The Centers: Circumcenter, Orthocenter, Incenter and Centroid

My geometry classes recently finished a unit on finding 'the middle' of triangles.  The word middle was thrown out constantly and I regularly had to ask for clarification- what kind of middle?  the middle of what?  In fact, I think I might make that the last question on the test- "Locate the middle of this triangle and defend your answer."  Those should be fun to read (don't worry, I have no intention of tricking my students, anything goes so long as it's justified accurately).  Last year I approached this unit with some incredulity- I have to get them to memorize the definition of circumcenter??  I don't remember ever even seeing this word before!  But we all got through it just fine.  Since then I attended conferences at the Education Development Center in Newton and the Park City Math Institute where we not only used these fancy words, but also talked about just how cool their properties are.

This year I started out much more organized.  I wanted the students to have a clear reference sheet with all the information we learned and discovered in one place, so they started the unit with this chart:

Triangle Segments

It's nothing fancy, but since I introduce the different segments on different days, and then they continue discovering characteristics as we go, it was hugely helpful to be able to say "make sure to add this definition/observation to your chart." Whenever they had a question I could refer them to their chart; this became especially useful as we did the following two activities:

The airport problem is a fairly typical perpendicular bisector/circumcenter problem that I copied from my textbook.  There are two characteristics of this handout that I like which aren't typically included.  First, I ask students to check that the airport they found is actually equidistant from the cities, but not in a way that just says "check your answer."  By asking students to convert their distances to miles we hit on scale factors and map skills, while hiding the question "are you sure that point makes sense?"  The second thing is asking whether that point really makes sense.  I'm hoping for an analysis of highways or whether there's enough open space for an airport there.  Most of the time I'm all for problems without context, but if we're going to use a a context let's really analyze it!

Airport Investigation

Finally, we try making mobiles.  The main goal of this activity is to discover that the centroid is the center of mass, but it has the added benefit of showing some of the properties of equilateral and isosceles triangles.

Balancing Triangles

I think I did a much better job of emphasizing how amazing it is that all of these lines are always concurrent and of comparing/contrasting the segments and points.  It's amazing what a year's time can do to change my enthusiasm for and confidence in a topic.  Next year I might add a variation on missclacul8's fire station problem (but with a lot less scaffolding) since the concept of regions didn't appear in the unit at all this year.  Who know's what I'll be thinking in another year's time though!