To open this class, I want to get students thinking about the functions they already know, so I post these four questions on the board:
I prompt students to discuss each of these questions at their tables (see slide #2 in ferris wheel ride).
I have purposefully left the first two questions open-ended enough that there's no presciption as to whether they should be answered with an equation or graphical representation. As a I circulate through the room, I listen to the conversations of my students.
How my students approach the word function will influence some of the moves I make in class today. Also along these lines, I am asking myself:
How many of my students notice that the third question refers to the "equation of a circle," but doesn't call it a function?
Concluding this segment of the lesson with a brief whole-class discussion, I ask students to tell me what they know about each of these questions. As we talk, I write the general forms of each type of equation, and the most basic sketches of each in the upper left-hand corner of my chalk board (see some functions). As we move on to discussing some of the parameters of a trigonometric function, we'll be able to refer back to these sketches as we try to make use of the structures that different functions share (MP7).
The notes for today's lesson come in the form of a Problem Solving Task that students will complete in pairs.
The Task: Each pair of students gets a sheet of ledger paper (11x17) and I tell them that I'm going to give them a series of prompts to complete. (Note to Teachers: before reading any further take a look at ferris wheel ride to get an idea of what this looks like).
Sketching a Ferris Wheel Diagram
Modeling Time vs. Height
The next task is to make a table of values - that other great function representation - that lists the height of a rider off the ground at each minute of their ride. I have given a minimal amount of time on each prompt up to this point, and I post this task when I see that the first few groups are ready for it. Some groups will still be finishing up their sketchs and their labeling, and it's fine for them to see where they're going next. One neat benefit of ledger paper is that there's room for two students to write at once - so if one student is still labeling the sketch on the left side of the paper, another can draw this chart.
This is where things really get interesting: I give students 5-10 minutes to discuss this task and to begin to fill in the chart.
As I walk around, I watch to see what they're writing, but I make no comments. I'm watching to see the order in which students fill in the table. Are they going row by row? Are they filling in 0 and 8 first, then the middle? The latter scenario of starting with the "easy" points reflects a strong understanding of the model, and indeed, this is what I see most students doing. The majority of students will fill in 0, 4 and 8 minutes first because they recognize these as the bottoms and tops of the ride. Most will also notice that at 2 and 6 minutes, a rider is "half-way up" the wheel, and therefore 30 feet above the ground. That leaves 1, 3, 5 and 7 minutes.
In general, there are two moves that students might make. They will either notice a linear relationship going up and down, and assume that height follows the pattern 10, 20, 30, 40, 50, and back down, or they will stop in confusion and wonder how to figure out the height of a rider at these odd-numbered minutes. Either result gives us a place to start on today's mini-lesson.
During the Mini-Lesson I will use Symmetry and Trigonometry to help students fill in the missing values. As students have been working, I have stayed a step or two behind them in making my own sketch on the board. When I see that all students have sketched their own circles, I draw a circle on the board. When they have all labeled their minute marks, I do the same. As they're digging into the table of values, I draw my own table of values on the board. When it seems like all students have at least filled in some of the rows of the table, and I can see that they've discussed but not really settled happily on solutions to the problem of the odd-minute points, I begin today's mini-lesson (see Photo to see what my sketch looks like).
I've already made my own observations about how students are filling in the table of values, but now I ask the class:
Have all of you been filling in your time vs. height charts in order, row by row? Or have you been using some other method.
I let students volunteer to explain how they started with 0, but then skipped to 4 and to 8 minutes.
To begin elaborating on my Ferris Wheel diagram, I start by drawing horizontal lines from the bottom and top of the Ferris Wheel, and I label these lines 0 and 4 minutes. At my students' prompting, I fill in the corresponding rows of my table, and everyone agrees that there are pretty straightforward. Next I draw another horizontal line from the middle of the wheel: this is the "2 minute line." I point out that even though I'm working on a chalkboard with an imperfect sketch, it's pretty clear that this line is halfway in-between my 0 and 4 lines. I ask if anyone has other reasons why the rider would be halfway from 10 feet to 50 feet at this point, and students talk about the radius of the wheel being 20 feet, the constant speed of the wheel, and other observations about the circle. We all agree that it makes sense to add 30 feet to the 2 minute row of the chart. Here, I also point out that there's symmetry on the wheel, and students are quick to point out that at 6 minutes, the rider is also 30 feet up. I point out that symmetry can be observed both on the circle and in the table of values. On the circle, the most important line of symmetry is a the vertical diameter of the Ferris Wheel; in the table, the ordered pair (4,50) makes a place where we "fold this table over." With both representations in mind, it's pretty clear that at the rider will be at the same height at minutes 3 and 5, and also at 1 and 7. The question is, what are these heights?
Next, I draw two more horizontal lines, from 1 and 3 minutes. This is the moment when it becomes very clear to students that the vertical changes from 0 to 1 and from 1 to 2 minutes are not equal. Now my students can see that this not going to be a linear function.
Although it's a little misleading in terms of what our variables mean, I draw a straight line from 0 to 2 minutes, and I ask students to compare the slope of this line to the curve of the Ferris Wheel. From minutes 0 to 1, the wheel rises gently, with a slope less steep than the straight line. Then, from minute 1 to 2, it rises more dramatically, "catching up" to the the straight line. We know that the same amount of time passes between each of these points, but how can we figure out the vertical displacement? I give students a few minutes to solve this problem.
