There Are More Hexagons Than You Think

Geometry is always fun to teach in elementary. Students feel liberated by not having scary three digit numbers flying around. Shapes you can bend, stretch, fold, connect and transform; not everyone is comfortable with bending, stretching, folding, connecting and transforming numbers yet. You can prove claims by cutting things and moving pieces around. Students have a lot of first hand experiences with the shapes world, they have a lot of ideas and opinions about it. They are not afraid to step into this world, they feel at home in it.


After my teaching partner discussed the attributes that separate the polygons from other weird flatland creatures, we decided to focus on reviewing our polygons vocabulary and refining our understanding of the attributes of different polygons.

We started with the lesson shamelessly stolen from Graham Fletcher Where is Poly? An Exploration in Geo-Dotting. Here is the picture of the prompt that we used.


Students engaged in a dots scavenges hunt, shared their ideas on the SMART board, everything was going pretty similar to the original lesson. Many students noticed that if you connect some of the dots, you can make a hexagon in the middle. When one student started connecting the dots, the running commentary was as follows: “It is a hexagon because it has six sides, six vertices, it looks like a circle, and all sides are equal.” This last part caught my attention.

I drew a triangle with obviously not equal sides. We all agreed that it is a triangle. I drew a wobbly quadrilateral. While the students have just recently learned this word, they identified that it is “quad-something because it has 4 sides and vertices”. I drew a bowtie looking hexagon. I have never seen my class agreeing so unanimously with anything before; this was NOT a hexagon.

I had a moment of weakness when I was about to turn around and announce that it is, and then explain how it works. Instead I told my students that we will investigate it later.


I took a picture and on my recess got to twitter. It’s not the worst way to spend recess. This was the idea I was looking for.

Instead of just taking a photograph I decided to make some sorting cards for my kids and it was a good excuse to spend my evening playing with the compass and pretending I’m working. I narrowed the set down to fifteen trying to find the hexagons that would cause the most confusion. I anticipated the responses for each one of them. I was close.

Hexagon Scavenger Hunt

Here is the picture of my hexagons and my students’ descriptions of some of the shapes. See if you can match them.


  • “A hexagon”
  • “It will be a hexagon if you push the sides out”
  • “It would be a hexagon but one of the sides is off balance”
  • “A triangle with a hole in the middle or an Illuminati”
  • “It’s a rectangle, that’s missing a piece”
  • “It’s a rectangle and a square together”
  • “This one is a triangle but a little piece missing”
  • “A pacman”
  • “A heart”
  • “A heart #2”
  • “Z”
  • “A part of a star”
  • “A star that is also a triangle with bent sides”
  • “Looks kind of like a trapezoid”


I gave students the cards and asked them to find hexagons, put them into a “hexagons” pile and try to describe what makes them hexagons. After about 10 minutes of logistical hiccups (“I need a pencil sharpener!”) and discussion, we put all the attributes on the board.

“It is a hexagon because it has six sides and six vertices and I think a hexagon only needs to have six sides and six vertices to be a hexagon.”

“This is a hexagon because hexagons usually look like honeycombs.”

“Hexagons always have even sides”

“If you cut it in half it will be the same on both sides.” (I think we are talking about symmetry here).

“If you cut a hexagon in half, it will have one, two three, four sides, it will be a trapezoid.” (Did not see this one coming, and kids really picked this one up, one of the last attributes they dropped.”

“The sides should not go into the middle, they should be either straight or go out.” (No tucked in sides)

“All angles are the same”

“All angles are 120 degrees”

“Inside sides don’t count” (I think this one was about the illuminati triangle)

“All sides should be parallel”

“It shouldn’t have sharp angles”

I find it fascinating how this conversation jumped all around K-6 geometry curriculum, and students made some observations that they don’t need to officially approach until grades 5-6. I think it takes care of some concerns that more student-centered lessons might not be the most efficient way to cover the curriculum. Actually, they are.

