Discrete Math - Day 8

This class started with a discussion of the second part of the pizza problem. Students were stuck on trying to use 5! as part of the solution, typically multiplying this by 6 to account for the sauce-crust combinations. I asked them what the 5! represented in the context of the problem. Most responded that it represented all the different ways that the five toppings could be created. I then asked if placing the five toppings on top of the pizza in different orders created a different pizza. Students realized that this did not change the pizza. I reinforced that 5! would represent all the possible arrangements of the 5 toppings but once they were placed on the pizza it was still, in essence, the same pizza.

I had students revisit the first part of the problem in which they calculated the number of 1-topping pizzas. I tried to get students thinking about how many 2-topping, 3-topping, 4-topping, and 5-topping pizzas could be created. At this juncture many students started creating lists to count these pizzas.

A couple of students caught on quickly to what they needed to count and found a pattern of 5, 10, 10, 5, 1 for the number of topping pizzas that could be created. They summed these and multiplied by 6 to account for the sauce-crust combinations.

Other groups started to get a better idea of what they were doing and I had these two students visit other groups to discuss what they had done.

At this point one group called me over to show me a connection to the pattern that they saw. They had included pizzas without any toppings and saw that the number of pizzas for each topping was

     1, 5, 10, 10, 5, 1

which happens to be the 6^th row of Pascal's triangle. They were very excited to make this discovery.

Students were ready to discuss the problem and the emphasis was that there were multiple ideas at play in the problem, which is more typical of what happens in counting problems. I showed how the number of 3-topping pizzas is connected to ₅C₃ since were are selecting a group of the items from the 5 available.

At this point I had the one group share their discovery of the connection to Pascal's triangle. I wondered whether it was a coincidence or if this would occur for 6 toppings instead of the 5 we worked with. Since pepperoni wasn't listed we added this and students confirmed that the counts matched the next row of Pascal's triangle.

I had told student's that Pascal's triangle shows up in unusual places and here it was showing up in the pizza problem. I don't go into the binomial expansion in depth but this would be one place where an investigation of binomial expansions could be inserted. I did lay out binomials and after writing the first few terms out I asked students to focus on the coefficients of the binomial expansions. They were amazed to see Pascal's triangle showing up. I simply stated that the coefficients in a binomial expansion can be characterized using _nC_r which is why there is a connection between Pascal's triangle and the pizza problem results.

Invariably students ask why they were never shown this in Algebra as it would have made life so much easier. I have taught Algebra 1 classes and introduced some of these ideas. My feeling is that if students are able to make connections and sense of the mathematics then it is appropriate to introduce the ideas.

We wrapped up this piece with students recording their thoughts on working with permutations, combinations, and general ideas about counting.

We then looked at problems involving the pigeon-hole principle. The sock problem is fairly easy for students to grasp and they quickly understand why five socks need to be drawn to form a pair when there are four colors available. It's easy for students to consider what the worst case scenario is when drawing socks. This makes sense to them.

The gum ball problem tricks students a bit because the inclination is that with three children and six colored gumballs they need to draw 18 to guarantee that all three children have the same color gum ball. After some more thought and discussion students come to the realization that they only need to draw 13 gumballs in order to guarantee that three match.

I cover the general idea of the pigeon-hole principle and ask students to consider of the sock and gumball problems what represents the coups and what represents the pigeons in each problem. I like to do this so that they restructure their thinking slightly to allow for more flexible solutions.

I then presented three scenarios and asked them to explain why the statements were true. The first two are fairly easy for students to explain. The third scenario provides a bit more of a challenge. Only a couple of students were able to reason through the third scenario. I asked students to think some more about this problem.

We'll work on a couple of more problems next class just to be sure that these ideas make sense for students.

Visit the class summary to read a student's perspective of the class and to view the lesson slides.