Hexagons, part III: Talking [Points] ’bout Hexagons

Apologies for leaving you hanging after my last post. The weekend. The football. These vital, important events sometimes push aside things like writing blog posts about hexagons. But I’m here now!

Let us recommence playing in our hexagon sandbox.

With the success of the Bongard Problems during the previous class, I felt empowered to try yet another new activity: Talking Points.

First we did an introduction to talking points:

And then we did a set of talking points specifically around our hexagon definitions:

During both I realized that it is pretty challenging to manage a group of students and make sure they are following the procedure correctly. My students are really good at dialogue, not so good at taking turns. My hat is off to @cheesemonkeysf on this one!

Once the hexagon talking points activity was finished we segued into some work with conditional statements. Nothing terribly innovative or exciting. Students did a quick warm-up deciding whether statements were true or false, then we discussed the similarities in all of the statements (they were all “If…Then”, they were all either true or false, they were factual). I gave a little lecture and defined some terms. Students did a worksheet to practice finding counterexamples and writing converses.

And then we were out of time. More of the same for homework.

Reflecting, one week in, on the idea of using the hexagons to introduce our geometric proofs unit I am uncertain whether or not it was a good idea. I think it remains to be seen how I tie the hexagons back in further down the road. If I stopped here and we never used them again, this probably wouldn’t be a plan I would use in future iterations of this class. But the unit has only just begun and so hopefully I will find ways to circle back around to hexagon proofs later.

I’ll be sure to report back about when and how I do that.

Hexagons, part II: Return of The Pakman

When we left our intrepid adventurers they were faced with the perplexing question:


The class was divided. Consensus seemed impossible. The original group was adamantly opposed to the idea; the rest of the class seemed open. Some students–*gasp*–seemed not to care one way or the other!

Ambiguity in a math classroom. It must have been a cold day in a very hot place…

However, that place was not San Francisco:

weather 1.6

Clearly *we* did not have school off because it was “too cold.”


Before re-addressing this delightful question, I began class by using Bongard Problems. This turned out to be a genius plan. The big idea of Bongard Problems is that there is a common defining characteristic that differentiates the left hand side from the right hand side images:


Thinking about what makes something a defining characteristic, rather than being a property of the object, helped to create a much more streamlined definition of each of our hexagons. We returned to the question from yesterday’s class.

We looked at our exemplar Pakman card again. We went back and forth, forth and back. Until finally we agreed that a Pakman was (drumroll please)…

A hexagon formed by reflecting a parallelogram across one of its sides.

Much rejoicing was had by all–though come to think of it, maybe that rejoicing was due to the fact that I said we didn’t have to go through this process for all of the other hexagons…

I passed out the remaining hexagons, printed on regular paper this time as I have just about used up my school’s entire supply of cardstock by now. Students determined whether any of the new hexagons belonged to an existing group and created new names and definitions for those that needed one.

Our final collection of definitions looked something like this:

The Hexagons

At this point we had officially run out of time. However by this point we had created a very rich and interesting sandbox to play in. Stay tuned for tomorrow’s final (at least for now) installment in the saga. In which the Hexagon Talking Points will make an appearance.

Hexagons: The First Day

The first day back from break can be a little rough.

Trying something new you’ve never done before can be a little rough.

Doing both at the same time?

Can be a magical roller-coaster of the good, the bad, and the who-even-knows-how-that-went.

On Monday I introduced the hexagons. Not having planned ahead enough to create a wooden set on the school’s laser cutter, I resorted to good old-fashioned cardstock.


I used the cards to separate kids into random groups and then each group named and defined their hexagon. This is the part that didn’t go so awesomely. Funny thing is, kids don’t know what a “definition” is. So, what they did was:

  1. Ask me if I wanted a definition of hexagon. [Thank you, but no.]
  2. Write a list of every single characteristic of their hexagon they could think of.

