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

Thank You Math Mistakes

Every year I dread teaching exponents like nothing else. I still do. It's one of those concepts (units) I have a tough time relating to, and if I have a tough time, imagine my students. For me, exponent rules and properties have been reduced to nothing more than a good puzzle. Not sure admitting that here is wise, but I do enjoy puzzles. However, to me, expressions with exponents don't necessarily lend themselves to having applied meaning for middle schoolers, or at least no simple contextual application I can relate with. This is where you jump straight to the comments and say something like,

"Andrew, exponents are seen in [insert awesome idea] and here's a link." or

"Mr. Stadel, students can relate to exponents when they [insert other awesome idea]." or

"Stades, I'm surprised you haven't done a 3 Act lesson on exponents using [insert other fabulous idea]."

"This year will be different." You ever mutter that to yourself? Well this year is and will continue to be. I shamelessly request you to temporarily stop reading this blog and visit Michael Pershan's website Math Mistakes. Seriously, type mathmistakes.org into your url address. The more I research teaching strategies and content, the more I'm starting to see the benefit of students learning math by identifying mistakes, correcting them, and justifying their reasoning while doing so. Inspired by Michael's post on exponents, here's the handout I threw at my 8th grade Algebra students today with the following directions:

The following statements are all INCORRECT.

1. Identify the mistake(s).
2. Correct.
3. Justify (show) your reasoning.

Me: I'll give you guys 3-5 minutes of individual time to work through as many questions as possible. Then you'll share and discuss your ideas with your group followed by a whole class discussion.

Students worked individually on the questions for a few minutes. When most students were at least 75% complete with the handout, I told them they had until the end of today's estimation song Can't Buy Me Love (in its entirety) to share their solutions and reasoning with their group members. I heard some great stuff. Next we did our Fish Bowl where students come up to the front, walk us through their work and reasoning on the board while we listen and watch both intently and quietly waiting to ask questions. Here's the four we got through today.

Students in the Fish Bowl identified the mistake as multiplying two by five, producing ten. Students also correctly explained how two to the fifth power is thirty-two. The fun part (for me) was seeing multiple representations. 

1. Common: 2*2 = 4, 4*2 = 8, 8*2 = 16, and 16*2 = 32.
2. Grouping two's so 2*2 = 4 twice. 4*4 = 16 and 16*2 = 32.

The mistake here was explained a few different ways, but mainly revolved around the student forgetting the negative.  So far, questions 1 and 2 have a lower entry point for today's task, considering these students haven't seen exponents since last year. Multiple representations looked like this:

1. Common: -2*-2*-2 = -8
2. Preferred: (-2)(-2)(-2) = -8

The classic. I always enjoy this example. Students explained that the mistake was made by multiplying -6 and -6, producing positive 36. Many students pointed out that the exponent is "attached" to the closest term (6) and not the entire expression (-6). Multiple representations included:

1. Common: -(6*6) = -36
2. Common: -6*6 = -36
3. Extra: -1*6*6 = -36

This was an extremely fun conversation to have in class. In fact, for some periods, we never established a conclusive answer to the question before the bell rang. Here's what they came up with. Some students simplified it to thirty-seven and many were convinced of this at first. Very few simplified it to one, but couldn't convince anyone why. Some students offered the following:

Joey: My fifth grade teacher told me anything to the zero power is one.

Raquel: There's some rule that says it's one. It's just the rule. 

Me: Who's convinced by their reasoning? No one? 

Here are the multiple student representations we saw:

1. Pattern: 37^2 = 37*37,  37^1 = 37, so 37^0 = 1 (this didn't convince many).
2. "Anything over itself is 1," some said. Therefore, 37^5 over 37^5 is one and you subtract the exponents (5-5) to get 37^0. Therefore, 37^0 = 1 (this convinced many).

I forgot to mention that before we did Fish Bowl, I asked students what was different about what they initially did with the handout. Here are a few things they said:

1. You gave us individual time instead of just going straight to group work.
2. We had to make corrections.
3. We had to try and figure out the questions without you telling us any rules.

The last observation was my favorite. This activity gave students a desire to listen to each other and want to know the answer to these questions. I wasn't at the front of the room blabbing out rules, properties, their names, and examples. I didn't provide students with guided practice. This combination of activities and strategies felt right. More to come tomorrow, but we can't ignore that there's something to this mistake idea. What do you think?

