##
* *Reflection:
Letters & Postcards, Day 1 of 2 - Section 1: Individual Time

This assignment may be better suited for the introductory problem for linear programming. Asking students to simply identify points by examination – after spending the previous few days working with a system of inequalities – seems out of order. That said, this problem certainly wasn't too *easy* for them.

Many students struggled to find potential solutions. The most common points first identified were pairs making 14, such as (8, 6) or (12, 2). After seeing a number of points falling in line, many students mistakenly assumed that *any* point on the line *x* + *y* = 14 was a feasible solution. It seemed that as they recognized the graphical pattern, they simply forgot about the rest of the conditions. In this case, I would point to one of the incorrect solutions and innocently ask, "Can you explain to me why this option is viable?" As the students began to explain, they would quickly recognize that it violated one of the other conditions.

Once students began to also identify pairs making 18, such as (15, 3) and (12, 6), they didn’t recognize the solutions in between. For example, after seeing that along with 3 letters Alice could send either 15 postcards or 12 postcards, they failed to see that she could also choose to send 13 or 14 postcards along with the 3 letters. Again, they were seeing the graphical pattern, but not attending to the *meaning* *in context*. In this case, I would say something like, "Oh, I see. In this case she writes to 14 friends and in *this* case she writes to 18. Good. Would it be okay if she wrote to 15 friends? How about 16?" With guidance like this, they were able to find the rest of the solutions (See the sample of Student Work).

A small handful of students opted to immediately set up a system of inequalities. One of these graphed the lines correctly, but needed help identifying the feasible region. Several others mixed up their axes and plotted the solutions inconsistently.

*I thought it would be easier than this!*

# Letters & Postcards, Day 1 of 2

Lesson 4 of 15

## Objective: 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.

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

*45 minutes*

#### Individual Time

*10 min*

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.

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#### Wrapping Up

*5 min*

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.

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: What is Algebra?
- LESSON 2: The Music Shop Model, Day 1 of 2
- LESSON 3: The Music Shop Model, Day 2 of 2
- LESSON 4: Letters & Postcards, Day 1 of 2
- LESSON 5: Letters & Postcards, Day 2 of 2
- LESSON 6: Choose Your Own Adventure
- LESSON 7: What Goes Up, Day 1 of 3
- LESSON 8: What Goes Up, Day 2 of 3
- LESSON 9: What Goes Up, Day 3 of 3
- LESSON 10: The Constant Area Model, Day 1 of 3
- LESSON 11: The Constant Area Model, Day 2 of 3
- LESSON 12: The Constant Area Model, Day 3 of 3
- LESSON 13: Practice & Review, Day 1 of 2
- LESSON 14: Practice & Review, Day 2 of 2
- LESSON 15: Unit Test