[Makeover] Low Arching Bridge: The Makeover

Once again, the task:

What I like:
I like the placement of the x-axis along the ground to represent zero height.

I like how this task reminded me of the low arching bridges along George Washington Memorial Parkway in Alexandria, Virginia.

What I dislike:
I dislike that the x-axis and the y-axis were already placed for us. The students have no say in this.
I dislike how the arch is already "modeled" by the given function. There isn't any chance for students to explore this on their own, especially if they had no say in the placement of the y-axis.
I dislike the answer to this question. It's hilarious. Get this:
The truck has to be dead center so that it will allow 0.23 feet of clearance on each side of the truck. Regarding number sense, what is twenty-three hundredths of a foot? No one talks like that, do they? After converting this answer, I could see myself telling the driver, “You have less than 3 inches to spare on each side. And that’s ONLY if you center the truck with the middle of the bridge." Let's look for an alternate route or someone might have to get out of the truck [not it] to guide the driver.

Things I'm intrigued by:
What was the reasoning behind the placement of the y-axis? Why isn't it dead center or along the right wall?
Why isn't there any sign on this bridge that says the maximum height and/or width of trucks allowed?
Is this a "one way" road?

Here's what I did:
*Disclaimer: I'm not pretending to nail this Makeover: I think it can be better. That's your job: so let's get it on and help me in the comments. I'll admit, the Makeover was more work than I anticipated and I'm tapped, but I'm happy to do it now during the summer. Thanks Dan for the Makeover challenge!

I found an accident report for a coach bus that crashed into this exact bridge (below) in 2004. There are many of these low arched bridges located along George Washington Memorial Parkway in Alexandria, Virginia. I've seen a few of them when we've taken our 8th graders to visit Mt. Vernon. I remember our bus driver telling us about this specific collision.

1) Show your students this picture, but don't tell them about the collision:

Allow students to make observations and ask questions (maybe Notice and Wonder). Tell them where this bridge is located if they ask. Don't tell them what the signs say. Have a discussion.

2) Now show your students this picture and ask:

Which of these (six) vehicles would safely pass under the arched bridge?

3) Have students make guesses and write it down. You're taking a chance, but at least one student should notice that some vehicles might pass safely using the left lane, but not when the same vehicle is traveling in the right lane.

4) Ask your students what information or tools they might need to help determine which vehicles can safely pass through this arched bridge.

• Bridge height(s)
• Vehicle height(s)
• Width of road
• Width of lanes

5) Find the vehicle heights we'll be working with. Depending on the time you have, students can use the internet for finding the average height of each vehicle. I did the grunt work for you with this slide:

6) Show students three heights of the bridge and street dimensions. They probably want to know what those yellow signs on the bridge say. Too bad! The picture is low quality and very pixelated. I'll admit, this might feel like we're now stringing the kids along, but let's offer them measurable dimensions, not some arbitrary equation that "models" the arch. Share the following:
Height of the bridge on the left side
Height of the bridge in the center
Height of the bridge on the right side

Width of the entire road (including space for lane lines and shoulder) and width of two lanes.

7) Offer your students Desmos or Geogebra. Plot the three heights. Use sliders to find an equation that models this low arching bridge. Here are three four scenarios I came up with in Desmos. I'm still not sure which I like best. You decide. I've linked the Desmos files for you to mess around with.
Where do you fancy the y-axis?

y-axis justified RIGHT:

y-axis justified CENTER of bridge:

y-axis justified LEFT:

Just for fun (the actual bridge):

Okay, I like both the center and the justified right. Placing the y-axis in the center of the bridge made it a lot easier to find an equation that modeled the bridge. Placing the y-axis on the right side of the bridge might produce negative x-values, but since distance is never negative, the absolute value of the domain will tell me how many feet away from the right side of the road the vehicle must be.