Mistakenly,

1227

*UPDATE: The next day goes like this.

Day 1 handout

Day 2 handout

Day 3 handout

Capturing Time (musically)

Recently, I had the idea to do a theme of "song lengths" over at Estimation 180. Inspired by a recent comment from Fawn, I chose Santana's Oye Como Va. At first, I opened up iTunes and took a screen shot of the music player.  I threw in an album cover and edited it to look like this, asking "How long is Santana's Oye Como Va?":

I can get away with directly asking the question at Estimation 180. How would you make your estimate? I'd make my estimate based on the time played so far (1:26) and the location of the playhead in the timeline. I absolutely love that students have to use time here, specifically 60 seconds in a minute. Furthermore, I'm hoping students use some type of spatial reasoning with the timeline, either as a fraction, percentage, proportion, or something else. But that's it. Can we go anywhere else with this? This task feels constrained. This doesn't capture the medium of music correctly. There's got to be more, right?

The more I thought about it, I was curious of better ways (or the best way) to capture time and music. Let me rephrase that. If I were going for a more perplexing approach and wanted to create a 3 Act task to share at 101qs.com, how would I go about doing that? I remembered that I own the djay app and experimented with a really lengthy Jethro Tull song titled, Thick As A Brick. This is where I need your help. I'd appreciate you checking out Act 1 and letting me know the first question that comes to mind. Or watch it here and leave a comment/question in the comments.

Based on some initial questions, I'm thinking of revising Act 1 where the virtual record player looks more like this. (notice the record?)

The virtual record player opens up many possibilities with this task. There's a white tape marker on the record for precise tracking when playing the track. I feel there's a lot more math opportunities here, but at the same time it feels a little contrived?
Am I over-thinking this?
What do you see here?
What are your thoughts?
I need some help. Thanks in advance.

Spin it,
339

Buses [Day 2]

Woah!!! We saw six buses today on our way to preschool. You might want to check out the Buses [Day 1] post from two days ago to understand the context of what's ahead. I was a little late to the bus-counting action on our way to preschool today so here's our first exchange.

Me: Have you seen any buses today?

Son (who turns 3 in a month): Yes. One.

Me: Was it a city or school bus?

Son: City. 

Me: Okay, we'll have to look for more buses today.

We immediately saw another city bus, bringing our total buses to two on the day. A little bit down the road was another city bus heading in the oposite direction.

Me: I see something on the other side of the road coming this way. What is it?

Son: It's another bus!! A city bus!

Me: How many total buses have we seen today? 

Son: Three.

Me: Good.

We continue along and a shorter city bus passes us. I don't point it out, but he spots it.

Son: There's another bus. 

Me: You're right. Wow! Now, how many buses have we seen?

Son: Hmph.

I pause and wait for him to process the question. Keep in mind, this is happening while I'm driving and he's in the backseat so I can't turn around to talk with him.

Son: Hmph

If I get a second "hmph" I know he doesn't have an answer and won't come up with one. Trust me, I've waited for long periods of time and will just continue to hear his cute little "hmph."

Me: Well, we already saw three buses and now we saw one more. What number comes after three?

Son: (whispering to self: one, two, three) Four.

Me: Good. So how many buses have we seen?

Son: Four.

Surprisingly, we hadn't seen a school bus yet. We were at our last red light and there it was in all its glorious mustardy-yellowy paint, a school bus. He exclaims, "A school bus!" Since we're at a red light, I turn around and ask how many buses have we seen. I get the two "hmph" count. I hold up one hand with four fingers up and the other hand with one finger up.

Me: We saw four city buses and now one school bus. How many buses have we seen today?

Son: Three

Did we just have a flashback (regression) to Tuesday?

Me: We did see three buses today, you're right. But, I think we've seen more. Count the fingers.

Son: One,... Two,... Three,... Five.

The light turns green and I have to go. I'm curious about him skipping 'four' and still landing on the correct number of buses. Seriously, what's up with that? I mumble to myself, "That's odd that he skipped four." Right as we're about to pull into the preschool parking lot, another school bus goes whizzing by in the opposite direction. He exclaims, "Another school bus!" Wow! We saw six school buses today, so I'm thinking we park the car and quickly review this last school bus.

