Note to Teachers:
The narrative for this section is long, because I'm trying to include all my thinking about today's whole-class discussion. I recommend starting by taking a look at today's Prezi and drawing your conclusions about the lesson, then reading my notes for all the details of how I enact this lesson. A lot of the lesson actually moves faster than all the words here might represent, but I try to be over-prepared for whole-class discussions, and this narrative represents a lot of that thinking.
Also, a note on voice: I slip back and forth between my "Better Lesson narrative" voice and my "teacher in the classroom" voice. I hope this provides a window into how I think when I'm putting together a lesson like this.
Opening: No Problem, Just an Idea
As students enter the room I have the first slide of today's Prezi projected on the board. There are no questions, just the beginning of a sentence: "If you take a ride on a Ferris Wheel..." with a picture of a Ferris Wheel and a picture of the graph with which we ended yesterday's class.
I direct student attention to this slide as I return the papers they made in class yesterday. I took a look at these ledger papers, but I did not mark them in any way. I wanted to see where students were in their understanding of yesterday's lesson, but I didn't want this to feel like an assessment. My feedback will come in the form of pointers I offer to students as they work today.
The first point I make to students is that I'd like them to carefully consider the shape of this curve and how the dots are connected. One thing I noticed when I looked at student work was that everyone was able to accurately plot the points, but the ways some of them connected the points led to some different looking graphs. (See the previous lesson for some student work samples.) I say that the graph in the photograph here (the axes are my work, but the graph is sketched by a student) is pretty much what it should look like. I ask students to assess their own graphs for sake of comparison.
Learning Target, Vocabulary Review, and Setting the Stage
Next we move directly into the learning target. I post the learning target and ask for a volunteer to read it. Then I ask the class if there's any vocabulary they need to know. (See my Strategy Folder: How to Introduce a Learning Target.) I give students the chance to shout out vocabulary as they write the SLT in their notes. To solidify our goals for today, I put up a slide with the key vocabulary, all of which students probably just read aloud. While this slide is up, I go back to the chalkboard, where I've left a note from yesterday's class that includes the slope-intercept form of a linear function, a quadratic in standard form, and the general equation for a circle. I ask students what happens to a line if I change the value of m or b. I ask similar questions about the coefficients in a quadratic equation or in the equation of a circle. I want to be sure students notice that they have some familiarity with these equations. Then I go back to today's vocabulary. I say that amplitude, frequency, and midline are represented by coefficients in a periodic function just like slope and y-inercept are represented by coefficients in a linear function. "Just like you could tell me what happens to a parabola as a (the coefficient of x^2) changes in a quadratic, today you're going to be able to see how different coefficients affect the graph of a periodic function. On the board, I write:
y = a sin (bx) + c and y = a cos (bx) + c
I say that each of these coefficients is related to one of these vocabulary words, and today we're going to figure out how. What I'm trying to do here is establish the fact that periodic functions have all the same features as other functions students know. They might seem a little more complicated than a line at first, but all functions share a lot of the same structure. Today's lesson really illustrates the connections between Mathematical Practices 2 and 7. By seeing the structure (MP7) that all functions share, we are able to frame our work. By reasoning abstractly and quantitatively (MP2) about the situation we're modeling, we can make sense of how the particular parameters of a periodic function affect its particular graph, which in turn allows us to even better see the structure of this function. I suppose we can think of this as a periodic teaching cycle that oscillates between these two mathematical practices.
In order to learn our way around these functions, we going to elaborate upon the mathematical model (MP4) that we already started building in the previous class. We've already sketched a diagram, made a table of values, and plotting points on a graph. Today we're going to write an equation that matches the graph of our Ferris Wheel ride.
Before we can get to the coefficients, the first thing we have to decide is whether we're going to use sine or cosine. I post the next slide, which includes one cycle of both sine and cosine and asks which one is which. Most students can differentiate between the two graphs either because they remember it from yesterday or because they get the concept of the unit circle. To further situate us within our work, I ask, "Why are these called periodic functions?" I want students to firmly grasp the connection between the unit circle definitions of sine and cosine, and the idea of "periodic phenomena" in general. I guarantee that students will give exactly the answers to this question that they and their classmates need to hear. It's the job of the teacher here to listen and to help synthesize this idea.
The next question is fun. Again I post the basic graphs of y = sin(x) and y = cos(x) and I ask, "Which one looks more like the Ferris Wheel ride?" My favorite response to this question is, "Sine does! Oh, wait, but...!" Students should recogize that neither one really fits our Ferris Wheel graph. The sine graph starts low then goes higher, but then it goes below the starting point. That can't work, can it? But the cosine graph is even worse: it starts at the top, goes to the bottom, then back to the top again. Now I reframe the question. If our goal is to make a graph that starts at the bottom then goes to the top and back, what is going to be more like this: something that also starts at an extreme, or something that starts in the middle? Will it be easier to simply flip something over, like we'd have to do with cosine, or to completely move the starting point, like we'd have to do with sine? (Of course we could do a horizontal shift of the sine wave, but that's not for today's lesson. I have purposefully ommitted that parameter from the general form I've shown students today.)
