Cent-ed Whiffle Balls

Want to know how to make Cent-ed Whiffle Balls? Here are the ingredients:

1. Bookmark this picture at 101qs.com
2. Do coin estimation with your students.
3. Go to the bank and withdraw a few dollars worth of pennies.
4. Get some Gorilla Glue.
5. Take whiffle balls from your son's collection (source of whiffle balls may vary). 

Show your students the picture from Step 1. Do the estimation task from Step 2. Show them the following slide! 

*If you don't know yet, we covered surface area of spheres in Geometry this week.

We just completed Nathan Kraft's Soccer Ball 3 Act lesson which was spectacular for volume of a sphere! (Nathan, post act 2 and act 3 for everyone NOW!) The Cent-ed Whiffle Ball is a simple task. You know you have a keeper when you hear the following come out of students:

"This is fun!"

"This is stressful!" 

Students first started this task by using a tape measure to find the circumference of their whiffle ball. Thankfully, I've finally won them over on using centimeters. Shooosh! Don't tell those people who like inches. Students then used the circumference to find the radius of the whiffle ball. Well done, kiddos! Next, students either used a tape measure or ruler to get the circumference or diameter of a penny, respectively. Ultimately, they wanted the radius of the penny. Then they got stuck.

"Mr. Stadel, what's the surface area formula for a sphere?"

Sweet! They want it. They need it. They crave it. I didn't write it on the board or give it to them on a handout. Here's where I wish I had an additional hour with these kids to explore this formula. Instead, I had a demonstration ready for them. I took our Nerf basketball we use for Math Basketball Review. I told students that I measured the circumference of the ball in order to construct a circle that has the same circumference. Before class, I cut out a second congruent circle and cut it into eight congruent sectors. I then played this game with students:

Me: How many of these circles will it take to cover the entire ball?

Student 1: Three

Student 2: Four 

Student 3: Three and a half

Student 4: Five

Me: Let's find out!

I pinned the sectors onto the Nerf ball with thumbtacks, covering a fourth of the ball.

Student 2: I was right! It's four!

Student 5: Cool!

BOOM! We had our formula: 4 areas of a circle with the same circumference as the sphere. Simply put: 4πr^2. Most groups immediately found the surface area of the whiffle ball and penny, dividing the two to get something like 88 pennies. One group of girls immediately came up to me and asked for their pennies. Before giving students their pennies, I drilled each group, asking them to explain their number and show their work.

Me: Now girls, if we've learned anything in here this year, we know that our answer on paper isn't always the actual answer. Have you accounted for everything? Look at this picture again (from the ingredients). Did you account for everything?

Devon: There's spaces between the pennies. 

Me: Yup. Why don't you go back and mathematically show me a different number of pennies, now accounting for those spaces.

I had this conversation with each group, or some variation of it. This is where the magic begins. Remember, students were allowed a maximum of six pennies. Here's what they came up with. I'll let the pictures do the talking:

 

 Chris asked for a compass to draw a circle having the same circumference as the sphere.

Elle found the area of a rectangle formed by six pennies. She then subtracted the area of six pennies to get the area of the space created by six pennies. 

Noelle used a parallelogram of pennies to execute the same idea as Elle.

Groups started coming back with revised numbers. They quietly told me their amount. Remember, there's a CASH PRIZE on the line! Im still not sure what that is yet. Groups came in with the following amount of pennies to cover their whiffle ball:
70 pennies
65 pennies
69 pennies
62 pennies

Good luck to them all. They are almost done gluing their pennies. Two groups are done and the other two are close. Here's a few pics!

This group used 71 pennies versus a theoretical 69.

I highly suggest you make Cent-ed Whiffle Balls in class! If not, here are the dimensions:

Whiffle Ball circumference: 28 centimeters
Penny diameter: 1.9 centimeters

Cent-ed,
1050

P.S. Help me make this task better.

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!

Not Drawn to Scale

I hope you'll allow me to vent for a bit. I have been encouraging my students to be in tune with the 8 Mathematical Practices by Standard of the CCSS for some time now. It's pretty safe to say that my students know I really favor Mathematical Practice Standard 6, Attend to Precision. However, some of the resources I occasionally use in class are beginning to play tricks with everyone's minds, including mine. Here's a resource I have, Cooperative Learning and Geometry by Becky Bride.

Don't get me wrong, I like this book. It has some great explorative exercises that have appropriately challenged my students. For example, look at this exercise to help students derive the 30-60-90 triangle relationships. Take an equilateral triangle, its altitude, and the Pythagorean Theorem to find out the special relationships between the shorter leg, longer leg, and hypotenuse. Great.