Me: Wow! We just saw another school bus. We saw a lot of buses today. We saw five buses and now we saw one more. How many total buses did we see today?

Son: Hmph. (x2)

I hold up one hand with all five fingers up and the other hand with one finger up.

Me: So we saw five buses and we just saw another bus. Count the fingers.

Son: One,... Two,... Three,... Four,... Six.

Me: You're right. What happened to five?

He giggles! I do too because it's contagious. Seriously, what's up with this? He answered the correct number, but skipped the number directly preceding it. TWICE! He's happy he saw so many buses today. I am too. He's content with landing on the correct number. I'm perplexed.

Hmph,
1140

BTW. Thanks Christopher Danielson, for continuing to inspire me to have (and cherish) these conversations with my son.

Dancing with the Functions

I'm honored and humbled to have been a (small) part of Christopher Danielson's online course The Mathematics in School Curriculum: Functions. There were some great tasks, discussions, and contributors. I now have a better misunderstanding of functions. However you interpret that last sentence, let me assure you that this two week course broke me down in order to give me a better perspective and idea of functions. Professor Triangleman moderated the course well, provided challenging tasks and opportunities that took me out of my comfort zone, encouraged us to think differently, and didn't hesitate to whip us into shape as you can see here:

He's referring to beating me down while informing the teacher's pet (Fawn) of his tactic.

Our choices for our final project were:

• a blog post,
• a lesson plan,
• an interpretive dance,
• a work of visual art,
• etc.

I thought writing a blog post was "too easy" in the sense that I could blog about anything ordinary at anytime. This class wasn't ordinary though, and I felt I'd rather try and give something back to the class, professor, and community in exchange for what I have received. No Fawn, not because I'm "too lazy." Therefore, I'm going to give you a lesson idea I have, based on an interpretive dance, which might be a work of visual art, all wrapped up in a blog post.

Interpretive dance really got me thinking. I thought back to the handful of dance lessons my wife (fiance at the time) and I took to practice for the First Dance at our wedding. My wife was a natural. As for me, well let's just say all the dance lessons in a lifetime wouldn't have helped. Here's a dance photo I have of us where it actually looks like I'm doing something worthy. Don't be fooled.

Don't worry, I won't torture you with video. Anyway, our dance instructor taught us many helpful tips and gave us a glimpse of dances like swing, salsa, the waltz, and the two-step box. We only did a few moves in our wedding dance, but it mainly revolved around the two-step box. We had a short song, thank goodness. I'm sure our guests would have taken their gifts back had they seen me dance any longer.

Here comes my lesson idea. I'd like to see the relationship between the number of steps taken in a dance over time. So let's make it a graphing story. Here's the first 30 seconds of a dance. Write a story for it. Even better, can you write the functions (along with any intervals, domains, ranges, etc)? Go here to Desmos to check your answers. I give you my interpretive dance.

Thanks to Sadie, Timon, and Michael Pershan for inviting me to their hangouts. I was honored to collaborate with you guys during one of the hangouts and bounce ideas off of each other. Thanks to Fawn for getting me in trouble, ratting me out to the teacher, and reminding me to submit my final project. Where would I be without you? Probably in class and not in the principal's office.

I would love some feedback on this lesson idea. Would you have your students dance? If so, what dance(s)? Would you have students come up with a function for each type of dance? What kind of relationships would you have your students look for? Would you consider "dancing" an applicable use of functions? I leave you with this clip. You must give these guys (Sean and John Scott) some crazy respect. They're insanely fantastic at tap-dancing. Just watch the first minute. Then make a graphing story.

Dance,
933

Buses

I just returned from taking my son (turns 3 in a month) to preschool. One of the many perks to Spring Break so far! On the way to preschool, my son spotted a city bus.

Son: Ohhhh! A bus!

Me: Right. What type of bus? 

Son: A city bus. 

A little background knowledge here: He loves trucks! Let me rephrase that. He loves anything larger than a car, has an engine, is big, makes a lot of noise, is big, does construction, is big, intakes diesel, etc. I've seen many kids share these interests, especially when the garbage man does his rounds in our neighborhood. You'd think the garbage man was passing out ice cream or something (and he doesn't need that silly Ice Cream truck music either). We have multiple truck books that get frequent use before nap and bedtime. We have a surplus of toy trucks and Legos, recently added to by the most generous, wonderful, and great Fawn. You're the best!

