The Ultimate Task for Vertical Planning: Stacking Cups

This past week, I submitted a speaker proposal for NCTM 2016 in San Fransisco. The proposal is for a Grade 6-8 Burst (30 minutes) with the exact same title as this blog post: The Ultimate Task for Vertical Planning: Stacking Cups. I figure if I don't get accepted, at least I can share my thoughts here and you all can help spread the word about my idea if you think it has potential. If it does get accepted, I look forward to giving an update a year from now at NCTM. Here's my session description:

Who says you can't use the same task each year? Come see why Stacking Cups might be the single best secondary math task to get teachers at your school, district, or state to see the importance and necessity of vertical planning. Use tasks that utilize connections from the previous year and extend the mathematics each year. Work smarter, not harder. 

Let's first back up a bit. I attended Alex Overwijk's session at NCTM Boston a few weeks back. I had already read his awesome blog post "Open Strategy Cup Stacking" and knew there are multiple teaching moments with Stacking Cups. I remember teaching Math 8 a few years ago and getting a lot of use out of Stacking Cups as you can see a couple times here and here. I was preparing for a training with math teachers from grades 6-12 and THAT's when it hit me: I could have a room full of math teachers from grades six through twelve and they all could:

• be working on this task
• see the different skills and tools necessary for solving
• know the expectation of each grade level

I've heard comments from teachers numerous times like, 

"Well, if they do File Cabinet in 6th grade, I can't do it in 7th grade with my students."

"If they've done Stacking Cups in Math 8, then I can't do it in Algebra."

"If the 5th grade teachers use Estimation 180 with students, then I can't." 

YES! YOU CAN! It's called vertical planning.

YES, YOU CAN! Instead, let's ask different questions like, "How can we use the same task to extend the mathematics each year?" and  "How can we make connections to prior learning from the previous grade level?"

Let's work smarter, not harder.

I will spend the rest of this blog post highlighting each grade level and suggested uses for Stacking Cups. It won't be complete or the final version as this is through the lens of one person. I'm confident, with your help and critique, we can make it even better.

Math 6
Question: How many cups do we need to stack (alternating) to reach someone's height?
We talk about rate. We organize our information on a number line, in a table, using a tape diagram, etc. We explore the rates using various models.

Math 7
Question 1: How many cups do we need to stack (alternating) to reach someone's height?
We continue the conversation started in Math 6 revolving around rates, using constant of proportionality. All of this can be represented in a table, as an equation, and in a coordinate plane.

Question 2: How many cups do we need to stack (consecutively) to reach someone's height?
We now shift our thinking a bit where there is still a constant increase with each cup, but there is an initial amount (the cup handle). Students explore how to write an equation to represent this situation and solve it.

Question 3: What would be possible dimensions of a box that would contain the cups to stack to someone's height? Which dimensions would be the most cost effective?
Imagine students understanding surface area and volume and how they're related to each other, especially if we model with mathematics, by identifying variables such as:

• cardboard cost
• delivery truck capacity 
• store storage sizes
• consumer trends with buying cups
• more

Math 8
Question 1: How many cups do we need to stack (consecutively) to reach someone's height?
Similar to question 2 in Math 7. However, we extend the mathematical understanding as we explore constant rate of change (slope), input and output, linear, and how our situation can be represented in the form y = mx + b.

Question 2: When will two stacks of different sized cups be equal in height and have the same number of cups in each stack?
We introduce students to linear systems using this task. Students can organize the information about each cup in a table. We can extend prior knowledge to represent the situation using graphs, equations, and functions.

*By the end of Math 8, it might be helpful to mention (at least informally) to students the significance of discrete functions.

Algebra
We tighten up the math (both questions) previously learned in Math 8. How can we extend the mathematics. Add more challenging situations like the stacks start on different objects like desks, boxes, etc.
Question 3: How many cups would we need to stack in a triangular formation to someone's height?
This questions really extends the mathematics for students, but we can still use the tools they've learned from previous grades. Maybe students start by organizing the data in a table. Maybe they graph the data and notice it isn't linear. Maybe we can use desmos with sliders or a line of regression to explore quadratics.