Here's where I start to beat my head against the wall. The book uses diagrams that simply shouldn't be used, especially in the context of 30-60-90 triangles. Look closely...

That's right, the 30 degree angle is opposite the longer (drawn) leg for questions 1, 3, and 4. My students get bothered by this contradiction. I do too. I have no problem admitting this to them. I'm honest with them saying, "I know guys. It goes against everything we strive to do in here. I encourage you guys to attend to precision and check for reasonableness. Yet, I give you this. I'm sorry. It says at the top 'not drawn to scale', but they should be drawn to scale. Right guys?!"

I think this about sums it up. Students will come up and ask about the dimensions they've solved for and whether or not they're reasonable. I'm proud of my students for making sense of their answers and checking for reasonableness.  I know something is a skew when my response to those students is,

"I never assume those things are drawn to scale." 

I feel rotten saying this to students. I feel like I've just provided them with a worthless and menial task. I've let them down. I feel dirty. Mr. Stadel's quality control group hasn't done their job. What message are we sending students? Do they think we're out to trick them? Do the directions read, "Find the mistakes?" They should. It's times like these that force me to (gladly) keep a closer eye on the content I provide my students with. Don't just throw some triangles at them with random angles and units. Make sure they're reasonable.

Have you ever felt this way? Have you ever been caught in this situation? What did you do? How do we avoid these situations again? How do we demand better quality content from publishers? How do we make sure we provide our students with content that matches the CCSS and Mathematical Practices? Maybe you're okay with these types of diagrams, so please explain why. I want to hear from you all on this.

nOt tO sCAlE,
939

Trashketball (2013 Pi Day task)

It all started with an episode of Suits on USA Network from January 31, 2013 (episode 213: Zane vs. Zane) where the opening scene has the two main characters (Harvey and Mike) playing a round of H-O-R-S-E trashketball in Harvey's office.  I jotted this one down on my digital "task ideas" list and knew it might have some potential later this year in Geometry. Here's Act 1:

Dan Meyer has thrown us some wonderful updates on 101qs.com. Head over to the Trashketball task where you will get all the goods when you sign in:
Act 1: video to wonder and notice about
Act 2: teacher notes, and visual data/information to help solve the task
Act 3: visual confirmation of the practical answer
Sequel: additional tasks to explore (especially for early finishers) and teacher notes

I was going to chip away at this task until I realized Pi Day was coming up. Needless to say, I started working a little quicker. Ironically, in calculating the answer to the task, Pi can actually be divided by itself or "cancelled." I grabbed (bought, not shoplifted) two trashcans from Bed Bath & Beyond. I found the exact trashcan from Suits. Woohoo!!!! That circular truncated cone trashcan is so dreamy and transparent. I also found a cylindrical trashcan for my Geometry class. As you can see from Act 1, it's not transparent, but it'll get the job done. Measuring each dimension of the can was simple. Measuring the diameter of the trashketballs is a different story. I'm open to suggestions here. You'll find this in the "Teacher Notes"

How do you find the diameter of a trashketball? Have your students come up with ideas. Test those ideas. Make conjectures.

I crumpled up 8.5"x11" paper and made it as compact as possible. I took a handful of trashketballs and put them down on a ruler to get a rough mental mean of the diameters. Then I traced the best-fitting circle to measure the best-fitting diameter of each trashketball. I took the mean of these five diameters.

An extension to the task would be to explore the difference one-tenth the radius makes in your calculated answer.

Seriously, I'm open to ideas. I quickly discovered that trashketballs are like snowflakes: no two are the same. However, I really started to perfect the form and process of making a trashketball. I'll admit, there's some buy-in with the trashketballs being perfect spheres. I'm okay with that. So maybe spend some time with your students perfecting the trashketball. Anyway, leave some ideas about measuring the diameter of the trashketballs in the comments, won't ya?

I'm looking forward to this task. My students occasionally play trashketball in my class with their scratch paper or class handouts (not necessarily mine) contributing to their idea of going paperless. I see this happening a lot on Thursday. Happy Pi Day!

Next up! The circular truncated cone trashcan. I'll start chipping away at having enough trashketballs for the circular truncated cone. Thanks in advance to the following people for helping with the volume of the circular truncated cone trashcan:
@mjfenton, @absvalteaching, @MaryBourassa, and @RobertKaplinsky.

