## Friday, December 27, 2013

### Piles of Tiles

I was at my parents' house for Christmas and came across this game (older than me) in a closet full of board games. Made by The Cootie Company back in the 70's, I give you Op Tile.

There's a lot going on here; game boards, tiles, dice, cards, bell-bottoms, shag carpeting. Instead of typing up the directions, amuse yourself with these:

The tiles look like plastic jello.

The cards offer some opportunities for strategy throughout the game.

There are many things I like about this game, even though I have never played it. I like the spatial reasoning component, the challenge of placing tiles depending on what you roll with the dice (and the order in which you have to place them), and the demand for strategy that the cards present. I didn't like reading through all the directions to discover all the nuances. Some parts of the game are not intuitive. However, I really like what is intuitive: placing the tiles on your game board in the best way possible to cover the most area (square units) of the game board earning the most amount of points.

I brought the box home to create some adaptations for my students. Here's phase 1: Piles of Tiles. Having recently blogged about weekly POPS, Piles of Tiles will become an additional option for the first P (Patterns/Puzzles) in POPS. In phase 1, I'd like my students to play around with the Piles of Tiles puzzle like this:

I'll pass out this sheet (maybe two) to students and have them cut out all of the figures. They can keep their cut-outs in a plastic sandwich bag.

The student game board will look like this.

Students can use their cut-outs to fill the 12x12 board with the specific tiles, found in the table at the bottom of the page. Once they have their solution, they can outline each figure within the game board and specify which section refers to figure A, B, C, and so on. They can use colored pencil or crayons to keep each similar figure the same color. For you detailed people, I made the grid so it prints each square unit as 1 cm by 1 cm. Therefore, you get a total of 144 square centimeters. AWESOME!

Students can stay organized and also submit the following to me:

All the Piles of Tiles goods (blank templates) can be found in my weekly POPS folder.

My goals (right now) are to get students to:

• manipulate shapes through rotations and translations
• build their spatial reasoning
• recognize there are multiple solutions
• organize their data
• have a better understanding of area

I'm open to suggestions or feedback, so please let me know.

Since we're on the topic of puzzles, ThinkFun has some great puzzles (as I've mentioned before). Go over to their site and check out these Big Games group activities to use with students. Many of the activities are puzzles you might have seen on paper somewhere, but they rewrote them as group activities to foster collaboration, spatial reasoning, and problem-solving amongst students.

Piles of Tiles,
1032

## Saturday, December 21, 2013

### J.T.A. my F.I.L.

Today I experienced a bittersweet-simultaneous-challenging-weakening-strengthening perspective on the following:

• There are some really amazing, big-hearted, significantly good-for-humanity people that come into your life (and the lives of others): make the most of the time you have with them.
• There can be a blurry (sometimes blinding) difference between how remarkable somebody was, and if given more time, what could have been. Don't let the latter blur the significance of the former.
• Life is more enjoyable celebrating it.

R.I.P.  J.T.A

## Sunday, December 15, 2013

### CMC North 2013

Here's a recap of my CMC North experience from last weekend.

Friday:
Fawn Nguyen and I flew up to Monterey Bay, checked in, and found the Fishwife restaurant for lunch. It was within walking distance of the conference grounds. Thank goodness because it was a little chilly outside. We enjoyed lunch so much we made a trip back there for the 4-6 p.m. Happy Hour so we could fine-tune our Saturday presentation. We checked in with the CMC people, and if you ever make it up to Pacific Grove for the conference, don't forget the badge they mail you. It'll cost you \$5 if you forget it, right Fawn?

Later that night, we attended the opening keynote by Dr. David Dockterman where he discussed the importance of a growth mindset.  Afterward, it was great to see and meet up with Fawn, Dan Meyer, Avery Pickford, and Breedeen Murray for some pizza. Good times. Good company.

