It’s time to review all of the identities in precalculus. And review stations are the way to go! Since most of this unit is skill-based, I used problems and situations from their review sheet and added a few other good questions. I gave them 5 minutes per question and most were able to complete the problem while discussing it. Quite a flurry of activity and good thinking!
I was really surprised how many student opted to use their iPad to do their work. I posted the questions with space to work, so most used this tool. Still have a few die-hard pencil-n-paper kiddos…gotta love their tenacity.
Needed a fun way to do some focused practice on the Trigonometric Angle Addition and Subtraction Identities in precalculus today. So the Musical Chairs strategy came to mind. Had my students move the big desks out of the way and used just a few of them to make tables that the 8 kids could walk around them. I created eight versions of a worksheet I created with Ultimate Algebra 2 by Kuta Software and color-coded them. I searched YouTube to get some teenager-friendly tunes and had them suggest a couple songs as well.
I played the music for approx 20-30 seconds as the kids walked around the tables (you can see the smiles on their faces, huh?!). When the music stopped, they worked the problem #1 on the sheet in front of them. Once most seemed finished, I started the countdown and then started the music again; interestingly, if a student didn’t finish, as they passed the paper, they continued to work the incomplete problem…without me even suggesting it. I repeated the process with the kids working #2, then #3, etc. on which ever sheet was in front of them.
When we finished the 8 problems, I had them walk one more time and then check/correct a couple of problems on the sheet in front of them. General consensus after the experience was that the practice in this way helped them become fluid and confident in their ability to apply the angle addition identities. And there was singing to the tunes along the way! Fun experience for all and EVERYONE actually did 8 practice problems! Whoo-hoo. How do you build in skill drill in a fun way?
I’ve talked earlier about check-ins here and here, but today I decided to try something new. I had the students complete a check-in in the allotted time. Since they are color-coded by highlighter markers, I had them switch papers with their table partner. Here are the two versions:
You can see the two versions are similar, but results are different.
Then the “yellow” papers regrouped together and the “orange” papers did the same. Then I asked them to check the work of their partner’s paper with the new group. Two things happened: they worked a new set of problems AND kid conversation happened. It was a wonderful way to have students experience all problems on both versions. More importantly, they asked for clarification about certain ideas while also deepening their understanding of the properties. Writing out suggestions for correcting mistakes also helped them solidify the main ideas.
What are some different ways you have kids “talk about math ideas?”
There are just times in teaching Precalculus (or any subject for that matter) when you get to use the beauty and structure of mathematics to actually reason and create an argument to justify a relationship. And when you can connect geometry ideas with algebra and trigonometry, it’s like the perfect storm of mathematics! Aren’t I right?! Well that’s what my students and I did today. We derived the sum and difference identities for cosine. And you know what?! They were with me AND they were asking pertinent questions.
The opener question was pretty straight forward, but I had a few students share how they proved the identity so they could review before the check-in. Interestingly, one student treated the identity like an equation, and many students recognized the error. They were really sweet to the presenting student when pointing out the error and the whole class benefited from the reminder of the difference between an identity and an equation (as per our class exploration/discussion and exit question on Tuesday; see student note).
After completing the opener and finishing the 4-problem check-in on simplifying trig expressions, there were only 20 minutes left when we started with the derivation of cos(u-v). We began with color coding the diagram to be sure we understood how various parts of the expression were related.
The we plowed through the algebra, with constant reminders of what we were doing and what the various parts of the solution represented.
Then we moved on to cos(u+v). I let them ponder some, and one student suggested re-writing it as cos(u-(-v))…and that was all it took to derive this one. I asked the students to work on deriving sin(u+v) and sin(u-v). Of course, I asked them NOT to go to the internet to see how someone else did it, but to use their own noggin. They were chuckling out the door.
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?
I used a “sneaky” opener today to get my students to derive the Pythagorean Identity in precalculus based on what they knew from our study of trigonometry:
Opener: How are the coordinates of any point on the Unit Circle related to the Pythagorean Theorem? Include a diagram to show the relationship(s).
Lots of good discussion between the table partners as they came up with the theorem…I think its great when students create the content rather than me tell them, don’t you? I was so happy that they remembered so much since we hadn’t done much trig (except for the “slid in” occasional problem during other topics).
We used what they remembered and relationships that had been eluded to during our initial study of the Unit Circle and trigonometric functions: odd and even relationships, the cofunction relationships, the reciprocal and quotient relationships. They were so excited about how “everything was related.” In particular, the idea behind cofunctions and the names of the cofunctions: sine and cosine, secant and cosecant, tangent and cotangent. They were pumped!
As we derived the other Pythagorean identities, we looked for patterns in their form and discussed why “memorizing” them wasn’t the best approach but rather deriving and seeing the patterns would reap longer-term retention. Hopefully tomorrow I’ll see how successful this introduction was for my kiddos.
How do you introduce the trig identities? What strategies do you use to develop students’ long-term retention and flexibility in anticipating, recognizing and using appropriate identity substitutions?