What I share here is likely to take more than one class period. Students will finish on different days and move on to whatever they do next. In the upcoming "work periods" keep in mind that this assignment is probably happening for some students in the background. Rather than splitting this investigation - and all that might happen - up into pieces, I want to share everything here.
In particular, a lot of the work on the Profit_Function_Extension will happen in the days following this lesson. As some students shift gears to the work of their choice, others will get all they can out of this task.
This unit is all about kids doing and understanding as much as they can at their own pace. If that involves reviewing the essentials of as many learning targets as possible, that's great. If it involves going as deep as possible into this task, as I'll describe here, then that's ideal too.
Students arrive to see that the agenda has been replaced by this assignment list. With just a few weeks left in the school year, this is the first time that I've deviated from putting a daily agenda in its usual place on the left-side whiteboard, and that's enough to get students to ask what's going on here. I explain that this is a current summary of the number of work requests I've received for each Student Learning Target. (To see how students request work, please see the third section of yesterday's lesson.) In the third column of the chart are the names of assignments that are available. You can see that this column is incomplete; I explain to students that in order to best manage my time, I'm preparing work for the most requested learning targets first, but that this list will be updated daily for the rest of the year.
On the center board is a list of the three steps students should take today. You'll see other notes that were written on the board as we worked through the lesson. I remind students what I said yesterday: that everyone must finish the Class of 2017 Fundraiser assignment as their "ticket" to move on to whatever they do next, "The best way for you to achieve what you want to achieve is to finish one assignment at a time," I say. "You cannot ask me for a pile of make-up work, but as long you keep getting stuff done, I'll be happy to keep giving you more opportunities to improve your grade."
The first step is to finish the Class of 2017 Fundraiser assignment from yesterday, and most students only have to put finishing touches on their graphs. Near the start of the lesson, or as I see that students are ready, I make a rough sketch of what the graph should look like. Then, I suggest that the graph seems incomplete, and I draw the dotted line you see in the photo.
I ask what kind of graph or what kind of function this is. Even though our quadratic functions unit has just concluded, enough students struggle to come up with that word that this is totally useful. "Don’t forget what you just did!" I say. Once we name this graph as a parabola, I encourage everyone to predict where the next root might be. Students venture their guesses, and I ask, "What will this root represent?" Eventually we're able to conclude that when the graph gets back to the x-axis, we've found the cost of a t-shirt for which the fundraiser will not raise any revenue. "Why will that happen?" I ask. "How can raising the price of a t-shirt result in us not raising any money?" Students recognize that if we charge too much, no one will buy a t-shirt. Now, students are more invested in making their predictions, because there's a real context.
I tell students to continue the table until it no longer makes sense, and the “model breaks down”. That’s a fun phrase to teach kids about, and it’s obvious when it happens, thanks to the negative # of sales. "We only need a few more points," I say. "This won't take you long to do."
Some kids try to find the last few points by reflecting existing points across the axis of symmetry, but then they see that it doesn't quite work out. To these students I ask, "Where is the axis of symmetry for this function, anyway?" I chose the initial parameters of this problem to avoid symmetry. It's neat to consider, and depending on how deep you want to go, this is a possible extension route to take: what would have to be true in a problem like this to ensure that symmetry happens?
As each student finishes, they receive the next part of this activity, which I describe in the next section of this lesson.
A Slight Jedi Mind Trick
Yes, students will have plenty of time to choose their own adventures over the course of the next few weeks, but first I'll push them to go a little further on the current investigation. The second step in today's to-do list says, "Get and complete Interpretation Questions," which is the next part of the "Class of 2017 Fundraiser". It is also the recommended work for anyone who requested work on SLT 1.4. I strongly urge everyone to give this assignment a try before they really start choosing their own work, and kids are either curious enough about the task or eager enough to master SLT 1.4 that they're willing to jump in.
I expect all students to complete #1-6. If they can get the rule for R(x) on #7, that's great, and we'll spend a little time exploring that in the lab tomorrow. Then #8 is an optional extension that leads to all sorts of rich conversations. Anyone who completes #1-6, gets a 2 on SLT 1.4. To get higher than that, students must show me that they understand the rule for the quadratic function, and then attempt the extension.
About the Assignment
The Interpretation Questions assignment consists of a series of questions about the Class of 2017 Fundraiser. My job is to stay out of the way until students need me. I circulate and encourage everyone stay on task, and I wait for students to ask questions.
This activity is satisfying because it places a linear function right next to a quadratic function, so students have to review and apply their knowledge of both. In addition to review, it provides a rich example of how each tool (linear, quadratic) might be of use at different moments in the same situation.