Beyond Algebra and Geometry:
I'll admit this is where I'm a little rusty and would need you high school pros to jump in and contribute. I think with the triangle stacking, it can be taken from quadratic to a divergent series. I've also seen high school teachers come up with the following representations:

Al Overwijk also stacked cups in a triangular pyramid which is awesome.

Let's keep this vertical planning going. If you would like a couple charges, here you go:

Go to your site and/or district and push for Stacking Cups to be a signature task at all sites and secondary grade levels. Help support your colleagues with vertical planning. Report back.

Look for other tasks out there like Robert Kaplinsky's Hot Dogs or Dan Meyer's Penny Circle or Mathalicious' Wheel of Fortune or Graham Fletcher's Water Boy that can be used with vertical planning. Report back.

Vertical,
432

Your Life Depends On It

On the drive home tonight, I was listening to RadioLab's most recent podcast entitled, Worth. (Thanks Justin) At the 17:45 mark, one of the producers shares her story about going out on the street and asking people the question, "What is a year of life worth?"

Take a second to think about this question.
...
...
...
...
...
...

• Would you immediately have an answer?
• Would you want to know more information before answering the question?
• What questions would you ask before finally coming up with an answer?

I totally see this as one of those questions that produces many more questions. Right?

• Would you consider the question uses Mathematical Practice 4, Model with Mathematics?

If you are curious, ask your students this same question, "What is a year of life worth?"

Just take a moment to learn from (and listen to) your students what they have to think of, share, argue, discuss, etc. I would guarantee students will raise so many questions and points. Is this a fake-world question? Can you imagine a question more specific to your content area that would spawn a rich discussion like this? I think Mathalicious does a fantastic job asking powerful questions in their lessons. Let's play a game here. Play along, won't ya?

The challenge: How can you take standards and concepts in your math class, and ask questions that almost get a similar response to the question above?

Let's call them, "Your life depends on it." questions.

Here are three images. What "Your life depends on it." question you would ask for each?

Again, what "Your life depends on it." questions would you ask? If any?
...
...
...
...
...
...
I'm not saying mine are awesome. I simply think in order to answer the question, students would want to know more information. Ask more questions. Make guesses. Construct a model. Use math to make strong predictions, and more.

What do you think here?

Worth.
537

Going Round In Circles

Whenever I start talking about circles with my students, I use this little wager.

I get students to pick one of the three choices and work the room, looking for a brave student I know will deliver my nachos. I talk up the nachos (and the circumference) as much as possible. Anywhere from 90 to 100 percentage of students will say the circumference is shorter than the height of the water bottle. Let's see if I win nachos or I let my students go to lunch early.

Okay, so double or nothing? I don't bring in this glass, but I do use a taller cup with a really small circular base. Where do you stand on the double-or-nothing wage? Did I give you enough information to take the bet? With a glass like this, you should get at least one student to keep you honest and ask which circumference of the glass you'll be measuring.

This little wager (activity) allows me a quick introduction and fun application of circumference. Somewhere I'll discuss vocabulary and formulas with students while giving them a graphic organizer they can fill out.

I'll usually do an activity where students measure the circumference and diameter of objects in order to discover the relationship of Pi. Stuff very similar to Fawn's Friday Bubbles. Note to self, use Excel (or a spreadsheet) to keep track of those measurements. I've also explored Rolling Tires in the past. This year, I brought the wheel to the class for a small activity. A physical wheel. The wheel from my son's wheelbarrow.

The small activity was for students to guess how many rotations this wheel (8-inch diameter) would make from one wall of my class to the other wall. Students were able to see how circumference can take on the meaning of a tire rotation, hence the graphic I made above. It was sweet to see students roll the wheel across my 21-foot long room and actually get 10 rotations like the math predicted. If you have a wheel like this, bring it in and do this activity.

