Showing posts with label Twitter. Show all posts
Showing posts with label Twitter. Show all posts

Video Error Analysis (Anti-Khan style)

Something I tweeted this week:
Crystal (colleague) and Lynda (fellow) wanted to know more about this. So here's the story:

The previous week, I met with one of my high school fellows who teaches Algebra to freshman. As with all my fellows, it's been an extreme pleasure to work with her because she's hungry for ideas and will take suggestions and run with them. It was so cool to walk into her class this past week and see her running with an idea, again.

She had already taught her students ways to solve linear systems; graphically, substitution, elimination, etc. On this day, she prepared six short videos of her solving linear systems and linear inequalities using Educreations on her iPad. Students were to watch the videos and do error analysis, reporting the following on their handout:
  1. Identify the mistake(s) for each question.
  2. Explain what should have been done.
  3. Fix the mistake and complete the question correctly.
Each video was between 60 and 90 seconds in length. We both discussed what we thought would be most effective for her students and short videos was a must. Have you ever noticed how the majority of Khan videos can be extremely lengthy? Sal Khan usually talks (and repeats himself) while writing things on his digital blackboard. To me, that's a waste of someone's time. It's like you watching me type this blog post while I reread every sentence two or three times, stalling so I can finish typing. Another thing I can't stomach in Khan videos is when he fumbles around searching for colors to write with. Lastly, I find it unfortunate that the videos rarely suggest the viewer to pause and consider what's happening. Here's an example. Sorry, here's a 9+ minute example:

I suggested my fellow pause the recordings often and write the equations "offscreen" when not recording. Then, press record again when she's ready to talk and/or write something important on her screen. She also took advantage of this offscreen time to select different colors in order to emphasize different equations, steps, lines, or shading (linear inequalities).
*See the video structure below with suggested notes and style points.

It took my fellow one prep period on a minimum day to create six videos, a supplemental handout, upload the videos to Educreations, and create hyperlinks on her Haiku page for students to access all the videos. That's super impressive. Talk about an activity with meaningful and HUGE return from an efficient investment in her prep time.

When debriefing with my fellow after class, she was completely ecstatic.
I asked her, "What elements made this awesome?"
She replied:
  • it was video and new
  • they liked figuring out someone else's mistake
  • the videos were short
  • students could pause, rewind, and start the video over
  • using Desmos to show a graph of the original equations at the end (comparison)
  • gave students the idea to use Desmos to check their work/answer
  • self-pacing
  • very little hand-raising or students drowning
  • the videos were easy to make
  • she passed out the handout and said "go" instead of modeling
  • the handout had a simple structure 
  • the students did most of work, not the teacher
I love this last element the most. The two of us talked about this specific element the previous week. Now she experienced it first-hand and it's an amazing feeling. As an observer, it was awesome to see the students working hard on a meaningful task and helping each other out so it allows her, the teacher, opportunities to calmly circulate and provide support where necessary.

Student engagement and interest were high. Discussions were plenty and authentic. Students were thriving using thinking skills in the "Analyzing" category of Bloom's Taxonomy or Strategic Thinking category of Webb's Depth of Knowledge. Here's a tip I suggested when I noticed some kids plowing through a video and hadn't caught the mistake: pause and make predictions. The video structure will explain pausing and predicting more.

Video Structure:
Part 1: She takes about 8 seconds to explain her plan
*All of this was written on the screen prior to her pressing record. Style points.

Part 2: Multiply the top equation by (-5) in order to eliminate the x-terms

*Here's where we need to ask students to pause and predict what the top equation will look like after being multiplied by (-5).
  • Model this for students.
  • Build "pause and predict" prompts into the video. 
  • Circulate the room and ask students to pause and predict.
SO valuable. Don't skip "pause and predict".


Part 3: Write the new equations "offscreen". Don't record yourself writing these equations.
*Notice the new equation is written in red ink. Style points!
**Pause and predict what it will look like when combining the equations
***Catch the mistake?

Part 4: Combine the two equations.
*Another great use of "offscreen" writing.

Part 5: Find the value of y.

Part 6: Substitute the value of y into one of the original equations.
*Yet, another use of "offscreen" writing here.

Part 7: Solve for x this time.
*Ask your students to check for reasonableness.
**Find an alternate way to validate (or invalidate) their conclusion.

Part 8: Insert a screenshot of the system graphed in Desmos.
*Mind grenade: the graph doesn't match the algebraic procedure.
**HUGE style points by inserting a visual representation of the correct answer.

For those of you who don't have 1:1 devices in your schools, no sweat. I still recommend you make a video of some sort. Borrow an iPad from someone. Create an Educreations video for error analysis. Use the tips and techniques mentioned here. Your videos should be less than 90 seconds. Play it to your class. Pause the video to have students make predictions and/or discuss possible errors. I guarantee you, good things will happen.