Back to our drive. We had just stopped at a red light and on the other side of the street he spots a school bus.

Son: Ohhh! A school bus!

Me: You're right!

Son: We've seen two buses!

Me: I know. Look what's coming up behind the school bus.

Son: Another bus.

Me: And what type of bus is that?

Son: A city bus.

Me: So how many buses have we seen?

Son: Three!

Me: That's right. We've seen 2 city buses and 1 school bus, so we've seen a total of...

I pause for him to fill in the blank.

Son: Three!

We're still stopped at the red light and have a little time before the green light. I turn around and illustrate this again with my fingers, because lately he's been doing really well identifying the numbers one through five on a single hand. Since we saw two different types of buses, I use two hands. I hold up on hand with two fingers up and say, "We've seen two city buses" and on the other hand I hold up one finger saying, "and we've seen one school bus. So we've seen a total of how many buses?"
I expect him to say "three" because it's fresh in his mind, but he pleasantly surprises me and I can see his eyes moving across my fingers and mentally counting the fingers to verify the word 'three' matches up with Dad's fingers. "Three." he says.

The light turns green and we're on our way. We have less than five minutes until we get to preschool. Of course, I'm keeping my eyes peeled for more buses. No buses. Shucks. However, we pull into the parking lot of the preschool and park. Before I exit the car to get him out, I turn around and want to try something. Simply for fun.

Me: So we saw two city buses [I'm holding up two fingers on one hand]. What if we saw two school buses [I hold up two fingers on my second hand]. How many buses would we have seen?

Son: Three.

Me: Are you sure? Count my fingers.

Son: One... two... three... four, five, six, seven, eight, nine, te...

Me: Okay, silly. [holding up two fingers on each hand again] If we saw two city buses and two school buses, how many total buses would we see?

Son: Three.

I left it at that. He's content with the concrete. It's not about future counting for him. It's about what he just experienced, what's relevant, what's applicable and what's associated with his interests. I made a small attempt at the abstract, just for fun. There's no need to push this any further. He's convinced we saw three buses and he's right. He doesn't care about a fourth bus that we didn't see. Plus, it's time for preschool. Man, I love vacation. I get to have conversations like this with my son. It doesn't get any better than that.

Buses,
1047

[UPDATE]
*Read what happened two days later in Buses [Day 2].

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

Race Car Math

If you've looked at any of Dan Meyer's Algebra or Geometry curricula on his blog, you'll notice he has "Race Car Math" throughout many of his Keynote slides. Since I couldn't put two and two together on this one nor find anything on his blog explaining it, I simply asked him.

My loving wife bought my son and me an RC Ferrari car for Christmas 2011 (featured in the 3 Act lesson Ferrari Ride), but I figured I'd invest a few bucks in a classroom RC car for Mr. Stadel's room. I made my way over to Toys "R" Us and spent less than $15 on this bad boy. The remote is about the size of a crayon box and the car is about the size of a cantaloupe. It doesn't go crazy fast. It's just the right size and speed for a middle school student, boy or girl. This might be one of the best $15 I've ever spent.

This week, we spent Wednesday reviewing linear systems in algebra and quadrilaterals in geometry. We have reviewed with Math Basketball numerous times this year and the ground rules are very similar. Check out Dan's Math Basketball directions if you need some guidance. Here's how I roll in my class. I toss a slide up with the following information. *The slide for my students is less text-heavy.

1. Work individually in your notebook (unless I announce "GROUP ANSWER" meaning students work with their group/table on their giant whiteboard).
2. Show all work.
3. Talking = DQ  (*talking during "group answer" is allowed)
4. One person stands with answer (all members of a group must stand before standing again).
5. 10 seconds with car:
1. Big box = 1 point
2. Medium box = 2 points
3. Small box = 3 points

When students are done solving a question you've thrown up on the board and most (if not all) groups have a representative standing, I usually call on the last person that stands up. They explain and/or give their answer. If any other person agrees with the answer, they sit down. I've already been circulating the room checking student work, so mentally I have a decent idea who has the correct work (answer) or not. If anyone is still standing, that means they disagree and have a different answer. I repeat this until every student sits down. If all groups have the same correct answer, I announce that. If there's one different answer, I (for time reasons) will demonstrate the correct solution while students watch in suspense to see who's correct.