8) Give students time to explore the functions, quadratics, sliders, domain, range, and so on. There's more. This task requires students to apply the heights of the vehicles in a specific manner. Sure, students can click and drag on the graphs in Desmos to find the heights of vehicles and determine if it safely passes, but what part of the car "safely passes"? The top left? Top center? Top right? Therefore, students have to now take into account the width of the vehicle. Let's go back to the original question:

Which of these (six) vehicles would safely pass under the arched bridge? And in what lane?

• Which vehicle(s) will pass safely in both lanes? 
• Which vehicle(s) will only pass safely in the left lane?
• Which vehicles(s) would have to go into the oncoming traffic lanes?
• Which vehicle(s) need to stop and turn around?
• Ask how far the vehicles will be from the right side curb when "passing safely"?

9) Tell students to look for a little more clearance than 0.23 feet (2.76 inches). You can read the accident report for all the details about the street and bridge. You'll find the clearance heights posted on the bridge and about 1,500 feet before the bridge.

Unfortunately, the accident report will also show the bus that collided with the bridge while the driver was talking on his cell phone. The bus ran into the bridge without even applying the brakes.

What you did or suggested:
Amy Zimmer emailed:
"Is it the new Daniel Craig James Bond that has the train scene where he has to duck just before he is about to run into the bridge when the good guy and the bad guy are fighting on a speeding train?" followed by "I would give lots of trucks and see which ones fit."

Everyone else's input can be found here:


If you've made it this far. I appreciate your determination and perseverance. Thanks for tuning in. I know this task can be better, so let's get it on in the comments.

Up next, Global Math Department presentation on August 13, 2013: Back to School Night: Ignite. Join the fun.

Under the bridge,
1230

[Makeover] Low Arching Bridge

We encounter many low bridges on our drive to Mt. Vernon each year when we take our 8th graders. Our bus driver must pay extremely close attention. This textbook problem reminded me of our experience. I'm playing along with Dan Meyer's Makeover Monday. Here's the task:

From McDougall Littell's Algebra 2 textbook (2007)

I'm curious how you would make this over. Next Monday, I'll post what you do and what I do to this task.

Find me on Twitter.
Email me your makeover.

Makeover,
1017

*[UPDATE]: Here's the Makeover task.

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

Guest Blog Posts at Estimation 180

Today, at Estimation 180, I posted some guest blog posts I collected from the beginning part of this year. I'm grateful, honored, and inspired by the stories they (or their students) shared. I hope you have time to check them out and share your story too.

Read their stories here. 

Snail's Pace

Last post, I shared a lesson (Woody's Raise) that included both Act 1 and Act 3. I asked you all to collaborate and design Act 2. Many of you came through like champs in the comments.
THANK YOU!

For this post, I only have an Act 1, leaving Act 2 even more open-ended. I'll admit, I only have Act 1 because I haven't invested the time necessary for Act 2 and Act 3. Here's my current Act 1.

I thought of this lesson many months ago while out walking in the morning, but wanted to capture it on video... no joke. So until that time actually comes along, I'll give you what I envisioned for Act 1, the video version. We start with Bill Conti's Gonna Fly Now (Theme from Rocky) as we take a couple close-up shots of the snail. The camera pans out to a bird's eye view of the snail starting at one side of the sidewalk, letting time elapse for about 15-20 seconds.

Back to the picture of the snail who has an increasingly long road ahead of him. I notice that he isn't taking the shortest path to the other side. I notice that there aren't any other snails to avoid. I notice the sidewalk is wet. I wonder what his path will be. Will his path be linear? curved? circular? other? I wonder what his rate will be. I wonder what the dimensions of the sidewalk are. I wonder if the Pythagorean Theorem could be used here. What do you wonder?

Head over to Dan Meyer's 101qs.com and enter a question (or skip it) so you can see my Teacher Notes for Act 2. You might need to log in. Thanks to Ignacio Mancera for linking a site with Speed of Animals. This will help assist our Act 2 adventure.

