Bankshot (I give 3-Act Lessons another try…)

Feel free to read this post, but the video I present at the bottom has since been revised. After you read this post and watch the video, I’d encourage you to read the comments. Then head over to Bankshot 3-Act Revised to check out how I interpreted the feedback.

I recently attended a workshop led by Dan Meyer (@ddmeyer) which I found to be incredibly valuable. In the workshop, Mr. Meyer broke down the 3-Act lesson design model that he describes on his blog. He demonstrated it and showed a few of his own examples as well as some of Andrew Stadel’s (@mr_stadel)I have been trying to make sense of the 3-Act model for a while now. I had continually felt confused by a handful of the aspects. I tried to do some activities but found that I was often either giving the students too much information or not nearly enough.

Bottom line, I was confused about the basic point of the 3-Act Model. It is designed in a way that maximizes engagement and allows you to raise the bar while being more inclusive (which is tricky business). The first act, you set the students up to be curious about the situation. You prepare a scenario where a handful of outcomes seem likely and ask the students to choose which of the outcomes they suspect will happen. This is a short amount of time. You get the students curious, you have them choose a camp and then move on.

In act 2, you start leading the students through the mathematical processes that will allow the students to rule out focus on the outcome(s) that seem to be the most supported by the math. This is where the students begin to explore the variables of the situations, determine the appropriate modeling mechanisms and choose which tools they are going to use. This might also be where you lecture them a bit if they are trying to use models that they are unfamiliar or uncomfortable with.

In act 3, you reveal the answer and the allow the students to make sense of any differences that “real life” has with mathematical modeling.

In an earlier post (the value of face-to-face) I commented on how powerful I find in-person, face-to-face interactions and how the #MTBoS, as powerful as it is, is unable to accommodate for this particular shortcoming. That is only more solidified in my mind now, after seeing how much better I understand the overall approach and value of the 3-Act model having gotten to interact with Mr. Meyer face-to-face, ask him questions, and hear his responses.

So… I decided to try again.

For my first Act I, I decided to go with a geometry/physics topic. Also, the Act II is still in production. So, it probably isn’t quite ready for implementation yet, but I want to see if I am actually making progress in being able to understand, deliver, and (hopefully) create 3-Act lessons.


Circles from Cedar Street

A sloppy bit of construction work makes for an interesting geometry question.

A sloppy bit of construction work makes for an interesting geometry question.

I was driving to the grocery store. This particular trip took me down Cedar Street. I drove past this manhole cover. It caught my eye in such a way that I decided to pull into a nearby parking lot and, when the traffic cleared, tiptoe out to the middle of the five lane road and snap a photo of it.

So, my mind instantly went straight to rotations. (Which is on my mind because transformations are Unit 1 of the Geometry Course that I start teaching in, like, two weeks.)

What if I started out rotations by showing this picture and simply asking the students how much of a rotation would fix the yellow lines.

My goals would be for the students to explore how to investigate an apparent rotation, learn how to visually represent a rotation, and struggle through the task of explaining it out loud to another person.

It would be okay with me if we made it to the convenience of using degrees as a descriptor of how much something is rotated. That would be up to them. I do suspect that after a short time of “about this far” and “about that much” they’ll like to find something to ease the trouble of explaining the transformation.

If you can think of a way to frame this learning opportunity better, please make me a suggestion. I feel like this is a good opportunity that I don’t want to waste.

Penny Circles

This discussion represents the final Makeover Monday Problem of the Year. I must admit, I found this problem to be easily the most engaging for me simply as a curious math learner. (Here’s  the original problem posted by Mr. Meyer.)

Now, I will admit that this is going to be more of a discussion of my experiences exploring this problem and, if you get lucky, I might come up with a recommendation somewhere toward the end.

The problem presents a rather interesting set up: filling up circles with pennies, making predictions, modeling with data, best fit functions. Lots of different entry points and possible rabbit holes to get lost in.

First change was that my circles were not created by radii of increasing inches. For the record, I can’t quite explain why I chose to make that change. Instead, I used penny-widths. Circle one had a radius of one penny-width. Circle two’s radius of two penny-widths. Circle three’s radius… you get the idea.

It looked like this:

I started with 4 circles with radii based on penny width.

I started with 4 circles with radii based on penny width.

Note: I understand the circles look sloppy. I had wonderful, compass-drawn circles ready to go and my camera wasn’t playing nice with the pencil lines. So, I chose to trace them with a Sharpie freehand, which… yeah.

Next, I kept the part of the problem that included filling each circle up with pennies.

It looked like this:

Penny Area 1

Then I filled them with as many pennies as I could

Then I filled them with as many pennies as I could

Here’s where the fun began.

I saw a handful of different interesting patterns that were taking place. If we stay with the theme of the original problem, then we’d be comparing the radius of each circle to the penny capacity, which would be fairly satisfying for me as an instructor watching the students decide how to model, predict, and then deciding whether or not it was worth it to to build circles 5, 10, or 20, or if they can develop a way to be sure of their predictions without constructing it.

But, I also noticed that by switching to the radius measured in penny-widths, the areas began to be covered by concentric penny-circles. So, we could discuss how predicting the number of pennies it would take to create the outside layer of pennies. Or we could use that relationship as a method for developing our explicit formula (should the students decide to do that).