Saturday:
Session 1:
The first session I attended was titled Using Formative Assessment to Create Equitable Practices by Karen Mayfield-Ingram. We diagnosed some student work and left with the following reminders:
1) Make a concerted effort to give students detailed feedback on any assessment.
2) Try and ask students questions that will encourage them to analyze their work better.

Session 2:
I showed up early at Larry Armstrong's session titled Flip Instruction to Transform Learning. I'm interested to know more about the flipped classroom. Not because I'm sold on the idea, but because my district is getting iPads the second semester and I have a feeling we will be encouraged to implement some type of flipped classroom model. Who knows? Again, I'm not sold on the idea, but I want more information just in case. Well, it's 5 minutes before the start of the session and the front of the room is empty: no computer plugged into the projector, no one is setting up, no Larry Armstrong. Nothing!
A CMC helper comes to the front to inform us Armstrong won't be presenting and we have time to catch another session if we want. I saw Brad Fulton come in at the back of the room and I got up to ask him if he'd present, but he said he was only coming to drop his stuff off for presenting at the following session. I turned to the CMC helper and offered to present my Number Sense session I gave at CMC South. She took me up on my offer and we had ourselves an impromptu Number Sense session. It was fun! For the forty to fifty attendees that stuck around, we had a blast doing estimation challenges and talking number sense for about 50 minutes. I will blog more about my presentation over the next few weeks. Stay tuned! I was honored when Rebecca Lewis, the Program Chair, gave me this token of appreciation for stepping in at this session. This proudly sits on my desk.

Session 3:
Shelly Lawson gave a great hands-on session titled Modeling Lessons Can Work for All Students - Yes, Even Yours.  I was excited about this one because it was geared toward 7th grade curriculum. You know you're in for a good session when you see a bag full of PVC pipes, a stopwatch, a meter stick, a steel ball, and some string. There were 5 different length PVC pipes, and four connecters; two right angles and two at about 135 degrees.

We had to construct a pipe so that the ball, when dropped, would make its way through the pipe at the fastest rate possible. Here's our contraption. It was the fastest because of my teammate's design. Great job Bob!

Shelly also introduced us to the Incredible Egg, Float That Boat, and the Penny Lab. All of these activities allow students to make measurements, mistakes, and formulate conclusions based on observations and data collected. I left with a packet full of hands-on activities that I can incorporate into my curriculum. The structure of the packet gives students pretty detailed steps and instructions. Personally, I will probably revise the activities to leave them a little more open-ended and less structured. Overall, some great activities. Email Shelly for a copy. See the link attached to her name above. By this time, I think I ran into Dittmer about 50 times. Really cool guy!

Session 4:
Fawn and I presented Hotel Snap to the CMC attendees. Fawn is awesome! Tell you something you don't know, right? Okay, chances are good you've already read her blog post on Hotel Snap. If not, go now and read it here. Fawn did an amazing job coming up with this task. You might be surprised, but I have nothing but great things to say about Fawn. Her task is challenging and can be used at numerous grade levels. CHECK IT OUT!
 Photo by Dan Meyer
Thanks to Brian and Dan for helping us with the calculations. Thanks John for helping us clean up!

Session 5:
My man, Max Ray, presented Becoming Better Reasoners: Supporting Students to Develop as Problem-Solvers. I was excited since this was my first time seeing Max present.  See how calm he is in this picture?

I love how Max is so patient and allows his students (us CMC attendees) to formulate noticings, wonderings, and other thoughts as we worked through some Math Forum tasks. I enjoyed this last session of the conference because we worked in groups to problem-solve a task, we analyzed student work, and Max charged us with asking students questions that would help them further their problem-solving approach by becoming better reasoners. Well done Max!

Saturday night:
Ignite talks!!! I could write a whole other blog post on these. I was honored and privileged to present with the following mathletes!

The Math Forum organized, hosted, and took over the Ignite talks this year. Thanks Suzanne for all you did! Watch them in January or February 2014 when they post the 10 videos online. It was a lot of work preparing my 5 minute talk, but was great fun! Fawn stole the night! The energy and inspiration from these talks was a great way to cap Saturday night!