Here's what I have so far if you can't get into 101qs.

What initial conversation(s) would you have with students?
How would you have students work with Act 2 information (dimensions, rate of snail)?
Is this a waste of time?
Should we (I) shelf this idea for now? (or even toss it in the trash can?)

Slowly,
333

Woody's Raise

We decided to get Netflix recently and I was excited to see that Cheers episodes are available. I occasionally put an episode on in the background while I get work done. I came across this episode that literally snuck in some math (money, raises, time, rate) right before the end of the episode. Sam Malone, the owner of the bar in the tie (played by Ted Danson), is talking with Woody Boyd, a bartender (played by Woody Harrelson), about a raise. Roll Act 1:

After consulting with my man, Nathan Kraft, I bleeped out a part of Woody's last line. The two of us discussed the tendency a bleep can have in implying some profanity was removed. So if this lesson goes horribly wrong, blame Nathan! All those toothpicks finally caught up with him. Here's how the exchange goes between Sam and Woody:

Sam: We were talking about your 50 dollar a month raise.

Woody: Sam, it was a hundred a month.

Sam is caught for trying to pull a fast one on Woody. Woody appears to let it slide, but something occurred to Woody. He turns to Sam and the exchange continues:

Woody: I think a hundred a month is too steep. I'll settle for [BLEEP] a week. 

Sam (without blinking): You got it!

I anticipate students noticing that the amount was bleeped out and wondering what was bleeped. I anticipate students not sure if Woody said, "[BLEEP] a week" or something inaudible? I anticipate students noticing that the studio crowd laughs while wondering if Sam was just made a fool by Woody. I would love to first have a leisurely conversation with students about who they think just got the better deal in this exchange, Sam or Woody? Or was there even a better deal to be had? If you've ever watched an episode of Cheers, you know that neither character has a strong IQ. If anything, Woody is portrayed as a real naive, gullible, and takes-you-at-face-value type of character. Sam is about a handful of points above Woody. So what about Act 2 after you take some guesses from the class on who just got the better deal from this exchange?

This might be the first 3 Act lesson in which I don't have any additional information for Act 2. In all fairness, this might not fit my previous rant on measurable acts, but I think the 8 Standards for Mathematical Practice are rubbing off on me (in a good way), especially Practice 4: Model with Mathematics.

I posted the Woody's Raise lesson on 101qs.com with very little in Act 2 because I'd love to know where the teacher would take this with his/her class. This type of teacher discretion can't be packaged in an online portal or catalog of video instruction. Here's what I threw out there for Act 2 (the first edition):

At what "raise" amount per week would Woody "settle" for the:

1. Better deal
2. Equivalent deal
3. Worse deal

I have many questions when thinking about Act 2. Here's a few:
Over time, when does Sam or Woody begin to benefit or suffer from this deal, compared to the $100 raise per month?
Do all months have exactly four weeks? Does that matter or should we use 52 weeks in a year?
How would you anticipate students representing Woody's better deal versus the worse deal?
What would this look like graphically?
What would this look like organized in a table?
What equations could you anticipate students writing? If any?
How does this deal apply to Woody's hourly rate?
In what classroom could you talk about the tips Woody might make? Remember this takes place in a bar. Middle school students? High school students? College? A workshop with teachers? I think there's a lot of fun to be had with this video clip. Let's Roll Act 3 and see what Woody would "settle" for instead of the $100 a month raise:

I'm posting this lesson because I'm thinking out loud. More importantly, I'm curious what you would do in between Act 1 and Act 3 with your students. How would it be different in an elementary classroom? Middle school classroom? High school classroom? Teacher workshop? What would your Act 2 be? Where would you take this lesson with your students? I believe this is a multi-dimensional lesson that can take on some great mathematics. Bleeping out that weekly rate in Act 1 really opens up Act 2 for some rich mathematical discussions and modeling. Toss your Act 2 in the comments. Thanks!

