##
* *Reflection:
The Music Shop Model, Day 1 of 2 - Section 3: Building the Model

By the end of the 45 minutes period, the class had just begun graphing the system of inequalities! The homework assignment was to complete the graph; tomorrow we’ll work out what the solutions set is and what it means. It’s only the 2^{nd} day, and I’m already behind!

For the graph, I’ve found it is *very* helpful for many students to make two simplifications. First, replace *B* and *G* with *y* and *x*, since students are more familiar with these variables. Second, replace the inequality with an equality. This allows students to identify individual points to graph and helps them to see the boundary line of the solution set. After the boundary line has been graphed, reassert the inequality and determine which side of the line represents the solution set. Finally, reinterpret this solutions set in terms of *B* and *G* to make sure that it makes sense of the given constraint. This is a longer process, but I found that many of my students were simply stumped by an inequality like *B* + *G* < 50.

In the final analysis, the primary struggle my students faced was keeping track of the meaning of the symbols being used (see Student Work). Many students would interpret something like “17 guitars” as “17*G*” rather than “*G* = 17”. They were thinking of the symbol *G* as a unit label, like "cm" or "sec", rather than as a variable standing in place of a number. They also had trouble choosing the operation that correctly captured the scenario. For instance, the total number of instruments might be written as a *product* rather than a sum. In this case, I say, "Suppose Jake bought 12 guitars and 18 basses, how would you use your equation in this case?" As they began to explain to me what they would do, it usually became clear that multiplying didn't make sense.

# The Music Shop Model, Day 1 of 2

Lesson 2 of 15

## Objective: In the context of a small business, SWBAT represent constraints as a system of inequalities and identify viable solutions.

## Big Idea: A system of inequalities is used to model a business situation that requires students to balance various cost constraints.

*45 minutes*

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After handing out The Music Shop Problem, the students should be given about 5 minutes to begin reading, thinking, and working in a quiet, individual setting. This individual time is critical in order to allow all students to engage with the problem and begin working toward a solution strategy. I would expect all students to complete problem 1 during this time and to begin making progress on problem 2.

The teacher's role during this time is to check in with as many students as possible to ensure that they are correctly understanding both the situation and the question. I expect that as students move on to problem 2, many will begin to struggle because this problem asks them to formulate inequalities to represent various constraints. Not only are they faced with the challenge of abstracting the situation, but there is the added challenge (for many) of working with inequalities. Be ready to assist, but be careful not to tell the students what to do!

#### Resources

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#### Building the Model

*20 min*

Now, give the class the opportunity to work collaboratively on this problem. Since it is very early in the year, I like to assign students to small groups (about 3 students per group); later in the year, I will begin allowing them to self-select their partners. Also, since I do not know the students very well yet, the groups will be formed more-or-less randomly.

The first task for each group is to come to a consensus on the equations & inequalities for problems 1 and 2. This is the stage at which the group is formulating the model (with some very explicit guidance) by creating algebraic equations & inequalties to represent the constraints in the situation. A good beginning is half the work, so take this step slowly! (**MP 2**)

The next task is for each member of the group to construct a graph of the entire system of inequalities in problem 2. For some reason, many of my students think that a good graph is a tiny graph, so encourage them to make it nice and big and to consider the domain and range carefully before beginning to plot any points. Some students may need a head-start, so I like to get the graph started and have a few copies on hand just in case. Finally, colored pencils may be helpful.

The third and final task for each group is to discuss the meaning of the solution set in the context of the situation. They must carefully decontextualize and interpret the meaning of an ordered pair in order to do this, and it will provide them with crucial practice communicating their own thinking and responding to the reasoning of their peers. (**MP 3**) Many groups will be tempted to look to you for "the answer", but be careful not to give it away just yet - they're perfectly capable of making sense of this problem and only need encouragement. (**MP 1**)

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#### Discussing the Model

*10 min*

A class discussion should follow in which student groups are called on to provide the inequalities, identify particular solutions, and explain the meaning of the solution set.

The teacher should focus on assessing the degree to which students understand the connection between the mathematical model and the real-world situation. To do this, you might ask a student to provide an example of an ordered-pair that violates one or another of the constraints. For instance, "Give me an example of an ordered pair that corresponds to purchasing too many instruments altogether? Too few guitars? Guitars and basses in the wrong ratio?" This kind of questioning tests students' understanding of the boundaries between viable and non-viable options. For strategies and tips on leading classroom discussions, please see my **Strategy Folder**.

By the end of this class period, all students will be able to explain how the mathematical model was formed, interpret a given ordered pair as representing either a viable or non-viable option, and identify the region of the graph in which the viable options are contained. You might choose to make one or more of these objectives a written exit ticket exercise.

#### Resources

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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