## Snowflake Math

[BTW: Mimi Yang’s remix is highly recommended.]

I’m about to give you what I’m convinced are good blueprints even though the house I built off of them today was pretty raggedy.

Here, three days before winter break, I wanted an activity that injected math into something mindless. I thought about snowflakes, you know, how you fold some paper, cut it here and there, and open it up only to discover you’ve recreated The Storming of the Bastille.

So here’s (what I’m convinced is) an awesome exercise in spatial intelligence for you and your students: predict what the snowflake will look like before you open it up.

I’m tempted to leave it there and let you decide how this oughtta shake out, encouraging you to please get back to me and let me know. Because what I did today didn’t have the same loose-limbed energy my best stuff usually does. This was second-rate but maybe we can spin something better out of it — you and me:

• I passed out a sheet of standard letter paper and some scissors to each student.
• I had them square the paper and fold it into fourths — now a smaller square.
• I put up a series of slides. Each one asked them to make one cut.
• They made the cuts and I said, before you open up the snowflake, sketch what you think the snowflake will look like.
• They sketched it.
• I walked around, observing, sometimes making comments.
• They opened it up and checked themselves.

Then, without passing out more paper, we went backwardsWorking backwards from a solution to the problem, incidentally, is the most reliable way to carry your kids a few rungs up Bloom’s Taxonomy..

• I gave them the result and asked them what cuts had been made to get it.
• I called up five volunteers to the board to show their solutions, most of which differed only slightly from each other, a fact which offered up some good conversations starting with words like “compare” and “contrast.”

Then I passed out this worksheet, which asked for eight visualizations, the second half doubling in complexity by adding one fold to the snowflake.

Typing all that here at the end of the day, it’s kinda obvious to me that this was too much even for my Geometry sophomoresNine of whom apparently read this thing so, hey, team, no disrespect.. The spatial learners had a blast but I didn’t manage to transcend that division and pull the other intelligences over the wall like my better stuff tends to. This thing lacked a certain scaffolding. In other words, buyer beware.

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### fgk

December 18, 2007 - 9:50 pm -

if i were to do this (and i’m not sure that this is the right thing to do) i’d include a reflection step (maybe on the back side of the worksheet)

#what did you try that didn’t work?
#what did you do that did work?
#what did you learn from another student?
#how would you explain your technique to someone else?

2. #### Mindy

December 19, 2007 - 3:32 am -

Just an FYI-This type of exercise is on the CogAT (Cognitive Abilities Test), a national normed test that we give all of our fifth graders. This is part of the non-verbal section and is a small part of helping us determine gifted services.

It’s always neat to find out which kids think that part of the test is fun. And which kids think it’s tortuous.

3. #### Sarah

December 19, 2007 - 6:41 am -

Last week we met with our mentor groups. I didn’t know until the last minute, so I ended up making Christmas cards to go to the local hospital with mine. While they colored, I cut out snowflakes, they were fascinated with them. “Make one for me!” They hadn’t made snowflakes before, so I taught those who wanted to learn. But I didn’t go into the math of it.

Today’s our last day and I expect maybe 5 students. The rest are on a field trip with the school. This means, the plan for today is to see who walks in and what they need to review. If I had a projector, I would totally steal this right now. Depending on who actually shows up, I still might.

I think I’d start this exercise even farther back. Do one “flake” to introduce/refresh symmetry with just one fold. Then do a prediction. Then move up to two folds. Then three. Then the four to make the snowflake proper. We’ll see how it goes.

4. #### dan

December 19, 2007 - 7:41 am -

Nancy, these snowflakes are intentionally dinky. I’m trying to limit the number and complexity of the lines of symmetry.

5. #### Angie

December 19, 2007 - 2:05 pm -

Depending on where you wanted to go and what you wanted to accomplish…you could also make it an assignment on giving and following directions. Like have one student describe how to fold and cut the paper to another student who isn’t looking and see if the final product matches the original. Another idea….have a snowflake already cut out and let students work in pairs to figure out how to fold and cut it to get to model.

6. #### jose

December 19, 2007 - 7:22 pm -

I noticed, in light of the “Call To Action” post, this post got lots of responses because you attached a narrative with the lesson plan. I’m glad you kept it in this format, since it really adds a dimension to what you’re doing. It’s like doing inter-visitations with other teachers, except on cyberspace. It’s a cool lesson, but you’re right about the multiple intelligences thing.