We also did these awesome lessons. And. I. Mean. AWESOME!
Pizza Pi by Mathalicious and
Penny Circles from Team Desmos and Dan Meyer.

There's so much to do with circles and so little time. 

Round and round,
945

Get Students to Argue in Math Class

I recently submitted my speaker proposals for both 2014 CMC conferences. One of my proposals is for the following session:
Title: "Get Students Arguing in Math Class with Number Sense Activities."
Description: Get students to productively argue about math situations. Participate in number sense activities requiring students to construct viable arguments, critique the reasoning of others, and use sense-making. Get ready to throw down.

I also had to answer a few questions justifying the session and connecting it to the CCSS and 8 Mathematical Practices. I provided the following connection:
The presenter will use number sense activities to get participants to construct viable arguments and share their reasoning like students. Using presenter-made tasks (Estimation 180) and other online resources appropriate for grades 3-8, attendees will be able to see the importance of student reasoning and creating productive discourse in the classroom. Teachers will also be provided with sentence frames and stems for all students, especially English language learners.

I'm really excited at the thought of this session getting accepted so I figured I would jot down a few ideas here and see what you all have to contribute. Even if I'm not accepted, I think every math class has to have students productively arguing at times. Doing estimation challenges with my students has been so beneficial for them to get better at the art of arguing. However, I know it could be better. I'm not sure if you've ever experienced it before, but it's a treat to stand off to the side or in back of a group of students arguing about a question in math. They have no idea you're nearby because they are so caught up in the argument. Don't get me wrong, it's not like they're swearing at each other and calling each other names. They are having a rich discussion, sharing conjectures, examples, counterexamples, etc. and I have the pleasure of spectating. I usually turn to an innocent bystander (nearby student) and whisper, "Awesome, look at them arguing. Isn't it great?" The student usually looks shocked that I'm happy their classmates are arguing. I love it!

I'd like this session to place a big emphasis on two mathematical practices:
MP 3: Construct viable arguments and critique the reasoning of others.
MP 1: Make sense of problems and persevere in solving them.

I plan to break these two practices down more in my session. For now, I feel I need to focus on three major parts to arguments: having excellent content, capturing the arguments, and indirect facilitation.

Content: There's a wealth of content available, but I think the more controversial the point of contention, the better. One of my favorite moments in math class was when we did Mathalicious' Datelines lesson. Students were arguing about which celebrity shouldn't date another celebrity because of the age discrepancy. Some students disagreed about the rule of (n/2) + 7, especially since I was teaching 14 year-olds at the time. It was awesome.
Here's my current list of resources that have given my students great things to argue about:

• Estimation 180
• Mathalicious
• Would You Rather 
• by John Stevens

My kids went nuts arguing about a similar question to this "Would You Rather" found here.

These don't have to be full-on lessons. They can be warm-ups, math talks, used during classroom transitions or to break up your direct instruction, etc. I'm really looking forward to using @MathCurmudgeon's site MathArguments180.com
Imagine your students arguing about which student should pack your parachute based on this data.

Another up and coming resource is Open Middle by Robert Kaplinsky and Nanette Johnson.

What resources would you add to my content list?

Capture: I need to capture these arguments for a few reasons. Students need to listen to other students argue, especially from different classes. My memory is very porous, and I can't remember what students say verbatim. Students can listen to the recordings and pick a side, or provide their own agreement or dissent. I'd also love to share student arguments with other teachers, especially at this session. How do I capture this?

I just downloaded Voice Memos for iPad onto my school iPad. I will test it out next week with students. Wish me luck. Here are the features I'm optimistic about:

• It will record in the background while another app is running.
• It was $1.
• I can pause the recording.
• I can trim audio clips.
• I can sync with Dropbox.

Have any tips for capturing student arguments?

