SWBAT solve real-world problems using linear programming and quadratic equations as mathematical models.

Practice makes perfect! Students make use of the structure of problems to solve them with reference to previous modeling problems.

5 minutes

First, give students about 2 minutes to read the Cattle Feed Problem on Modeling Practice and Review.

Next, ask for students to explain the situation in their own words. The situation is complicated, so listen for evidence that all students understand that the owner needs to purchase a *mix* of the two types of feed, and that the mix must provide the correct balance of nutrients per pound. Finally, point out that the ranch owner is also interested in minimizing his cost - just like Alice sending her letters and postcards! This big hint should help everyone to see that linear programming is the appropriate tool for this problem (**MP 5**).

Once it's clear that everyone has made sense of the problem, its time to let them get to work! Since this is a review, I will assign students to groups of 3 right away.

25 minutes

Students will be working in groups to solve the Cattle Feed Problem, and I will spend my time circulating among them to ask and answer questions, offer tips, and provide clarification. For more general thoughts on group work, please see my Group Time strategy video.

Since this problem is intended for review, I will encourage students to share solutions, discuss their answers, and explain their thinking to one another as much as possible. When I see than the majority of the class has completed one part of the problem, I will ask one group to quietly write their solution to that part on the board. By the end of the 25 minutes, the entire problem should be solved on the board for all to see.

I will also refer back to the Music Shop Problem and the Letters & Postcards Problem as much as possible. It's important for students to see that although the context is very different, the problem type is the same, and so the solution strategy is also the same.

A number of groups will probably have trouble creating inequalities to model the constraints in this problem. In this case, I'll suggest that they have a look at the first Hint Card, which should help get them on track. (Please see my **Hint Cards Strategy **video for details on how I use these in class.)

Once students get over this initial hurdle, the next sticking point will likely be the cost equation. There may be some confusion over the meaning of the various variables. In particular, some students may be confused about the meaning of "x" - is it the name of the feed variety, or the number of pounds purchased? Further, some may be confused about the relevant time interval. Be sure to help them understand that since the nutritional requirements are given in units *per day*, their cost and constraint equations should also be understood in units *per day*. The second Hint Card will help clear up both of these difficulties.

15 minutes

Once the Cattle Feed Problem has been solved and its relation to previous problems has been explored, students will begin working on the Motorboat Problem.

The situation in this problem is much less complex, so there should be less difficulty making sense of the problem. However, many students may not recognize the similarity between this problem and projectile motion. The Hint Card for this problem will explicitly point students back to the What Goes Up lesson and help them to think of the current situation in terms of acceleration, velocity, and position.

Once students have a created a quadratic equation to model the motion of the boat, the rest of the problem should be relatively easy to solve. That means creating the equation is the heart of the problem, so be careful not to give away the solution! Focus your attention on helping them to make the comparison to projectile motion explicit, to see that structurally the problems are the same and can be approached in the same way.

Make every effort to ensure that all students have created a valid mathematical model (i.e. equation) before class ends. If they have not yet used the model to answer the question, that will be tonight's homework.