Whoever stood and got the answer correct comes to the front of the room to represent their group in driving the car. If a person got it wrong, they are to stay at their desk and study the board so they can write down the correct solution or talk with someone around them for the correct solution. Since students must drive the car along an L-shaped path, they can follow it to the finish line. Here's what's at the finish line.

Students can drive the car into the largest box worth 1 point, or two other boxes with narrower openings worth two and three points. The 3-point box is the narrowest. In order for the team to get their points, they must enter the box (cross the plane) with the two FRONT tires before the end of 10 seconds. I count down. Ready. Set. GO!

We had a couple of groups somehow get two tires on the same side of the car in the box, but it didn't count. Set up your own rules. Whatever they are, stick to them. Many kids said they liked this better than math basketball. I can't blame them. You should see some of their shooting form. On second thought, you might not want to see their basketball form when shooting a nerf basketball. Scary!

My favorite exchange came from Chase in geometry who scored three points for his team with ease.

Student 1: "Chase, you're really good at that."

Chase: "Yea, I'm part of an RC car club."

Student 2: "Really?"

Chase: (sarcastically) "Yes, after school I practice driving RC cars."

Student 3: "Really Chase? Wow! Do you really?"

Chase: "Yes, I'm part of an RC car club." and sat down.

Chase remained straight-faced the entire exchange. Hilarious! I'm not sure if some students were still convinced he was part of an RC club, but he's not. Since this was my first time with Race Car Math, I'm happy how I kept fine-tuning it throughout the day to make it more efficient and student-friendly. For instance, with larger classes I said we would solve two questions first before racing the car. If they got both questions correct, their team would simply double the points of the box their car entered. If a team only got one question correct, they would get the box's points at face-value. This also allowed students to send up who they thought could best drive the RC car. For round two, that person could not drive again.

I found that the classroom dynamic and energy to be better when I did more "group answer" questions and students collaborated on their giant whiteboards. It's a win-win. They get to stand up, talk to each other, and collaborate just like any other day in class. Plus, I get to listen to them problem-solve, argue, agree, and cheer each other on.

*[Update] Here's an idea for the to-do list: Another way to play is have students accumulate correct answers for a sequence of approximately three questions. Each group earns 10 seconds with the car for every correct answer. Set the boxes up like goals on a soccer field, maybe 15 feet apart. If the group got three questions correct, they have 30 seconds to drive the car between boxes to score as many points as possible. If you test this one out before me, let me know how it goes.

Ready-Set-Go!
411

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

Couch Coins

Today, I had 25 minutes to get as far as possible with Couch Coins and a second grade class at my school. I'd like to debrief on a few things, but here's Act 1.

During the Summer 2012, I found a money/coins concept in a Second Grade Everyday Mathematics book similar to Couch Coins. I was inspired by PES' Coinstar commercial and ran with the concept. The intended question is: "What coins will my wife get?" On my way into work knowing that I would soon be surrounded by seven and eight year olds, I announced to Twitter that I'd be doing this 3 Act lesson with second graders today and wondered how it will go, asking for their thoughts. The response was great. Robert Kaplinsky, Christopher Danielson, and Sadie Estrella all chimed in offering that money can be challenging and to be careful of the "fewest coins" part of the task. In other words, finding the total value of the coins and half the total might just be challenging enough for second graders. Chris Lusto (in true Lusto fashion) provided some comedic relief:

Act One: Which one of these kids do you think has to pee? Act Two: Watch them squirm. Act Three: Reveal. (Act Four: Clean up.)

Why just 25 minutes? I had to bail after 25 minutes to go teach my own students (8th graders) or else I would have had about 50 minutes with these second grade kiddos. So what transpired in those precious 1500 seconds? I asked the students what they wondered and noticed about the Act 1 video. I had them write down at least one thing they wondered and one thing they noticed. I asked them to share, starting with "notice." Keep in mind my time was limited here so I only took a few...

-The coins were moving on the chair like magic.
-There were just coins.
-There were a lot of quarters.

