Geometry & Trig & Coherence

The year I taught Geometry and Trig in the same year, I probably learned the most I ever have about vertical coherence (across grades rather than within a course/grade).  There were so many connections.

For example, we spend so much time in Geometry working with congruent triangles, so much time proving triangles congruent.  This proof process is important for truly understanding the unit circle definitions of the trig ratios.  When I taught Trig previously, not at the same time as Geometry, I would always just have students tell me that two triangles in a unit circle were congruent, but we never took the time to explain truly WHY the triangles were congruent.  Understanding why the reference triangle for a 120 degree triangle and a 60 degree triangle are congruent is a powerful tool.  Students could tell me that the triangles were congruent, but not how they knew.  None of them made the connection to the triangle congruence that they all spent so much time with in Geometry.

This coherence is a two-way street.  In Geometry class, we need to spend more time on topics and items that will likely enhance understanding in later classes.  Instead of doing a bunch of congruent triangle proofs like this...

to proofs of things that will help their conceptual development later like this....

I'm not saying we need to abandon problems like #2, but rather shift the balance.  Most books and curricula that I've seen have put a heavy focus on triangle congruence like #2 that will not likely show up again for many HS math students.  If we shift the balance to do the majority of problems that students will likely encounter in their future, we will help enrich their comprehension when dealing with more sophisticated math.  

Conversely, in Trig, we need to make sure that we are spending the time to go through the proof of why sin(Θ) = -sin(180 + Θ).  Using problem #4 above, we can help students make the explicit connection of why this identity MUST be true, even when we don't know the value of Θ.

There was another area where I saw this overlap, and again, was underutilized in the Geometry class, which was Translation Symmetry or Frieze Patterns.  We spend a lot of time on reflection and rotation symmetry, but not so much on translation symmetry.  We also spend much of this time on 2-D shapes and next to none on applying this to functions.  It is really no wonder why students struggle with even and odd symmetries, as well as periodicity, in Trig when they are asked to apply symmetry to functions for the first time, while at the same time learning about trig functions.

Look at this example from my class several years ago

While the definition is good, all of the examples are 2-D shapes, no graphs.  Adding a parabola, absolute value, or cosine graph would help students understand that this definition applies to graphing as well as 2-D shapes.  The connections between algebra and geometry can be strengthened by figuring out WHY parabolas are symmetric.  

Students were then asked to calculate values for x equal distances to the right and left of the min/max x-value.  They began to notice relationships, were asked to explain those relationships, then connect to the symmetry of the graphs.  

In my opinion, this is much better than marrying algebra and geometry in the traditional sense, i.e., "a segment has length 2x - 5 and the distance from its midpoint to its endpoint is 3x - 10, find the length of the segment".

One of my goals this year will be to see where else in our curriculum we can focus on coherence throughout the grades and courses.  I'd love more ideas on where we can strengthen the coherence of our curriculum, especially in high school where the connections can frequently get lost between the courses.