Sunday:
Dan Meyer gave the first keynote of the morning titled Fake-World Math. Dan is the man! His presentation was fantastic. I don't want to post any spoilers here, but he talked about the importance of modeling and how CCSS defines modeling. If I could summarize some important points, here's what I want to reflect on for future reference.
Although important to the math classroom at certain times, the following is NOT modeling:

1. I do, we do, you do
2. Using concrete manipulatives
3. Graphs, equations, functions
Furthermore, Dan stressed that the real world is not always greater than a math problem. Vice versa, a math problem is not always greater than a real world situation. He emphasized that strong modeling starts with a simple question and allows students to identify variables, formulate the necessary model to organize and solve the task, and to validate conclusions a la something along the lines of a 3-Act task. He reminded everyone that modeling tasks are out there: Dan, Robert, and here. My summary can't do justice to his presentation, content delivery, and charisma. If you get a chance to see Dan live, DO IT!
*For the record, my initial estimate for Dan's Super Stairs task was over by fifty seconds. Bam!

Dr. Timothy Kanold gave the second keynote titled The Art of Teaching Mathematics: Inspiring Students to Learn. CMC North 2013 ended with Bill Withers' Lean on Me.

Before Fawn and I flew back to Los Angeles, we had brunch with some great math amigos (pictured below).
 Left to Right: Stadel, Nguyen, Meyer, Murray, Pickford.
We also said hi to some fish at the Monterey Bay Aquarium.

Thanks Dan, Avery, and Breedeen for driving us around too!

North,
819

## Wednesday, November 27, 2013

### Intervention strategies

@TmathC tweeted about possible presenters and sessions at Twitter Math Camp 2014 next year.

Selfishly, I wanted to think of a session I could present on so I could justify to the boss (my wife) that attending #TMC14 was within our means. Instead, I looked through the list found here and saw something missing: a session dealing with intervention strategies or techniques for helping students who struggle in math class. Then it dawned on me, whenever I attend CMC South, I rarely see sessions dealing with intervention for students who struggle in math. Why?

Intervention, to me, is not reteaching, relearning, or repeating the same lesson to students by yelling it at them in a louder voice. By the way, don't tell Sadie you're reteaching. I'm right there with her on the Blame Game. Let's face it, every student comes to our class lacking some type of prerequisite skill, some more than others. It's not about blaming the previous teacher, previous curriculum, the "apathetic" student, the "unsupportive" home, or any other scapegoat. I know I've let students down in the past (and currently) and feel bad as they move to the next grade level. However, I want to be a better, more effective teacher, especially for students who typically struggle in math class.

I doubt I'm the only one who could benefit from more intervention strategies and techniques. I believe every teacher who actually cares about their students would appreciate more intervention strategies no matter where they teach, what they teach, or who they teach. Being at a new school this year, I really could benefit from more intervention strategies.  Most of my students this year need intervention badly. They need help with numerous elementary concepts. I need more strategies to help students become better math students. I need more strategies to help students increase their number sense.

I'm not looking for a silver bullet. I'm not looking for someone to tell me to reteach it using similar worksheets, but change the values of the coefficients, or numerators, or integers, or percentages. I love the #MTBoS and all the great resources, but I feel it lacks this crucial element: intervention. Maybe it's just me. Maybe I'm absolutely ignorant of some rich resource somewhere. I'm asking for help. Do you know someone who writes about intervention? Are there reputable intervention programs/sites online? What intervention strategies do you use that are effective? Please share. Go to the comments and list blogs, sites, or people I need to follow who have intervention strategies I can use. Thanks in advance.

