Deep Unit Circle Understanding

Chris Bolognese started a new Honors Trigonometry unit a few days ago and posted a new problem set, welcoming feedback.  Clicking on the document image gives you access to Chris’ problem set.

This post is my feedback.  I wrote it as a conversation with Chris.  it

FEEDBACK…

Question 2 suggests you have already discussed special angles (or your students recalled them from last year’s algebra class).  What I love about this problem set is how you shift the early focus away from these memorized special angles and onto the deep symmetry underlying the unit circle (and all of trigonometry).  Understanding deep structure grants always far more understanding than rote memorizations ever does!

HINTS OF IDENTITIES TO COME…

The problem set’s initial exploration of trig symmetry starts in parts a-d of Question 1, asking students to justify statements like $sin(x+\pi )=sin(x)$ and $cos(2\pi -x)=cos(x)$.  The symmetry is nice, and I hope you refer back to this problem set when your class turns to its formal exploration of trig identities, helping them see that not all proofs of identities require algebraic justifications.

CONSIDERATION 1:  Put some creative power in your students’ hands.  Challenge them to discover additional algebraic/transformational identities like Questions 1a-1d.  For example, what is the relationship between $sin \left( \frac{\pi}{2} + x \right)$ and $sin \left( \frac{\pi}{2} - x \right)$ ?  There are LOTS of symmetry statements they could write.

CONSIDERATION 2:  I would shift Question 1e to Question 4, as 1e is the basis of the Pythagorean identities you explore in the later group.

CONSIDERATION 3:  I suggest dropping Question 4d and asking students to discover another equation showing another Pythagorean relationship between trig functions.  That cotangent and cosecant have not yet been used is hopefully a loud, silent hint.

CONSIDERATION 4:  They’re not ready for this one until they see sinusoidal graphs, but another trig pre-identity I love is graphing $y=cos^2 (x)$ and asking students to write an equivalent equation using only translations and dilations.

Unlike the symmetric relationships in 1a-1d, I don’t think you can actually KNOW it is true without algebraic relationships.  I use problems like this to set up and justify the transition to identities.

VERY CLEVER SUMMATIONS …

I don’t recall ever seeing something like your Question 3 before.  In my opinion, is the gold nugget in the assignment, especially with your students’ prior exposure to special angles.

CONSIDERATION 5:  In some ways the large number of addends in  gives away that there must be a simpler approach than the problem suggests on its surface.  What about something that suggests a direct solution if you don’t invoke the symmetry, like ?

CONSIDERATION 6:  Closely related to this, why not shamelessly take advantage of the memorized values to see if students notice the symmetry to notice a simpler approach?  I suggest , with sine or cosine.

CONSIDERATION 7:  My prior two examples tweaked your initial  problem.  Like Consideration 1, why not challenge your students to develop their own summations?  I bet they can develop some clever alternatives.

In Question 3, you had some nice explorations using degrees.  Unfortunately, I’m not aware of many equally clever early questions involving radians that aren’t degree-oriented in disguise.  Here are my final suggestions to address this gap.

CONSIDERATION 8:  Without using any technology, rank

sin(1), cos(2), tan(3), cot(4), sec(5), csc(6)

in ascending order.  Note that all angles are expressed in radian measures.  One pair of expressions is very difficult.  (A softer version of this question ranks only sin(1), cos(2), & tan(3).)

CONSIDERATION 9:  One pair of expressions in Consideration 8 is very difficult to rank without technology.  Which pair is this and why is it so difficult to rank?  Exchange angles or functions to change this question in a way that makes it easier to rank.

Thanks for the fun and thoughtful problem set.