The three lessons preceding this one in the probability unit have opened in the way, with students answering 6-8 questions about the probabilities of pulling different cards or combinations of cards from a standard 52-card deck.
Similarly, today's opener (on the second slide of the lesson notes) is about drawing cards from a deck. This time, however, it's just one problem:
If you have a deck of 52 playing cards, is it more likely for you to draw two cards that have the same rank or to draw three cards that have the same suit?
The purpose of using this problem is to show students how to apply what they saw during yesterday's lesson about counting, combinations, and permutations in a more advanced context. I expect this problem to be just a little bit beyond what my students can already do; the main reason for that is that they haven't quite seen a problem like this. By giving students a few minutes to think about the problem, and then showing them how to break it down, I hope to help students synthesize what they've seen so far before setting them off to solve problems on their own.
So that's what I do: as students arrive, the problem is projected on the front board (I should note that desks are back to being arranged in groups of four, after yesterday's lecture-ready pairs). I tell my students to work together to try to come up with an answer to the problem. After a few minutes, I say, "Tell me everything you can about this problem - and if you can't come up with a complete solution, you should at least decide among your group which outcome you think is more likely. Is it more likely to draw two cards with the same rank, or three cards with the same suit, or is the probability of each about the same?"
I let everyone work for 5-10 minutes: longer if everyone is engaged, less time if everyone is floundering. Usually, students are on the right track, getting parts of the problem right, but they need to see how to put it all together. I ask for everyone's attention, and I say, "I'm going to show you how to break this problem down." Slides #3-#9 are designed to help do that.
First (slide #3) we we have to figure out the size of the sample space for drawing two different cards or for drawing three. Yesterday, we saw that there are 1,326 distinct pairs of cards that can be drawn from a deck, so we know that part already, "but how did we come up with the number?" I ask. Students should recall that we used the combination 52C2. So to figure out how many distinct sets of three cards can be drawn, we'll use 52C3, yielding 22,100 combinations of three cards. We all acknowledge: that's a much bigger sample space! I ask anyone if they'd like to reconsider their predictions.
Now that we know the size of each sample space, we have to calculate the number of distinct desired outcomes for each scenario. The questions are posed on slide #4, but we'll have to break these down a bit more. To get started (on slide #5), I say, "Let's consider just one rank, like queens. How many queens are in a deck?" We note that there are four, and then I ask, "How many different pairs of queens could we make?" If students want to list all possibilities, I'll go there with them, and I'll invite someone to the board to write all possible pairs of suits. Then, if no one else brings it up, I'll note that we can use the combination "4 choose 2" to get the same result: there are six different pairs of suits. Next - and this is the part that many kids just have to see once, because it's a new twist on what we've been doing - we acknowledge that there are 13 different ranks, and six possible pairs at each rank, for a total of 78 different pairs that might be drawn from a deck.
We follow the same logic (slides #7 and #8) to compute the number of sets of three same-suited cards we might draw. I give students a few minutes to try this on their own, following the same process we just used. There are "13 choose 3," or 286 sets of three cards in each suit, which we multiply by four suits to get 1,144 different desired outcomes.
Finally, we put it all together to compute the probabilities: we have a 78/1326 (.059) chance of drawing a pair and a 1144/22100 (.052) chance of drawing three cards with the same suit.
By the time we're done with the opening problem, I hope that students have had a chance to solidify some of what they've seen about how to use combinations. I tell them that, and then I switch to slide #10 of the lesson notes to review the Student Learning Targets that everyone is currently trying to master.
I say, "You've all demonstrated that you're confident calculating the probabilities for simple events," as I point to SLT 4.2. Then, "Yesterday, you learned about permutations and combinations, and you've been working on computing the probabilities of compound events." I pause to let students think for a moment before pointing to SLT 4.4. "This learning target is also about compound events," I say, "and you've already seen the addition and the multiplication rules. It's just that we haven't named them yet."
