Similarity is one of my favorite units in Geometry.  I love it because it provides a fun opportunity to get outside and use our knowledge of Geometry to do the seemingly impossible.   Let me ask you this:  Could you tell me the height of this tree using just a mirror and a yard stick?

Guess what?  My students can and did!

It turns out that if two pairs of angles of from two different triangles are congruent, then the triangles are similar.  And similar triangles have proportional sides.  Combine that with our knowledge of how mirrors work, that your ingoing line of sight bounces off the mirror at the same angle, gave us this sketch as we planned in the classroom:

We then headed out in groups to measure various tall objects around campus.  Some great number sense happened as the groups measured and calculated.  “Wait a minute, that tree can’t be 3 ft.  That doesn’t make sense,” I heard one student say.  Upon looking at their paper they had measured their height in inches but other distances in yards.  A quick fix to make sure everything was consistent fixed the problem.

In hindsight, I wish I hadn’t dictated what objects they would measure.  I wanted to give them the freedom to choose their own objects and the space to not be on top of each other.  But, it would have been a fun discussion of measurement error to see why our estimates of the same object differed.  Maybe next year!