We looked through all the attributes, and found that there is only one hexagon that fits them all, the honeycomb one. Then I told the students that there are more than one hexagon. They went off back to their groups to decide which attribute they might consider dropping to allow more shapes to join the hexagons ranks.

After the second round we had 3, the regular one and the slightly crooked “honeycombs”. The attributed of equal sides was dropped first.

Here is when the real fun started. A few shapes were so resembling triangles and rectangles that reasoning through attributes was coming into conflict with the shape intuition. This triangle is just missing a little piece, if I put it back it will be a perfectly fine triangle again, even with six edges and vertices it is way closer to regular triangle than to the regular hexagon.

These conversations made me wonder: If our geometric vocabulary is usually develops through the context of regular shapes, do we subconsciously allow the attribute of “regularity” to take over everything else?


We made a few more rounds of dropping attributes and adding more and more hexagons to the family.

“I have an idea! I think they are all hexagons!” This idea was appearing around the table groups and spreading around the classroom. One student came to me, “I think the pentagons might be like hexagons, all different. And other shapes too.”


Thoughts and Observations

My goal was to have a conversation about the properties of the polygons without me being on the spotlight. I did not anticipate hexagon being a topic for such a passionate debate for 8 year olds that couple of groups required my intervention to direct students from speaking at each other to speaking to each other. While everyone had some aha moments, I am not sure everyone is equally convinced, but I think I can work with it. I also loved how students started extending their ideas to other polygons, and I plan to use their ideas to shape my next lesson.

Moving Forward

I think I want to make a WODB with 3 pentagons (2 “weird ones”) and one funky quadrilateral. Or maybe bring in a hexagon friend. This will be another fun filled evening because I don’t remember how to construct a regular pentagon with the compass.

Special Thank You to

Graham Fletcher for Geodotting lesson, it was engaging, interactive and brought up a lot of vocabulary

Christopher Danielson for giving me an idea when I didn’t know where to take my kids confusion

Simon Gregg for continuing the conversation and helping to organize my ideas into lessons

Zak Champagne for a great session Talk Less Listen More in Effective Practices for Advancing the Teaching and Learning of Mathematics online course in Mount Holyoke College. You are responsible for my “shut up don’t say it” moment that lead to this lesson.



Everything Is a Shape But I Don’t Know What Shape Is.

It all started with the tweet from Sarah Caban a few days ago.


I decided to have a chat about it with the Kindergarten class, ask my grade 3 students, and one of our grade 5 teachers kindly agreed to ask the same question to her grade 5s. Here are the answers and some context from three different grades. At first, I planned to throw it all into a google doc and share with Sarah. Kindergarten kids’ ideas got me excited. When I started reading the other grades’ responses, I realized it’s worth sharing with anyone who might be interested.


Students in kindergarten have already learned some shapes vocabulary and identified different shapes in their environment. I started with pointing at their shapes wall, “I see you have been learning about shapes, we are learning about shapes in grade 3 too. Can you tell me what does it mean – “a shape?” A forest of hands.

“It’s a heart!”

“It’s a diamond!”

“It can be a circle.”

“A small triangle can be a shape.”

Students were pointing at shapes around them.

After recording all their “shapes” ideas, I asked, “What do all these things have in common? Why do we call them all shapes?” I was surprised how quickly kids considered the question and were all ready to discuss their ideas.

“Shapes are everywhere”.

“Everything is a shape but I don’t know what shape is.”

“We are made out of shapes”

Then someone mentioned that some letters are shapes. But not all. Students did not seem to have a consensus on that point, and the discussion continued around particular letters.

“C is not a shape, but you can make it a shape.”

Our kindergarten teacher who kindly invited me to talk math with her kids today, ask the student to explain what he means.
“You need to draw a straight line to make it a shape.” He drew a line connecting two “ends” of the letter C together. “Now it is a shape.”

Grade 3

Grade 3 students have just started talking about polygons and sorted their polygons and non-polygons into two groups. Their thoughts about shapes were likely affected by this recent activity. Students answered the question on the post-it notes and we did not have time to discuss it yet.

