Won’t be long before I have to change the post title to Dr. Meyer’s Taco Cart. This lesson went really well today.

**Act 1**: We watched the video clip. Their guesses:

Me: That was fun. Kinda split in the middle there with your guesses. But that question of who gets there first only gave you two choices, Dan or Ben. What other fun questions could we ask?

Student 1: The length of the road.

Me: I did say “fun.”

Student 2: What their walking rates are?

Me: That’sfunnerthan “length of the road”?

Nathan: What is the fastest route?

Me: What do you mean by that? Can you come up and show the class what you meant?

Nathan came up and traced out the blue path with his finger, “They can walk to the cart like this…”

Me: Oh, okay, what is the fastest route? Yeah, right, because what if neither of the guys’ routes was the fastest? This will give us a bunch of different guesses. Or, a similar question, where on the road should Dan or Ben enter to reach the taco in the shortest amount of time?

I gave each student this working placemat (Dan’s Act 2 slide) for work space and the road strips to throw down a guess. They needed to line up road-on-paper perfectly with road-on-strip before marking their guess so that both marks indicated the same position.

Their guesses:

**Act 2**: The questions began.

Me: Okay, so now figure out how much time it takes to walk the route that YOU chose.

Student: Can we have the dimensions?

Me: Which one?

Student: All of them.

Me: Wrong answer. Try again.

They yelled out for the legs, the sides, the road, the hypotenuse.

Me:I’m just gonna give you one of the sides. Just one. Ask wisely.

For whatever reason, they agreed it should be the hypotenuse. (Dan did not give the length of the hypotenuse on his slide. I purposely put a white box there pretending like maybe they could ask for this length too.)

I gave them the hypotenuse as 650.0 feet. They stared back at me, faces scrunched up as if they were begging. They knew they needed another length to use their trusty little equation of a^2 +b^2 = c^2. I also gave them the walking rates on sand and on road.

I walked around the room, peeking to see what they were doing on their papers. One student plugged the rates into the equation. He wrote this:

The 2 and 5 in the bottom equation came from the walking rates. The 105625 seemed to have come from multiplying the distance (650 feet) by the time to walk that distance on sand (325 seconds), then dividing this number by 2.

Half of them were just quiet — daydreaming, thinking.

Almost 10 minutes had passed. I said, “I gave you the hypotenuse, correct? But before I gave you the hypotenuse, don’t you have the hypotenuse on your paper?”

Five seconds went by, then…

Gabe: Proportions!

Lauren: O my God, I hate that when I think too hard!

Nathan: I thought about measuring!

Janelly: Me too! But I was afraid to.

Maia: Do you measure with a ruler?

Janelly: No. Measure with your toe.

Ha!! *Measure with your toe!!* How can I not love these kids. So, they got busy with their rulers.

Maia: Should we measure in inches or in centimeters?

Someone: Centimeters! We never measure in inches in here.

Me: Not never. But why might we want to measure in centimeters for this problem?

Someone else: It’ll be more accurate.

They measured carefully and checked each other’s proportions. And because I only gave them the hypotenuse, their calculations for the legs weren’t all the same, but close enough. We respected the margin of error when using a ruler.

It was another day that I didn’t want this class to end. But with only five minutes left, I had to wrap up the lesson.

Me: What’s the chance that your guess happened to be the fastest route?

Various students: Not likely. Way off. I think mine is perfect — I picked exactly half-way on the road. More to the left, I think.

Me: So, after you figure out the time for your route tonight, I want you to pick another point on the road. Do two sets of calculations.

Nicole: Like do it again for a different location?

Me: Yes. And if you chose your second point to be on the left of your original guess, then why? Or to the right, why?

Thomas: Will you tell us who got closest tomorrow?

Me: Sure. I have your original guesses already [on the strips]. And guess what? The student with the closest answer gets a Del Taco lunch!

Students: Really? What about Taco Bell? I love tacos! I hope mine is closest! Are we taking a field trip there?

Me: Fine. Taco Bell or Del Taco, your choice.

**Act 3**: We’ll have to wait until tomorrow to see everyone’s work and answers for their chosen paths. I’ll reveal this again with the green line drawn in to indicate the best spot to enter the road. But it would be so cool to get some numbers right near the minimum time. Maybe I’ll bring up differentiation. Maybe we’ll plug something into Wolfram Alpha.

What made this lesson work for my kids:

- Nathan’s question of “what is the fastest route?” allowed for a lot of entry points than just “who would get there first, Dan or Ben?”
- Giving them just one side of the triangle was really the best thing I did for this group of kids. Because once they figured out the other legs, they just tapped away on their calculators — already comfortable with the equations they’ve worked with before.
- The strips provided me with an easy way to display student guesses.
- Printing out Dan’s Act-2 slide as work placemats for students allowed uniform access to the beach scene and made measuring the triangle sides possible.
- I only thought to ask the kids to pick another point to figure out for homework just seconds before. Glad I did because we just doubled our data points!
- The promise of Del Taco. No, I can’t just give to the kid with the closest answer. I’ll make sure I bring in enough for the class. This fun lesson deserves a fun closing.

Thank you, Dan.

[Updated 11/07/12]

## One Comment

I really love what you did with this lesson. You took what I would consider an extension problem (i.e., finding the fastest route) and really opened it up by having your students create data for various guesses. I’d be intrigued to see the plotted data (e.g., time elapsed for trip with respect to position of mark on strip) and possibly generate a fit curve from that to estimate the position of the mark on the strip that minimizes time elapsed. (Does that make sense?). :)

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