7. #### Ben Chun

December 20, 2007 - 4:11 am -

Did you have any problems with orientation? That is, did you give a standard procedure for which folds to make in what order so that kids were always holding the folded paper the same way when cutting it? Just curious.

I think having the object in their hands is probably the best way to help them see what’s going on. But did you ask the question, “Why does making one cut in one place have more than one effect?” It seems obvious, duh, you’re cutting more than one sheet. But getting them to think about how those multiple sheets got to be under the scissors at the same time is kinda the essence of thinking about what the effect will be.

Maybe before you let everyone go wild on the examples, ask for a little writing (or at least verbal formalizing) about how cutting folded paper in some places affects the corners of the unfolded square, but cutting in other places affects the center. How can we know the difference between those places? It might be that forcing them to put it into words will bring along some of the other intelligences.

8. #### dan

December 20, 2007 - 9:38 am -

Orientation was an issue. In that first slide I tried to point out where all the folds should meet but miscommunications persisted. In the end, it didn’t matter what or where they cut, so long as they were trying to predict what the result would be.

We also asked the question, “you cut out one triangle, why are there four triangles cut out of the product?”

Some said, “You folded it four times,” which wasn’t correct. Then you point out how, when you cut a triangle out of the “major” fold, you get a diamond. where are the four triangles? etc.”

You and me had pretty much the same strategy running except, since I’m functionally illiterate, I didn’t bother having them write anything down.

Next year.

9. #### Jason Dyer

December 20, 2007 - 10:04 am -

I am going to attempt this with my students after winter break. (Transformations and symmetry will be our first topic.) I know I’m going to start with just one fold (as Sarah is suggesting) and not go past two for asking the end result. (At least for the required part of the assignment — I have a couple students I can ramp up the difficulty on.) I’ll give the instructions for a full fledged “pretty” snowflake as a grand finale.

I do believe orientation (as Ben Chun suggests) might be an issue, for when I first tried it myself I was temporarily confused as to which corner went where. (You have the “folded” effect but it isn’t enough.)

I’ll let you know how it goes. Thanks for posting your lesson plan!

December 21, 2007 - 5:57 am -

I agree that the narrative provides not only the opportunity to replicate the project better, but also the opportunity to discuss what could be done to improve it (since you asked).

One thing that I was left wondering from your writing is how did they do with the first part? The predicting?

The big jump to me is giving them the end product with no paper…how about giving them the created snowflake, paper, and the time to try to make one like it as your second step. That would allow the opportunity to trial and error and process (concretely) what is happening. It would help the kids who struggled with the first task and deepen the understanding of all.

Students who get it fast could be given more complicated ones to try. Lots of different stuff going on at this point. Then pull them all back and discuss what they have noticed. How do you make a certain portion? What happens when you make a cut, etc. Writing is good…don’t do enough of it myself…but the discussing lets them all process together. THEN give the worksheet. As you have carefully crafted the examples on the worksheet it could reinforce and extend what kids have done.

My thoughts anyway…

I should probably stop here, but two more thoughts are in my head. 1) I think the number of responses to this post is correlated to the request for improvement combined with transparency of practice. I would love to see more blogs with this tone. (Enough so that I want to teach kids again NOW to try it.) and 2) Interesting (but not really surprising) that your students read you blog. How much thought about a student audience goes into your writing?

11. #### dan

December 21, 2007 - 11:59 am -

Zero. The long rants about misbehaving, ungrateful kids has never been my territory, otherwise this revelation would’ve shaken me a little more.

12. #### Jason Dyer

January 10, 2008 - 8:28 am -

I ran this yesterday. I made a slight tweak to the worksheet that made the assignment easier, and I ended up undershooting on the difficulty. I am working up a webspace and maybe even a new blog to post this stuff.

13. #### Jason Dyer

January 11, 2008 - 1:41 pm -

I have the blog now (check link in my name), and I will post all about my snowflake lesson this weekend.

I invited my students over, so feel free to say hi.

14. #### Jason Dyer

January 30, 2008 - 3:54 pm -

“This weekend” turned out to be a month due to some bizarre technical difficulties and then my forgetting. Sorry! Here’s to hoping someone else takes up the gauntlet.