Intervention,
309

## Wednesday, November 20, 2013

### Weekly POPS

Have you ever tossed a puzzle at one of your students? Y'know, the puzzles kids can play around with using their hands and minds? It's crazy, right?! It's fascinating to watch a student display a wide range of behaviors: curiosity, engagement, perseverance, frustration, and an earnest desire to know the solution if they get fatigued and stumped. I have a couple bins full of puzzles in my classrooms that do this to kids. Students rarely have a chance to play with them, but when they do, they go bonkers in a good way. Occasionally, you'll hear a triumphant yell when someone solves a puzzle. Other students look in disbelief. It's hilarious. My collection of puzzles ranges from the Bedlam Cube (now known as Crazee Cube), to Cannonball Pyramids, to the Rubik's Cube, to Tangrams, to ThinkFun puzzles, to other miscellaneous puzzles I've picked up over the years. A few weeks ago, I was driving home and wanted to know if there was something I could do in math that had a similar magical effect on kids.

Have you ever tossed a puzzle at one of your students? The ones on paper that require logic and critical thinking? Those are crazy too! Kids can really get into them. Around the same time I was thinking about the power of physical puzzles, my school wanted to revamp our weekly intervention/study-hall period. I thought students could benefit from working on logic puzzles, patterns, or Get to Ten. I went to a resource called The Colossal Book of Short Puzzles and Problems by Martin Gardner in which Fawn recommended. I came across this Billiard Balls gem in which I remade for my students:

So I got to thinking and thought of some inspirational people/things from my PLN. I've always wanted to incorporate Fawn's Visual Patterns into my classroom more, especially with it's beautiful new makeover. Fawn is also known for her weekly problem-solving tasks. I've also wanted to incorporate more PoWs from the Math Forum. The Math Forum has an abundance of problem-solving tasks that range in difficulty across grade levels. Sign up, yo! Last but not least, Dan Meyer had impeccable timing and recently wrote a very invigorating post on [Fake World] Conjectures that has created quite the buzz in the comments. Personally, he struck a chord with me as he ended it saying:
Find those puzzles in the real world, the fake world, the job world, or any other world - it doesn't matter.
His post and quote made my day (with a smile).

The result of all these crazy things: Weekly POPS.

POPS stands for:

• Patterns (or puzzles like the Billiard Balls above)
• Order of Operations (Get to 10 or Get to 24)
• Problem-Solving

Patterns (or puzzles):
I will include a pattern either from Visual Patterns or one I create. As you can see from the handout below, it's similar to Fawn's form. I am adding a section for students to describe the pattern in their own words. If I decide not to do a pattern that week, I'll do some type of puzzle like the Billiard Ball puzzle above.
Order of Operations:
Students are to use the four given numbers and mathematical operations, symbols, and/or notations to get to the values of ten (or twenty-four). As you can see from the handout, students need to write the expression and evaluate it correctly using Order of Operations (or PEMDAS).
Problem-Solving:
Definitely one of the most important parts of the Weekly POPS, problem-solving. Right now, I'm finding old PoWs from the Math Forum's library to share with my students. As you can see from the handout, be sure to include the Math Forum's copyright information when photocopying. I'm looking for students to organize their work, demonstrate their solution strategy, and think critically.
My goal with Weekly POPS is to get students to really think critically and problem-solve. Why? because they so desperately need it. It's challenging, demanding, and necessary. There's a slight puzzle feel to POPS. Students have really been into it this week.

Students receive the POPS every Friday and have a week to complete it. They'll turn it in the following Friday and receive a new POPS. I've invested a lot of time in class this week going over my expectations, but will use Monday and Tuesday next week to show my classes student POPS that are exemplars, average, and sucky. I told them, "You earn a zero on your POPS, it's the same as POOPS."

I look forward to this adventure with my kids. Here's a folder with the POPS I've created so far. Feel free to join in the action. If this link is broken, please notify me and I'll fix it, unless your name is Fawn.

[UPDATE]: Check out Piles of Tiles that can be used in place of patterns. (12-27-2013)

POPS,
951

## Monday, November 11, 2013

### Why I Blog (maybe)

Kate Nowak challenged us to share a few thoughts on blogging as she prepares to be a featured speaker at NCTM. I have a few minutes to share so here it goes.

