I begin this lesson by recirculating the exit slips from the previous day’s lesson. As I hand back the slips, I ask the students to respond to the Poll Everywhere Question: Of the comments that I have made on your paper, what stands out the most to you?
I am going to try asking my students a question of this nature for a few different reasons:
#1 The students have the opportunity to learn from the misconceptions of their peers. (MP3)
#2 Although it is impossible to talk about everything that I saw on the exit slips, it is easy for each group to highlight one thing that they noticed. (MP1)
#3 With all of the comments on the screen, it is easy for us to pick out re-occurring themes. (MP7)
#4 It keeps me accountable for leaving meaningful feedback that might not only help the specific group, but also the whole class!
#5 If all of the groups with one of the five problems have the same issues, it might be a great platform to call an optional workshop for those students! (MP3 AND MP4)
While identifying WHAT the students are trying to maximize/minimize can be challenging for them, (for example: MINMIZE COST, MAXIMIZE VOLUME, etc.), constructing constraints is often a helpful first step along the way to seeing the larger picture. Although the examples I have provided above seem simple to grasp, the students will struggle because the problem is so multifaceted. For example, don’t be surprised if the students respond with the following misconceptions in the Chicken Feed problem:
- we are trying to maximize egg production
- we are trying to maximize nutrition
- we are trying to minimize waste
Although these things are true at the surface level, what we are REALLY asked to do in the problem is MINIMIZE COST! All other aspects of the problem (the fat, carbs, and protein info - - the limit of 5 oz. of food) are CONSTRAINTS on what we are able to do when it comes to minimizing the overall cost of mixing our feed. This is why first identifying the constraints with the students is beneficial (MP4).
At the onset of this section of the lesson, I ask the students to re-identify their constraints in the problem and think about ways that they can express these constraints mathematically, and/or graphically. While providing the students a few minutes to work together with their partner, I rotate the room and engage in conversations. I anticipate that most students will initially seek to model the constraints with equations. If this is the case, then I will encourage the students to explore a graphic representation of the constraint, so that they might see that equations do not really apply. In most of the cases, the constraints are actually modeled by inequalities, or more complex representations. This is easy once the students graph it and are prompted to think about all of the possible selections. It also makes for great individual and whole class discussion moments.
As the students work, you may need to provided them with hints (in terms of how many constraints are present in their problem). Don’t forget about the “obvious constraints” that none of our values can be negative! The students almost ALWAYS overlook this! FOR EXAMPLE:
Storage Bins: 2 obvious constraints, 2 other constraints
Chicken Feed: 2 obvious constraints, 4 other constraints
Sporting Goods: 2 obvious constraints, 5 other constraints
On the other hand, the investment problem and the lumber problem are a little bit unique. These problems require the students to recognize that they have several variables to deal with (either 3 or 4 depending on the problem). Nonetheless, they can still set up the constraints with these variables, but graphing them will be next to impossible without a sophisticated (3D) graphing utility. Although visiting the graph in 3D is great to do with the class, so show its unique features and why it is three dimensional, it is beyond the scope of what we are asking the students to do in this problem set.