[Makeover] Low Arching Bridge: The Makeover

Once again, the task:

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

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

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

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

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

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

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

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

2) Now show your students this picture and ask:

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

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

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

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

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

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

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

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

y-axis justified RIGHT:

y-axis justified CENTER of bridge:

y-axis justified LEFT:

Just for fun (the actual bridge):

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

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

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

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

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

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

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

Everyone else's input can be found here:


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

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

Under the bridge,
1230

[Makeover] Low Arching Bridge

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

From McDougall Littell's Algebra 2 textbook (2007)

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

Find me on Twitter.
Email me your makeover.

Makeover,
1017

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

Snail's Pace

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

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

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

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

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

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

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

Slowly,
333

Styrofoam Cups

Tuesday, I was on my way to BTSA and my subconscious screamed something at me. Find Dan Meyer's Stacking Cup lesson. Seriously, go read his post right now. I hadn't read this post for over a year now and I had to get my lessons ready for the next couple of days as my Algebra classes finished up Pixel Pattern. I was on the road dreading the idea of sitting through a couple hours of BTSA, so I asked Dan if he had the link to his lesson since it wasn't in my bookmarks (that was silly of me) and he came through like a champ! Seriously, check out his post. I'm promoting his blog post more than anything further I have to say here.

First, by all means, spend about $10 and do the lesson with your kiddos. This is one of those 3 Act lessons that just screams "hands-on" activity with your kids. It's tough to capture the overall excitement and energy with a video. If you can't do the "hands on" with your kids or you want to be environmentally friendly, here's my version of the Styrofoam Cup 3 Act lesson: a cheap backup.

It felt most natural to stage this so the cups stacked to the top of the door frame. Even then, I'm not convinced my Act 1 screams the question I'm looking for, "How many cups will stack to the top of the door frame?"

Enough about me and the video, to my classroom with the students. Dan's got a great script for you to follow, so do it! One of my classes was actually able to finish writing their rules before the bell on Friday so we had time to actually stack cups. Check out their rules and predictions for stacking cups to my height.

We started stacking with the lowest number and went from there. The kids went bonkers. Each group thought they were the best, but knew that they all couldn't be correct. When we revisit the lesson this next week, we'll be discussing where groups went wrong in order to learn from those mistakes. Watch Styrofoam Cups - Act 3 Stadel to find out who won. But I recommend you watch the door task also.

Styrofoamed out,
813

Bouncy Balls

Today was a good day. Yesterday wasn't and I'll leave it at that (strictly speaking of school). One of those days where I couldn't find a wall fast enough in order to bang my head against it not once, but multiple times. It's a great thing that I get to end my days with my wife and son. My students are having a blast finding Felipe, our classroom Elf on the Shelf, each day. With a waterfall schedule, my last class of the day gets to hide him for the next day. It's a fun little activity for the kids to burn some energy off until the holidays. We did our seasonal estimate today and I started my Algebra classes with this video:

Yes, this video is cruel. Not necessarily perplexing, but enough to hopefully generate some curiosity? So, which ball will bounce higher? Give me a thumbs up if you think the 2012 Super Ball will bounce higher. Give me a thumbs down if you think the 1976 Super Ball will bounce higher. Give me a thumbs middle if you think they will bounce the same. In all three classes, there wasn't a strong majority, but if I had to estimate I think most students voted that 2012 will bounce higher. And... I don't tell them, show them, or even hint to them. I know, cruel. Enter this picture:

This lesson snuck up on me as I was collecting balls. I forgot to get the following balls: ping pong ball, racquetball, tennis ball, and one of those pink spongy balls. Okay, let's get this out of our system; middle schoolers and the word, "balls." So here we go, "balls, balls, balls, balls, balls, balls, balls, balls."

"Now, look at the balls and quietly, to yourself, make a guess. Guess which ball will be the best. In other words, which will bounce the highest? Now, guess which will be the worst? Don't say anything. Write that in the top corner of the handout you are about to receive. Don't share it with anyone." Students were looking at the screen, scoping out the different sizes, shapes, and textures of each ball. I saw some students writing the golf ball as the worst and the lacrosse ball as the best. Some were putting the Super Balls as the best. "Now, share your guesses with your group. Does anyone want more information about these balls besides just a picture." Trust me, it was very difficult not to work "balls" into the conversation as much as possible. Seriously, it can be fun to see them squirm, grin or laugh at times like these. I refrained from making comments like, "Don't worry guys, you'll get your hands on these balls soon enough." or "We're not playing with the balls people. Simply dropping them and seeing how high they bounce." C'mon people, "Make sure you handle the balls with care." You get the point.

I'm a slow learner. You've probably heard me say this before. This might be one of the best parts of my day. When introducing projects this year, I've made the mistake of displaying the handout on the screen first, having students read parts out loud, and throw in some pointers before they get their supplies. You can predict what happens next. Students get their supplies and start exploring the project in the wrong way or ask me questions to parts I already reviewed. What's the typical response? "Read the handout again." or "I already went over that. Ask a classmate." And you know with each student that you see not following directions or that comes up to you and asks what to do next simply gets more irritating with every time. For example, when we stole from Fawn Nguyen's Barbie Bungee, I'd see students simply letting Barbie hang freely from the top of their meter stick and measure that distance with every rubber band they added. They weren't dropping Barbie and measuring the lowest point she extended to. So it dawned on me, once again because I'm slow. "Everybody, you are to read the entire handout with your group first. When you've done that, come up to me and explain what you are doing. If you accurately tell me in your own words the objective and directions of the project, you may grab a ball and start collecting data. If you can't, I send you back to read it again." Money! It worked like a charm. Students knew what they were doing. They knew the correct steps. They knew what increments and how many drops per ball. It was great. I still have to do the Barbie Bungee project with my Algebra Honors class and will see how well they do with reading the directions.

Here's what students did today. Students were to use one ball at a time to drop the ball from 10 cm to 1 meter using 10 cm increments and at least three drops from each increment. Once they completed that, they were to exchange their ball for another ball and do this for a total of three balls. Hint: no one was allowed to use the 1976 Super Ball until they collected data for two balls first. Plus, I don't let the 1976 Super Ball out of my site. I've had that since I was a kid and do you know what those guys go for on eBay? They kept track of the rebound heights and were to make observations. Were there any constant changes? If not, was there a close average change? Our goal is to predict the rebound height of a drop from 3 meters and from our balcony of 5.7 meters.

At least I didn't play this video to intro the lesson.

Balls,
1026