Now, I asked what they wondered. I followed each student question with, "Who else would like to know the answer to that question?"

-Why are the coins moving out of the couch? 10
-How did the coins stack like they did? 7
-How much is half of the coins? 10
-How many coins were moving on the couch? 3

Not one student asked anything about the coins my wife should get. NO PROBLEM! You can see there was a tie between two questions and don't ask why the numbers are so low; the class had 26 kids. I told the students since most of the questions revolved around the animated coins, I would reveal the camera magic at the end while we focus on the other top (main) question: How much is half the coins?

Time for estimating the right answer and guessing a number that's too low and too high. The second graders really enjoyed this part. One student felt 10 cents was too low because they saw quarters. One student was very proud of his $99,999 being too high. Once we got our estimates, I asked the class to read and revisit the main question: "How much is half of the coins?" I typically ask my students to reengage with the task/question before moving to Act 2 so we are reminded of our task.

Here comes Act 2: I asked the students what information would help them answer the main question. Many raised their hands. One student said, "We need to know how many of each coin." This was immediately followed by agreements voiced as "yea" or "that's what I was going to say." I displayed this information and we had less than 10 minutes to work. The second grade teacher broke the kids into groups.

Students were chatting, drawing pictures, using tally marks, adding by grouping, and using other strategies. One student asked, "Can we get out our money bags?" My response, "Shyea!" (translated yes). It was fascinating to see them operate for that short amount of time. I wish I had taken pictures. Sorry everyone! I had to leave.

This morning before the lesson, the teacher and I talked about the expectation of her students and the original task. She knew her kids were capable of finding the total and half. However, she also thought that finding the fewest coins might prove to be a challenge. The second grade teacher and my pals on Twitter were right. I think Act 1 deserves an edit saying, "My wife wants half of the money." This allows the sequel to be, "What coins would my wife get if she also wants the fewest coins?" and this can be used for the early finishers.

This experience reminds me that I need to record one of these 3 Act lessons so you guys can help me get better at doing them. There's still a ways to go, but I'm loving the opportunity to work with other teachers and grade level students. I've learned so much from these other teachers. I have a great respect for elementary teachers. I also love seeing how elementary students remind me that learning can be a blast. They're not going through puberty. They're energetic. They're so much fun! Not fun enough for me to pursue a multiple subject credential and teach primary though. I love my middle schoolers. I received an email from the teacher this afternoon saying:

It was hard, awesome, fun and different, cool, fantastic, interesting, the best, made us smarter, fabulous, I liked it, and magical!  Those were a few of the comments from my class about the math lesson this morning:)  Thanks for spending some time with us this morning!  It was fun for me, too!!

Soon, I will also be posting about my recent experiences in a 4th grade (Back Box2) and 5th grade (iPad percentages) classroom. Thanks for reading!

Cha-ching,
1119

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

Quotes of the Week [QOTW]

Have your students said something that completely moved you?
Was it insightful?
Was it relative?
Was it an epiphany?
Was it a proclamation?
Ok, let's not get carried away.

I hope you know what I'm talking about. Those comments become even more powerful when students see you acknowledge them and they didn't even think you were paying attention. I'm talking about those insightful things students say while they're collaborating with their peers, discussing solutions, or completing tasks. They blurt out something that catches you off guard (in a good way). The first semester has come to a close and I'm reflecting on student quotes. Quotes of the week: QOTW.

My students have said some great stuff and I was lucky enough to start telling myself to capture it on our front whiteboard. Why are there so many opportunities to hear what they say? Because we do a lot of group work and collaboration so they're bound to say something sensational. It's not about me. It's about them. They don't think I'm listening, but I am.

There's a section now carved out on my whiteboard for student quotes. It happened out of happenstance. This wasn't planned. I didn't find this idea somewhere on the wild internet (although it'd be cool if someone started a site for student quotes). A couple of students said something toward the beginning of the year and I wrote it on the board to share with all my classes throughout the day. They went nuts. Students were quoting the quote. I snapped a picture of it to make room for the next big quote and away we went. I'd like to share some of my favorites with you as I shared them with my students today.

"There has to be an easier way!" This is the one that started it all! In response to solving a weekly PS (Lucky 7’s) given to me by Fawn Nguyen, a group of students was filling their pages with numbers as they worked through exponential rules. Shawn continued the pattern for a long time on his paper, badly wanting to figure out the nth term in the pattern, lifted his head and let out this gem. The rest is history.

