The making of a 3 Act Math : La pelle mécanique

I have been trying to come up with an engaging and authentic task for a couple years now involving the ambiguous case of the law of sines.  I have a bunch of ideas and knew I needed help from experts.  So, I took a chance and sent a message to Dan Meyer.  I know, the man must be busy, but he answered my message and agreed to help.

I sent him the following picture from a textbook and explained to him that I think if we showed a video of a backhoe digging a trench but stopped the video and asked the students what they noticed and what they wondered, we could get a good 3 act task going involving the ambiguous case.  In my mind, I see the students asking :

How long is the trench?  How deep is the trench?  How far is the shovel when it starts digging?  What is the maximum height of the shovel?

Then, we can come in with some measurements in Act 2 and let the students go to work.

Before asking Dan Meyer to spend time helping me, I figured I would test it out with my class.  Luckily, our school is having a garden built and the man building it is my sister’s father-in-law.  He happened to be using the backhoe that day and so I made this video :

I sent it to Dan Meyer to see what he thought and I showed it to my students the next day.

They asked the questions I thought they would :

I didn’t have an Act 2 ready at this point – as in exact measurements – so I improvised and used the measurements from the textbook question.  We agreed to neglect one of the joints in the backhoe and assume it was straight.  I plan on adding in that joint when I rework this task.  The students got to work and this is what I loved about it :

Five different groups solved the problem in five different ways!

I feel like at this point, I was onto a great question because of the numerous entry points and different strategies the students would use.

Here are the solutions :

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So, that’s where I am at with this lesson.  Again, I didn’t have an official third act where a video revealed the solution but we are working on that.  I am hoping to continue collaborating with Dan Meyer and have something more concrete soon.