The Counting Principle and Permutations
Chapter 4 - Day 9 - Lesson 4.7
Use the multiplication counting principle to determine the number of ways to complete a process involving several steps.
Use factorials to count the number of permutations of a group of individuals.
Compute the number of permutations of n individuals taken k at a time.
Activity: Too many choices, too early in the morning
We try to start activities with a question that students can start right away without the need for new information. We want our activities to begin with students accessing their prior knowledge and slowing extending it further and further (the Zone of Proximal Development).
We being with the scenario pictured at right. This is a display from Dunkin’ Donuts. We was to find the number of ways we can order our coffee. At this point, students do not know the multiplication counting principle. So to answer this question many groups started by creating tree diagrams or tables. Some students were able to reason through the process and came up with a sort of modified counting principle.
In the group’s work at below, you can see they began by finding the number of ways to order a small original. There were 8 ways. They extended this to reason that there are 32 ways total to order a small; multiply this by 4 (S, M, L, XL) to get the total ways, 128. Many groups did a variation of this, mostly because they felt creating a complete tree diagram was too much work. When we went over the work with the groups and identified the counting principle, many students groaned, and said, “I can’t believe it was that easy!” We love it when this happens! It means we’ve created a need for the material. You should go over the answer to the question and explain the counting principle before the groups move on to the rest of the activity.
Now that the students know about the counting principle they should be able to apply that to the remaining questions. Make sure to add in proper vocabulary when going over the answers. We like to add in additional instruction in the margins with a different color.
Today’s activity is so effective because it creates a need for the students. We were at a conference once where Dan Meyer described the need to “break students’ tools.” The meaning of that is to create a situation where the students can begin by using the tools they already have, but then you modify somehow so that the tools are no longer effective or efficient.