1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?
I was hooked on blogs when I discovered I could use things others had created. I discovered that my teaching could improve by learning from others who were humble and honest about the mistakes they made. A close friend of mine who teaches science sent me Dan Meyer's TED talk and my blog reading increased from zero minutes a day to about 240 minutes a day: time-suck!
2. What keeps you coming back? What's the biggest thing you get out of reading and/or commenting?
I keep coming back because I get the opportunity to observe other teachers (even if it's a snapshot) without being in their classroom. Anytime I've been in another teacher's classroom, I've learned so much. It doesn't matter if the teacher was good or bad, I was either learning what to do or what not to do. Reading other blogs is a similar opportunity for professional growth.
3. If you write, why do you write? What's the biggest thing you get out of it?
I write to document things, reflect on things, and share something I'm proud of. If I had time to blog about everything I do, I would. However, it would become white noise for those who take the time time to check in. Therefore, I write about things that stand out to me: great experiences, cool lessons, or things I'm proud of. It's not intended to be perfect. I hope to keep it raw. I'm not the best writer. Somehow, I happen to write a few things that a few people are interested in. That positive energy keeps me going.
Specifically, I blog at two places now: Divisible by 3 and Estimation 180. Divisible by 3 is reserved for all the things I've mentioned above. I blog at Estimation 180 to keep people updated on new estimation challenges I posted and/or any stories behind the challenges. I enjoy when a story is attached to a mathematical experience.
4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to's? Stories?
From my experience, I enjoy a keynote speaker who is both inspiring and interactive. I like the sessions where the speaker is well-spoken, engaging, inspiring and challenges me as I leave the session to become a better teacher. However, I don't want the entire 60-90 minutes to be spent by them just showing me pictures, videos, quotes, or other things that are one-directional. I also need to be challenged by actually being forced to do math, tasks, or be given the opportunity to explore what they're presenting on. Make it feel like a classroom where I get a healthy mixture of inspiring talk and engaging tasks. Kate, I'd love for you to share the good, the bad, and the ugly about blogging. Then have me get out my internet device (phone, tablet, or computer) and either explore a list of blogs or even create a blog. Put up a list of blogs that do different things. Have me explore them. Give me some questions/tasks:
1. Compare and contrast the blogs.
2. Which blog would you find yourself using most often, least often, never?
3. What kind of blog would you like to create?
4. What suggestions do you have for these people blogging?
5. ...and so on

Give your attendees a Google Form to fill out so you can document the answers. By this point, you've already shared with me your inspiring thoughts on blogging and you're getting me involved. Get me to experience blogging while we have time together. Don't make blogging a homework assignment for your attendees.

Hope that helps. Good luck Kate!

## Sunday, November 10, 2013

### 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.

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

## Monday, September 23, 2013

### Manilla folders [Number Sense]

I must share this short story with you about the number sense I've been witnessing for the past few weeks with many of my students. Be warned: not for the faint of heart.

Today, a few 7th grade boys came in after school today and wanted to help with some things around my room: tidy up desks, organize whiteboards, etc. There was a box of 100 manilla folders that I had just brought back from the office. I needed 80 of the folders and would later return the remaining folders to the office.

