This demonstration is fairly simple and gives students a visual reference for the derivation of sine and cosine graphs. I like the fact that it addresses standard HSF-TF.B.5, while giving students a clear picture of sine and cosine beyond the right triangle they’ve previously associated them with. You will need a large circle (a paper plate works ok, but something colorful is better against the whiteboard) which you are willing to mark. I use a large plastic plate which takes either overhead markers or whiteboard markers. I’ve included a brief video of this demo in my resources so you can better see how it works.
In a nutshell, I mark x and y axes on one side of the whiteboard, extending the x axis across the board. I hold the plate against the whiteboard and mark it as though it was a unit circle, with a starting point of sine (0,1). I then carefully roll the plate along the x axis, marking where the starting point has moved to, until I’ve completed at least one rotation of the plate. You may choose to have a student help with this part of the process, either by turning the plate or by marking the points. It is important to be precise with this exercise or the impact is greatly diminished. I’ve tried this as a student seat activity but because it’s easy to get off the center line, it is less successful in conveying the concept than when I do it as a demo.
Student Discussion: When I’ve completed the rotation, I ask my students what labels we should use for the x and y axes. This may seem like a simple idea, but many students won’t make the connection between periodicity and circles without guidance. I use leading questions if necessary to get students to say what the labels should be, rather than just labeling them myself or telling them the answer. I value this process even though it may take a few extra minutes because it gives my students ownership of this information and strengthens their ability to understand why the axes are labeled as they are, which is a part of MP4. For example, I might ask what point we started with or what point we were plotting along the x-axis to get students thinking about the y-axis being the sin(x) values. To elicit a label for the x-axis I might ask what the plate represents to get my students thinking about the unit circle we started with. If necessary I can then ask how far the plate turned across the board compared to what measure we give the distance around the unit circle. When I’ve gotten these responses I label the axes (x – radians, y – sin(x)), then connect the points to show a sine curve.
To finish the discussion I ask for a volunteer to indicate the amplitude and period of the graph. This is important for our next discussion of transformations which often change either the amplitude or period. This demonstration can be repeated for the cosine function if desired. I try to very specifically use language like “parent function” so that my students become as accustomed to the language of math as they are to their own slang…I even refer to this language as “mathlish” which my students think is just one more sign of my relative eccentricity as a math/science teacher.
For this section you will want to print copies of the Calculator Directions handout. Once my students understand where the graphs of sine and cosine come from, I have them use their graphing calculators. For most of my students, this means explicit instruction on how to input the functions and set up the viewing window. I do not consider this activity as meeting any specific content standard although it addresses MP5 in helping students learn when and how to use their calculators, but it is absolutely necessary for my students to be able to complete the rest of this lesson and it’s much easier to teach the whole class through direct instruction than to try to troubleshoot each student individually.
The other math teacher at our school is very traditional and rarely has students use graphing calculators, nor does she teach the trig sections of geometry or algebra because they’re at the end of the textbook. I cannot change this, but I can address these deficits for my students before they head off to college! I’ve included a Calculator Directions handout in my resources if your students also need this instruction. If your students to not need this, you can proceed to the next section.
I begin by giving each student a copy of the handout and asking them to follow through the steps with me. Invariably some students will work ahead, which is ok with me as long as they are following the steps accurately. If someone asks how to do something before we get to it, I just assure them that we will get there, rather than jumping around to accommodate individual students. I find it helpful to put my graphing calculator under my document camera to complete this lesson so students can see the buttons I’m pushing as I go through each step and can see the result on the whiteboard (my projector screen). It also works to use a TI projector if you have one, but this activity can be accomplished without any projection options, by walking around and making sure each student has successfully completed each step before moving on. I will not walk you through all the steps in this narrative since they’re on the handout, but I will reiterate that it is very important to make sure each student can successfully complete each step before moving on to the next activity or you will find yourself facing some very frustrated students!
For this section you will want to print out copies of the Playing with the Numbers handout. I generally have students work individually on this section because I want each student to look for the patterns independently. I give out the handout and make sure the students see the chart for recording their observations (I print the handout as a double-sided paper to reduce the number of papers I have to handle and to keep the equation sets and observations together for the students to use as a future reference. This means that at least one student will ignore my initial directions about the chart and either begin writing their observations on another paper, or will complain that they can’t record their observations because they don’t have any extra paper. Usually a classmate will set them straight, but if not, I just show them the chart and move on…this is not worth making an issue of because it’s not critical to understanding and would definitely distract the student and probably a few others from the work at hand.)
