Day 107: Identity Crisis? No Way!
Today in Precalculus, we began in earnest our journey into Trigonometric Identities. I think this is traditionally a dry subject so I wanted spice it up a little by having some interesting and engaging activities with thought-provoking questions to get the juices flowing! Here’s my agenda for the day, but it morphed, depending on the questions and “holes” my students exhibited.
I started the “trigologists,” as they call themselves, with two expressions to simplify as the opener. I wanted to see if they recalled the Pythagorean and reciprocal identities while using them appropriately to simplify expressions. I use the “pass the pen” strategy to get the work posted where each student put one part of simplifying process and then passed the pen. Once both were done, we went back and gave justifications for each step.
Interestingly, at this point, I asked if the beginning expression sinx – sinx cos²x was equivalent to sin³x. I thought this was a quick, no-brainer type question, but it wasn’t for one of my classes. My students weren’t sure. They were caught up with the “=” meaning equation instead of “equivalent.” We explored various “expressions” to find simplified “equivalent expressions” but it took going back to point-slope form of a line to slope-intercept form to get the “oh, those ARE equivalent expressions.” We also looked at the graphs of the two expressions to see that they were the same, discussing that graphing doesn’t establish proof of equivalency.
Now we were cookin’ so we looked at a few more simplifying expressions to work on the skill, still using the “pass the pen” process. They were feeling confident, so I wanted to look at the interrelated goals “determine whether an equation is an identity, explain why an identity is NOT an equation and a graph does NOT prove an identity.” Using variation of an idea I found through the Georgia Department of Education, I had the kids graph 4 function expressions on the same set of axes and asked which pairs might form an identity and which pairs would form an equation. Included in my questioning was probing again about why certain combinations made an identity and others made an equation. At that point, I knew they knew the difference. I had a student show the verification of the identity.
Once the foundation was established with the kiddos, we did a card sort created by Marsha Foshee on her blog Math-Termind via the 2013 Twitter Math Camp (TMC13) archive. I tweaked the size (the slips just seemed too small for high school fingers) and color-coded each problem. Students worked in their groups of 4 to solve each identity and match the justification. I chose to give them the starting line and they worked collaboratively from there.
It was great to hear the comments, “Oh, oh, you would…and then use the Pythagorean Theorm,” “Yah, yah! That works out!” “Why would you factor out…oh, I see, to get the Pythagorean theorem,” “I think the next step would be…” “Oh, it’s that sin²x = 1 – cos²x, and then…” “I’m feelin’ a factorization comin’ on.” I love to hear the “OH” exclamation because that tells me that a synapse has been connected in the brain! And there were soooo many OHs today!
How do you know that the “connections” are made?