On slides #11-14, I post the rules and tell students to write these in their notes. Each rule is followed by an example problem or two, and I also ask students if they can think of any times that we've used these rules in the last few days. We'll have a brief conversation if students start to share ideas, but I don't force anything.
Then, I summarize by saying, "Just like with permutations and combinations, the most important part about using the addition rule and the multiplication rule is for you to be able to tell when each is the best option. I know you all know how to add and multiply probabilities -- today you should pay close attention to when you should add and when you should multiply."
Today is the fourth lesson of this short unit on probability, but in most classes it has taken 4 or 5 class meetings to get through the first three. At this point, students have run two experiments to help them understand the purpose of sampling, digested a lot of new information during a lesson about combinations and permutations, without ever actually naming the fundamental counting principle, and applied the addition and multiplication rules without formalizing them until just now.
In other words: it's time for a problem set! As I noted in the previous unit, problem sets are a way for us to meet students where they are, by thoughtfully choosing problems that will help them solidify and extend what they already know. Spending a work period on a problem set is a way of slowing down: not in a way that abandons urgency, but in a way that allows everyone to say, "Hmm, ok, I have learned a lot of new stuff, and actually, I do know how to use it!"
So with that all of that in mind, here is today's problem set. It is organized by learning target, drawing on each of the three SLTs that I just reviewed with students.
The first six problems are about simple probability, but they also draw on the kind of elementary number theory that students should know as they leave high school. Students must recall what they know about multiples, perfect squares, and prime numbers to answer these questions. Problems #5 and 6 provide the opportunity to tell students about the prime number theorem, if anyone wants to go that route. That's another use of problem sets: with so much incredible mathematics to squeeze into any given school year, I always look for ways to expose students to bigger ideas that they might explore beyond this class. If a student or two decide to investigate the prime number theorem independently, then that's a win!
The next six problems are counting problems in which students will apply what they know about combinations, permutations, and yes, the fundamental counting principle. Problem #7 requires the outside knowledge that a batting line-up consists of nine hitters - I included this as an example of how cultural knowledge can affect our ability to answer math problems, and because I have a bunch of baseball players in class who will be excited to tell anyone who doesn't know what they need to solve the problem.
The last eight problems are applications of the addition and multiplication rules. Note that throughout the problem set, I use problems specific to my school and some teachers there. There are so many boring probability problems out there (and I know this problem set is not immune to that), but it's not to hard to play around just a little bit and have enough fun to engage the kids.
I give everyone about 45 minutes to work on the problem set. They should complete all problems on loose-leaf paper. They can work together or in groups, but I tell everyone that I will assess them on the work they submit with their name on it. I look for opportunities to teach mini-lessons as needed. I might have small groups of students come stand with me at the board while we discuss a problem, or I might ask for everyone's attention to have a quick whole-class discussion about a problem. If I know that one student has had great insights about a problem, I'll refer other students to her for help. Students probably will not finish the set during class, so I assign what's left as homework.
At the end of class, I run a quick check-in quiz. Like I've noted previously, this class is designed for students who have struggled with mathematics before. My goals are to expose students to engaging new ideas, and also to help them prepare for their exit exams. It's the most fun to focus on the first goal, but it's not too hard to spend just a little time to achieve the second.
Just like the problem set, today's quiz is dynamic. I'll use Problem Attic to choose 5-10 multiple choice problems related to the three learning targets on the problem set, and the difficulty level depends on my experience of where the kids are at. If they demonstrate mastery by choosing the right answers, I'll give them credit for that. If they have a hard time answering these questions, I'll have a roadmap for how to remediate.
I like to prepare this quiz on timed Powerpoint slides, like I did for the opener of the Moore's Law lesson in Unit 3; this strategy works just as well at the end of class as at the start. The advantage of creating timed slides is that I don't have to make copies, and I can quickly swap problems in and out to meet individual classes where they are. Included here is an example of such a quiz, but it's just one of the several I've created for closing out this lesson. More importantly, I want to share how I create these quizzes, so you can use this strategy to meet your students, wherever they are. Take a look at this video, and let me know if you have questions.