As students begin working, I walk around the room, checking to make sure that everyone is getting the equations entered correctly and fixing problems with setting up the viewing window. I ask questions to help students who are “stuck” trying to determine what changes they see or who can’t figure out what they want to write. For example, I might show a student struggling to find a pattern how to turn on just two graphs at a time, then ask them to look for a difference between just those two graphs. (Make sure that one of the two graphs you keep on is always the parent function – either sin(x) or cos(x)) When they can see that, I turn off the second equation and turn on a different one, again asking the student to compare just the two graphs he/she can see. Eventually, he/she should be able to come up with a common change and make a generalization about this change. I believe communicating coherently about patterns is a critical aspect of MP8, so I try to help my students build this skill. If you have a student or students who move at a faster pace than the majority of the class you can offer them this Playing with Numbers Extension Challenge
to keep them engaged. (Be aware that some students will hear you working with a struggling student and decide that they need more attention. When one of your strong-but-lazy students tries this ploy I suggest that they try looking at just two graphs at a time, but don’t walk them through it step-by-step unless they really seem to be struggling.)
Expanding the learning
There is also a video narrative that goes with this section in my resources. I use it to explain some of the pedagogy involved in this lesson and why I structure it the way I do.
When all the students have successfully completed all the equation sets and completed the chart- I can tell this by my observations as I walk around the class - I tell them they will get to work with a partner for the next activity. This is the preferred method of working for most of my students, so it’s a sort of reward for all the hard work they’ve just completed. I already have a system in place for pairing students up, established at the beginning of the year, so all I need to say is “work with your right shoulder partner” and everyone knows what to do. If you’d like to see my system, you can find it in my strategies folder as the video "partners".
As soon as the students are ready I tell them they are going to be playing a game using the charts they’ve just created and their calculators. I ask for a volunteer and use that student to help demonstrate the role of each player in the game. For this game I intentionally select a volunteer that I know will be able to make an accurate guess. If you would rather not put a student on the spot, you can have the entire class contribute to guessing the impact of the transformation. I act the part of student A and have my volunteer be student B. The rules for the game are simple enough that I don’t print them out, but I’ve included a copy in my resources to make it easier for you. I assume the role of challenger to start the game and enter a transformed trig equation into my graphing calculator. I tell Student B the equation and he/she then guesses how the transformation will change the graph. If he/she is correct he/she earns one point. Play then switches with Student B creating the equation and me guessing the change. Play continues until one player reaches 20 points or time is called. This "game" helps students reinforce their understanding of the patterns involved in tranformations another key aspect of MP8.
As students are playing I walk around the classroom observing which teams are working well and which ones are struggling. Teams may struggle if the challenger is making the transformations too complex or if either member has inaccurate information on their chart. I can fix these problems by suggesting simpler equations or asking the pair to review each other’s chart for accuracy. Sometimes a struggling student just needs a little boost, like being told whether the transformation affects the amplitude, the period, or both, but without being given the specific effect. It’s also possible that an individual student is really not visualizing the changes, especially if they’re one of my students who is very inexperienced with the graphing calculator. For that student I have found that it works well to let them use their calculator to enter the challenger’s equation, as well as the parent function, with the additional directive that they must accurately describe the transformation they see in order to earn a point. Some students are very willing to play by different rules based on ability, but others become belligerent about “fairness” so to avoid that conflict I may reassign students if I know that one will probably need help or if I notice one struggling.
After 10 minutes I call time and ask if anyone earned 20 points. I then have one of those students to come up as my next volunteer “expert”. For the second part of the game, students will again be working in pairs, but instead of guessing how a graph will change given an equation, they get to guess how an equation will change from a given transformed graph. It still surprises me how much my students enjoy these games, even as they cement their understanding of transformations of trig functions! I think most people like to be challenged, but sometimes students have a hard time seeing a purpose to just learning something new. I think this kind of activity gives the immediate purpose of doing well at a game which I know I respond very positively too, myself. I compare it to the spelling bee or MathCounts competitions giving incentive to students to push beyond their normal comfort zone and/or level of interest.
I give students another 10 minutes playing this second part of the game, and up the challenge to earn at least 20 points.
After 10 minutes, I call time again and ask if anyone earned more than 20 points. I congratulate these “experts” and assure them that their skills will come in handy over the next couple of lessons. I also tell them that I’m expecting them to help me teach anyone who is still struggling with transformations. To close the lesson, I give the students a homework challenge to create one or more transformed equations that they think will stump me. I give the following parameters:
After handing out notecards and making sure everyone understands the directions for this homework assignment I allow them to use any remaining time to begin plotting how to stump me.