the tug of war challenge. They were given permission to adapt their robot, but it had to take its power the same way that the other robots did – meaning that they had to redesign the power supply (from a 9 volt battery to direct input) and reprogram the robot. I have to say that I was impressed with what they accomplished in such a short amount of time. They were able to field a tug of war robot that did pretty well against the robots that didn’t move – but not so well against those that did move. After all, it wasn’t designed to pull anything. Here’s what happened in a rematch (that’s the little Maze Runner at the top of the frame):

Was it Worth it?

I mean, was it worth segregating the 9th graders from the rest of the school during these 12 weeks. Was it worth making them work on these challenges designed by adults, instead of letting them propose and work on their own project ideas? Was it worth trying to teach them some foundations of project management and project planning? Ultimately, the answers to these questions won’t be known until the end of the year. We will have to see how these 9th graders integrate themselves into the existing projects, or how successful they are at proposing and completing their own projects, or how they are able to continue with teacher-designed challenges. Are there things that I would do differently? Of course there are. This is the first time we’ve done this. Iteration and improvement is what Baxter is all about. Am I proud of what I saw the 9th graders do on December 4? Yes. I was skeptical – I’ve been teaching for a long time, after all – but, yes, as a group, they rose to the challenge. They might have slacked off earlier during the term. But when it mattered most, they worked hard – and continued working hard – until the last moment. And they learned about themselves from this experience.

That’s the point of Flex Friday.

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Intro to Statistics (Unit 2)

During the first few days of the new unit students explore relationships in the data we had collected about the class.

Back to the Class Data

Looking at class data gives us the chance to ask questions about relationships among the variables. Here are some questions my students came up with:

  • Is arm span really equal to height?

The easiest way to dig into this question is to look at a scatter plot of the data. So, we plotted the variables, along with the line height = arm span.

11-22-2015 Image001

We noted that two people were on the line, two others were very close, and the rest were either above or below the line. What do those points above the line mean about height and arm span for those people? What about the points below the line?

  • If my hand span is longer than my wrist circumference, then shouldn’t I be able to wrap my hand around my wrist and touch my pinky to my thumb?

11-22-2015 Image009

One hundred percent of students had longer hand spans than wrist circumference, but only a couple of students could wrap their hands around their wrists.

  • Is the age (in months) related to any other measure?

11-22-2015 Image010

It would seem that none of the other variables is a good predictor for age in months. It also seems as if age vs height has a negative association. Huh?

Digging a Little Deeper

If the line height = arm span doesn’t describe, or predict, that relationship well, then what would do a better job? We added a “movable line” and adjusted it until it looked about right.

movable line


Our line predicted that height = 0.85 * arm span + 26 cm. Wait, what? Height is 85% of arm span? And what is that +26 cm all about? It made for an interesting conversation, especially this question from a student: “How can a person who has an arm span of 0 cm be 26 cm tall?” Which prompted: “What does an arm span of 0 cm even mean?” I certainly don’t have definitive answers to these questions. What I can do is encourage the curiosity, the conversation, and point out that the relationship we discovered is for these measurements. Does it make much sense to use our calculated relationship to make predictions about heights for arm spans that are relatively far away from the data we collected?

Correlation, Causation, Outliers, Influential Points

All of these topics follow from this initial discussion about the class data. Ultimately, students once again find their own variables of interest and complete an analysis demonstrating what they’ve learned. This time topics included unemployment rates, marriage rates, divorce rates, distances & temperatures of celestial objects, height & weight, obesity rate & life expectancy, and mean snowfall & mean low temperature.

Once again, the variety of topics that interested my students is greater than what I could have come up with. More importantly, because they chose their own variables, they were interested in analyzing the data and answering their own questions.

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