In this section, I'll guide students through an area function problem so that when we do the main problem of this lesson, they'll be familiar with important terminology and practices. I start by giving the students a copy of Geometry Area Function Problems.
First I have the students read the problem and start to make sense of it. Then I have them do a pair share. The A partner will describe the context of the problem and given information. The B partner will then explain what the problem solving aspect of the problem is. In other words, what the goal of the problem-solving is.
Next I'll do a quick whip around the room getting students predictions for what dimensions will maximize the area of the corral. I'll make a dot plot on the board to represent the distribution of predictions. Next, I'll model how we define variables in a situation like this. I'll write on the handout "Let x be the depth of the corral" and then I'll label each of the three vertical segments on the rectangle with an x. Then I'll explain that we need to define the width of the corral in terms of x so that we don't introduce a new variable. I give the students a minute or two to think of how we might do this.
When the time has passed, I'll do some call and response to get to the expression for the width of the corral. I'll say "How much fencing did we have altogether?" [30 meters] "How much have we used so far?" [3x meters] "So how much do we still have to use?" [30 - 3x meters] "How much will be on each of the longer sides then?" [(30-3x)/2 meters]. Of course it doesn't go that smoothly, but we get there.
Now that we've defined the length and width in terms of x, I ask the students to write the Area as a function of x. We eventually get to A(x)= 15x-1.5x^2. I'm careful to show and explain the notation so that students are comfortable with it.
Next I'll write on the handout "Practical Domain of x:" and ask students to do the same on their papers. Then I'll explain to students that not all values of x are mathematically possible in this scenario. I ask them to think about the lower and upper limits of x. Eventually through some explaining we get to the fact that the practical domain is (0,10). Since this is hard for some students to understand, I would usually do a pair share at this point asking students to explain why the domain is (0,10).
Next, I'll ask the students to choose some values strategically within the practical domain of x and create a table of values of the Area. As they're doing this, I tell them to keep in mind that they are looking for the value of x that will maximize the area. I give them 5 minutes or so to make headway on this. Then I pull up Desmos and give a demonstration that is described in the following screencast.
Finally, after students have viewed the demonstration, I'll ask them to sketch the graph of the function and then write an explanation of how it helps us to solve the problem and what exactly it tells us.
In this section we'll be working on the parallelogram area problem on page 2 of Geometry Area Function Problems. First I take some time to explain that even though the diagram shows a static parallelogram, we have to envision it as a dynamic situation in which theta is allowed to vary. I also reiterate our purpose, which is to describe how the area of the parallelogram changes as a function of theta.
Once students understand that, I'll direct a pair share asking what the practical domain of theta should be. I ask them to discuss how they came up with the practical domain until they agree on what it should be. Then I call on a few students randomly to give them a chance to share what they're thinking. Eventually we get to the practical domain being (0,180). Anything else is just fodder for discussion and critiquing the reasoning of others. Once the students have shared, I ask them to write on the handout what the practical domain is and their explanation for why it is what it is.
Next, I'll ask students to envision (without doing calculations) how the area of the parallelogram changes as theta moves through its practical domain and then sketch their conception of the graph. I ask that each student does this independently so that we get students' true thoughts as they are at that moment.
Then I'll ask the students to compare their graphs with at least four other students' graphs and determine if they want to make any adjustments (e.g., form) or improvements (e.g., scale, labels, continuity). After that process has taken its course, I'll call a student up to show their graph and explain why they think it makes sense.
Next, it's time for students to write the function A(x). I allow my students to work independently on this because they have learned all they need to know in order to complete the task. I'll be walking around checking in with students to see how their progressing, providing corrective feedback if necessary. When students have successfully written the function, A(x)=50sinx, I'll push them on to the next task: creating a table of values and graphing the ordered pairs. When they've done this, it will be time for them to write about whether their initial conception was supported or refuted by the actual graphing of points on the function.
When most, if not all, students have finished, I'll bring the attention to the front of the class and we'll discuss the form of the graph. We'll discuss how, from the looks of things, the graph of this sin function is not much different than the graph of the quadratic function we saw in the first example in this lesson. I explain to them that this doesn't really give us a fair depiction of what a sine function is really like. The reason why, I explain, is that we've looked at only limited domain of the sine function. It's kind of like how people thought the Earth was flat at one time because their domain was limited at the time. In the next section, I'll be showing students what sine functions are really like and how they can describe this situation perfectly if we expand our notions of angle and area.
In this section, I'll be demonstrating for students how the trigonometric functions like the sine function in this example actually allow us to describe periodic phenomena that continue cyclically theoretically forever.
I created the Parallelogram sine function.gsp, a sketchpad file, to help with that. See the following screencast to see the sketch in action.
After that demonstration, it's time to look at the graph of the sine function in a way that hopefully will help students to connect the graph to the situation it is modeling. The following screencast gives a glimpse of that process.
In the previous section we looked a a "child" of the sine function. In this section we'll look at the parent function as well as the the cosine parent function in order to make some generalizations about these functions and compare and contrast them.
I have access to a classroom set of chrome books so for this lesson each student will have a chrome book to work with. I start by handing out Comparing Sine and Cosine Functions and DoubleBubble compare contrastMap. Then students will have 25 minutes to work. I'll walk around checking student progress and giving assistance as needed.