Facilitation: Here's where I need to do a better job. For many of my students, English can get in the way of them articulating their point. I'd like for students to listen better to each other and respond accordingly. I want to hear what they have to say. I want them to be a contender in their disagreement, but I don't want them to be held back because of language deficiencies. Therefore, I need to provide them with sentence starters and stems. Fortunately, these can be used with any student. Here's a few:

• My opinion about this is _____________.
• I could argue that _____________.
• I disagree with your statement that _____________ because _____________.

Have any stems or sentence frames you're already using with students to help them articulate their thoughts?

Argue,
339

CMC South 2013

Here's my recap on CMC South 2013... or how I developed a man-crush on Robert Kaplinsky.

Thursday night:
I drove out to Palm Springs and met up with Karim and Fawn. We found a nearby casino, math-chatted, and then landed at the roulette table where $20 lasted me for a good chunk of time. I broke even and called it a night.

Friday:
Fawn and I spent the first session reviewing our presentation in the lobby on my laptop. Poor Fawn, I think she was regretting her decision to present with me.

We reached a point where we couldn't think of anything more to review so we walked over to Robert's session at the Hilton. Lucky us, we were able to sit with John Berray, Avery Pickford, Breedeen Murray, Karim, and John Stevens.

You can read a better recap of Robert's session at Dan's blog. Robert did a fantastic job. He really did. Therefore, I guess I should explain my man-crush now, right? I think Robert has such an edge right now that the #MTBoS, math community, and those in education can all benefit from. Follow him on Twitter, use his lessons, read his blog, and get him to your school for some trainings or to observe... because he's totally willing. He has an edge because he's still in the classroom working with both students and teachers in his district. He's eager to ask questions that will make him better while at the same time will share the mistakes he's made and how's benefited from them. He delivered a stellar session that was informative, humble, resourceful, fun, engaging, honest, and necessary. Seriously, we need more Robert Kaplinskies out there working with both teachers and students. Follow him on Twitter, use his lessons, and read his blog. Lastly, if you get a chance to meet Robert and spend five minutes talking with him, you'll quickly see that he's interested in what you do and is more than willing to be in your classroom so he can observe and learn. He eagerly wants to improve his craft in education, learn from others, and find ways to be more effective at what he loves doing. Very inspiring!

Fawn and I presented after lunch. The title of our session was:

It was such a pleasure to collaborate and present with Fawn. She came up with this great math task and I look forward to presenting with her again at CMC North. She's probably dreading the thought of presenting with me again! Not to spoil anything for our CMC North session, we had attendees use snap cubes to build something that involved income and a building cost. The task required strategy, problem-solving, collaboration, and a lot of calculations. Groups were required to present their calculations, information, and reasoning on their giant whiteboards. How vague was that description? Don't worry, we'll post more after CMC North. In the meantime, here are some pictures from our session:

The man! Robert Kaplinsky!

Avery Pickford: our winner!

Some attendee quotes a la Quotes of the Week style

Thanks to all those who helped enter calculations, pass out things, clean up or kept us on track during our session.

We were able to make the last half of Jo Boaler's presentation after we cleaned up our session and headed back to the convention center. I look forward to taking her class in April 2014 and my take-aways from her session were:

• Get rid of timed tests because "Math should never be associated with speed."
• Give students more diagnostic feedback and tell them "I'm giving you this feedback because I believe in you."

Keep an eye on her site www.youcubed.org

Saturday:
I rolled in casually late to Michael Serra's Polygon Potpourri session. No need to describe the fun found here as Dan already recapped it.

CMC South wouldn't be right without attending a Brad Fulton session. He's interactive, funny, and engaging. Next time you get a chance to see him present or give a workshop, be there! His focus was on Math Talks. Fawn shared a little insight on his session here. However, she took it a giant step further and generously created a space for us at mathtalks.fawnnguyen.com to share student thinking and giving students a voice.

