Hexagons, cont.

Friday, we dove into a modified version of Christopher Danielson’s hexagon activity as partially described in my previous post. Instead of coming up with classifications for each individual shape, I asked students to group the shapes. Not a lot for them to go on. Here’s what I saw…

Types of Groupings

  • Favorite Numbers
  • These pieces fit like a puzzle
  • They’re long and skinny
  • They look the same

Not a lot of math vocabulary and it was pulling teeth in response to my guided questions trying to elucidate SOMEthing a little more precise.

Number of Groupings

  • Just enough so that each hexagon is in one group.
  • Until I asked them to mix the hexagons up again and make new groups

This number of groupings makes me really anxious for quadrilaterals. I know students struggle with the idea that a shape can be more than one classification. Any tips or suggestions?

ALSO this makes me realize that I haven’t done enough work to foster group talks in my classroom so far. I’m looking at blog posts like these to help me. Any others you would suggest?





Hexagon- the first spark

This was my warmup today:
Draw a hexagon.
Draw a six-sided shape.

I followed up in our discussion with “Is a hexagon a 6-sided shape?” and “Are all 6-sided shapes hexagons?” Tomorrow they’re going to be in groups to categorize the shapes from Christopher Danielson’s Hierarchy of Hexagons project. These are my probably incoherent notes:

1st Period: Tenth graders were taking the ACT Plan, so they were not writing down warmup. Still had time to have a lively debate about hexagon- I had asked them to draw a hexagon and a six sided shape. Then I asked if they were different? Lots of people said yes, so we had to look up the definition of a hexagon

3rd Period: 4 9th graders only due to testing. Had them do the warmup- they focused on the definition of hexagon as having 720˚ internal angles. Were able to prove a polygon a hexagon by reasoning using the right angles. Then inferred that this example meant their previous notion of what a hexagon was needed to expand. WHOA. Then we worked on groupings- group the hexagons from Chris Danielson however you want. All 4 of them put each hexagon into one category only- I asked them if they thought a hexagon could be in more than one category. They were not convinced, but I could also tell they were rapidly losing interest. Next time I would try to push them more to write down more categories. Pics (disclaimer: one board is mine).

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5th period: Got hung up on the terminology “irregular hexagon”. Big argument (unsettled) over if “irregular hexagons” could be called “hexagons”. S: “Why do they have the name irregular hexagons then?” Good question. First time seeing some people draw a six-sided shape as a 3-D rectangular prism or cube- different interpretation of the word “side”. Reminds me of Chris Danielson’s video “One is one. Or is it?

Photo Oct 30, 12 53 30 PM

6th period: Much less debate than 5th period. More students came up to board. Used online definition of hexagons, was able to use that to prove other shapes were hexagons. The 3D shape came up again. I felt good that we forced the need to define hexagon, especially with the 2-D “flat” and the fact that the sides didn’t need to be any particular length.

Many students had an idea of how a hexagon was “supposed” to look, probably based on a vast majority of experience with a hexagon being regular and not as much, or any, experience with irregular hexagons. Makes me realize how important it is to show a bunch of different examples to use to test any definition. Also makes students start thinking about how to go by definitions instead of just looks.

Small Groups

Was able to do small groups with proofs today. I let them self-select into my facilitated group while the rest of the class tried on their own and checked against answer key posted online.


Identified where individual students had trouble

Automatic buy-in because of the self-select

Students who were working independently were excited to try the proofs on their own

Let students get to really talk about ideas- especially one student who hardly participates suddenly jumping in all the time

Not awesome

I know there are some students who should have been in my group who weren’t. This is totally on me because I should have snagged them anyways, but just didn’t have the energy to fight that battle today.

Only got through 1 proof

Still have a nagging feeling I was doing too much talking. Maybe they weren’t quite ready? Fine line where productive struggle separates “blindly following teacher’s method” and “floundering in the dark”

On a related, but separate note, I just came across this article/research paper by Jo Boaler regarding her research on math education. This is a must read!! http://nrich.maths.org/content/id/7011/nrich%20paper.pdf

Group Presentations

Today in Geometry we did group presentations. I struggled with this because I had not given very much feedback to the groups on their proofs prior to their presentations. The bad news was, an overwhelming majority of the groups’ proofs were glaringly incorrect. The good news was, this process allowed to me to take specific notes about where the groups’ misconceptions were- did they mix up words and symbols? Did they know how to use a certain piece of given information? Super enlightening. On the other hand, can I justify that if a proof was correct, the entire group knew how to do it?


In dance, we often have to stop and find our center. I need to do that with teaching as well. I’m in my third year and it is feeling scarily like my first year all over again- noone wants that.

Where I’ve lost focus:
Creating quality lessons that highlight problem solving

How I’ll get it back:
Make a point to check on blogs DAILY- other teachers around the country/world inspire me. I need inspiration to feed my growth! MTBoS is like Miracle-Gro for math teachers.
WRITE MY OWN BLOG. As Dan pointed out, it’s been a long long while since anything came out of here. Force accountability by having someone check my blog every day!
Get friends in industry to give highlights for qualities/traits that they have found most useful in their jobs. Shoot a quick video and show to students- rewrite goals for math class. What is most important?