Cheers,
1026

Back to School Ignite Talk

Man, I love a good Ignite talk.
5 minutes.  15 seconds per slide.  20 slides.
Concise.  Succinct.  Compelling.

Why not do my own version of an Ignite talk at Back to School Night next year? I get 10 minutes with parents and would love to change it up a little this coming year. Trust me, after surviving last year, I think the parents deserve a better, improved, and more reassuring version of Mr. Stadel. I'll explain that last sentence in some upcoming blog posts that I'll use to debrief about the 2012-2013 school year. If you're not sure what an Ignite talk is, let me introduce you to my man, Steve Leinwand.

If you like that, check out more Ignite talks by Annie Fetter, Dan Meyer, Max Ray, and Phil Daro. These are my go-to talks when I need a math pick-me-up. Do the math, that will be a little over 20 minutes well spent, being inspired by some key people in our math community. Seriously, check out those four talks.

I'm brainstorming in this space, so feel free to share some input please. At Back to School Night, I'll start by giving a brief 30-60 second introduction of what an Ignite talk is and how they work. I'll give an Ignite talk for 5 minutes, covering any of the following things:

• Standards Based Grading (SBG)
• Noticing and Wondering
• Common Core State Standards (CCSS)
• 8 Standards for Mathematical Practice (8 MPs)
• Problem Solving
• Homework (or No Homework)
• Find my Mistake
• High School Placement Tests
• Desmos
• A healthy balance of mathematical application with procedural and conceptual understanding
• Problems of the Week (PoWs)
• Quotes of the Week (QOTW)
• 3 Act lessons
• Reassessments
• Estimation (Number Sense)
• Tangram Tuesdays
• Whiteboarding

This leaves approximately 4 minutes for parents to ask questions or something else... Have any suggestions for those last 240 seconds?

Who's with me? Does anyone else want to do a Back to School Ignite talk? There's already been some interest generated on Twitter and I started a Back to School Ignite list. Shout at me if you're in. Or is this a really foolish idea? Seth Leavitt, my new online colleague and EnCoMPASS Fellow asked if I'll post it online. I don't see why not. Maybe we can create a space for Back to School Ignite talks.

*UPDATE: Each item listed above does not correspond to its own slide. I simply listed ideas that could possibly work their way into the presentation. Some support each other. For example, when talking about the importance of problem solving, I would mention resources such as 3 Act lessons and The Math Forum's PoWs. Feel free to add to or subtract from the list.

Ignite,
930

#MTBoS is a Math Warehouse

This morning, I read Dan's post and Kate's post and listened to a few minutes of last night's Global Math. The following are my initial thoughts; raw, unedited, and mostly incoherent.

I picture a store opening in my town. All I know is its title. Let's call it Math Warehouse. I drive by it, maybe a friend told me the warehouse has been open for some time now. They suggest I should check it out. I did a little window shopping. One day, I was at a store nearby and found myself with a few minutes to spare. I ventured into the Math Warehouse without any expectations. I see that there's no membership to be a part of the Math Warehouse. Just like any other store, I can come and go as I please. I can talk to those that are in the store. I can browse all the items available to me. I can get recommendations. I can go to certain aisles that have resources and tools I might need for my classroom, students, and teaching craft. I have no expectations. All I know is that the Math Warehouse is a place I can go as a math teacher to find resources from people who are willing to share. I can't use it all. Not all of it applies to me. Not all of it can be consumed in a lifetime. This Math Warehouse became the greatest thing. It instantly became my favorite store. It is the MTBoS or place we refer to as Math Twitter Blogosphere.

I entered this online community of educators with no expectations. All I know is that I feel I owe this math community a large greeting card of gratitude.  The kind of greeting card that makes you feel all good inside. My students, my classroom, my craft of teaching has grown immensely because of you and everyone else that has given me feedback on math education. I've had opportunities to meet like-minded teachers, both online and in person. This is a cool experience. Honestly, I've never given it any thought as to how it should be run or what direction it needs to go, because I am just a small rain drop in the beautiful rain that this online community showers us with. I'm a little saddened that so much thought is going into these discussions. Granted, I don't know everything, but I don't like hearing where this place needs to go. This online community will do what it needs to do. There are things I wish I was better at and ways I could give those greeting cards to more people.

