Geometry

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.

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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:

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My thoughts:

+’s

  • 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

∆’s

  • 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?

Materials:

Go to the dollar store. Not Target.

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