The first three questions ask students to interpret the work they've already done. I learn a lot and I enjoy reading what kids write. Here's an example of what a typical student's work looks like. I encourage students to "think like real people, and not to give purely mathematical answers." Here's an example from a student who did a good job of that - look at her answer to #2, for example.
Review of Linear Functions
Problems #4-5 are about creating and then using a linear function to model the relationship between the price of a t-shirt, x, and the number of t-shirts sold, n(x). When SLT 1.4 was originally introduced, it was only about linear functions, and I want to be certain to hit that here. Remember that the purpose of this unit is to review key ideas from throughout the year, and to apply them in a modeling context.
Some students will recognize the function immediately, and others will need a review. I'm ready to teach a mini-lesson about this. One way I structure the review is by putting a few review examples on the board. On the left, you can see the table of values for which we're trying to write a function. On the right are two simpler linear functions, and I guide students through each of them in hopes of clearing the cobwebs and dusting off these skills. A third example is structured just like the one we're looking for, starting at a higher number and then decreasing each time.
I wait a little while and check in with individual students to make sure they've got it. If anyone is really stuck, I help out directly, and make a note that the next step for these students is to get more practice writing linear functions from tables.
The fifth problem is a review of using function notation in context. Again, I help students as needed, and I keep track of who will need some extra practice on this.
The Revenue Function
We've already identified that the revenue function is quadratic, but what about writing the function rule? To move in that direction, students answer some questions about the graph on #6. The task is complicated by the fact that the axis of symmetry and the right-hand root are not whole numbers. It's useful to see what kids do with this. My goal is to make sure that students can identify these vocabulary words in context, so I tell students that I'll accept approximate answers. Once students have the correct algebraic rule for this function, we can go back and assess those estimates.
For example, here is some great work, but notice that this student is still just a little off on finding the second root and the axis of symmetry. It will be useful to ask her to try to use her rule for R(x) to get more precise values.
If kids can spot the relationship in the table - that R(x) is simply the product of x and n(x) - they're quick to find the rule. This is one of those things that I wish every student could do -- it's an unexpected approach that requires a different kind of thinking, and there's often little overlap between the set of students who notice it and the set of highest-achieving students.
Using Technology (Tomorrow)
When it comes to writing the function rules for R(x), students will have the opportunity tomorrow in the computer lab to match a function through these points.
When it comes to functions and modeling, problem #8 is the richest task here, and all sorts of good stuff happens when students get to this point. Like the rest of the assignment, this part starts with a reasonable idea: that we're not going to get these t-shirts for free. If anyone wants to sell anything, they usually have to account for the cost of producing the items they're going to sell.
That's enough to get kids going, usually by making a new table, or simply adding a column to what they've already got. From there, students can graph the profit function, and I encourage them to plot the Revenue and Profit functions together. An even more complete picture emerges when both graphs extend a little below the x-axis.
When they work to write an algebraic rule for the profit function, students really start to develop a deeper idea of how functions work, and essentially come up with the idea of composing functions on their own. After giving them time to play on their own, I provide a brief overview of the definition of profit. I elicit student ideas as we recognize that can use the revenue function that's already written, and that the cost function is just $3 times the number of t-shirts, or 3*n(x).
From there, students see that they can use substitution to write P(x) in terms of x. If they stop there, that's good enough, but of course I love the opportunity to my hands dirty with anyone who wants to simplify things. Looking at work and showing them how to pay attention to every detail - I'll sit with students and edit their work (the blue pen is mine) - it's a blast to watch kids make this knowledge their own.
To close the lesson I want to illustrate the idea that there is a range of results for which we might consider the fundraiser a success. When we look at the graph, it's clear that there's actually a range of pretty good results that extend to either side of the vertex. Some students have already danced around this idea on their own over the last two days, saying things like "$10 is pretty good" or "Whoa, that decreases quickly!" as they fill in their tables or create their graphs.
I start by roughly sketching the parabola on the board, and I ask where the parabola is increasing and decreasing at the fastest rate. Then, I ask where it’s changing slowly. This is an important feature of parabolas, that near the vertex they change the least. I ask the class what they’d consider a successful fundraiser, and we come to agree that anything more than $2000 would be a pretty good pull. What’s nice about this example is that there’s a nice clear range of outcomes in the $2000 range, as prices range from $10 to $14 per shirt. Would it be nice to max out and make $2040? Of course! But no one’s going to complain if the fundraiser nets $2000.
I don't usually go quite this deep, but if you really want to analyze it, the range of possible outcomes that are within 2% of the optimal revenue covers a per t-shirt cost of more than 16% above and below the ideal cost. It’s nice to know when you have some wiggle room.
Students love this idea, and some include it in their own work.