Swish,
1140

Wooden Balance Game Pt. I

Let's play a game! Actually, you're welcome to invite your students to join in the fun here as well. Here's what you do:

1. Watch the video below.
2. Check out the specs.
3. Submit your order.

1. Video:

2. Wooden Solids and specs:

Make estimates of the dimensions.

What do you notice? What do you wonder?

3. Submit:  goo.gl/naDhr

Good luck! I'll tally your submissions for the week and stack the top configuration.

Balance,
327

Best Halves [Square]

A few months ago Dan Meyer reached out to Timon Piccini, Chris Robinson, Nathan Kraft and me to participate in what would eventually become his Best Midpoint, Best Square, Best Triangle, and Best Circle series of 3 Act lessons. I was honored to be part of a stellar group and great lesson. I love the potential of these lessons and can't wait to use them with my geometry kiddos later this year. Currently Dan and Dave Major have kicked it up a notch with some great interactive play/learning for better best squares, also providing us with an interactive teacher's guide. Check it out: I nearly cried tears of joy upon reading their two posts: Dan and Dave.

Recently, I've had conversations with Fawn Nguyen about fractions and although fractions aren't the spotlight of my Algebra and Geometry curriculum, I'm still fascinated by them and in turn want to help students build their number sense or spatial reasoning. I had an idea to extend Dan's Best series into the realm of fractions and emailed him for his blessing, hoping I'd do it justice. Here's what I came up with so far:

You might notice it closely resembles Dan's format with very few stylistic differences. "If it ain't broke, don't fix it." That's my motto here. I called on Dan and a few other comrades to make an appearance and compete in this first installment of Best Fractions. This first installment: "Who drew the best half?"

Thanks to Dan, Fawn, Sadie Estrella, and Shauna Hedgepeth for taking the time to contribute. They were great sports! I still don't know who drew the best half yet.

I see a lot of geometry potential here: area, perimeter, midpoints, distance, coordinates, polygons, etc. I'd love to target primary grades with this activity as well (not just secondary), finding an entry level that elementary kids are capable of exploring. I'm not too sure calculating the area of trapezoids would be appropriate for a 4th and 5th grade classroom, but I might be wrong.

I'm not pretending to nail this 3 Act lesson and I'd love some feedback on how you would apply this in your class or make it better. I'm still working on the Act 2 information and will gradually chip away at it over time.  I gathered enough information from the contestants to keep me busy for the next year. I plan to release other installments of Best Fractions, specifically the best half, third, fourth, and fifth of both a square and circle. Just imagine the fun with circles: area, sector area, arc length, degrees, percentages, and more. Stay tuned!

Test it out on your students in the meantime and give me some feedback. Click here for directions and handouts to use with your students.

Best,
420

Transversals, Tape, and Stickies

Today in Geometry, we're discussing two lines, a transversal and the angle relationships formed. We did a few minutes of word wall pics and direct instruction of Corresponding angles, Alternate Interior angles, Alternate Exterior angles, and Same-side (or Consecutive) Interior angles. Then students were presented with the following setup on my walls. I used three strips of masking tape to create the lines intersected by a transversal and numbered stickies. I was able to set up 3 stations since I have a small group of 8th grade Geometry students this year.

Students worked in groups and were instructed to start with the two parallel lines and the transversal. It's a lower entry point as opposed to the three lines intersecting to form the triangle (which my textbook chooses to introduce this concept. Silly publishers). Groups are given the following handout and need to place the stickies in the correct places, based on the given clues. Work together, GO!
Handout and solutions here.

If you have limited space, create 1-2 stations and have groups rotate as other students are completing a task at their desk. Put a timer on the board and tell the students to get as far as possible within the allotted time. When the timer finishes, I'd take a picture of their work, reset the stickies, and let another group tackle it, resetting the timer.
Here are possible solutions. Let me know if you find any errors.

It went well. There was a lot of tension in the groups. Some kept moving stickies around because they disagreed. They disagreed because of the overall connection, not because of getting the relationship wrong. It was so fun to hear them get so excited about this activity. We ran out of time and the quote of the day came from a girl, "That's upsetting me." She wanted to finish. She wanted to know the answers. She wanted to figure out the puzzle. Many other students had similar feelings. I love it!

What I learned: Don't make the groups too large. Go with about 2-3 students (4 max) per group. Use really good stickies. The orange ones you see in the pictures were old and had lost their stickiness. If groups are struggling too much, encourage them to find a set of angles that has the least amount of possibilities.

Transversals,
1239