After lunch, Fawn and I went to Avery's session. Dan did a recap and Avery did two posts: part 1 and part 2. Check it out! Unfortunately, I had to leave early to go get my materials and set up my presentation. My take-away was working with the Locker Problem and Avery sending me this resource. I've never tried the locker problem, so it was a bunch of fun to play with. He also shared this gem which he demonstrated with snap cubes using the document camera (very cool!):

I presented last at CMC and wanted to thank those of you who were able to make it, even if I was your fourth choice after seeing Dan Meyer's sold-out arena. The title of my presentation was:

Overall, I think my session went well. I wish I could give this talk again, after a little more refinement. I could've used ten more minutes to better connect with the 8 Mathematical Practices. However, there was a lot to share, interact with, and discuss. I learned a great deal from my attendees; especially on rating the level of difficulty with estimation challenges. My attendees got a sneak peek at Days 176-180 of Estimation 180 and I hope you had as much fun as me. I look forward to sharing more with you about this session in a later post.

My CMC regrets: I wish I could've attended the sessions by Karim, Breedeen, and the dynamic duo of Matt Vaudrey and John Stevens. I've linked their names to their sessions. I look forward to CMC North. Maybe I'll see you there.

South,
927

Garage Jams are my #TMC13

Divisible by 3 is usually used for math talk. It's summer and this post has nothing to do with math. It's just a post about something I love to do, play live music. While so many fun people are enjoying their time in Philadelphia at Twitter Math Camp (#TMC13) this week, I've had the pleasure to jam with my nephew and edit some of the video we captured. For those of you looking for math stuff, my next few posts will definitely be filled with math... trust me.

I set my Flip camera on a tripod in the corner of my garage for capturing my nephew on drums and I taped a GoPro camera (thanks Karim) to the headstock of my guitar for those guitar licks. Here's a quick soundcheck:

My nephew and I get together every so often to jam. We mainly rock out to songs by our favorite artists or songs from the bands I was in during college. Last summer, he added drums to my PEMDAS song. Between the two of us, we'll choose a few songs, practice them individually, come together, talk about a few transitions or endings, count off, and then rock out! Everything here is our first take. I'll start you off with Jimi Hendrix's Spanish Castle Magic found on his album Axis: Bold As Love.

As you can see, we're not trying to nail these songs note for note. We enjoy adding our own style, sound, or feel to the songs we cherish while maintaing the integrity of the original. It's just drums and guitar in case you're wondering why there isn't any thumping bass or screaming vocals. I'm using the audio from the GoPro camera which ended up capturing the sound decently. As for the following songs, the camera definitely picked up more drums than guitar in the mix. That's okay because we're not out to release this stuff for a record deal. This is just pure fun and I'm using this creative space to post it. Up next, Rage Against the Machine's Guerrilla Radio found on their album The Battle of Los Angeles.

Tom Morello comes up with these simple, yet powerful riffs while adding some slick effects to his overall sound and solos. It's always fun and challenging for me to figure out what he's actually using and playing. We had a few other jams on 2013-07-23, but this will be the last one I share here. You'll find updates inside this Vimeo Album. We both enjoy Incubus and have jammed to many of their songs. This last song was a last minute decision, but I think it turned out alright. Here's Blood on the Ground from their Morning View album.

It's always a blast getting together with my nephew. He's off to college in the Fall and our opportunities to get together and jam will be fewer. Maybe we can jam via Google Hangout! It's been a pleasure to watch him improve at drumming over the years and really have a wonderful ear and talent for music. As promised, this post has nothing to do with math. I'm not rambling about how jealous I am of those at Twitter Math Camp because live music truly has a special place in my heart. I'll return in a couple of days with some thoughts as I prepare for my CMC South presentation in November.

Jam,
1203

CMC - South: What could be?

November 2 and CMC South are only a few days away! I'm excited for a few reasons:

1. World of Nathan Kraft is flying in (if Hurricane Sandy isn't too absurd this week) from PA to attend. I'll pick him up and we'll carpool to Palm Springs for the conference.
2. Fantastic Fawn is literally missing her favorite football 'game of the year' to attend and cause some trouble with Nathan and me.
3. Mullet King Matt Vaudrey is both attending and presenting.
4. Recent doctorate and newly appointed Mathalicious brain Matt Lane is making the trip.
5. The infamous Dan Meyer is presenting (need I say more?).
6. ... and there will be a Tweetup on Friday with all these fab people, hopefully with John Berray too.

