I start this class by reviewing conditional probability (from the previous lesson).
Review Problem 1: Screen Shot 2014-06-15
Review Problem 2: Screen Shot 2014-06-15
I like Review Problem 1, because seeing 1 die on the table does impact the probability of seeing 2 ones, and this is a seemingly counter-intuitive scenario. Students insist that the die on the table doesn't alter the die under the table, but it does.
I remind students that we are already half way there. We already have a one and only need one more. So our chance is 1/6 instead of 1/36.
What I like about Review Problem 2 is that is brings the idea of mutually exclusivity into the mix. There are four colored pegs to choose from and we want to know the chance of getting any two colors in a row.
So we can reason that you can get a match in the following ways:
RR, OO, GG, BB
For each pair, there is a 10/40 * 9/39 chance of getting a match. If we reduce and wait to multiply, students will see how simple this problem really is:
(1/4*9/39 + 1/4*9/39 + 1/4 *9/39 + 1/4*9/39)
We can factor out the 9/39 and get
9/39(1/4 + 1/4 + 1/4 + 1/4) = 9/39(1) = 9/39
Although it is admittedly a little goofy, I use the Law and Order image to introduce the idea of order and permutation.
To start, students choose 4 of the faces in the image to reorder. They label the faces A,B,C or D (or with whatever symbology they want.) I circulate and see what orders they come up with. In the process of ordering they have lots of questions like, "how do we know if we have reached all the possibilities?" I always suggest that they build a tree diagram to support their list.
The goal is to let students slowly process the 24 possible orders.
If groups finish early, I offer a few extensions:
1) How many orders could we find if we used all five faces in the image?
2) How many ways can we order the letters in the word TRIANGLE?
3) How many orders in question 2 start with the letter T?
4) If we have the set of numbers (1,2,3,4,5,6,7) and use letters without repetition, how many numbers can we form between 200 and 500, inclusive?
The main goal is to connect the tree diagram to the idea of a permutation. I want students to see that a tree diagram is an expansion of a permutation. We are literally looking at a model that shows the multiplicative growth of ordering. I have a students present how they arranged the faces from the image. To do this I simply give them screen shots of each face in a notebook or ppt file and then let them drag the faces onto their tree diagram.
We finish by introducing the factorial notation 4! and its meaning in this problem.
The conversation goes something like this:
"We started with 4 faces in our tree diagram and for each face we had 3 other possibilities and for each of those we had another 2 possibilities and then finally only 1 other possibility. Is there a faster way to use this idea to find the total number of orderings?" Students are quick to recognize that multiplication would count our total because we use the pairing language of "for of the 4 first choices there were 3 others." This type of phrasing denotes 4 x 3 or 12. Then we finish with the factorial language.