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.

Background

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”

Lesson

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.

## 6 thoughts on “There Are More Hexagons Than You Think”

1. So good that you held back the first time, got all those hexagons ready, and then pushed their hexagon ideas when they had hexagons to get their teeth into!

I think we teachers have created the problem we then have to solve – with those wall charts that have regular hexagons, with our pattern block hexagons and suchlike. It’s natural that everyone should assume hexagonality equals regularity. As if with young children we were to call only oranges fruit always, they know there are other kinds of sweet plant things that aren’t mentioned so they assume they are not ‘fruit’.

Part of me wonders whether, in this situation, I would have abandoned the word and got on with properties… Perhaps to say, ‘What have all these shapes got that’s the same?’ Hoping for, ‘They’ve all got six sides and six vertices.’ And then get on with properties. Can you make a wodb with four of them where you could argue that each one is the odd one out. Make it, and annotate your wodb with reasons for each ones differentness to the other three. Can you group them in some way, splitting them into two groups? Or more groups? Can you make a different kind of 6-sided shape? I’d be hoping at some point there’d be a disagreement about something, and or a generalisation about properties that we could write out and put on the wall. ‘A 6-sided shape can’t have more than 3 inside vertices,’ or something like that.

(There could be a little reveal at the end, just to link in with the name. Maybe show a list of Greek number prefixes? Mono, di, tri, tetra, penta, hexa, hepta, octa… and what -gon meant – corner.)

Now that you’ve expanded the students’ idea of what a hexagon is, I’m sure you’re going to do something along these lines and more. I think I’d also want the students to physically or digitally construct them as well. There should be a reason for this… Maybe to make a larger version of their favourite hexagon, giving it a name and a description of properties?

I’m looking forward to hearing how you take the students, and the hexagons, on!

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2. Thank you for your comment, Simon. Conversations with you influence my lessons a lot lately.

I felt a bit overwhelmed yesterday, so many different pathways we can take. During today’s WODB, I noticed that students really focused on composing and decomposing shape, trying to see if they can somehow make a regular hexagon or a rectangle, or a square.

“I think I’d also want the students to physically or digitally construct them as well. There should be a reason for this… Maybe to make a larger version of their favourite hexagon, giving it a name and a description of properties?”

I love the idea of physically constructing some hexagons, or maybe extending it to other polygons. I have revisited Van Hiele model while analyzing WODB, and many of my students make sense of the polygons by comparing them to some real life objects or by trying to cut them and rearrange them into “friendly” shapes. Some students are regarding particular attributes (equal sides, angles, parallel sides), but I think constructing, analyzing and comparing their own creations might be a way to go at this point to encourage more students to start shifting from visualization towards analysis.

I don’t shy from technology, but I find that somehow geometric constructions are more beautiful when hand made. I am contemplating if I should use wooden blocks or paper to continue the exploration. You can fold and cut paper which is a good thing. Wooden blocks are more “stable” and less destructible. That twitter post you shared about constructing polygons from other polygons is really stuck in my mind, I wonder how I can make it work.
Maybe students can construct a new polygon, describe its attributes, identify its mathematical name and give it a “nickname.” Maybe we can then have some WODBs with students creations too. Maybe we can sort them.

My current question is: Do I want to give some constraints regarding one of the attributes to focus students’ work? Do I want to allow some free play first and then make a decision if the constraints are needed?

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3. There are so many great directions you could go here – and you’ve mentioned some great ones already. I like giving students big bits of card and letting them make shapes with those. I liked your idea of using tangrams too, and now I’m wondering, how many ways are there of making a hexagon out of tangram pieces?

Thinking constraints… … Could a minimal constraint be that there has to be something special, something to say about their hexagon? (By the way, I think no-one mentioned mirror or rotational symmetry in the wodb – that might give them more options too.)

So, I might prompt – think about how regular or irregular you make it. How many accute or obtuse or tucked in angles it has, how many parallel sides, how it is made of other shapes, how many sides the same length; try to make a hexagon different to everyone else’s…

I’m enjoying following your work Lana – it’s good when I’m teaching young ones to still be able to think about what older students are doing – and I feel privileged to have a part in it all! 🙂

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1. That! Hexagons with tangrams! There is enough focus to make it “problem-solving” and open enough to really explore different attributes. I think I have tangrams hoarded somewhere in my classroom.
Also just realized that I’ve been helping gr5s to work on snowflakes compass constructions for their water inquiry, and just yesterday we got to regular hexagons with the compass. I might find a way for my gr3s to get together with gr5s for some hexagons action at some point.

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