So, if I am designing this activity, I feel like the original learning targets are too focused and specific. I want the students to be able to use inductive reasoning to predict. Simple as that. If they find that using a quadratic model is the most accurate, then cheers to them, but this problem has so much more to offer than a procedural I-do-you-do button pushing a TI-84.

First, I’d start with them building their circles using a compass and some pennies. Build the first four and fill them in. I’m quite certain that the whole experience of this problem changes if I leave the students to explore some photos of the my circles. They need to build their own. Plus, that presents a low entry point. Hard to get intimidated lining up pennies on a paper (although, I have plenty of student intimidated by compasses).

Next, I think that I would be if I simply asked them to tell me how many pennies they would need to build circle five and to “prove” their answer, I believe we would see a fair amount of inductive reasoning arguments that don’t all agree. The students battle it out to either consensus or stale-mate and then we build it and test. It should be said, though, that as they are building the first four and filling them in, I’d be all ears wandering about waiting for the students to make observations that can turn into hooks and discussion entry points.

Then I would drop circle ten on them and repeat the process. This is where the students might try to continue to use iterations if they can figure out a pattern. Some will model with the graphing calculators (in the style the original problem prefers). I’m not sure that I have a preference, although I would certainly be strategic in trying to get as many different methods discussed as possible.

Then, when they have come to an agreement on circle ten, I’d move to circle 20 and offer some kind of a reward if the class can agree on the right answer within an agreed margin of error. Then, we’d build it and test (not sure of the logistics of that, but I think it’s important).

All that having been said, I see an opportunity to engage this in a different manner by lining the circumference of the increasingly-bigger circles with pennies so that the circle was passing though the middle of the pennies and seeing how, as the circles got bigger, how the relationship of the number of pennies in the radius compared to the number of pennies in the circumference. I suspect it could become an interesting study on asymptotic lines.

It just seems like this particular situation has, as Dan put it, “lots to love” and “lots to chew on.”

“Truly wonderful, the mind of a child is”

photo credit: Flickr user "sw77" - used under Creative Commons

photo credit: Flickr user “sw77” – used under Creative Commons

I agree with Yoda. (The title of this piece is a quote of his.)

This morning, as she was finishing her cereal, my daughter was completely taken by the reflection of her cup on our table. She was passing her hand under the table and above the table and, as is her custom, talking the whole time.

Then she fired out a wonderful question.

“Daddy, why is the reflection upside down?”

My daughter asked a fantastic question. Would the average geometry student be able to answer it?

My daughter asked a fantastic question. Would the average geometry student be able to answer it?

In my head, I thought that my students should definitely be able to hammer out a pretty reasonable response to that question by the end of the first unit of geometry.

And thanks to my daughter, they will.

The Value of Face-to-Face

Today I got an opportunity to facilitate at EdCamp Mid Michigan, which was just the second time I served in any manner of leadership role at a teacher professional development workshop. The format was designed to be casual and conversational. Facilitators opened the conversation and the participants contributed for 45 minutes or so asking questions, telling stories, stating concerns and helping each other.

Sam Shah, a math teacher to whom I have received tremendous support developing a calculus class for next year recently posted a fantastic piece about the power of community. I encourage you to read the piece and reflect on how powerful a member of this sharing and learning community that you are. Each contribution matters, each “like”, each comment, each bit of dissent. We impact each other and we take it back to our classrooms.

And while I can completely embrace Mr. Shah’s post, I would like to offer a bit of a “yeah, but…”

Today’s EdCamp was a great microcosm of that greater community. There weren’t as many people, but each person was expected to contribute, because each person brings value to the table. They bring experiences, questions, concerns, anecdotes, advice… all of these parts are necessary for the community to flourish. The “mathtwitterblogosphere” or (MTBoS as it has come to be known) is a similar community. Some do a lot of writing. Some a lot of reading. But it is inclusive. (Shoot, if they’ll welcome me, they’ll welcome anyone.)

But EdCamp included one part that has been missing from my experience with the MTBoS: eye contact. That’s the one missing piece. The overwhelming majority of the teachers that I have communicated with through twitter and my blog are people who I have never met face-to-face. And while the MTBoS does it’s very best to facilitate conversations among folks all over the world… (I say that as though I have forgotten how remarkable it is that such technology even exists)… I wish that more could be done to create opportunities to get a chance to break bread with so many of the fantastic folks that I am meeting through twitter handles and avatar photos.

With that eye contact today, I tried (as Mr. Shah describes similarly in the aforementioned piece) to describe the value of the MTBoS to some math teachers who hadn’t explored the community much. I’m pretty sure I did a poor job. You see, one of the most important functions of face-to-face interactions is the power of facial expressions. Truth is, I am grateful for conversation because I often don’t know how well or poorly I’m explaining something until I see the faces people make when they are listening to me explain it. As I have moved out of my 20’s and into my 30’s, I am finding that I am doing more and more explaining to other people. The ability to look someone in the face and converse is one that I find incredibly valuable.

Bottom line: Thank you for all the support that you’ve provided MTBoS and thank you EdCamp Mid-Michigan for the support you’ve provided. I am thankful to be a part of both communities and hope that I’ve been a meaningful contributor. Today simply reminded me that while it is fantastic to embrace the resources and contributions (and amazing that it is even possible to do so) it is equally important to embrace the community that exists nearby me, too.