Monday, May 6, 2013


Last week, my geometry class entered the room with the following directions waiting for them:
  1. Fold your paper in half.
  2. Put a point in the center of the paper on the fold.
  3. Draw a circle (using a compass) with a 10 cm radius.
  4. Cut out the circle.
*Toss the trash. Keep the scissors.

We had just completed Dan Meyer's Popcorn Picker the previous day and I promised the class I'd bring in popcorn for a job well done. My local store didn't have a bag of pre-popped popcorn so I bought a 10 oz. bag of Pirate's Booty instead. Oh darn, right? That stuff is insanely awesome. Stop reading this and go buy a bag if you've never dabbled in the addictive powers of Pirate's Booty. As the students are cutting out their circles, I say:
If you can make me a cone, I'll fill it with Pirate's Booty. All you need is a tiny piece of tape and I don't want any folding to form your cone. Figure it out.
Student 1: What size?
Me: Any size. 
Student 2: I'm not sure how to do this without folding it.
Sean: Use the scissors. He did tell us to keep the scissors. Cut the circle on the folded line.
Some quickly figured out how to cut the radius and overlap the paper to form a cone while others needed to see their peers do it. Most cones took on the form of your typical snow cone. However, Chase came up to me last and held out this slightly bent circle that barely resembled a cone. Sneaky, yet I was secretly hoping someone would do this. I admire his ingenuity for creating a cone that maximized his Pirate's Booty. We enjoyed our snack as we did our estimation task, flying from Boston, MA to Philadelphia, PA. After finishing our estimation task, I tossed this tub in front of the kids and said:
Don't get weirded out by this, but partner up with someone and measure each other's hat size in centimeters. In other words measure the circumference of their head and write all those numbers on the board. 

I started seeing numbers like 22, 24, 24, 22, 23, 25, 22, etc. being written on the board. I'm thinking, "You knuckleheads. I said centimeters."
Me: Ughh, guys? What are those numbers?
Students: Our circumferences.
Me: Measured in what?
Students: Inches.
Me: Did you not hear me say centimeters?
Students: Ohhhhh!
Me: That's fine. Leave it. Most of you are done.
Sean: But Chase and I just got done measuring in centimeters.
Me: You two rock! Go back and get quick measurements in inches and add them to the board.
We got our two last numbers and I wanted to tell them that based on their inability to measure in centimeters, we'll be making dunce caps instead. I wisely passed on that joke and told them:
Find the average (mean) class hat size. We're making cone (party) hats and we're going to be cone-heads. Go!
The class average ended up being 22.5 inches. I held up my two hands and told the class I wanted the cones to be about "so" high. I measured the "so" length of my hands to be about 9 inches.
Me: How much paper will we need to be cone-heads?
I wish I could tell you that my students worked diligently and strategically to figure this task out without any hiccups, hurdles, roadblocks, or challenges. I'd be lying. They struggled. The closest anyone came was Chase who asked if we could use the Pythagorean theorem. Like you need my permission, Chase? Ha! This felt very similar to Fawn's recent post When I Got Them To Beg. They needed some strong guidance. As Fawn would say, "They beg. I win."
I'll give you a nutshell walkthrough of the activity:
  1. Use the average circumference of the class' head size to find the radius of the cone-head hat.
  2. Use the radius (3.58 in.), desired height (9 inches) and the Pythagorean theorem to find the lateral height. 
  3. This lateral height (9.69 in.) is also the radius of the circle we need to cut out, but we don't need the entire circle to make one cone. We only need a portion of it and we're not going to overlap the paper like the cones we made for our Pirate's Booty.
  4. Therefore, we need to figure out the lateral area of the cone. We use πrl or π(3.58)(9.69) and come up with an area of 108.94 square inches. 
  5. We need 108.94 square inches of paper from the circle that has a radius of 9.69 in. and we figure out the total area of said circle to be 294.98 square inches.
This is where I really challenged the students to finish this. What do I do with all these numbers?  
Devon: We could divide the two areas so we know what percentage of the [9.69 in. radius] circle we need.
Me: Go for it!
We get 37%.
Me: Now what? How does this help us figure out what to cut? I don't need the entire circle. What do we do?
Sean: We can figure out what 37% of 360 is and create that angle within the circle.
Me: 360 what? Where'd you get that?
Sean: Well, there's 360 degrees in a circle and 37% of it will tell us what angle we need to make.
Me: Go for it! 
Student: (blurts out) 133! 133 degrees.
Me: Okay. What does that mean?
Nick: We need to make an angle of 133 degrees in the circle with the radius of 9.7 inches and cut it out.
Me: Okay. How many cone-heads can we get from one circle?
Brace yourself. This is one of those moments when students blurt out answers before thinking:

