SWBAT describe patterns and relationships in a profit maximization problem and to identify the problem.

Look at a different type of problem-- one that has important similarities to and differences from the constant sum and difference problems -- to allow students to ask the question themselves.

40 minutes

30 minutes

To start this off, I just tell the story of the Prom Tickets, with details and embellishments, to the class. I don't give them any handouts or even display this on the projector, because I think it feels more like a story and is more engaging for students than looking at another handout. I provide copies of this document for students after I tell them the story and they have started thinking about it.

Now the fun starts. Students get started. They ask questions and I don't answer them -- pretty much no matter what students ask me, I tell them, "I don't know, what do you think?" or "I don't know, how could you investigate this?" This is a great day to have computers with internet access so that students can use desmos.com/calculator to create data tables and try to find function rules to fit their data. Obviously, there is no need to tell them about how to do this, but I often ask them throughout the day's investigation: "Is there a way you could use the computer to make this easier?" or "Is there a way you could use the computer to test this?" (**MP5**).

Students are the ones who figure out how to think about this problem. This takes *way longer* than it would for you to tell them how to set up the data table, or even how to set up the function. In fact, it make take two or three days for students to fully understand how to use functions to solve this type of problem. This is all essential to making **MP1** a part of the class: students need time to think, without you telling them how to think about the problem, and they need to know that they have time to do things that might not work out. So I constantly tell them during the day's investigation: "You will have the whole week to figure this problem out--give yourself time to think. Don't rush. See if you can check your own ideas without asking me."

10 minutes

The closing of this lesson can vary depending upon what students actually figure out during the investigation. Usually, different groups of students have come up with different ideas. One nice way to close the lesson is to assign partners on the spot, based on students who have figured out different things or who have different ideas. To do this, I take a quick minute while students are still working and just jot down pairs on a piece of paper. I try to make pairs of students that have different ideas about the problem, but who have developed a similar amount of knowledge. (For instance, two students who have both made data tables, but organized them in different ways, or two students who have both attempted to write function rules but who have different rules.) I have found that students get distracted when I write these pairs on the board, and I want them to keep working while I figure out the pairings, so I write them on a piece of paper and then I display this using the document camera (or, if this isn't working, I just read their partners out loud.) I tell them that they have 60 seconds to find their partner, and then they need to learn as much as they can from their partner about the problem in the next 5 minutes.

After this, I ask students to write a check-out about what they learned about the problem so far, and what they still don't understand. You can have students share this with the whole class, or you can skip the writing and have students just share with each other. I like to ask them to write, even though I know that it is difficult for them to communicate their ideas through writing, just because I like that accountability. I stand at the door as students leave and I read their check-outs and ask them quick questions about what they wrote as they leave the class.

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