In the previous lessons, we have learned...
I begin class by asking the students to come up with list above. (This may take a little prodding, because it is a non-traditional way to start a class period... typically, in my class, we begin with a some sort of activity. This time, however, I am asking them to jump right into the lesson by recalling what we had talked about in the previous day.)
Next, I reveal to the them that there remains one transformation of the function that we have not yet encountered. I encourage them to discuss in partners what they think this transformation will be.
After a couple of minutes, I ask the class if anyone came to a conclusion. Most groups are able to tell me that we have still yet to shift the graph horizontally, so that is where we are going today. With the ability to do this, I tell the students, we will be able to rewrite ANY sine wave as a cosine wave and visa-versa.
After this short discussion, I send the students out on to the interactive graphing site DESMOS to try to predict how to alter the equation so that it shifts horizontally. Yesterday, the scaffolding revolved around using a slider applet to see the behavior of the graphs (in connection to amplitude, period, and vertical shift). Today, however, I am asking the students to build upon that knowledge of the general form to reach educated guesses on how to shift the trig function horizontally.
I allow the students to work for 5-10 minutes on Desmos as they attempt to reach a conclusion on how to alter the equation.
As we transition into the second part of the lesson, I ask my students to remain on Desmos and attempt to find the equation that I have displayed in Desmos #1. The students should recognize that this particular graph is not horizontally shifted.
After all students have had an opportunity to play with the graphs, I ask the students who were successful to share their secret with the class. If the students do not do so on their own, I probe them to use appropriate terminology such as "period" and "amplitude" - this is difficult for high school students. No one wants to sound "dumb," but may times students are equally as scared of sounding "smart"!
Desmos Simon Says
Now that the students have grown accustomed to using Desmos to manipulate the graphs of the trig functions, I explain to them that we are going to play a game of Simon Says.
RULES FOR SIMON SAYS
1) Everyone will start with the function f(theta)=sin(theta) OR f(theta)=cos(theta).
2) I will tell you to do things like "shift the graph 2 to the right" ... or "change the amplitude to 4"
3) After each set of instructions, I will ask you to compare with your neighbor's graph before I reveal the results on the screen. I, too, will be graphing my instructions so that everyone can compare their answers to mine.
4) If you incorrectly move the graph, you are out for the round but you MUST continue to graph the remaining functions with the rest of the class.
5) The winning student(s) will receive free cookies at lunch from me!
The students really enjoy this competitive activity. It forces them to use MP3 and MP7, and it also brings to the surface common mistakes among students. If 2/3 of the class gets out on one particular transformation of the graph, then it is a great indicator that I need to go back and re-teach that element of the transformation!