Where I might struggle:
Too much negative re: chatting/classroom management –> As lesson quality increases, student engagement increases, student redirection decreases.
Too much to do –> Focus on priorities, keep running list. Lesson planning/blogging comes first!!

Ok, I think that is good for now. Any comments/suggestions/encouragement/feedback/accountability is more than welcome!

Hallmark ain’t got nothin’ on us.

We did an awesome project today inspired by Fawn.  Coming up on area of composite shapes, I needed a way for students to practice the composite-ness and the “splitting up” skill. A popular text book problem asks about pools and tiling/grass/etc around the pool in the yard, so I thought about having them draw a pool and do all that jazz, but I couldn’t really get that idea together.

A popular text book problem asks about pools and tiling/grass/etc around the pool in the yard, so I thought about having them draw a pool and do all that jazz, but I couldn’t really get that idea together. As it turns out, there’s this kinda big holiday called Easter Spring coming up, in which Hallmark has secularized another Christian holiday for a profit. So I had my kids make greeting cards at the end of class yesterday (after they worked on a “traditional” worksheet on area of composite shapes in groups) and photocopied them after school. I was pretty vague about what we would be using them for on purpose, which drove my students a little nuts. Some of the kids even came up to me after school yesterday to ask what the cards were for. Mostly, though, they liked the elementary school throwback to arts and crafts.

Today, the entirety of the class was devoted to this one question: How much of your card is left uncovered?

Here’s what my awesome students came up with:



My thoughts:


  • Different ways of seeing- most students broke up the shapes and then subtracted those areas from the card area, but some broke up the non-covered areas to calculate it directly.
  • Differentiation!!!!!!!!! Some students took a long time to figure out how to split their shapes, some students were able to do some but not all calculations, some were ready to start splitting into smaller shapes to be more precise, and some students were even able to finish all calculations and write a paragraph to explain their reasoning
  • Foreshadowing of calculus: the smaller we split our shapes/the more shapes we calculate, the closer we can get to the actual area
  • Using the document camera after about 5-10 minutes of work to show classmates’ examples helped tremendously
  • I was able to have some students put fewer shapes on their card and other students put more shapes on (more differentiation!)
  • Copy the cards! So helpful to have a paper version to draw on so they could still use their cards as cards
  • Most students engaged


  • Some students had a hard time getting started- I ended up suggesting that they start by finding the covered area instead but a lot of them still didn’t understand what to do until their classmates presented their beginnings. Is this a problem? What other scaffolded questions could I ask to help students get started?
  • How can I/ should I make the connection to calculus more explicit?


Go to the dollar store. Not Target.

  • Rulers
  • Calculators
  • Coloring supplies
  • Colored notecards
  • Foam sticky shapes
  • Stickers

Why you should still have an analog clock

Teaching ACT Math this semester is actually really fun- I get a chance to dabble in teaching all kinds of math. The main difficulty with it is that I teach 9th graders. Who are very interested in college, but the ACT is just so far away…

I try to incorporate sample problems into the Warmup/Do Now/Bellringer/pickyourflavor. Today’s was an especial gem that led to a lot of rich conversation.

Screenshot 2014-01-26 10.46.37

We talked about drawing a picture and then some kids admitted they were stuck. One student volunteered that she split the clock into 4 90˚ chunks, and then split that in half to get 45˚ but she still wasn’t sure of the answer. Another student took that idea and said he split the 90˚ into 3 chunks since there were 3 hours so each hour is 30˚. Since we were looking for 1 o’clock, he reasoned, it is one 30˚ angle (choice B)

Then another student volunteered that instead of splitting into 4, she split the 360˚ circle into 12 and got 30˚. She had a hard time coming up with a reason for why it was 30 and not some other multiple of 30 when I had asked her earlier, though.

Finally, I introduced the idea of proportions (1/12 = ? / 360) and showed how it was just like what they had done.

We were really riding a high as a class, one girl was like everyone in here is so smart, and I was like, maybe I we can just take this gig a little further. So then I asked them what time each of the other answer choices would show. They told me 45˚ would be 1:30 because “it’s a whole 30˚ plus 15˚, which is 1 hour plus half an hour”. This was my proudest moment of the day today- maybe my constant number sense modeling (a la Jo Boaler) is finally taking root in their freshman brains!

NCTM14 -> New members?

NCTM 2014 left me ON FIRE. (Almost literally, it was so hot walking around on Saturday.) But seriously, it left me hungry for more: more PrBL in my classroom, more student conversations, more teacher online collaboration, more ed conversations, more all of it.

One conversation that I am glad to see continuing is the one centered around getting younger teachers in the classroom to participate in NCTM. An NCTM rep said the average member of NCTM is 57. Whether or not that statistic is true, here is what I’m interested in exploring from that conversation.

