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:
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).
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?
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.
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.
What lies ahead?
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.
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
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."
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.
