Discrete Math - Day 7

Today was a continuation of looking at permutation and combination problems. The basketball team problem is a nice follow-up to the egg problems from last class. The inclination is for students to consider that this problem is exactly the same.

Students made some nice visual representations for this problem and came to realize that things were different. For others, I asked if the teams of Fred, Jane and Joe and Jane, Joe, and Fred were the same. Once students realized they were the question was how do you account for the duplicate counting? Students worked through these issues with some interesting looks at the problem.

Before discussing their solutions I asked students to consider the similarities and differences between the basketball problem and the egg problem. Students came up with several ideas such as both problems involved 5 items and that factorials were involved. They realized the problems were different in that the egg problem involved arranging items while the basketball problem involved grouping items. I want students to consider these since most counting problems involve combinations of these things and they need to take into consideration what elements are at play in any given problem.

Afterward students presented representations of and thinking about the problem. In doing this I have students who took the wrong path but had some elements that could be built upon present first. One student considered the number of different ways that 3 people could form a team and realized there were 6 orders for the team. The student didn't know where to take that idea but it was an idea that tied into the solution and was worth bringing out. Other students showed the lists they created and this helped present a visual connection for what was happening in the problem. Finally, students talked about getting to their result. The final presenters showed how they started with 5! and wrote it out as 5 x 4 x 3 x 2 x 1. They then thought about the egg problem and realized they had 5 x 4 x 3 orderings of teams. They then found that each three person team had 3 x 2 x 1 orders and they needed to divided through to get a final count, they then wrote

   (5 x 4 x 3) / (3 x 2 x 1) = 10

This was a nice representation that connected directly back to the egg problem but also differentiated what was happening with the duplicate teams. It also allowed a connection back to the first presenter's thinking.

The class then worked on the Pizza Problem. Again, visualizations were a key component of the thinking. Students created different list formats and also created tree diagrams. Students also explained their reasoning in calculating the number of pizza combinations with several different justifications for calculating 2 x 3 x 5.

In choosing students to present, I try to pick an order so that the presentations build upon one another and a story is created about this problem. This enables students at any level to contribute to the discussion and shows that their thinking is a viable way to approach a problem. I constantly remind students that I'm not looking for an answer but am interested in the journey they took to get to an answer.

After working through part (a) of the pizza problem I had students start on part (b). I asked students to think about what is happening in the problem. This was a homework assignment that we'll discuss next class. This problem presents new challenges as it is not simply one time of counting problem. It presages the type of problems that will be encountered shortly.

Visit the class summary for a look at a student's perspective on the lesson and to see the lesson slides.