“A shape is something that has vertices and edges.”

“A shape is random line but with curves and edges.”

“A shape is a structure that most things are made out of. It is something that makes up everything.”

“A shape is something that you can outline.”

“Shape is something that you can use for a pattern.”

“A shape is something that connects. A shape means a thing that has no opening.”

“Shape is an object.”

“A shape is something to represent slots, key slots and other items.”

“Shape is 2D which means it’s flat.”

“A shape is something that is 2D or 3D and it needs a face.”

“A shape is something with vertices and everything is a shape.”

“A shape is something that has to have vertices/faces/edges and more than 2 edges or vertices.”


And the one that really made me pause and contemplate.

A shape is a 2D or 3D object that is drawn or held by humans.” – I thought bringing in the point that shape is something manipulated by people was really interesting.

I am also curious about this.

“Shapes are everything you see in life. You can find anything and it falls under 1D, 2D, 3D and 4D shapes.”

Grade 5

Grade 5 students worked on describing 3D and 2D objects and sorting quadrilaterals in the beginning of the year. They explored points, lines and did a lot of “hands on” geometry work.

“A shape is a thing that makes everything.”

“A shape can be a polygon and it can be irregular.”

“A shape is some sort of structure.”

“A shape is a structure of an object with all connected lines. Sometimes shapes have lines with holes.”

“A shape is a form of geometry that represents objects in their form. They are the appearance of an irregular or regular object.”

“A shape is something that can have sides, vertices, straight edges or none, so they can be anything like a soccer ball.”

“A shape is something like a symbol of math.”

“A shape is geometric combination of lines and curves making a closed region.”

“A shape is a closed 2D object with nothing inside it.”


I found it interesting to observe how the intuitions unfold with age, to see how students would use new and more sophisticated vocabulary to grasp the same concept. How they start looking at new mathematical concepts and to compare them with their idea of “shapes”. How they try and test new learned attributes to define a shape. When I started organizing these notes, I realized how much this simple question can actually tell me about my students’ mathematical thinking.


When other teachers look at all this students’ thinking, what do they notice? Just like these kids use different structures to define “a shape”, how are our “teaching structures” that we use to interpret students’ thinking are the same or different. Not sure it makes sense. “Everything is a shape but I don’t know what shape is.”

Proof and Evidence: Reasoning About Subtraction Part 2


Last week I shared how my grade 3 students were preparing to prove their conjecture about subtraction: If you add +1 to both numbers in subtraction, the difference will not change. I really appreciate all the feedback and conversations that happened around the topic on twitter. It was also interesting to read blog posts from Jenna Laib , Simon Gregg and Sarah Caban about their experience working with proofs and representations.

My class finally finally got to our proof lesson last Monday, it was an extra long lesson, students came up with many great models and their teacher ended up with many questions. All this made for an extra long post.

Towards Lesson 4

We ended last week with building and sharing different ways to represent subtraction. I promised that on Monday we will work on Proving the Claim. I admit, I was nervous regarding the level of frustration and confusion the task could cause, and I was not certain how to prepare my students for it. So I did the only thing that came to mind; I scared them.

Lesson 4: Proof and Evidence

I started with warning my students that this is going to be the hardest part of the whole process. They came to the point where they need to prove their conjecture for all positive integers. The only time my students attempted proving was in the beginning of the year when we used snap cubes to reason about sums of odd and negative numbers (the lesson inspired by this post in TCM blog).

I reminded my students about this experience, pointed at the selection of tools and manipulatives and sent them off to work. I sensed an air getting heavy with confusion just as I anticipated. Before it ascended to panic, I made a suggestion to rebuild the models that students used last time and then see if they could tweak them to prove our conjecture. I tried really hard to not impose my thinking on my students, so I was mainly wondering around asking students how their models work, recording their explanations and asking for clarifications. The sense of direction was all that was needed, and I will leave the rest of this post to my student’s work. I wish I could post the videos, but my school board is very particular about not sharing students’ voices or pictures online so I will have to stick to the transcripts.