Style points,
1209


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#PuzzleMath ideas

Tonight, my son wanted me to work with him on his new puzzle.

I don't know your strategy for doing puzzles, but I find all the corners first and then start putting the border together before I start working on the inside. Look at that box again. Would you be able to determine the dimensions (in puzzle pieces) of this puzzle by knowing the total number of pieces?

That was my first question:
A) If you know the total number of puzzle pieces, could you think of the all the possible dimensions (in puzzle pieces) of the puzzle?

Answer:
This puzzle will either be a 1x35 or 5x7.

Then came the next question:
B) Estimate the actual dimensions (in puzzle pieces) given the picture on the box?

Answer:
I'm going to go with 5x7 because five and seven are the only factors of 35 that would reasonably make the rectangular picture on the front of the box. The puzzle should be 5 pieces high and 7 pieces across from left to right.

With a box of 35 pieces, these questions aren't too ridiculously challenging. However, I know there are crazier puzzles out there in the world; puzzle with 500, 750, 1000 pieces, etc. That's where I called on Twitter to help out. Like a champ, the #MTBoS came through and hopefully will continue to come through with #puzzlemath.

Below are some of the tweets I received followed by additional math questions I'm curious about.






Additional Questions:
C) If you think you know the dimensions, could you determine:
  • The number of corner pieces
  • The number of border (non-corner) pieces
  • The number of inside pieces
If so, could you write a rule for any of these?

D) Knowing these quantities, say you randomly choose a puzzle piece, what are the chances it's:
  • A corner piece
  • A border (non-corner piece)
  • An inside piece
  • The exact center (if possible) piece
Please add to the collection of puzzles and questions by tweeting with the hashtag #puzzlemath.

Puzzlemath,
1050
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Stretch

I am reluctantly pressing "publish" for this post. However, please know that the rawness and honesty in this post is aimed at making each and every one of us better at what we (both individually and collectively) do to support our students.

Kate wrote a great post the other day about some teachers coming out of TMC14 feeling inadequate. I support Kate's attitude and conclusion:
We are all good at some things and suck at other things. One thing we all share is the recognition that we all have work to do, and that we can all get better, and that focusing on that is worth our time.
I believe we need inadequacies and need to feel them at times because they make us better at what we're striving to be: the best teacher for our students. We don't need inadequacies to feel inferior to other teachers or generate some type of MTBoS worth. Here's how I think we all need inadequacies.

I started learning and exploring how to play the guitar in high school. I stunk. My family probably got tired of me playing Deep Purple's Smoke on the Water and Metallica's Enter Sandman all day. The first two riffs (and eventually songs) I learned. However, I practiced. A LOT. When I wasn't playing basketball, I practiced guitar. When I was supposed to be doing homework, I practiced guitar. There were guys at my high school who played guitar and were better. It made me practice more. It made me want to be better at something I loved doing.

When I got into college, my focus on guitar playing was similar. I was a lot better by this time, but still practiced a lot. When I wasn't working or going to class, I practiced guitar. When I was supposed to be studying, I practiced guitar. Then I joined a few bands and we practiced a lot. Not only did I continue practicing by myself, but now I practiced with others. That's awesome. We got better together! I would also jam with other guitarists who were better than me. Sometimes they were better so I learned a lot. Sometimes, I was better so I got to share some things and could relate. Every time I jammed with someone, it was a chance for me to improve at something I loved doing.

I loved going to concerts or watching videos of my favorite guitarists like Jimi Hendrix, Eric Clapton, or Warren Haynes. I wanted to cut my hands off many times because there was no way I would ever be as good as them. However, it only made me want to learn from them, steal some of their licks (guitar moves/techniques), and be the best I could be with their help and inspiration. I remember meeting James Hetfield from Metallica and was star struck. I thanked him for his inspiration. That's all I could muster up the intelligence to say. If I could jam with him I'd probably mess up A TON! But I'd never turn that opportunity down, because I'd learn a lot and he'd push me to get better.

Once during college, I was in Chicago at Kingston Mines blues bar hanging with my cousin. This blues/funk band, Charlie Love, was up there laying down some great songs. I went up at their break to compliment them and they invited me up to do a funk jam with them. I was completely honored and humbled at the same time. Here is this tall white guy trying to play funk with the Chicago blues/funk band and I did not play as well as I could have. However, I was grateful to meet them, I learned a lot from watching them, and it again made me want to go home and practice until my hands fell off.

For me, this connects so well with where I am as a math teacher. I am grateful for many other math teachers who have inspired me. There are many times I feel inadequate. Maybe I've met some of these math teachers and I feel like my brain shuts down. The best I can utter is some number and ignorantly nod my head in agreement. However, meeting inspiration and hanging out with inspiration has made me want to become a better teacher for my students.