There are some appealing conference speakers this year. I wish there were more on Standards Based Grading. I think the two presentations I'm most looking forward to (besides Dan's) are Standards Based Grading to Evaluate Mathematical Practices by Lisa Miller and Take Your Places by Brad Fulton. I've seen Brad present before. He has got some great ideas and is hilarious with a capitol H. Sorry Mullet Vaudrey, I truly wish I could be at two places at once. Video record yours and post it, will ya?

I'll admit, there are parts of me that would get a kick out of the following:

1. Nathan checks his tuba at the airport, pays the outrageous fee, and carries it around CMC testing out the echos. Heck, we are in the desert, but there is a small mountainside nearby and we could test the echo off of that sucker. 
Tuba Echo from Nathan Kraft on Vimeo.

2. Matt Vaudrey needs to sport a mullet the entire CMC (if he hasn't grown one out by now) and have his camera ready to take pictures of some locals with mullets. I'll bet Matt a beer that by the end of Friday, we spot at least 3 mullet-y locals worthy of your ratio lesson for this year.
3. Fawn, better bring me that box full of avocados she owes me from File Cabinet. I'm tempted to pack my OG Nintendo and Tetris so we can have a little Tetris showdown. That way, she can't cheat on her Xbox (Okay, she might still kick my butt). At least I'll stand a better chance, right? If she doesn't bring me the avocados, maybe she'll bring me all her Brad Fulton books so I can steal, I mean copy them.
4. As for Dan Meyer, I'm looking forward to meeting him and actually looking up at someone for a change. I'm speaking of height here people. I look up to everyone mentioned in this post! However, wouldn't it be fun if our Twitter group heckled him during his entire presentation?

Stay tuned for a full report from CMC and some fun contributions to Estimation180.com

CMC to be,
930

Quick note: re-focus and fun

I've think I have finally come down from the natural high of finding and incorporating Dan Meyer's 3 Act lesson format. I'm addicted and plan on using parts of my summer to compose more valuable lessons for my students. I love it! However, I love many other things in life and I feel like I might have lost focus on a few of those elements that make my life so grand. It's time to re-focus. Allocate my time better. Watch this video:

Benga-I Will Never Change, although it might appear random, sums up where I currently am. Yes, it could be edited to use as a 3Act lesson. Yes, it could be used as an estimation question for my students... yes it could be used for something mathematical. Lately, my brain has been so heavily trained and geared toward finding that math moment, math media, and math invitation that I've lost focus of some other things.

This video is just fun!

It emulates a Soundcloud wave form. It's great to see the peaks and valleys. It's great to just sit back and enjoy. I appreciate it from an artist's standpoint: the staging, the filming, the editing, the overall production time. It's just fun to watch.

As the end of the school year approaches, this is a great opportunity to allocate my time more efficiently so that I can better contribute to my family, friends, students, community, online colleagues, and world. I have met many resourceful people such as @fawnpnguyen, @nathankraft1, @wahedahbug, and many others on Twitter and have the pleasure to call you online colleagues and friends. I even look forward to meeting many of you one day. I was fortunate enough to meet Karim from Mathalicious.com while I was in DC last week with my 8th graders. He talked about his week off and shared some amazing experiences. He hadn't had a break in close to two years since starting up Mathalicious. That's incredible! He needed an opportunity to recharge his battery. I hear ya brotha!

This online community of math brains has been a blessing. I plan to contribute, share, take, and participate in a meaningful way that is both beneficial and fun. Does this sound reasonable? Am I on to something here? Am I on point?

Want another fun video... check this guy out! Diego Stocco is the man with sounds. Is there math involved? Absolutely. Is it fun? I think so.

I'm out,
1047