Students’ Proofs


“Music is actually a lot like math”.


“How do I know that it always works? Because it never ends if we just keep adding by 2s, even, even, even”.

The next one got me really excited.

“All of the bottom is white. I added this way as a handle or to add more of these squares. Green is the difference and the red is the subtraction amount. Red and green are the main number. The black adds one more to the total number and subtracts one more.” -“Why do you add one cube? In our conjecture you add +1 to the first number and +1 to the second, and you just add 1?” – “Because this one cube adds 1 to the main number and the other number, like the minuend AND the subtrahend.”

There was also a trio of “story models” that showed that if you subtract 1, the conjecture will also work. I really enjoyed the narrative part.

“This is 10 – 6. The white is 6. I made it like this because white is like extinguished water and that’s why I also have blue which is the remaining water. And that equals 4. The original number is 10”.
“I made this like cheese. This is like an entire square of cheese. Five of these got stolen by mice cause they like cheese and now there is only 4 bits of cheese left. 4+5 equals 9, 9 is the original number”.
“This is the leaf, burning. This is an entire leaf of 8. This one works like 8-4=4”.

Students’ Reflections:

The lesson took longer than I expected, so I asked for a more open reflection with our general prompts: questions, challenges, observations, discoveries.


I knew this one was coming!

My Notices and Wonders:

I wish I could say that all of my students experienced the same degree of success with these lessons, developed multiple models for proof and expanded their range of subtraction strategies and their understanding of the operation. I am afraid it is not the case. When I was analyzing my lesson, I got back to thinking about my goals. I wanted my students to

  • explore different representations and contexts for subtraction
  • gain deeper understanding of subtraction and its properties
  • attempt and experience proving
  • construct effective arguments
  • explain our ideas in words, models, diagrams and mathematical symbols

Even students who were not able to come up with a working model on their own, had a chance to discuss and review other models. They had a chance to reason, to try and to model. They might not have gotten the result but it doesn’t mean that they didn’t get the experience. I wonder if they count it as their success.

The Best Intentions..

For the “We All Fall Down” edition of the blogging initiative! I skipped the last two prompts with the valid excuse of report cards writing, but I couldn’t skip this one.

The Fail

I finally get my students to calm down from all the excitement and ask, “So… do you see the problem? How can we solve it?” I look around and to my horror realize that no, they don’t, because there is no problem.

The Premise

I really bought into Dan Meyer’s “If math is the aspirin then how do you create a headache?” idea.  I always felt that teaching is close to storytelling, to creating an engaging narrative. Then after watching some of Dan’s videos and reading some blog posts, articles and discussions around, it appeared very intuitive. Of course, I won’t be interested in solving problems that I don’t have. I won’t spend my mental energy or waste my memory capacity on that. If I don’t care, I won’t learn.

All this seemed so obvious and easy that the real challenge got obscured: How do you plant and grow the problem? What seed do you throw in? What notices and wonders do you nurture?

The Lesson Plan

In early September, I planned a lesson to introduce standard units of measurement. The idea was to let kids experience how standard units allow us to have common language to communicate our measurements. Students were asked to measure different items in the classroom with… whatever they felt like. Then we were to discuss the length of their desks and to realize that we can’t come to a consensus because we all used different units.

The Lesson

Students were very enthusiastic and measured a lot of different items around the classroom, shared their measurements and observations, recorded them. Then the moment came to discuss our problem and to see how we all need to use same units… “What problem?” When I looked around I saw twenty-five happy eight year olds who have been having time of their life measuring the classroom with everything from pencils to their heads. They didn’t have a problem, they had a blast. They did not care that we all had different measurements for our desk. It was exhilarating to find out how many different ways we could measure our desk! I wanted my kids to need metric system. Well, after my lesson they didn’t want anything to do with metric system.