I'd love to follow more people, but following and paying attention are two different things. I'll admit, I limit my follow group mainly so that I can pay attention to those I have filtered to provide my students, teaching, and classroom with growth. If someone is doing something, I'm confident I will eventually hear of it and consider its application. I'd love to comment more on people's posts, even if it's a "thanks for posting." I'd love to know all the great things that are happening in my twitter feed or blogs I don't know about... but I can't. It's beyond my human capacity. I heard some stat once that sticks in my head. Take all the hotel rooms in Las Vegas. If one were to spend a night in each hotel room, it would take multiple lifetimes. All the content available to us at our fingertips, mouse-clicks, and eyeballs is beyond what our brains, classrooms, lesson plans, curriculum, and pedagogies can handle. It's a warehouse.

I'm not interested in shaping where this online community goes. I'm happy to be a part of it. Let me rephrase that. I'm extremely grateful to be a part of it. Maybe I'm blowing this out of proportion. I'm happy for you all. I'm thankful. And I'm one to just let this be a good shopping experience at our Math Warehouse, if you will. Come and go as you please. Take what you need. Leave some ideas behind. Share, share, share!

Lastly, I'm at Drexel University working with some other great teachers on assessments and rubrics. One of The Math Forum leads, Wesley Shumar, shared this with us this morning:

Community is the result of the process.

The process he was referring to was the sharing of ideas, resources, strategies, and other educational components. We have a community here that is the result of the process. We share, we provide feedback, we listen, we retweet, we blog, we share, we favorite, we comment, we create, we borrow, we improve, we share, and as a result we create a community that will define and direct itself through the process.

Best,
108

CST and SBAC Questions

I compared a few questions from CST and SBAC. CST stands for California Standards Tests and these are questions that were on previous STAR (Standardized Testing and Reporting) tests given to students in California and have been released to the public. I'm using recently released practice questions from SBAC (Smarter Balanced Assessment Consortium) that have been designed with Math 6 and Math 7 Common Core State Standards (CCSS) in mind.

I'm curious what thoughts or questions you might have. Leave them below. Thanks.

QOTW 2nd Semester - 2013

In January, my Quotes of the Week post highlighted student comments captured during our first semester of the 2012-2013 school year. I'm here to post a few captured from the second semester. It's hard to compare them to the first half of the year, so let's not. Instead, let's just enjoy the comments, observations, or questions that students gift us with, enriching the mathematical climate of the classroom.

I experimented with Kelly O'Shea's Mistake Game in Algebra about midyear. Emma is presenting to the class about identifying linear functions, given three points. Justin (from the audience) claims that Emma’s graph is NOT linear, but says there's a slope to her three points. Without skipping a beat, Emma unleashed this response at Justin. Way to go Emma! Talking smack with math vocabulary.

Students were exploring x- and y-intercepts using Desmos.com. After graphing a few lines and writing down the ordered pairs of each intercept, the wheels started to turn inside of Lisa's head. She saw the structure and pattern of the intercepts containing zero. The follow-up:

Me: So Lisa, why are you seeing all these zeros?

Lisa: Because we're learning about intercepts today. 

Me: What are the intercepts of the lines? 

Lisa: It's where the lines hit the x or y axis. 

Me: So what's up with all the zeros? 

Lisa: If it hits the x-axis, the y value is zero. And if it hits the y-axis, the x value is zero. 

Me: Bingo! 

Mark needed to graph a straight line as part of his task. That week, we had just spent an entire class period doing a Jigsaw activity so that the students could explore the 8 Standards for Mathematical Practice. Mark is proud to be using Mathematical Practice 5, Use Appropriate Tools Strategically.