As I circulate, I see some students finding the 45/45/90 triangle that is the key to solving this problem. I see others considering the lengths of the arcs between each pair of points: for these students I try to get them to see that the arcs will have the same length, but it's only the vertical change we're looking for. I see other students using rulers to measure the distances between these horizontal lines, then using proportions to estimate what those distances represent. Some students are simply making their best estimates without the help of any tools, but aren't sure how to verify their answers.
Knowing that a few students have already found it, I ask the class if anyone can find in this diagram a right triangle that might be able to help us. I try to get some of my more confused students involved by asking what we know so far. Are there any angles we know? Well, we know we've got that central angle of 45 degrees from when we sliced-up the circle into 8 1-minute increments. Are there any sides we know? We know that the radius of the circle is 20. What is the relationship between these? Is it possible to make a right triangle that uses the central angle and the radius. In the photo, you'll see that I've used blue chalk to highlight the triangle I want students to see. I've drawn the height as a dotted blue line, once as the vertical leg of a 45/45/90 triangle, and again off to the right, as the distance between the 2 and 3 minutes lines.
Before we continue, I go back to the symmetry of this diagram. If we can figure out this distance, what else will it tell us? It will be the same as the distance between 1 and 2 minutes. Additionally, once we know this value, we'll just have to subtract it from 20 feet to get the distances between 0 and 1, and 3 and 4 minutes. Finally, as we've already seen, these values will match those on the left side of the diagram.
The key here is that I'm trying to embed this problem in as much context as possible. Most of my students can quickly calculate the length of a missing side of a special right triangle. Fewer are able to find such a triangle and apply this knowledge. So before we get to solving the problem, it's important to figure out all the ways that it will help us to do so. After this stage is set, I set students to task. I recognize that I've written a lot here; I've done so because I'm trying to cover all that I try to show students as we work on this task. The conversation takes maybe 10 minutes, if I include all the thinking time I'm allowing for. Once students get back to work, I'm able to circulate and see how much they were able to understand. I give them space to talk to each other and to fill in/revise the rest of their charts.
So What If I Keep Riding? (Or at least continue the chart?)
As students finish the work of filling out this chart, I post the next prompt:
Write a sentence or two describing what will happen if you continue this chart.
I have chosen this wording because I want to leave it to students to interpret what the chart represents (a ride on a Ferris Wheel), and what its continuation will represent. An alternate wording would be to say something like, "What would this chart look like if the rider continued to ride the Ferris Wheel?", but I like the mixing of representations and the translating back that happens when I ask about the chart.
How is this related to sine and cosine?
I preemptively post the very general question, "How is this related to sine and cosine?" We have not yet made any sort of explicit connection between periodic functions and the sine and cosine function, but there's certainly a trail of breadcrumbs that the enterprising student should be able to pick up on. I'm just curious what will think about this, and it's always interesting to see how they phrase their understandings before I reference these connections.
Make a Graph + What Happens Between the Points?
The next task is to plot the points from our time vs. height chart on a graph. Students can make this graph directly on the ledger paper, and there's graph paper available if they want it. Plotting the points is pretty trivial, although I make sure to point out the symmetry that we've seen on the circle and in the table once again.
I do my best to help students make two more observations that I find much more interesting: first, that we're not going to have straight lines in-between these plotted points. We go back to our initial visualization - at what pace is a rider rising as they take a spin around the Ferris Wheel? (Of course, such a question makes some room for a future study of Calculus, and this is a theme I try to hit a few times in this unit.) The other thing to consider is what happens if this graph continues.
Look at These Graphs: Which one is more like the Ferris Wheel ride?
Finally, I ask students to pick up a graphic calculator and look at the basic graphs of y = sin(x) and y = cos(x). What they see will depend on whether their calculators are in radian or degree mode, and on the viewing window they've set. I tell students to stick with radians, and then to consider at least changing the y-scale to reflect their knowledge of the range of possible values for each function. Just to give students a view of what they're looking for, I put desmos.com up on the projector screen, and plot these two graphs.
After students have seen these graphs, the final question of the day is this: which one looks more like our Ferris Wheel graph? I ask students to answer this question on their ledger paper, and I pair it with a "coming attraction" for tomorrow's lesson: how can we write an equation for a function that will match what we've made for the Ferris Wheel?
A brief note on the direction in which a Ferris Wheel turns:
Invariably, a student will ask if it matters which direction we have the Ferris Wheel turning. In reality, the direction a the wheel turns depends on which side we watch it from -- all Ferris Wheels go both clockwise and counterclockwise at the same time! For our purposes, I have my wheel turning in a counterclockwise direction, just to match the unit circle.
Tonight's homework is to begin work on a new Delta Math assignment that takes students through some details about the graphs of trig functions. I tell students to pay attention for key vocabulary, and that we will use their work on this assignment to expand our knowledge of this topic in the coming classes.
There is no formal closing to today's class, but I will collect the work of all student pairs as an Exit Slip. These documents provide great evidence of student thinking. It's great to see the kinds of problem solving that students did, to read their responses to the written prompts of today's assignment, and to see how they incorporate today's notes into their own work. This is pure formative assessment. Going into the next class, these papers will tell me precisely who and what I need to focus on.