SWBAT maximize or minimize a quantity using linear programming.

Formalize the concepts from yesterday's Lego problem.

20 minutes

Yesterday’s work involved a lot of exploration and students were introduced to the vocabulary of linear programming near the end of the lesson. The big idea was that we can use systems of inequalities to maximize the profit of the cat furniture.

To begin class today, I give students this worksheet and have them work on Questions #1-4 to recap the work that we did yesterday. The **main purpose** of this is to give them some more exposure to the vocabulary of linear programming. Terms like decision variables and objective function may be new to students, so they will probably need to review them again today. After students work on these for about 5 minutes, we will quickly discuss as a class to review the terms.

Next, I present **the Fruit Grower problem** to students and have them work on this with their table groups. I explain that they should look at Questions #1-5 for guidance if they get a little lost. I will give students about 10 minutes to work on this problem.

In my experience, students can easily define the decision variables, and the graphing of the constraints is usually no problem, but they often have **difficulty setting up the inequalities**. One thing I will try to get them to focus on is what each inequality should represent. In the Lego problem, one inequality represented the yellow Legos and the other inequality represented the green Legos. In this Fruit Grower problem, the inequalities will need to represent the acreage, the days for picking, and the days for trimming. If I can get students to recognize that, they can usually finish up the inequalities fairly easily.

**Teaching Note**: When students graph, it might be most efficient to use a graphing calculator or Desmos so that they can find the intersection points easily without getting bogged down by the algebra of finding the intersection points. Also, since the numbers are fairly large, it might be difficult to precisely graph these.

10 minutes

Once a majority of the class has completed the Fruit Grower Problem (from this worksheet), I will pull the class back together and will **choose one group’s work** to show to the class and we will critique their work and reasoning. I will choose a different group to try to make sense of what their work is.

For the graph of the feasible region, I stress not getting hung up on specific number. We can **make a sketch like this** that is perfect for this instance since we know all of the intersection points. Finally, we can test those intersection points in the feasible region to see which combination will maximize profit.

Like the Lego problem, I will also ask students what the best combination would be if the profit for an acre of crop A was really high (like $950) and the profit for crop B was really low (like $3). I really want students to know that the **objective function is what determines** what combination will be most profitable.

Another interesting situation to consider is what if the **profit for the two crops is the same **(Question #9)? Students will find that the points (60, 90) and (75, 75) will each produce the same amount of profit, because in both cases there will be 150 total acres. If that is the case, could we produce 150 acres of crop A with no crop B at all? Students will see from the graph that we cannot because we will not have enough days for picking.

20 minutes

Here is an assignment for students to work on to cement this knowledge of linear programming.

**Teaching Note:** One of these problems does not involve maximizing or minimizing. Students are asked to only produce a system of inequalities. I discuss more about this assignment in the video below.

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