Groups were given a warm-up one day where they had to collaborate and stack the highest and strongest tower using 100 snap cubes. I determined the winners by kicking the desks they stood on to see who survived Earthquake Stadel the best. While cleaning up, Logan is noticing that he could do a better job next time. This is exactly what we can do in the math classroom: learn from our mistakes, come up with a revised problem-solving plan, model, collaborate, and persevere.

The class was doing a cocktail of mixture problems one day. They were collaborating with their groups and one group had your typical coin question. Something along the lines of:

I have $9.75 from X amount of nickels and dimes. How many of each coin do I have?

Upon solving the question on their whiteboard, Sara quickly realizes that you can’t have 1.95 of a coin. She was the first to notice it and say something to her classmates about this contradiction. The group first tried looking for their mistake and then tried solving the question again. Good job girls!

Midyear, I showed all my classes the 1st Semester QOTW slides. For the first month following these slides, Charles was trying to turn every little thing he said into a quote just so he could be up on the board. As I've mentioned before, these can't be planned or contrived. These quotes just come out naturally in the regular happenings of classroom interactions.  Students were doing some individual work one day, and Charles actually let a good one slip out. He was still working individually on the task and didn’t want his efforts spoiled by someone blurting out the correct answer. How many times have we been there, either on the receiving end or the one ruining it for others?  

This is Jenna's response to the following video:

I asked the class for their interpretation of the video or any of the drawings? Was there a drawing they could relate with? Without blinking, Jenna says, "It's like graphing stories." I totally agree. It had been awhile since we explored graphing stories, so Jenna just confirmed what a great impact that concept had on her. Very cool! 

Arielle is a former QOTW winner. Students were exploring exponent rules by finding mistakes. Michael Pershan and I have done some extensive blogging about this. Anyway, we had reached the final day or two of learning through mistakes and Arielle raised her hand and shared this gem with everyone. It's a beautiful observation in my opinion about the importance of math mistakes, learning by making conjectures, and students coming up with the rules on their own instead of me spoon-feeding them. 

Two quotes in one class period! That's a first. We were playing Race Car Math and one of the review questions asked students to factor a polynomial. Eli, a boy a few words, caught his group-mate incorrectly factoring the polynomial. I love the “Dude!” part of his quote.

Also during Race Car Math that day, Kailey was so excited to be on the board after correctly graphing a parabola, following the flowchart we put up in class. Our class flowchart is shown.

Last, but not least, Chris closed out the year with, "I think my number lied to me." In geometry, we were reviewing volume of a sphere with questions similar to Nathan Kraft’s volume of a soccer ball task. Chris set up, and used the incorrect proportion when solving. When arriving at an unreasonable answer, he realized his numbers lied to him. Way to check for reasonableness!

It's truly been a pleasure having the QOTW section carved out on my whiteboard for student quotes. As far as I'm concerned, this will be a staple in my classroom for the remainder of my career. I look forward to next year's quotes!

QOTW,
717

CCSS Workshop 2013

I did my first workshop this past week (5-30-2013), discussing the transition to Common Core State Standards, with a focus on the 8 Standards for Mathematical Practice. I'm working with K-5 elementary teachers, a couple of middle school math teachers, some support teachers, and a couple of administrators in the room. I'll admit, I'm not as eloquent a speaker as Steve Leinwand, but there might be a few times where I actually make sense. I'm throwing this video out there for some feedback. I don't expect you to watch the whole thing so I've provided some chapter markers you might be interested in. At the end of the notes for each chapter, I reflect by creating a wish-list of moves I would've done differently. They come across as rhetorical questions, but feel free to chime in. However, here are a few angles I'd appreciate you considering:
What did I forget to talk about?
Where could I have been more explicitly clear?
Where coud I have used a better strategy? ...and so on.

