SWBAT to create functions to model volume and find the maximum value of the function.

A classic maximization problem is used to investigate relative extrema.

10 minutes

Start this lesson by putting the task (Activity - Maximizing Volume) on the board for students to see. Don't give them a paper copy yet. Ask a student to read the problem to the class.

Give students a couple minutes to think about this problem on their own without doing any calculations or figuring. I like to have students put their pencils down and calculators away so they can just think about the concept of the problem; I don't want them to get bogged down with the numbers and calculations.

Here are some suggested prompts to get them thinking:

1. What do you think about this problem?

2. What might you need to do to solve this problem?

3. What are the important constraints in the problem?

4. What is the first thing you are going to do to solve the problem?

After thinking silently for a few minutes about the prompts, have students turn to someone around them and discuss their answers.

After that, it is productive to have a class discussion about their responses to share ideas with the rest of the class. Don't try to clear up any misconceptions yet. In this lesson it is fruitful for students to think through things and discover mistakes on their own.

Most students will not know how to get the answer right away, and that is okay!

40 minutes

After our initial discussion in the Launch, I randomly assign students a partner to work with and I give each pair a copy of the problem with the rubric for their final draft (Activity - Maximizing Volume). My goal is for students to write a clear explanation about how they solved this problem. I want to see their thinking.

It is a good idea to give students a specific audience to write for. In this case, I tell them that they should be writing for a precalculus student who was not in class for this activity. Defining the audience gives them a good idea of how in depth they need to go; I’m not looking for them to explain every single nuance, but they should be giving clear explanations of their process.

Visually, it can often be difficult for students to make sense of how the two dimensional square is made into a box. It will be helpful to have students construct the box to see how it is assembled. If time is a concern, you may want to have a few samples for students to manipulate.

To solve the problem, students understand the need to write an equation. Even so, many often have difficulty deciding what the input variable should be. In this case, you scaffold their thinking by asking them what changes as you make different sizes of boxes. Ask students what the dependent variable in this problem will be. Most will recognize that it is the volume. Then, you can ask them what the volume depends upon.

Once students realize that the volume depends on the side lengths, see if they can decide what determines the lenght of the sides. Hopefully, that will be enough for them to see that the length of the cuts dictate the lengths of the sides of the box.

I want students to turn in a thoughtful and coherent explanation for this problem. Thus, they should be writing out a final draft after they have completed their work. This final draft should include diagrams, explanations, graphs, and anything else that was used to solve this problem. If they used a graphing calculator to compute something, I want them to explain what they did and why they did it.

5 minutes

After students have had an extended period of time to work on the task with their partner, I want to have some time at the end of the day for some reflection about the math that they have been working on. As an Exit Ticket, I want them to answer the question below.

**Exit Ticket Question**: What was one thing that you struggled with during the work on today's task?

I will collect these tickets as they leave to get a sense of their progress. I will look for trends in their responses and use them as the starting point for tomorrow's discussion as we continue the problem.