If The Piece Doesn't Fit, You MUST Acquit!

In math class we have started investigating triangles.  My good friend, running coach, and teaching partner, Erika, suggested to give them a bunch of straws, some vocabulary, and let them go off and running.

Who am I to say no to such a wonderful structure?

We reviewed some geometric terms: Acute, right, obtuse, scalene, isosceles, and equilateral.  Then we gave them a challenge: create a triangle with each of the vocabulary terms: one for sides and one for angles.  They realized that they had to create nine total triangles.

Students got started by building an equilateral acute triangle.  It was a solid beginning with students easily conquering that task:

Many thought they'd finish all of them inside of 10 minutes... then they tried the next one.

Feeling confident, they went on to another triangle on their list: equilateral right.  Students used the straws and tried to build it, but no matter how they arranged the straws they just couldn't manage:

If the piece doesn't fit, you must acquit!

Students became frustrated and annoyed. There was some fantastic and frank mathematical discussion amongst themselves. They discussed lots of options.  Some said it was impossible, others argued that can't be the case, but then had second thoughts... Is it possible?  After conferring and discussion, they decided such a shape was not possible to build because "there will always be a little piece of triangle missing on one side, and that side will always be longer."

Why do these teachers constantly try to trick us?

Students continued on to build the other triangles, obtuse scalene, isosceles right, but then  got stumped with 'obtuse equilateral'.  The students went back to their previous thoughts and ideas that were built from 'right equilateral' and concluded that such a triangle could not be built.  Students also made amazing observations:

"...when I built an isosceles triangle, it looks like there are two angles that are always the same too.... So maybe when sides are the same length the angle is the same degrees?"

Those two acute angles look eerily similar

We followed this lesson up with one that used protractors.  Students have begun to confirm similar thoughts and hypotheses, as well as showed that the angles of triangles "always seem to add up to about 180 degrees."

I loved all of the discussion and discovery this lesson gave the students.  Erika and I facilitated discussion, but we never clued them into the 'impossibility' of building a right equilateral triangle.  They came to this conclusion on their own and were successfully able to argue (in a middle school way) why it wasn't possible to build such a shape.

I'm excited to see how they apply this knowledge to the rest of our geometry unit!

Collaboration for the win!