Opener: 0:00-10:10
I use an estimation task (Day 127) to kick things off, providing teachers with the handout Michael Fenton created for estimation180.com and my students. The handout has been through many revisions and I think we have a final version that's a winner.

I use the whiteboard to write the "Too Low", "Too High", and "My Estimate" of a few teachers, asking for reasoning along the way. We watch the answer and discuss.
8:15 is a precious moment where a teacher asks, "Wait a minute! What was it (the song length answer) really?" I love how a teacher is demanding more information. I wish I spent more time expressing the importance of her question. I feel rushed because of all the workshop content I have prepared. I wish I had allowed the other teachers to address her question more. I do with my students, why didn't I do this here?

Active Notebook Part 1: 10:15-12:55
Teachers glue their Estimation 180 handout to the inside cover of their workshop Blue Book and the Table of Contents to the first sheet while listening to Can't Buy Me Love.

Preview of Workshop: 12:55-20:50
I attempt at working on the following question throughout the workshop,

What's our role as we reshape the classroom with the Common Core State Standards?

I share with teachers what today will not be and what today will be. On a parallel universe I share with teachers what I perceive the CCSS to not be and what I perceive the CCSS to be. Then I share a few personal items from this past school year. Seeing that I'm doing a workshop with multiple grade level and content teachers, I'm expressing the focus of the day to be the 8 Mathematical Practices and what we do in the classroom. How do we help facilitate the learning?

Estimation Task #2: 20:50-26:45
We use Day 129 where teachers see that the song length is also the track length. Listen to their reasoning. I love this! Hearing this reasoning from teachers and students is one of the many joys I get from doing these daily estimation tasks. However, I wish I did a better job (23:50) of getting the teachers to justify their reasoning and "argue" a little more than I did. I wish I created a little more tension. Check out (25:00) the excitement of the teachers as they watch the answer.

I introduce the language of creating a task that has a low-entry point and could see that many teachers had no idea what I was referring to. Not their fault. However, I like their reaction when I translate "lower-entry point" to being easier. I wish I explained "low-entry point" better. I wish I had explained it as creating a task where most, if not all students have an equal opportunity to engage with the task, regardless of their mathematical proficiency. I wish I expressed the importance of providing students with a task where math vocabulary and thinking come as a natural result of solving the task.

Math Tools & Two Uses: 26:45-33:55
I ask teachers to glue a picture of a math tool to the front of their Blue Book and write down at least two uses for each tool. They have two minutes to complete this task. After We Will Rock You, I ask the teachers for their uses instead of telling them. This serves two purposes. Yes, I have a list of uses that I anticipate them to come up with, but I want to hear from them first, selfishly providing me with additional uses that I didn't anticipate. Secondly, I'm using this activity to illustrate how students need to provide the answers in the classroom. I wish I could have given this more time in the workshop where the teachers actually used the tools in the manners they suggested. I also should have had the teachers write down all the uses we came up with.

Introduction of 8 Mathematical Practices 33:55-1:04:00
Again, I'm feeling rushed for time! Yes, I said it, "Let me talk about these 8 Math Practices real quick." Real quick? How silly of me. These practices are not something to gloss over. Don't worry, we spend ample time doing a Jigsaw activity so teachers are out finding information about them. I provided the teachers with the handouts found at Jordan School District's site. The practices are presented in a manner representative of the grade level you might teach. Thanks Fawn for this link.  I could have explained and facilitated this a lot better, but the redeeming value is hearing teachers that were appreciative of this specific activity because they were "forced" to explore the practices instead of just receiving a handout with the information embedded. If I were you, I'd skip the section (39:30-52:00) unless you want to hear some of the teachers talking in their groups as they rotate around the room to their four different stations. I'm proud of the rotation table I provided and how teachers travel together according to their "math tool." DON'T miss the "perfect high-five" at 38:10. I love that a teacher commented that "perfect" is subjective. I say this all the time to my students.
At 52:30 I give teachers time to regroup and complete the practices by receiving information from the other teachers in their group. I recap (1:00:00) and then show teachers how to create a pocket (1:01:00) inside their Blue Book so they can store a Quick Reference "teacher" version (I referred to as "adult") of the 8 Mathematical Practices.
I wish I gave more time for reflection. I wish I reviewed each practice with the whole group by having them share out loud something they learned. During the Jigsaw activity, I was told that my time with the teachers was cut short by about ten minutes so I had to hurry things along. Arghh!