Imagine there was an opening at your school and you could hire your teaching colleague. Would you turn down the chance to work alongside:
This is a snapshot of the many teachers who have inspired me and continue to raise the bar for me. I wouldn't turn them down because I might have some inadequacies. They would make me a better teacher for my students. Imagine if you were the teacher after receiving students from any teacher who inspires you? Imagine if you're the teacher before sending your students to a teacher who inspires you? Wouldn't you want to be the best teacher for your students? Isn't that healthy? Who wins? I would hope your students. 

I recently offered a keep-your-head-up comment somewhere saying,
"Think of the skills you will acquire when making changes."
My challenge to you (and myself), if you're feeling inadequate or inferior at any time is to:
  • take risks
  • be brave
  • tap into your influences and inspirations to stretch yourself
  • be the best teacher for YOUR students, not the MTBoS.
Stretch,
210
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Little Mr. Sunshine

I'm honored to receive some rays from the sunshine of Shauna Hedgepeth and John Stevens. Like Christopher Danielson, I don't forward those silly emails of inspirational PowerPoints, cats in meadows, Bigfoot sightings, and other pyramid emails. However, I am compelled to do my part (or some of it) and respond to the two math tweeps who threw sunshine my way. If you're not familiar with sunshine, click on any of the three links above and read their posts so I don't have to retype it.

Act 1: Acknowledge the nominating blogger(s)
Hedge is totally awesome! She loves her students. Don't even argue that. Email her some thoughts, ideas, and/or jokes and her replies have a sincere, lovable vibe to them. She makes me laugh with all her A.D.D. talk, but I'm not sure how true that is. Wait, yes, I do. Just read one of her blog posts. They're awesomely funny. Where she might be intimidated by my height, I think I would be equally intimidated (if not more) by her love and knowledge of stats. Stats is definitely one of my shortcomings as a math person. I look forward to meeting her at NCSM in April 2014.

John Stevens reminds me of a professional juggler. Not necessarily a clown juggler. However, recently he did dress up in a chef's outfit during a presentation. This guy juggles and contributes to many cool parts of math education.  Check out his blog for iPad resources, his CUE Rockstar status, presentations, #CAedchat and his up and coming Would You Rather site. When you meet him in person, he's a very personable and interested guy. Sit down and have a beer with him. Keep rocking John!

Act 2: Eleven Facts
1) I've never broken a bone (knock on wood). Never.
2) I don't drink coffee. However, I love iced tea.
3) I'm a philosophy major.
4) I've only used my passport once and that was to Canada. I hope to travel to other countries in the future.
5) I enjoy doing yard work.
6) I can't remember the last time I've slept passed 6:30 a.m.
7) My iTunes library would take 36 days, 22 hours, 55 minutes, and 5 seconds to play from start to finish.
8) I'm actually terrible at estimation (just kidding).
9) My In-N-Out order is:
1 Protein-style Cheeseburger with grilled onions
1 Animal-style Double-Double
1 light-well french fries
1water cup (with water in it)
10) I love/have the board game Othello, but I have no one to play against.
11) Newcastle and Rogue

Act 3: Answer questions from the people in Act 1
Hedge's Q's:
1) If you had to pick one area/concept of math that is your "jam", what would it be?
Spatial reasoning.

2) True, but there's at least one student that sticks out in my mind that I feel I failed. Do you have one?
Yes. Actually, many students from my 2012-2013 school year. See this recent post for a visual. I went overboard with too many questions, not enough validation and too little direct instruction. Avoid this cocktail.

3) Twenty years from now, what's something kids will probably remember about you (phrase, moment, habit, characteristic, etc.)?
That guy sure took a lot of pictures standing in front of stuff so we could estimate its height. He worked hard and expected us to work hard.

4) What's something that you'd like to "fix" about yourself in your current job?
I'd like to learn more Spanish.

5) Name a movie title that describes you and why.
Tommy Boy. I don't need to explain.

6) Which tweep would you love to have a conversation with over a beverage?
My top 3 tweeps I haven't done this with are Hedge, Christopher Danielson, and Steve Leinwand. By the way, just one beverage? What kind of nonsense is that? I'm excited as it looks like I'll meet Hedge and Triangleman at NCSM this year! WOOOOOHOOOOO!

7) If you couldn't teach your specific subject, what else would you teach?
Home economics and culinary. I'd learn how to be a better cook and I could eat.

8) Everybody has a song they can dance/jam out to. What's yours?
Venice Queen by the Red Hot Chili Peppers.