"Do it really neat so no one writes any bad stuff." After getting our new whiteboards and whiteboarding for a few days, students walked the room dishing out some harsh criticism to each other and their work. After addressing this criticism with them, students realized the importance of keeping their work clean, organized, and neat. Before beginning their task, Will verbalized his desire to be neat as to motivate his group. Good idea!

"We're actually learning." Yes, girls we’re actually learning. That’s because you’re actually thinking on your own while exploring math and not being told some procedure to regurgitate back to me. This was the result of my geometry class exploring parallel and perpendicular lines in a coordinate plane. These two girls were struggling for a day or two without any intervention from me and on the third day they had their shining moment. 

"That's upsetting me!" A quote is only as good as the context that goes with it. If you look at this quote, it could be your typical math student after doing the typical math question, resulting in typical frustration. However, Elle was working with her group on my Transversals, Tape, and Stickies task where they were given limited clues and had to identify twelve angles created by three intersecting lines. The bell rang and as she was heading back to her desk, let this one rip. She wanted resolution and was upset she didn't complete her task before leaving for the day. She came in the following day and conquered it with her group! Tenacious!

"Is that the opposite of PEMDAS?" In solving equations using inverse operations, James asks if the procedure is basically the opposite of PEMDAS (order of operations). Why, yes James it is. This was an "a-ha" moment for him. I couldn’t let this one escape.

"I plugged mine in. It worked! It's ALIVE!" You know those stories where someone says, "You had to be there." This is one of those stories. Elijah was checking his answer to an algebraic equation. Sure he could've just got a number for his answer and stopped, but he didn't. This is Elijah plugging in and verifying that his answer is the only solution. His excitement that the solution worked is hard to capture with an EXPO marker, but he took on the persona of a mad scientist, a la Frankenstein. I didn't write it on the board, but his "It's ALIVE!" was followed by "MWOOHAHAHA!" I love it!

"That doesn't make any sense." Another quote that could be any math student at any time. We've all been there. We've all heard this before, but what's the story here? Sierra said this after doing her calculations for our Stacking Cups task. She received some weird number of cups to stack as tall as Mr. Stadel. She immediately points out to her group that it doesn't make any sense. I love how students might be getting numbers, but they're checking those numbers for reasonableness before applying them. Back to the drawing board she went.

"We're demanding more information." The classic case of eating your own words. This past week we were exploring both Fawn Nguyen's and Dan Meyer's infamous Graphing Stories. All my classes began asking for more information as we progressed through Dan's videos. I continually praised them for demanding more information. We were working on the MARS lesson "Interpreting Distance-Time Graphs" I stole from Fawn and the students wanted more information as they wrote a story for Tom. I repeatedly refused any help by saying, "no" or "be creative" or "use the information on the page." Sean quickly replied, "But, Mr. Stadel we're demanding more information." You got me Sean! He practically jumped out of his seat when he saw me writing his quote on the board. FUN, right?

Expect a blooper's reel when doing this. you'll have the clowns that want to force something or think they're saying something sensational. That's my George. "I like colors." I don't think so George. You can stop now.

I can't make this up. I'm not paying my students to say this stuff. It's not contrived. It's natural. It's authentic. This board reminds my students that I'm listening. The more I can capture these and write these up, I believe the safer it is for my students to take risks, share their thoughts, and explore math. It's all them, but remember every quote has a story. So keep listening!

QOTW,
909

Tip Jar

Vimeo has a feature where their users can add a "Tip Jar" option to a video. Yes, viewers and users on Vimeo can monetarily tip other users to show their appreciation for a video. Today, I activated that on most of my videos.

I started typing this blog giving some examples in life where I gladly tip, reluctantly tip, and refuse to tip specific services in life. I changed my mind as I'm not here to cause waves, offend people, or get into an argument about tipping when the decision to tip a service is completely subjective.  Bottom line: I gladly tip others for their services when the service was completed in an efficient, professional, and satisfactory way, the service was something I can't do on my own, or they're sharing some passionate artistic talent that touched my heart in a compelling way.

