Today’s lesson still focuses on matrix multiplication, but it will have a completely different feel to it. Instead of the entries in the matrix representing units sold or dollars, the entries will be only 1’s or 0’s. These 1’s or 0’s will represent the presence or absence of an available flight between two airports.
I start class by giving my students the worksheet and talk through the first couple problems together so that they understand the diagram. After this discussion I have them work with their table groups on the front side of the worksheet. For question #5 students are to create a matrix that makes sense based on the diagram. I am always surprised that many of my students will come up with the correct matrix without me telling them anything about it.
The correct matrix will be a 5 x 5 square matrix (as shown below) that represent every possible flight as shown below. It can be read from row to column, so a “1” in the N column and A row means that there is a direct flight from Norfolk to Akron.
A common matrix that I will usually see will be one that lists the cities along the rows, and then the columns will represent number of departing flights and number of arriving flights (as shown below). This is great organization, but you can ask your students what information is missing in the matrix that is present in the diagram. Since you cannot see which specific flights arrive or depart, we will want to add that to the matrix somehow.
If a student is completely stuck, just getting them to think about the dimensions of the matrix is a good start. A lot of times my students will decide that the matrix should be 5 by 5 after some prodding, and then will be well on their way to finding the correct matrix.
Now that the communication matrix is set up, the next phase is for students to make sense of it once we raise it to different powers. I tell students to use their graphing calculator to find the matrices for #6 and #7 on the worksheet and then to work with their table groups to interpret the new matrices.
Many of my students correctly conclude that A2 will give the number of routes with two flights from one city to the next. Some students will think that it gives the number of routes with up to two flights. The distinction between exactly two flights and up to two flights is important for the rest of the worksheet, so I have students verify their conjecture. Students will often list the possible flights to prove or disprove one of the conjectures. I discuss more about this in the video below.
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The rest of the worksheet can be completed as a whole-class discussion. I will alternate between a few minutes of work time and a few minutes of class discussion for the rest of the problems.
Finding the correct matrix for #8 can be difficult for students. Many think they should use A3 to get the correct answer, so it is important that they leave class knowing why A + A2 + A3 is needed since the question says “up to 3 flights.”
I love this lesson because it really shows that matrices are a powerful tool for planning and analyzing purposes. Sure, the situation we looked at with the five airports was pretty simple, but what if we wanted to analyze the network of flights for 80 airports? It would be close to impossible to quickly find possible routes with two flights from one airport to another. I discuss this to my students so that they can see the bigger picture than just what we are working on in class.
Here is a homework assignment that students can do to summarize the concepts from today.