I ask the students to start by writing out the formal definition of a permutation:
Then we explore the meaning of this definition by looking a few simple scenarios:
1. How many arrangements of the word SEA can we create? Write out the possibilities as a list and tree diagram. Use factorials and permeation notation to solve.
To help students with a list or tree diagram, I suggest that they start with a letter as an "anchor." I started with S as my anchor and then permeated E and A. Then I shifted to A as my anchor and permeated S and E. Finally I used E as my anchor and permeated A and S:
With factorial notation, students are quick to write 3! and they are also quick to recognize that the notation would be 3P3.
Then when we plug this into our formal definition, we get 3!/(3-3)! which gets us 3!/0!
Here we encounter 0! and We discuss the reason that 0! must equal 1. It is always a fun discovery.
2. We try to arrange the letters from the word BOAT into two spots. After setting up the lists and notation, we get 4!/(4 - 2)!= 4!/2! = 4*3
Here the goal is to help students see how factorials cancel each other out (a theme we return to in the summary)
For this part of the lesson, students work in partnerships on the remaining questions from the permutation template:
I circulate and continue to ask students how to evaluate question in permutation notation. Basically I walk around with a dictionary and ask students to pick out words to use. If the words has repeated letters, I either ask them to pick another one or (if they are ready for the challenge) I show them how to deal with repeated letters.
I will review any question from any student, but question 13 really seems to be tough for them. Even though it isn't strictly a permutation question, they stumble on the process of setting it up.
With three prizes and each prize having a choice, students often try to set up the problem where you choose from all three prizes at once. However this makes the problem difficult and an attempt at a tree diagram futile.
Instead we discuss the useful approach of thinking of each prize as a new "event" and having one taking place after the other. Once we order the prizes, students are able to create the tree diagram and make sense of the answers: there is one way for people to only select cash and 2 ways to not select any cash.