Here's my exchange with the student we'll refer to as Albert:
Me: I need 80 folders from that box. Albert, think of a quick way you could get those 80.
***Let's pause. How would you (the reader) quickly get 80 folders from this box?
Albert: I could count in 5's.
Me: Okay. Any quicker than that?
Albert: By 10's.
I get what Albert is trying to do. He doesn't want to count to eighty by ones. To that point, I would say his initial two responses made sense and are practical, in the mind of a 7th grader. I thought maybe I'm asking the question incorrectly, so I try it again.
Me: Right. That would be a good way to organize the folders to keep track of them as you count. Albert, I'd like you to think of a way to quickly get those folders out of the box.
I can see the look on his face is one of confusion. Not that look like he's trying to figure it out, but that look like he has no idea what I'm asking. So after a minute, I mistakenly ask another question (in hindsight, I wish I would've stopped the conversation and let him do his thing).
Me: How many folders are in the box?
Albert: 100.
Me: How many do I need?
Albert: 80.
Me: Is there a way to get me the 80 without counting all 80?
Albert has no idea. I like this question because now it's specific. His challenge is to get me 80 folders without actually counting all 80. After a minute. He needs some prompting.
Me: If the box has 100 and I need 80, how many will be left?
Albert: 30.
Me: So 80 plus 30 is 100?
Albert: No wait, 20.
Me: Okay, so I will send 20 back to the office. How can I get 80 folders out of the box without counting all 80?
Albert has no idea. This exact exchange continued for another round. I'd like to say that Albert eventually came to an efficient way on his own, but he didn't. I tossed 100 up on the whiteboard. We subtracted 80 and wrote 20. I thought after Albert saw the 20 on the board, he might realize to count 20 folders from the box, take out the remaining folders and switch them with the 20.

This is quite common. The number sense (or lack thereof) my students have (or don't yet) is quite fascinating. I have a lot of work ahead of me this year. One thing is for sure. I can get better at asking shorter questions. I can get better at looking for these learning opportunities. I can get better at not looking for one answer when asking questions of students. As Max Ray would say, "2 > 4."

Folders,
918

## Saturday, September 14, 2013

The first full week of school is done. I can breathe... a little.

I'd like to share a lesson we (my awesome 7th grade teammate and I) came up with this week when reviewing multiplication and division of fractions with 7th grade students. It actually went quite well and was fun. Simply titled, "Your BF!"

If we said "fractions" to students, a majority of them would instantly shut down. Instead, I titled their notes at the top with "Your BF!" and it looked like this:

Of course, some kids, only reading the title, responded with:

Then, some kids continued to read and saw it. That word. That terrible, miserable word. That word that has a paralyzing affect on students. Almost like you just told them their dog was run over by a car, or that Christmas was cancelled, or [insert some terrible event you dread]. That word is FRACTION.
I quickly responded to their whining, groans and moans by saying, "Let's make this a little friendly competition. Let's see who can draw the best half. Use the rectangle I gave you to draw your best half."

All of a sudden, it's a game. There's a challenge.  No rulers or measuring devices allowed. They must freehand the fraction. As they're working on their BF (Best Fraction), I announce, "I'm looking for two volunteers who think they've drawn the best half."

Immediately hands shoot up. Kids start to yell at me, "Pick me Mr. Stadel!"

Everyone feels so confident about drawing a half. I yell over them, "I'll put it up on the document camera and you all can be the judges today."

"Oh! Pick mine."
"Me! Me! Mr. Stadel, pick me!"

I pick two students and they walk their papers up to the camera. The class is critiquing each classmate's halves. Some students, from their desks, quickly blurting their half is better. I then display mine (having used a ruler) just so they could see an ideal half. This is where I wish I took pictures of student work. We put it to a vote. I noticed some kids just voted because their friend is up there or they might not like one kid, or whatever. Therefore, in hindsight, I should have specified that they vote for the fraction who's line looks closest to the middle and splits the rectangle into two equal parts. Even more hindsight, I should have asked students what they think would constitute a half. Next time!

Repeat the competition with a best third and a best fourth:

We compared three student papers for the third and four student papers for the fourth. After our competition, we transitioned into the visual representation of multiplying a half by a fourth.

At first, this visual seemed a little confusing to some students, but we walked through it together. When circulating the room to check for understanding, I could see better understanding on the second example in which they completed on their own.

