I begin today's class by asking students to solve the equation on slide 2 of the flipchart - Solving Quadratic Trig Equations. My goal is to get students thinking about how to solve quadratic equations using inverse operations. I am prepared to remind students that:
If possible I will avoid providing assistance to students on this warm-up problem. I want to assess who can and who cannot solve the problem independently. However, I plan to go over the problem with the class.
In the middle part of today's lesson, my goal is to have students review solving quadratic equations before we move onto solving quadrtaic trigonometric equations in tomorrow's lesson. After guiding students through part 1 of the Student Activity: Solving Quadratic Trig Equations, I will have students work through exercise 1-5 in part II focusing only on the Algebraic form. then students will present their solutions. Here is more detail on how this went my first year teaching it: Solving Quadratic Trig Equations Day 1, Video Narrative, Quadratic Trig Equations
Struggling students should hopefully be supported in their teams today. If you find that you have teams that hit a wall and are doing nothing, you may want to ask some prompting questions to get students there. For example, many of my students tried to solve the quadratic equations by doing inverses. But then they always get stuck when they can’t combine the variable with the squared variable. So I ask things like:
“Where did you get stuck?”
“Why can’t you do that?”
“What are some strategies from the brainstorming session you could try?”
Most of my students quickly recalled factoring, but many did not recall the quadratic formula. I had it on the board already so some students took that as a hint too and made the connection.
For my students that kept leaving off the negative solutions when directly solving (exercises 3 and 5) I asked them “How many solutions are we supposed to have?” This is one thing they KNOW! I really drilled it in the polynomials unit.
The most challenging problem for my students was exercise 2. For the middle part of the lesson today, give that to your stronger teams. They may even need a hint: FACTOR.
If students are ready for an extension, have them try the trigonometric form of the Algebraic form they were working on. I had one student who got it perfectly! I plan to use him as a tutor tomorrow.
A few of my higher level students had some fun figuring out exercise 2 in the Algebraic form. I observed many different solution methods throughout the classroom. Some students factored out the x and then solved using the zero product property (that is how I would’ve done it!). Others added x to both sides then divided by x to obtain y=1/2. They incorrectly left their solutions at just that, without accounting for possible values of x. And then some students just sort of used their logic and number sense to figure out the solution.
Here is an example of a student’s work who took this logical-type approach: Interesting Student Solution Method. When he presented, many of my students were confused at how he obtained that y could equal 1/2 by his work. I stepped in and wrote what was in red after the student expressed that he was having difficulty explaining it to everyone. Interestingly he understood, but couldn’t explain his thoughts to the class. I wish I would’ve had more time to really draw his thoughts with some questions to help him explain it to the class. But due to time, I stepped in and explained what I wrote in red. This students was able to do a great job explaining that if y did equal 1/2 then we would know that x can be any value. He also did a great job explaining that y could also be any value if x=0. I think this really helped to clarify for many of my students what the solution they were obtaining really meant. Some students had the misconception that y=1/2 AND x=0 instead of the fact that y=1/2 OR x=0.
To close out today’s lesson, I will ask my students to identify questions they still have about solving quadratic trigonometric equations. First, I will have students write down one question they still have (page 4 in Flipchart). Then, I will encourage students to discuss questions in their teams (page 5 in Flipchart). If questions go unanswered, I will step in and make clarifying suggestions. I want to mak sure that students arrive in a confident place before the end of class (page 6 in Flipchart).