The time for the lesson was almost up. No way to salvage the lesson materialized in the last 10 minutes of the class. I spent this time mumbling about standard units of measurement who nobody cared about.

I started typing in how I fixed the lesson next day building on my failed one, and then decided I should leave it out. I’ll share if someone is interested. The most important impact of this fail was that it made me more sensitive to my assumptions, made me question the questions I ask my kids and the prompts I give to them. It made me experience that I can’t force a problem on my students, it has to come from them.

The Question

What would you have done to save the lesson?

Proof and Evidence: Reasoning About Subtraction Part 1


It’s the third year that I spend some intentional time in my classroom working on a “making friendly numbers” subtraction strategy: 234 – 19 = 235 – 20. I was not happy about my previous two attempts. Some kids I think just bought into it, some kept confusing it with addition (take from one number and add to another); in any case, I felt like students were not able to figure out why this works, and they either took my word for it or tossed it away. In year one, I gave it to them. In year two, I made a lesson with Cuisenaire rods. I offered my students a great model they didn’t really care for at that moment.

A few students brought this strategy up this year during our number talks. “Why does it work?” – “I don’t know, my teacher told me it always does.” It was time to try again for the third time.

Original plan was a bit blurry: get the number string on, get students to notice the rule. Then model it with the snap cubes in two towers; let students work with partners to change the number of cubes in each tower without changing the difference.

Intermission 1

The new idea came after Mike Flynn’s session about exploring multiple representations, a first session in Effective Practices for Advancing the Teaching and Learning of Mathematics course in Mount Holyoke College. I tossed my original plans and decided to try and follow the example that was shared in the session; let students notice the pattern, come up with the claim and then prove the claim using their own representations.

Lesson 1: Developing the Claim

First lesson was a brief number talk. 3-1, 4-2, 5-3. Students noticed that if they add “1” to both subtraction numbers, it seems like the difference stays always the same. I also started recording the steps of mathematical thinking that we will follow. My worries during the lesson were that a few students did not seem to notice a pattern. I hoped that the Testing lesson will help.

Lesson 2: Testing the Claim

The plan was to ask students to spend 15 minutes testing the claim with as many examples as possible. Then things went sideways. “Can I try negative numbers?” We don’t teach integers operations until grade seven. I teach grade three. I panicked then said, “Go for it.” Then there were groups of kids engaging in some interesting discussions.

  • It doesn’t work! (-5) – 3 is not the same as (-4) – 2!
  • You are not subtracting -3, you are subtracting positive 3.
  • If (-5 )– 3 = (-8), then what is -5 – (-3)?

Not sure what I was thinking when I briefly introduced integers earlier this year. You can’t give eight year old kids a new math toy and expect that they won’t play with it. I’m bracing myself for multiplication and division of integers in the spring.

One student discovered that if you subtract a number from itself, you will always get zero. From that moment on, he forgot about original claim and was testing his new discovery. I think we will be proving it next. It was all fun if not quite what I anticipated. At the end of 15 minutes we all agreed that the rule seems to always work.

Intermission 2

The plan was to now ask kids to use any models to prove this claim. And that’s when I realized that I didn’t allow my students to play with different tools and to develop their own subtraction models. We built numbers, we did not build “subtraction.” I felt we are not ready. I decided to let kids to explore different representations for subtraction first.

Lesson 3

“Today we will be testing the tools that we will later use to prove our claim.” Students worked in 4 groups to build a model, draw a diagram and write a story for 9-5=4. I had a selection of different items ready for the students to pick; from ten frames and balances to dominoes and snap cubes. First ten minutes were terrifying. Students used manipulatives to build symbols 9,5 and 4. I told that with their manipulatives they are trying to explain it to someone who might not anything about subtraction or symbols for numbers, but can count.

After all groups seemed to have come up with some working representations, students interviewed each other to find a representation that was different from their own and reflected on it in their math journals.


Closing Thoughts and Plans

These few lessons made me realize that when I ask kids to explain their thinking, I often give them my tools and they end up modeling my thinking rather than developing and explaining their own. They need to find their own tools and develop their own reasoning and I need to give them more opportunities to do it. On Monday we will move on to our proving.