Now student sentiment has shifted. The cosine graph is like an upside down Ferris Wheel graph. "Alright," I ask. "So how do you flip over the graph of a function?" I move back to the functions we already know. "If I wanted an upside down parabola, what would I do to its equation?" As much as possible, I'd like to embed this new knowledge in what my students already know, and they know that a negative sign is all it takes to flip a parabola. I move to the computer at the front of the room and call up desmos.com. I type y=cos(x), press enter, and ask where the negative sign is going to go. Students may suggest that it goes in front of the x, so I put it there. The graph doesn't change. I tell students to file that away for later: we're going to want to figure out why that didn't change anything. So where else can we put that negative sign? At the prompting of my students, I put it front of cosine, and that does the trick!
Midline and Viewing Window
I go back to today's Prezi and return to the photograph of yesterday's Ferris Wheel graph. "Now we're going to adjust the graph of y = -cos(x) so that it looks like the graph of our Ferris Wheel ride," I say. I tell the class that we're going to begin with midline, and that this word is just what it sounds like. "Where is the middle of our Ferris Wheel?" I ask. "Where is the middle of y = -cos(x)?" After students answer these questions, I say, "It feels like we have to slide this entire graph up, so that its midline moves from 0 to 30." Moving back toward the general equation (y = a cos (bx) + c) that's on the board, I ask, "Which of these coefficients should we change?" Students immediately have ideas, and many of them are right. But just to continue connecting this to prior knowledge, I go back to the linear and quadratic functions. How do I "slide" the graph of a line up and down? How about a parabola? Students see (what's funny is, it really feels like some of them are noticing this for the first time) the structural similarity (again, think about that MP2 <--> MP7 cycle!) between a linear and quadratic function, and how whether we call it the "y-intercept" or the "constant", the value that causes a function to shift vertically is in the same place. Now it's abundantly clear that we just need to add "+30" to the end of the equation we're building, and that's going to slide the midline up as needed.
Of course, as students follow along on their graphing calculators, this makes the graph disappear from the default viewing window. It's not a great leap to understand that we're going to have to change the y-maximum to a value greater than 30 to see this graph, but I say that while we're at it, we might as well prepare our viewing window to match the graph that we're trying to produce. We use the Ferris Wheel graphs from yesterday to settle on an x-axis that goes from 0 to 8 and a y-axis that goes from 0 to a little more than 50. What we now see is a graph that wiggles across the center of the screen, but doesn't go nearly as high or low as it needs to go.
"Now we need to make this graph go as high and as low as the Ferris Wheel," I say. "See if you can change anything else in your equation to make that happen." Most students are game for the challenge and they're eager to play and to try some things. For those who aren't sure where to begin, I point to the general equation on the board to show them how we've changed the "c" value to match our midline. We still have two more coefficients to change. I encourage students to pick one, to give it a value, and to see what happens to the graph. (There are dozens of ways to get to this point of playing with the parameters on periodic functions - this is just one example that I'm documenting here. In my experience, the method I'm sharing here involves the right amount of scaffolding for my students.)
Students are eager to discuss what they see with each other and with me, so I spend a few minutes circulating and giving them the chance to share. Most of them are pretty quick to figure out which coefficient needs changing, but it's the value that is a little more confusing. More than half of my class gets to the point where they input y = -40cos(x)+30 and see a graph that goes beyond both the top and bottom of the window. I ask where the 40 comes from, and they tell me it's the diameter. The key here, I tell the whole class, is that although anyone who thought about using the diameter has the right idea, we actually have to think about this number in relation to the midline. "What's the height of the middle of the Ferris Wheel? What's the midline of our graph?" I ask. "How high does the Ferris Wheel go above that middle line, and how high does our graph have to go?" This line of questioning leads students pretty cleanly into a definition of amplitude; the most confusing piece of this definition is usually that when they think of the "height" of a graph, most students are inclined to think of this as the distance from the bottom of a wave to the top, but that's not it: it's the distance from the midline and to the top or bottom. Having this experience helps students make sense of this distinction. We replace the 40 with a 20, and everyone is satisfied that we're one step closer to our goal.
While this is all happening, I also like to point out that the amplitude of a periodic function is directly related to the radius of its generating circle. I take a moment to go all the way back to the start of the course: we learned about the role of the hypotenuse in a right triangle, we saw the relationship between the hypotenuse and the radius of a circle, and now we're seeing the relationship between radius and amplitude. Structurally, we can see now that the amplitude of a periodic function shows up in the place that the hypostuse shows up when we're solving for the length of a leg in a right triangle. Structurally fascinating stuff!
Side Note: AM vs. FM Radio
Somewhere in today's lesson, I also like to tell a brief story here about why AM and FM radio dials are called what they are. It's great. Kids love to know this. You should have this yarn in your back pocket as you get to this point!