9) Who would you love to see karaoke at TMC14 and why?
Steve Leinwand. First he, wouldn't need a microphone. Second, he can probably rap faster than any rap artist, alive or dead. Third, what jam would he pick? Definitely NOT a ballad.

10) What's one thing (item, app, software, etc.) that you love so much that you can't imagine doing your job without it?
Apple's Keynote or Pages. Don't even ask me to use PowerPoint.

11) If you could job shadow one tweep for a week, who would it be and why?
Michael Pershan. Having met Michael and been intimidated by his fountain of intellectual curiosity, I'd love to see how he facilitates a classroom. Runner-up: Fawn Nguyen. Maybe I could steal her Brad Fulton books when she's not looking.

John's Q's:
1) Why do you teach?
Am I supposed to be teaching? I'm still learning and I expect my students to learn with me. This might help explain.

2) If you didn't teach, what would you do for a living instead?
Playing music in crappy bars while making guitars at a guitar factory.

3) Money being no obstacle, where would you like to visit? Why?
Europe. Most of it. See my passport comment above.

4) Kids always ask who your favorite student is.  Describe the characteristics of yours.
My favorite student is the one who perseveres, is open to making mistakes, takes responsibility along with the risks, and can see that math isn't about printed number on a paper.

5) What is your favorite board game and why?
See fact #10 above.

6) What is the most frustrating component of education right now?
Grades.

7) Would you rather buy a Mac or a PC?
See the answer to Hedge's question #10. Mac!
John, you had to sneak in a "Would you rather?"

8) What is your favorite book?
Early John Grisham stuff.

9) If you had to choose blogging with no way to share it (ex. via twitter) or tweeting with no way to elaborate (ex. via a blog), which would you choose?
Blogging.

10) Who is your hero?  Why?
Wow! Too many to name. My grandfather (hero for being a role model of a man). My sister (a family and best friend hero). Dan Meyer (a math hero who saved me from ruining my teaching career).

11) What is the most exciting part about your job?
I get to make mistakes. I get to create my own lessons. I get to explore number sense and logic with non-adults. My job doesn't define who I am.

The Sunshine stops here. Thanks for reading. I'm not torturing 11 other tweeps. It was fun.

Sunshine,
243

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Daily Something [WCYDWT]

#WCYDWT

What can you do with this?

Why would a teacher use this?

How would you use this in your class?

What would you add? subtract? replace? other?

How would you use the information from student performance on this?

How could this be used as a pre-assessment? an assessment? an intervention?

Take a few minutes to complete each day. What do you notice?

Hint:
*  **  ***  ****  *****  ******  *******  ********  *********

Answer as many or as few of these questions... or feel free to add your own.

This:

Thanks in advance!
843


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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
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CMC South 2013

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

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

Friday:
Fawn and I spent the first session reviewing our presentation in the lobby on my laptop. Poor Fawn, I think she was regretting her decision to present with me.
We reached a point where we couldn't think of anything more to review so we walked over to Robert's session at the Hilton. Lucky us, we were able to sit with John Berray, Avery Pickford, Breedeen Murray, Karim, and John Stevens.

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

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

It was such a pleasure to collaborate and present with Fawn. She came up with this great math task and I look forward to presenting with her again at CMC North. She's probably dreading the thought of presenting with me again! Not to spoil anything for our CMC North session, we had attendees use snap cubes to build something that involved income and a building cost. The task required strategy, problem-solving, collaboration, and a lot of calculations. Groups were required to present their calculations, information, and reasoning on their giant whiteboards. How vague was that description? Don't worry, we'll post more after CMC North. In the meantime, here are some pictures from our session:
The man! Robert Kaplinsky!
Avery Pickford: our winner!
Some attendee quotes a la Quotes of the Week style
Thanks to all those who helped enter calculations, pass out things, clean up or kept us on track during our session.

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

  • Get rid of timed tests because "Math should never be associated with speed."
  • Give students more diagnostic feedback and tell them "I'm giving you this feedback because I believe in you."
Keep an eye on her site www.youcubed.org

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

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

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

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

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

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

South,
927
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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
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[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?






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

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[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.
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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


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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:
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

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#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
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Dancing with the Functions

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

He's referring to beating me down while informing the teacher's pet (Fawn) of his tactic.
Our choices for our final project were:
  • a blog post,
  • a lesson plan,
  • an interpretive dance,
  • a work of visual art,
  • etc.
I thought writing a blog post was "too easy" in the sense that I could blog about anything ordinary at anytime. This class wasn't ordinary though, and I felt I'd rather try and give something back to the class, professor, and community in exchange for what I have received. No Fawn, not because I'm "too lazy." Therefore, I'm going to give you a lesson idea I have, based on an interpretive dance, which might be a work of visual art, all wrapped up in a blog post.

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


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

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


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

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


Dance,
933
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