I am learning more and more about how this whole problem based learning shtick works, and I have to say I love it.  Today was an example of how much I love it, and how worthwhile it is.  I have been struggling to motivate the use of the Pythagorean Theorem in a natural way so that students can jump in.  Enter Mr. Piccini's Summer Purchase...

I started off the "story telling," by discussing how it was time for my wife and I to upgrade our T.V. I showed them the picture and asked them: "What problems do you think I may run into?"  I realised after that this was a little too much of the "Guess what is in the teacher's head" but the question was so natural they needed to ask "Will the TV fit under the shelf?"  We talked about what it meant for a TV to be 40".  Almost half the class knew that it meant the diagonal.  I asked what do we notice about the shape of a TV is when cut down the diagonal.  Clearly there were two triangles, and thus began our exploration.

For the next part I stole the great investigation from Dan Meyer found here.  This worked so well it was unbelievable. Students were very quick to see the relationship between the sum of the small areas and the large area.  Students enjoyed the manipulative nature of this exploration, and the result was clear and accessible to the students.

We did some practice and then came back to my T.V. the next day.

We started the next day with whiteboards, and I had students write what information we knew and what information we needed to know.  I reinforced to students that I did not want to renovate my wall because I am lazy, but I am not too lazy to move my speakers.  They said then that we needed to know the height to the shelf. So I gave them this...

Depending on how hard you want to make this problem you can do either cm or inches.  Inches is the easiest.  Since students recognized that we need information on the diagonals, we need to talk about what a good diagonal measurement would look like for this space.  For that I broke out geogebra, to show students how sizing TV's looks.  I used these two geogebra apps.

We talked about what the different shapes of TVs look like, and what your average wide screen would look like.  Using the geogebra animations the students discussed how the proportions always stay the same.  At which point we could talk about ratios.

At this point students needed guidance on how to find the dimensions of the space, but it came quite naturally to talk about the largest possible TV that I could fit in my living room.  Students found that this was a 55" TV, and as it turns out...

If you don't believe in collaborating online via blogging etc. let this section be a testament to this.  Neither of these extensions were my own thoughts, they were completely lifted from others, awesome!

• From Karim in the comments there is also this: "What is the largest standard TV (4:3) that could fit in this space?

Online collaboration for the win!

This was such a simple execution, but it gave the students a real challenge, and a real-world application to try from the get-go.  I had a debrief with my students in one class, and they said they loved how visual, real, and engaging it was.  I never expected to hear that.  The only complaint was from students that finished really quickly.  I was not ready with sequels, and in fact, I still can't think of any. I never thought that this specific problem would have as much engagement as it did, but once students got rolling they were hooked. All in all I was happy today, and I sure beat the socks off this version of the problem.