The math is done. Now we start cutting. Here are the kids in action and our stockpile of cone-heads.

As a bonus, I had some ribbon lying around so students made chinstraps since some of their head circumferences were beyond the class average. Someone suggested we use rubber bands for the chinstraps so they could be just like party hats.
Me: Are you kidding? Do you remember who's in this class? You think it's a good idea to give some of these guys rubber bands?

Let me tell you, those cone-heads looked awesome! I told them they could wear them for the rest of the day. I'd send an email to their teachers explaining our learning and that students are expected to respect the wishes of their teachers. If other teachers want them to take the hats off in class, they better follow directions. Furthermore, if any foul play happens, their cone-head is to be confiscated and I issue an automatic detention. We didn't have any problems. Now, go make some cones!


*BTW: Don't use white paper!!!


  1. I LOVE this, Andrew!! I had to read it a couple times to follow along (I'm slow but you know that already.) Totally stealing this when we get to cones later this week. We'll see what stumbling blocks my kids will encounter. (My kids are better than yours. Don't forget that.) I'm meaner though, thinking they each have to make their custom cone hat.

    You're brilliant, my friend. And thank you for the mention, of course.


    1. I like your idea of custom hats. I forgot to mention how my kids also needed to figure out the actual distance of 9.7 inches. Silly billies... Now do you see why centimeters might have been easier to use here?

  2. Those hats look really familiar:

    My kids have been struggling a lot too...especially on a sequel I'm doing with Dan Meyer's toothpicks. I feel your pain.

    Before you showed your method to finding the number of degrees, I tried to get it on my own. I did just about everything the same with one exception. Instead of finding the area, I found the circumference of a circle with a 9.69 inch radius (60.86 inches). I knew that I wanted to use 22.5 inches of this circle to make my cone. So I found the percentage of the circle used by dividing 22.5 by 60.86. I got the same 37% as you. This seemed more intuitive to me and I'm wondering if any of your students did it similarly.


    1. Those elf hats are stylish man!

      Man, your circumference approach makes so much sense! No one thought of that, including me. I was fixated with "surface area" of the cone, especially since I had to figure out mow huch butcher paper I needed (and didn't waste) when preparing for the lesson.

      I truly love your circumference approach and will share it with my students. Thanks Kraft!

  3. Wow Andrew. There is some great learning going on in your classroom. You said something that is worth reflecting on:

    "I wish I could tell you that my students worked diligently and strategically to figure this task out without any hiccups, hurdles, roadblocks, or challenges. I'd be lying. They struggled."

    Something I have come to learn is if someone is supposed to be doing something challenging and you ask them how it's going and they say it is going fine, then either it wasn't as challenging as you thought, they are not doing it, or they aren't being completely honest.

    Learning and growing are challenging but beneficial processes. Something in American culture makes us fear struggle but we need to embrace it. You should be proud that you gave students an opportunity to do a problem that they enjoyed so much that they embraced the struggle and learned something in the process.

    Great job.

  4. Great lesson... Thanks! I' ll definitely do it with my students and share the outcome... I think they will struggle a little bit more with the procedure (they WILL measure in cm, though!)