After all of this, here are a few favorite moments:

1. When moving onto the best thirds, I heard a student say, "I don't get this." They must have looked at their neighbor's paper. Within seconds they responded with, "Oh, I get it."
2. When showing students my half, I accidentally showed them the rest of my paper with the ideal thirds and fourths. They responded, "Ohhhhhh!" Like they just saw the answer key or it ruined the fun for them to see the answers. That was funny.
3. One student's paper was not picked, and he turns to a friend and I overhear him say, "He should have picked my thirds. Look at that sexy fraction." I stopped class and he thought he was in trouble. Nope. We had a good laugh as I shared with the class that I've never, in my ten years of teaching, heard a fraction described as sexy.
4. When doing the visual representation of multiplying, I said, "Add this to your notes." A student replies, "Oh, we're taking notes?" He was participating and didn't even know he was learning.

Here's the handout, which also includes an example for dividing fractions.

This lesson idea spawned from my Best Halves idea.

336

## Friday, August 23, 2013

### NEW JOB!!! and some fraction ideas

I recently accepted a new teaching position with a middle school where I'll be teaching 6th and 7th grade math. I was fortunate to be at my last school for about 10 years exploring 7th and 8th grade math: Pre-Algebra, Algebra 1, Algebra 1A, Algebra Honors, and Geometry. I'm extremely grateful for the opportunities, experiences, friendships, and professional growth opportunities the school afforded me. As I advance in my teaching career, I'm very excited about my new position, new school, new students, and new everything. There are many differences between my previous school and my future school... and I welcome them wholeheartedly.

As my future school transitions to Common Core, I'm giddy at the thought of exploring so many wonderful concepts in 6th and 7th grade math. However, I will be working with students that have typically struggled when it comes to understanding math. Therefore, I had a few ideas about fractions I thought I'd like to explore with you.

I'll include all the visuals here, but feel free to go to my "fractions test page" at Estimation 180 to get the full experience. Please offer me some feedback. I'd like to pursue these "fraction" ideas with other items; some easier, some more difficult. Is this something you could use? Is this something worth pursuing?

Question: Where would the cylinder be one-third full?
(Image 1)

We're estimating here. I did not provide any choices because I want students to formulate ideas on their own. Look at their screen and move their finger up and down the screen to find one-third. Come up to the board at the front of the classroom and put a post-it note on the board.

Offer some choices: When ready, click on the image for choices.

Notice I said, "when ready"? Did you have students discuss? point with their fingers? place a few sticky notes on the screen at the front of the room? or something else to get students invested? Because of the restrictions at Estimation 180, this image will currently serve as the next viable step. Now students have a choice. I'm not the biggest fan of this, but it's something. Were there students who were way off because their sticky or initial guess didn't even fall within the given range?

Make a choice and demand reasoning: Why did a student choose "C" instead of "D"? Have students try and convince each other. Argue! Egg them on a little bit. Have students choose a line in which they think the cylinder will reach one-third its capacity.

Do some math? I provide you with the capacity of the glass: 1,170 milliliters. Find one-third of that. Encourage different strategies in your class. Doing the math won't tell students if the answer is choice A, B, C, D, or E, but it might help with later parts of this activity.

Reveal the answer: a really short video.

I have additional video for two-thirds, fourths, and a full cylinder (when using thirds or fourths). I haven't inserted the choices, added a counter, or other after effects. Would this be something you'd be interested in? Please let me know.

Two things:

1. I also set this up as Red Dot (Active Prompt) activity and it'd be fun to see how students would approach this activity without multiple choice. Then, show the class their results before watching the answer (video).
2. I'd love to see Dave Major make a slider so students could slide a bar up and down the cylinder. Using a computer or tablet, students could place the bar where they want and without a given range of choices. Then we could see who was actually correct.

What feedback do you have for me? Again, is this something you could use? Should I prepare more at Estimation 180? Would you like to see the remaining fractions and other ideas?

NEW,
723

## Wednesday, August 14, 2013

### Collaboration: Virtual vs. Face-to-Face

First, I don't like the "vs." in the title, but decided to leave it. Don't think of it as a competition or that one form of collaboration is superior to the other. After reading this post, you can interpret the "vs." however you like.  I'm going to do my best to keep this post short even though I thought I could fill it with many links.

