SWBAT write a system of linear inequalities and use the system to answer questions about balancing time and cost in a real world context. SWBAT leverage mathematical practices 1, 2, 3 and 4 as classroom norms.

Linear systems are useful mathematical models for situations with a several of constraints. Time is money!

10 minutes

Since the Letters and Postcards problem is similar to the Music Shop Problem, I plan to provide little or no introduction. As I hand out the problem to the students, I will mention that since everyone did so well with the previous problem, I thought we'd try another. This problem is set in the "old days" before email, text messaging, facebook, and Twitter, so it might be a good idea to talk with the class for a while about letters, postcards, and stamps. This is a good opportunity for some laughs and for being impressed at how quickly communication has changed!

Now I let the class know that I'd like them to take 10 minutes on their own to read and make sense of the problem. (**MP 1**) Their aim should be to complete as much of part 1 as they can before the 10 minutes is up.

After the students have had a chance to read the problem, and as they begin making their graphs, I'll circulate around the room to answer questions and to make sure that everyone is on the right track. From time to time, I'll stop to ask a particular student to explain to me how he knows that one of the points he's indicated is a viable solution; I'll *always* ask about an incorrect solution, but I'll sometimes ask about a correct one, too. (**MP 2 & MP 4** - *attend to the meaning of quantities, not just how to compute them*) My goal is to assess how well the students have understood the problem, and also to help them to correct their misconceptions if they have any.

30 minutes

5 minutes

During the final few minutes, I like to check in briefly with the class. Everyone has been working hard for 40 minutes, and I'd like to get a sense of how they're feeling.

So I'll ask, "I know you aren't finished yet - and we'll keep working on this problem tomorrow - but what questions do you have right now?"

There's no telling what kinds of questions might come up, and it's not necessarily important to answer all of them, but I want to reassure everyone that questions and difficulties are part of the process.

I expect many students to still be struggling to create equations that define the boundaries of the feasible region, especially since they are explicitly instructed to do so! I'll need to help students understand that an *equation* can be thought of as a description of *where* certain points can be found - in this case either above, below, or on a certain line. This is not an easy concept, but it's a vital one for linear programming.