Not exactly what I had in mind…

But I went with it. Students wrote up their characteristics on chart paper and posted their “definitions” around the room. Then I had everyone do a Post-It activity where they drew a hexagon based on the written description. Another flaw: everyone said they remembered what the hexagons looked like from the grouping shuffle at the beginning of class, so there wasn’t much gained from this part of the class. Oh well.

Next step was to create Venn diagrams of each group’s hexagon and the hexagon to their left. The goal was to discover if it was possible to create a hexagon that fit into the intersection of the two circles. I don’t think any group was able to do this based on their definitions. We repeated with the group to the right. Same story.

That’s when things started to pick up.

I grabbed a marker and set up shop at the hexagon in the front of the room, the Pakman (aside: this group took freaking forever to come up with a name for their hexagon). I asked if there was any way to simplify their list of categories into one succinct description that contained the essential characteristics of a Pakman without additional or redundant information.


After some debate and discussion we settled on the idea that a Pakman’s essential element was that it was a hexagon composed of two congruent parallelograms.

I ended the class by asking:



And, just like the class itself, this will be continued tomorrow…

I Notice Awesomeness

…because there really is no way to have too many posts about the “I Notice/I Wonder” process.

My Math 3A (geometry) students had never done an I Notice/I Wonder before. That is the beauty of this process–it’s practically bullet-proof.

Here’s what they noticed and wondered about the side-length ratios of a 30-60-90 triangle that we played around with in GSP:







In which I write a blog post comprised primarily of tweets from other people. I think normally we call this “being lazy.”


At Asilomar this year I gave a talk on math electives, which was kind of fun:


Also I Ignited…[Alternative Title: Help–I’m On Fire!]

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But unfortunately my Ignite talk didn’t include any unicorns.

Instead, I talked about raising the value of mistake-making in the math classroom:

Some people were kind enough to tell me I did okay (even without the unicorns).

Holy crap, there were a lot of people…

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All in all, it was a fun time. See you next year!



Review Days

As some of you already know, my school operates on a trimester system. So while most of you are looking forward to Thanksgiving turkey and trips to see family and then coming back to finish out the semester, I am gearing up for finals this week. We don’t have a special schedule or anything, but I am giving finals at the end of the week, hence the title of this post. [Note: I am still doing all that looking-forward-to-Thanksgiving stuff.]

Last Thursday/Friday was my first day for review in Math 2 (one of my classes doesn’t meet on Thursday; the other doesn’t meet on Friday). I have a typical, fairly boring, packet of review problems that I was planning on handing out to students so they could do some diagnostics and figure out what they needed to work on for the next few days.

Then the Xerox machine decided to go on strike.

It was bad, people. It was very bad.

Finally, after both of my classes had met *ahem*, the Xerox machine decided it was okay and I was able to print off my review packets. I passed them out in class today. But I have skipped over an important step in my story. The chronology of this blog-post is out of whack!

What did I do with my bonus 80 minutes for which I had nothing planned, you ask?

I did concept maps:





I don’t have pictures of this part, but my genius move was, after students had worked on the whiteboard concept maps, I had them make a “final draft” on chart paper…but I had them revise a different group’s map. I couldn’t think of a good third-round that would enable them to have hand’s on for all three maps, but 2 is better than 1.

While this definitely didn’t take the whole block, it did take a good chunk of time and it helped students to focus on what concepts were in each unit so they could develop their own study guides. I wrote on the agenda that it was “Choose Your Own Adventure: Study Guide Edition.” No one commented on the reference, so I assume this generation of students all had horribly deprived childhoods. All in all, it worked out to be a pretty productive review day. And I don’t particularly like review days, so that’s saying something.



Inspiration from & Annoyance with Google Images

There should be a way to “report” images on Google, because this is not a rhombus:




And this one really isn’t a rhombus:

However, this is super cool:

And it gave me the idea, what if you had students write definitions of a shape on Google forms and then threw the text into a Wordle. Then you could go back and look at them together and see what jumps out.