One thing I know is this:
My face-to-face collaboration has improved as a result of my virtual collaboration.
Monday, I met up with Fawn (@fawnpnguyen) in Los Angeles to prepare our CMC session for when we present both in November (CMC-South) and December (CMC-North).

Tuesday, I met with the other math teachers at my school to plan out our year as we transition to Common Core State Standards. I wish I could share their twitter handles and/or blog addresses, but those don't exist. Hopefully, they one day will.

Tuesday evening, I presented at Global Math Department. I was grateful to test out a Back to School Night presentation a la Ignite style with Jessica (@algebrainiac) and Amy (@zimmerdiamonds).

Wednesday (today), I have a chance to reflect.

Meeting with Fawn is always a blast. It's both fun, productive, and interlaced with our typical banter and joke-slinging at each other. We usually collaborate via email. However, we both know there are so many virtual ways to communicate and collaborate on anything math-related. We both cherish the online math community as a professional learning network and have greatly benefited from it. But in person. Let me repeat that, IN PERSON! (face-to-face), I feel so much more can be accomplished because of the immediacy and tangible elements a virtual collaboration can lack.

Meeting up with my colleagues at school on Tuesday was great too. Our 7th grade teacher and I went off on a tangent during the afternoon as we talked about the Estimating Celebrity Age activity. We had a blast as we tried to decide a winner if you made this activity an in-class competition. We came up with about four different ways by hashing things out, giving counterexamples, and coming up with strong arguments. Two math teachers totally in the thick of Mathematical Practice 3: Construct Viable Arguments and Critique the Reasoning of Others. It left me thinking that teachers need to take these 8 Math Practices to heart, not just in the classroom, not just with students, but when collaborating with other teachers, virtually or face-to-face.

Global Math Department could not have been anymore beneficial. The online math community showered us presenters with constructive feedback and suggestions, even some jokes. This virtual collaboration was exciting too. We did our presentations and people were honest about the appearance, content, delivery, layout, format, timing, etc. With such a large group of people attending, the Global Math Department has a solid format and structure to allow the presenters to say their piece and receive viable feedback from a respectful community.

Again, this is not a competition. These two types of collaboration are so important and can truly benefit each other. I can safely say that collaborating with others virtually, through this online math community, has helped me improve my face-to-face (in person) collaboration. I'm excited for the school year to start so I can utilize both, once again. Summer has been great, but I miss that face-to-face interaction.

• This year, collaborate like crazy with my school colleagues. Go beyond anything I've done in the past.
• Invite others at my school to practice our BtSN presentations to each other ahead of time. We could help each other by providing constructive feedback.
• Continue collaborating formally/informally with all of you in this digital-virtual medium known as The Internet, and our online math community.
Collaborate,
357

## Sunday, August 11, 2013

### Global Math Department: Back to School Night; Ignite!

This Tuesday, August 13, 2013, stop by Global Math Department to find a few teachers testing out an Ignite presentation for their Back to School Night. Read more about the initial Ignite idea.

I received a wealth of feedback from many of you and I appreciate the honesty. I know you have my back! The Back to School Ignite idea might end up being a great way to deliver information to parents on that night, or it might end up being a complete catastrophe. That said, I'm extremely grateful for Chris Robinson to offer us a slot this Tuesday so we can give our presentations a trial run. The interested (brave) teachers are:
Each teacher will present for 5 minutes, using 20 slides at one slide every 15 seconds. We'll be open to constructive feedback, opinions, comments, suggestions, questions, jokes, and more. Your honesty and input will help improve our Back to School Night presentations.
Hope you can make it!
9 p.m. EST
6 p.m. PST

A sneak peek?

Ignite,
1014

## Monday, August 5, 2013

### [Makeover] Low Arching Bridge: The Makeover

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.

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 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?

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