To this point, we have successfully made a graph that starts where our Ferris Wheel ride starts, that goes as high as the Ferris Wheel, and whose midline matches the middle of the wheel. We have also adjusted the window to fit one trip around the wheel. The problem is that the ride doesn't appear to stop where we want it to. It looks like it ends a little after 6 minutes, then starts over again. We want to reach the bottom of the wheel exactly 8 minutes after beginning the ride.
Fortunately, there's no mystery about which coefficient we need to change. By process of elimination, we know that b is the only one left. The question is: what do we do to it?
Again, I let kids play with this by trying different values, even though it's not quite as obvious how things work. At the very least, kids can get the concept that a greater b value results in more waves. Most students don't think of b values between 0 and 1 on their own, so I drawn their attention back to the parabola. We know that a greater coefficient on x^2 results in a narrower parabola, but what if we want a wider one? Now students recall that 0.5 and 0.25 will do the trick there, so they begin to try the same in this context. They see that such values will prevent us from fitting an entire ride into 8 minutes. In fact, after their initial excitement over seeing how to make "longer" waves, morale dips a little as the task of making the ride end at exactly 8 minutes begins to feel impossible. Here we need to summon a little perseverance and make sense of this problem (MP1).
So I direct everyone's attention back to the graph of y = -20cos(x) + 30. I trace the projected image of this graph from desmos, as I retell the story once again. "You get on the Ferris wheel, ride it to the top, and then back to the bottom. According to this graph, where does the ride end?" We look at the graph and see that it ends a little after 6 minutes. "A little more than six minutes," I rub my chin and put on my most curious face. "The whole ride takes a little more than six minutes. You're at the top - halfway through the ride - at a little more than 3 minutes. What does this make you think of?" Some students are beginning to catch on. I say it again: "A whole trip around a circle takes a little more than 6 minutes. What is it about a circle and a little more than 6?" Now more of the students are getting the idea: one period of this graph has a length of 2π - a little more than 6. To make this point as clearly as possible, I type x = 2π into my Desmos window, and it's easy to see that this is where the ride ends.
I record this note on the board: "if b = 1, then the period is 2π". "What if b = 2?" I ask. Students graph it with this new insight in mind, and see pretty quickly that "if b = 2, then the period is π," which I add to the board. Beneath that, I write, "if b = 0.5, then the period is" and I leave it blank for students to fill in. It takes a little negotiation, but students can soon see that the period is 4π in that case. So what is the pattern, I ask? I give students space to explore on their own as they come up with a rule. I make a table, and I ask, "What kind of relationship is there between b and the period of this function?"
It's great to give students time to come up with a relationship between the value of b and period on their own, and I'm willing to give them as much as 10 minutes to do. I assist them in gathering more data, and I try to lead them in the right direction. If they don't have the answer after 10 minutes, however, I don't sweat it, because that's not the real point of the lesson. I ask for volunteers to share what they got, and I direct the conversation toward the conclusion that there's an inverse varation between these two values, and that b * period = 2π. Once we have this relationship, it's no problem to plug in period = 8 and find that b = π/4. I type this into Desmos, and just like that, our graph looks exactly like we set out to make it. I take a few extra moments to make sure students can write this equation on their TI-83's, coaching them to make proper use of parentheses.
These Defintions Are Still Informal
At no point in today's lesson have I explicitly defined the words amplitude, frequency, and midline. I barely even touched the word frequency, substituting period into this part of the lesson instead. This is ok, and it's really the point of doing this sort of work. I've just used these words within the context of the lesson, and I've laid groundwork for a more formal look at these words in the next lesson. For now, I want students to have the experience of seeing these words and how they are used in a problem solving context. There will be time to formalize.
If everything has gone smoothly and I have time to do it, I also like to have students use the STAT function on the TI-83's to enter our table of data from yesterday's class (L1 is time in minutes, L2 is the height of the rider in feet), then use STAT PLOT to make these points appear on the graph. We haven't done this since the Fall semester, but most students are pretty quick to remember it, and it's satisfying to see how all the points do indeed match the graph we just made.
With this example problem under our belts, I pose this challenge to students.
Write a periodic equation representing the height of a rider over time on the world's larget Ferris Wheel.
I say they can get started now, if they want to take out a mobile device and look up the necessary background knowledge.
Some students will be very curious about the modeling challenge, and if that's the case, I'm happy to let them work on it. Other students will want to go a little more in depth on the nuts and bolts of periodic functions.
For these students, there is a currently a Delta Math assignment about graphs of periodic functions, so I project it on the board and give students a brief overview of what this looks like and how it will help them.
I point out the key vocabulary from today's work in the context of the assignment and tell students that we will continue work on this over the next few class periods. They should try to learn as much as they can by working on these exercises between now and the next class.
With five minutes left in class, I ask students to shut down their computers. As everyone packs up, I tell them to make sure they know what their next steps are. Are they finishing up on the modeling challenge? Will they work on Delta Math? If so, what are their goals on that between now and the next class?