I'm not putting my lessons, videos, or pictures on Teachers Pay Teachers. I don't work for a textbook publisher who pays me to do this stuff. I'm not selling this stuff to teachers, schools, or curriculum writers for profit. I've simply put it out there (on that wild internet) for others (teachers) to use, enjoy, and most importantly use with their students for learning math. Please don't think of this as a tip jar at a restaurant or specialty food service. Think of my Tip Jar as that open guitar case in front of the person pouring their heart out on the street giving you a few seconds of raw talent to brighten your day. I might sing out of key a few times, forget the right chord, or might have a string out of tune, but I'm sharing this stuff because I'm passionate about it, love doing it, and enjoy seeing other students learn math. If you feel obliged to tip, my gratitude will be eternal. If you don't tip, I still love you for taking the time to check out my stuff and possibly use with your students. That's the best tip you could give me!

TJ,
540

Styrofoam Cups

Tuesday, I was on my way to BTSA and my subconscious screamed something at me. Find Dan Meyer's Stacking Cup lesson. Seriously, go read his post right now. I hadn't read this post for over a year now and I had to get my lessons ready for the next couple of days as my Algebra classes finished up Pixel Pattern. I was on the road dreading the idea of sitting through a couple hours of BTSA, so I asked Dan if he had the link to his lesson since it wasn't in my bookmarks (that was silly of me) and he came through like a champ! Seriously, check out his post. I'm promoting his blog post more than anything further I have to say here.

First, by all means, spend about $10 and do the lesson with your kiddos. This is one of those 3 Act lessons that just screams "hands-on" activity with your kids. It's tough to capture the overall excitement and energy with a video. If you can't do the "hands on" with your kids or you want to be environmentally friendly, here's my version of the Styrofoam Cup 3 Act lesson: a cheap backup.

It felt most natural to stage this so the cups stacked to the top of the door frame. Even then, I'm not convinced my Act 1 screams the question I'm looking for, "How many cups will stack to the top of the door frame?"

Enough about me and the video, to my classroom with the students. Dan's got a great script for you to follow, so do it! One of my classes was actually able to finish writing their rules before the bell on Friday so we had time to actually stack cups. Check out their rules and predictions for stacking cups to my height.

We started stacking with the lowest number and went from there. The kids went bonkers. Each group thought they were the best, but knew that they all couldn't be correct. When we revisit the lesson this next week, we'll be discussing where groups went wrong in order to learn from those mistakes. Watch Styrofoam Cups - Act 3 Stadel to find out who won. But I recommend you watch the door task also.

Styrofoamed out,
813

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

Estimation 180 update & RSS

Head over to Estimation 180 and throw the blog in your RSS feed. I usually update the site once every week or two and the RSS feed will alert you of those updates. Don't worry, they're not daily and it's definitely not s-p-a-m, as I despise s-p-a-m.

Huge update of estimates today starting at Day 78 and going through Day 91 (over halfway done). Home Depot was just so fun.

*Can you spot something suspicious on Day 86?

Enjoy!
519

Bottomless Mug

I found this glorious sign a few weeks back at Bruegger's Bagels and ended this post with saying I'll work it into a 3 Act.

As promised, here is the 3 Act lesson.

Act 1: My question: How much money could you actually save?
Other popular questions can be found at 101qs.com. I like getting to that initial question because many of the others will be answered along the way.

Act 2 info would look like this for my area, but the cost of a medium sized cup of Bruegger's coffee might be different in your area (for a limited time, of course). Check their website.

Now you know the cost of the mug, but I find the 3 days, 5 days, and 7 days per week (the rate where you live next door) very intriguing. In solving this one, my natural tendency was to round that cup to $1.90. Be careful, that difference could buy you a bagel or two. Anyway, have fun with this one. My wife and I just celebrated the birth of our daughter on New Year's Eve day. I don't drink coffee, but I do enjoy iced tea and this Bottomless Mug Club is starting to look rather appealing now.

Personally, I like the sequel tasks more than the original task. Sequel tasks include:

• On what day in 2013 would you break even if you get coffee 3 days/week, 5 days/week, everyday?
• When would be the last day to buy the mug and still save at least $1.89?
• What could be the prorated price of the mug if bought in January? February? March?... 

Have a sequel to add? Toss it in the comments.

Happy 2013,
1050