Find My Mistake: 1:04:20-1:13:00
I made an executive decision to skip the model lesson I had prepared for the worksop. I'm glad I didn't skip the Find My Mistake segment of the workshop. I'm very adamant about teachers finding the mistake quietly here. I encourage them to share with each other before we review with the whole group. I give props to Michael Pershan for mathmistakes.org. You can hear kids in the hallway, alerting me that our workshop is coming to an end very soon. We listen to each other make corrections or talk about the misconception and why us teachers are good at knowing the content we teach. We're constantly telling students what their mistakes are and telling them how to fix it. Let's switch that role. Make the students find, correct, and tell each other what the mistakes are, especially items that use algorithms. Remember, most of the teachers here are elementary teachers. I also point out that I haven't been jumping for joy every time a teacher gets an answer correct. Instead, I try my best to throw it back on the class for what they think, allowing them to critique the reasoning of others. I love how estimation and number sense is addressed with teachers on how to help encourage students to avoid these mistakes.
I hit a nerve (1:10:45) when I was asked, "What about simplifying?"
I could very well be wrong here, but my current understanding is we (as teachers) are to allow for multiple representations of the correct answer, unless explicitly instructed otherwise. In other words, ten-eighths is just as acceptable as five-fourths.
I wish I reviewed with the teachers the importance of doing an activity like this quietly and individually first, before group discussion. I wish I expressed how much I love group work and collaboration, but need to remember both teachers and students need that quiet time FIRST. The worst is being in a group where one person dominates the conversation and you don't have time to think or worse, problem-solve. I wish I had covered this with the teachers.

Summary: 1:14:15-1:19:00
I provide teachers with a fill-in-the-blank handout to glue to the inside of the back cover. Again, watch their reaction when we get to "low-entry" point. I hope I drive it home when referencing the "Cent-ed Whiffle Ball" task I recently did in Geometry. I remind myself and the teachers to listen to the students. I'm constantly working on allowing students to finish their thoughts. Don't cut students off or finish their sentences for them. I ask teachers to create a couple of goals. I provide the teachers with a list of resources found here. I like how they (at least some of them) want another in-service/workshop.
I wish I emphasized the importance of knowing the 8 Mathematical Practices better and to use the summer to better prepare for next year. I wish I had more time to pump up these points in the summary. I wish I had more time!

Unleash yourself in the comments if you will.

Thanks!
729

Cent-ed Whiffle Balls

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

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

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

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

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

"This is fun!"

"This is stressful!" 

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

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

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

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

Student 1: Three

Student 2: Four 

Student 3: Three and a half

Student 4: Five

Me: Let's find out!

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

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

Student 5: Cool!

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

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

Devon: There's spaces between the pennies. 

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

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

 

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

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

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

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

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

This group used 71 pennies versus a theoretical 69.

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

Whiffle Ball circumference: 28 centimeters
Penny diameter: 1.9 centimeters

Cent-ed,
1050

P.S. Help me make this task better.

Amnesia

Gadzooks! You wake up tomorrow with amnesia!

Someone significant (spouse, child, significant other, neighbor, etc.) in your life informs you that you teach math. You get dropped off at your school (place of work) and are led to your classroom. Your principal can't find a substitute and your colleagues run to the photocopier to make you some busy-work handouts. You can't remember any mathematics and your students are 10 minutes away from entering your room, oblivious to all of this.

You look down and see this on your desk.

What do you do?

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