### [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,
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