Here is the template I made to plan my future Mathematical Reasoning Lessons.


How do you lead students from drawing the picture to drawing the diagram? From building numerical symbols to building a model?

My Favorite: From Sharing to Discussion

My grade 3 students love sharing: stories, artifacts, ideas. They are much less interested in listening – to what others have to share. Which kind of defeats the purpose. Math time is not an exception.

I believe that discussions play important role in elementary mathematics classroom and all students need to become comfortable with constructing arguments, defending their reasoning, questioning and proving. I try to design my lessons and routines so that they would include these skills as their natural component. And yet sharing doesn’t turn into discussing as often as I hope it would.

The background:

There were some things that worked in the past for me: Circle counting (I learned about this routine from Sadie Estrella here ) encouraged students to listen to each other and WODB (from Christopher Danielson and Mary Bourassa) seemed like a scavenger hunt to my students who were interested in “who found what else and how it can help me to find something else”.

But I am always on a lookout for more ways to engage more students into productive and attentive mathematical discussion, and my favorite “aha” this week was this “one new way”.

About a month ago in December, I joined a Global Math Department webinar lead by Andrew Stadel (@mr_stadel) , Clothesline Math: The Master Number Sense Maker.

The lesson:

So as January came around, I decided to start my first day of school in grade 3 classroom with Clothesline lesson: looking at numbers, addition and subtraction equations and arranging them on the number line; first in the right order and then in the right spot relative to each other.


Students worked in groups first to put the numbers in order, then each group moved one number on the large clothesline at the front. Everyone agreed on the order but more precise locations were already causing a lot of disagreements. Usually, I would ask students to come up one by one to move one number and justify their reasoning and then have some discussion routines to roll. The same kids usually want to come up. The same kids usually want to have a discussion. The same kids want to argue about making the location more perfect. And all these kids do not constitute a majority of my class.

Using some webinar ideas, this time I asked a student to come up, move one number and remain silent. Other students had to try and “read her mind” to figure out the reasoning. Whatever the reason, this shift worked. Kids who are usually not comfortable presenting did not hesitate to come up, to move the numbers, and then to listen to others try and figure out their reasoning. “Not quite”, “On the right track”, “That was what I was trying to do”, “There is more to it” – the list of responses developed on the go. At the end, they found it easier to summarize their explanation after listening to all these other people trying to “read their minds”.

The “favorite” stuff:

The best conversations happened around some solutions that most of the class quickly identified as incorrect. They still had to try and figure out what was the reasoning behind it instead of jumping straight to disagreeing.

Student 1 comes up and puts the card 5+7 between 0 and 10-5.

Me: “What do you think is the reasoning?”

Student 2: “Maybe 5+7 is close to 0 because 5 and 7 together are smaller than 10 and 5?”

Student 3: “Maybe it’s smaller because 7-5 is 2 and it’s very close to 0. But one is minusing and another is adding, so it might be important.”

Student 1: “Can I revise my thinking?” Moves 5+7 past 10-5 and 20-10. I did not think that I need to add.”

What worked well:

-New students joined the discussions.

-More conversations happened without my interference.

-Students listened to each and “revised their thinking” based on each others feedback.

-Students who find it challenging to explain their reasoning felt more confident when they had the “help” of others working not to challenge their ideas but to understand them.

My Questions now:

  • How can I adapt similar routine to other activities?
  • What other strategies and routines other teachers use to encourage authentic discussion?

I appreciate your ideas and suggestions!

The first post is the first post

I suppose this is an embodiment of my one and only New Year’s resolution.

Multiple times during this school year I had an urge to share the project or the question and I’ve reached the point where I want to create the space to do it. I am not certain how often I will be able to update; maintaining the required class blog generally drains my blogging energy. But I have a few projects coming that I am excited about and I hope some of them will end up here.

Happy Holidays!