Cone-heads

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:
ONE!
THREE!
TWO!
NO WAIT, TWO!
YEA RIGHT, TWO!

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!

Cone-head,
1014

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

More Tangrams Please!

This week in Geometry, we did the 3 Act lesson Hedge Trimmer. I'll debrief about that another time. Students needed to find the area of some isosceles trapezoids along the way and I didn't give them access to the area formula for trapezoids. Instead they needed to be resourceful and figure it out on their own. Well, that didn't go too well at first [cue the whining]. Many students had trouble breaking the trapezoid into 3 polygons: a rectangle and two triangles. Their warm-up the next day was to play around with tangrams for the first 5-10 minutes of class.

Me: Use all seven pieces to make any one of the following polygons. Do your best!

I drew a square, rectangle, trapezoid, parallelogram, triangle, and circle. I'm just kidding about the circle. However, I should have drawn one. That's funny. My 8th grade students were terrible at this. I use "terrible" with all the love in the world, knowing this is a learning experience for them.

Me: Have you guys ever messed around with tangrams?

Class: No!

Me: WHAT!!!! Are you guys serious? No one has ever let you mess around with tangrams before? Well, I'm glad we're doing it now. You guys need this. Seriously? You guys have never messed around with tangrams.

Class: Nope.

Me: Okay, well keep trying. [as I began scraping my jaw off the floor]

My request to you all: MORE TANGRAMS PLEASE!

Especially elementary teachers, more tangrams please. Have your students mess around with them. Sure you can download some app onto your tablet or find a web-based site to simulate tangrams, but please do your best to get actual tangrams into the hands of your students. Math formulas come and go for math students. However, if they can visually break apart polygons into more recognizable polygons such as rectangles and triangles, I believe their mathematical proficiency greatly increases. My goal is to get these 8th graders to play around with Tangrams once a week for the rest of the year. At least one of my students was eventually able to put together a trapezoid (top left), which quickly turned into a parallelogram, which quickly turned into a rectangle.

Me: How'd those other shapes come so quickly?

Sean: I just moved this one larger triangle to different spots.

I took a picture of his first configuration so I could share it with the class. I figured I'd give the class a chance to redeem themselves and copy his rectangle configuration.

More tangrams please! 

Repeat after me:

Thanks for listening.

Tangrams,
1104


BTW: Cheat sheet for displaying student work immediately:

1. Sign up for Dropbox.
2. Have the Dropbox app on your phone.
3. Take picture(s) of student work.
4. Allow the app to upload your camera photos.
5. Sync your computer with Dropbox.
6. The pictures arrive on your computer in seconds.

Wablammo!

Mistakes to the Half Power

Today, after completing Day 128 of Estimation 180, we briefly reviewed the first four questions of the handout from Day 1 of exponent mistakes. Read about Day 1 here. We then attacked the next four questions and each of my four Algebra sections had different ideas, explanations, successes, and challenges. It was beautiful.

Question 5 brought the best conversations of the day. Many students thought that 100 to the half power was 5000 because they went 100*50, thinking 100 should be multiplied by half of itself (50). There were two classes with one student in each that thought the answer was 10, but each student had a difficult time articulating their reasoning. Those two students weren't necessarily convinced with their answer either. I took the lead on this one after hearing this from a couple of students:

It can't be 5000, that's too high. It's gotta be somewhere between 1 and 100, but it isn't 50.

I wrote the following on the board:
100^3 =
100^2 =
100^1 =
100^0 =

Me: Do me a favor and evaluate each of those four expressions.

Respectively, students gave me:
100^3 = 1,000,000
100^2 = 10,000
100^1 = 100
100^0 = 1

It helped that we already proved a zero exponent simplifies to one. PHEW! So I stood there with a nondescript look on my face and asked students to make observations. They got my "What do you wonder? What do you notice?" face.

Me: Let's talk number sense here everyone. Where would we place 100 to the one-half power?

Many students quickly placed it between 100^0 and 100^1. So how is that 5000 looking? Does that make sense? Some keen observers (only a few throughout the day) noticed a pattern of decreasing zeros. One million decreases by two zeros to get ten thousand. Ten thousand decreases by two zeros to get one hundred and one hundred decreases by two zeros to get one. No one really said it was because they were dividing by 100 each time, but I let it slide. I didn't want to take anything away from some of the lightbulbs lighting up in class.

Sheena: Since you take away two zeros each time, and 100^1/2 is between 100^0 and 100^1, you only take away one zero to get 10.

Me: (to the class) What do guys think?

Class: Yea! Totally! That really makes sense.

Me: Is it enough to convince you all? Will this work every time? So let's try this:

I asked students for factors of 100. Students gave me:
50*2
10*10
4*25
10*100

Me: Which one of these is a perfect square?

Class: 10*10

Me: So let's rewrite 100 as 10^2 and keep the half power.

Class: Woah! It's ten to the first. So it's ten!

Me: So what can we conclude about something to the half power?

Student: It's the square root.

I toss up a couple expressions for confidence: 49^1/2. They shout, "seven!" or 64^1/2. They shout, "eight!" Now, if you want to really mess with some of your kids, throw 27^1/3 up on the board and ask, "So what is something to the one-third power?" You might get lucky with a kid that says 3. Maybe walk them through that one. For my honors kids, I would ask, "What number on the board is both a perfect square and perfect cube?" You know you have a smarty pants when they say 64. That's a gem right there. Don't expect that often in 8th grade. That's my Elijah! (the closest I'll get to Fawn's Gabe.)

Question 5 stole the show and our explanations for questions 6-8 were not as spectacular. That's fine. However, I think I'll switch the order of the questions and put this question #6 last (after current question #8). I found that the flow of explanations from students for current question #8 (quotient) really helped explore/explain negative exponents, making question #6 easier for students.

In one of my classes, Arielle came out of nowhere and gave us this gem. I immediately put it up on the Quotes of the Week board.

That's right! We're exploring these rules and students are defining them through observations and patterns. I think students have a better understanding of these properties and rules when given incorrect solutions (mistakes). In case I don't have time to wrap things up with you about tomorrow, the handouts from this week will be at the bottom of this post. Michael Pershan recommended I tell students that some solutions are incorrect and some are correct. I tried that out for Days 2 & 3. We only had about ten minutes to start today's handout. After their abbreviated individual time, I put these two things up on the board:

1. Share with your group the one solution you feel most confident about.
2. Select one question you want to know the answer to most.

I had students stand and vote once (by raising hand) for the question they wanted to know the answer to the most. I kept a quick tally. Most classes picked question #8. Still standing, I told students to either face the hallway if they thought the question was correct or face the windows (opposite the hallway) if they thought the question was incorrect. I take a quick tally. Sit back down and have students share out loud. I'd resume Fish Bowl if I had more time. Try this sometime. Fun. 

Tomorrow: more group work, less teacher. 

Half-power,

1002

Day 1 handout

Day 2 handout

Day 3 handout