A lot of young teachers I personally tried to cajole/blackmail/etc into coming to NCTM wanted to come, but couldn’t find the $$$. So, how can we make that more accessible to them? Give a “new teacher” discount” a la student discount (how do you define/prove “new teacher”)? Give a first-time attender discount? Give scholarships? A quick look at the NCTM page gave this single conference scholarship for first-time attenders as well as a similar one for prospective teachers. There’s also some other grants and scholarships but most of them have a requirement that you have to have been teaching for at least 3 years (which disqualifies me until next year). What if MTBoS banded together to come up with a scholarship or 2? I’d be willing to chip in a few bucks.

Geometry Instruction [MTBoS #2]

Lately I’ve been reflecting on feedback from my students- it’s the end of quarter 1 so a natural opportunity arose. Many asked for me to have “more examples”.

I’m not sure if “more examples” means:

  1. More lessons that are traditional “I do one example while students furiously scribble into notebooks, we do one example where I cold call from the class, you do 20 of these silently and independently” OR
  2. More examples/experiences on how/where to apply concepts we are learning (this one’s a bit more nebulous for me)

If I’m not doing enough of 2, then I’m failing as a teacher.

I lean towards pretty much flat out refusing to do much more of #1 because I never want my students to think that math is about mimicry and that I can somehow magically model every geometry problem they would/could ever encounter. Also, geometry in TN is not a state-tested subject, so there is not a lot of pressure towards an end-of-course performance test. This allows me with a lot of flexibility to make my curriculum a little more… shall we say progressive? I resonate with and align my core teaching values with many of the values that I’ve observed in the rest of the MTBoS re: quality math tasks.

Enter Dan Meyer.

No one I respect thinks students should discover all of geometry deductively. … To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.

Which has forced me to reevaluate and define the difference between meaningful struggle (what Dan would refer to as perplexity) and frustrating struggle [it sounds so obvious written out- I guess that’s why I should blog more often?]. If I have to present my students with something in order to allow us to engage in more meaningful struggle, I am not robbing my students of the opportunities to make deductive conclusions. I am supporting them by preventing frustrating struggle that leads to disengagement and shutdown.

Enter Chris Danielson and the Van Hiele levels of Geometry Instruction. These are not something that I’ve paid any attention to when planning my lessons, which should probably will definitely change. Like Chris said, high school students come in, or maybe should come in?, around a level 1.

Frustrating struggle [current]: My questions/examples are aimed towards getting students to remind themselves of properties (which maybe 1 person in a class can currently provide for us).

Meaningful struggle [goal]: I need to purposefully and consistently provide the opportunity for students to make deductions about properties and shape relationships based on experiences (data collection -> observations) AND for students to learn how to defend those relationships (whether presented or deduced) through quality math talk.

Now, to bring it back to Dan’s comment. How do you decide what is appropriate for students to discover deductively? Is a traditional “I do, we do, you do” lesson ever appropriate? If so, when?

Transformations and Chalk [MTBoS Mission #1]

The very first activity I did with my kids to start congruence was to go outside and get our hands dirty (literally).

Our medium: sidewalk chalk.

This was less of a problem and more of an activity, but it definitely sparked some great conversation. I started out with an activity for each rigid motion (reflection, rotation, translation) and then had discussion.



Photo Aug 29, 9 01 18 AM


Have a line drawn on the parking lot. Each student draws and labels their own point. “Walk directly to the reflection line and count the number of steps it took you to get there. Now, keep walking the same number of steps and draw your new point. Label it with the same letter and an apostrophe- this is read letter prime”.


What kinds of paths did we take?

What about the distance from the line?

Are the two points we drew the same? Why or why not?



Photo Aug 29, 9 01 09 AM Photo Aug 29, 9 01 22 AMDirections:

Make a point where you are standing. Now draw a circle around you and your point. Where you are facing, make a little mark on the circle and write 0. Now turn halfway around on your point. Where you are facing the circle, make another mark and write 180. Now turn halfway between your two marks. Make another mark labeled 90. Now, turn halfway around the circle, mark that 270.

Now stand on the line of your circle. Walk on the line until you get to the 90 mark.


Did we move from our point? What is the point that we are standing on?

What do these numbers represent?

What angle got us to halfway around? What angle will get us all the way around?

Whose 90 is to the left of their 0? to the right? What are these directions called in this context?

Is this still rotation? How can we know?

Where is our reference point?

Can I rotate a negative angle?




Simon says move (scholars ask move where?)

Sorry, Simon meant to say move to the right (how much?!)

Sorrrrrry, Simon really meant to say move to the right 3 steps.

Simon says move backwards 5 steps.


What clarifications did you need to follow my directions?

Did you end up the same distance from the people around you? Would you have expected to?

Why didn’t you?


The discussions were really awesome and we ended up with this sweet set of notes. We were even able to come up with perpendicular paths for reflections (a little prompting from me) and it was great to be able to refer back to the activity throughout our congruence unit.

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I would love to hear your feedback- are there questions I should have asked but didn’t?