Building Cat Furniture: An Introduction to Linear Programming
Lesson 12 of 16
Objective: SWBAT find maximum revenue using a linear programming model.
Launch and Explore
I love today’s lesson because it takes an abstract math concept and makes it very tangible to the students. Today we will be working with Legos to build cat furniture and we will analyze the constraints of the problem in order to maximize revenue. Students will actually be building cat furniture (the Cat Tree and the Dual Napping Station) in order to find out every possibility and will find the profit for each case. After this initial investigation we will use a system of inequalities to try to find a more efficient method. This is a lesson where it is really important to do the activity yourself so you are prepared for the issues that may arise.
To begin, I give students the task worksheet and having them work through the problem. Each group will need a bag that contains 8 green Legos and 12 yellow Legos. If you do not have Legos readily available, you could cut out pieces of colored paper to represent the blocks. The important thing is that students have something tangible to move around and manipulate in order to build possible combinations of cat furniture. I have students work in groups of two or three for this introduction to linear programming.
After students have had sufficient time to work (and play with the Legos) I will bring the class together to share our thoughts on this problem. I will choose a group to share one combination of Cat Trees and Dual Napping Stations and then ask the rest of the class if it is possible with the given materials. I will continue this process until all of the combinations have been exhausted.
Then I choose a group to explain which combination would produce the most revenue and have them say how they came to that conclusion. Question #3 on the worksheet asks students to consider if the choice would change based on the price of each piece of furniture. This is a great conversation to have and my students will usually intuitively understand that if one piece of furniture is really expensive and the other is really cheap, it won’t be worth it to use the materials to produce the cheaper product.
This activity with the Legos may have been fun, but it was not an efficient way to find maximum revenue. Now we will try to come up with a way to mathematically find the most revenue without having to find every possible combination of the two products.
I give students this worksheet and have them only work on questions #1 and #2. I don’t expect that most students will be able to just come up with the correct inequalities on their own, so I have built in some scaffolding for them to first think about what the variables will represent and whether we should use equations or inequalities.
Questions #3 and #4 have a blank that can be filled in with the word “inequalities” once we establish it from question #2. I find that after discussing the first two questions, students will understand that this has something to do with systems, but may not know the exact process.
A hint that I will give while students work on #3 is that one inequality will represent the yellow blocks and one inequality will model the green blocks. Something I often see is that students will try to create an inequality to represent each piece of furniture, but it doesn’t make sense with the given variables.
Once students graph the two inequalities, I ask them if they notice anything. The first thing they almost always notice is that the intersection of the two lines is the combination that produces the most revenue. I tell them to consider the last question from the activity when the price of one piece of furniture is much higher than the other and they see it is a different intersection point. Finally we talk about every point inside of the shaded region and how they represent different combinations of cat furniture with the given restraints, but the vertices will always be using the most materials.
In the video below, I discuss how to transition from the graph to the vocabulary of linear programming using Desmos.
To finish the lesson, I explain to students that we just used a process called linear programming. The main point I want my students to know is that it uses a system of inequalities in order to to maximize or minimize some value.
The Wikipedia page for linear programming gives historical background about how it was used in World War II to reduce costs for the army. I will give a little information about this so that students can appreciate the historical significance of what they have just worked on for the lesson.