(Complete the example. Practice practice practice. Quiz. Quiz. Done until end-of-year review time.)
That formula, with the combination and the n's and p's and r's and q's, is like a bouillon cube. Very useful, but not intended to be consumed raw. (To rip off an Ellen Kaplan analogy.) As awful as that slide is, this is what is going on in classrooms across America, because the sobering volume and ill-conceived spiraling of content means we are rushed beyond reason. I know Algebra 2 teachers whose ulcers live in fear of a snow day for the loss of instruction time.
So here's my idea and how I intend to deploy it in class. I was indeed fortuitously inspired. I didn't think of this just because Bernoulli trials are coming up.
Please bear in mind this anticipated dialog is boiled down, like bouillon, and there will be much waiting, hemming, hawing, frowning, clarifying questions, etc etc. And, it will go differently in all three classes.
"When I go visit my mom, it's my job to set the table for dinner."
"I was visiting for five days last week, and on Thursday, this happened:"
"My mom said, 'Hey, did you do that on purpose?'
'No' I said. 'That's weird.'
Why do you think we were surprised?"
(The bowls and plates match.)
"Now, my mom thought this was a miracle. A once in a lifetime Fiestaware occurrence. So, what do you think is my question?"
(What are the chances you choose matching plates and bowls.)
Here, someone will assert that I subconsciously tried to make them match, skewing the chances of a matched set. I will acknowledge this but ask if we can, for the sake of learning, pretend I was choosing plates and bowls randomly.
"So, listen. I wasn't so sure this was so improbable, because remember, I set the table for five days."
(It would be more likely the more times you did it.)
"Right. So in order to convince my mom this wasn't so unlikely, what would I need to calculate?"
(The probability of choosing two matched sets one time in five days.)
Write this on the board.
"I'd like you to discuss possible approaches with your partner. You may not be able to solve the problem yet. That's ok. I'm going to give you four minutes to discuss. I want you to make progress. What would we need to know to make progress? Write it down. And if you can't, what questions do you need answered that would help you make progress? Write them down. Is there an easier problem, that you can solve? Write it down. Go."
I don't think I can script any more after this. It depends on what they come up with. They are likely to be able to, in order of increasing difficulty/decreasing likelihood:
- calculate the probability of choosing one matched set
- calculate the probability of choosing two matched sets, performing the experiment once
- calculate the probability of choosing two matched sets on a particular day during the five days
After the four minutes, I would ask each pair what was their finding, or what is their question, and write all these on the board. And then I would go from there. With the intention of deriving the formula for Bernoulli trials that day or the next day, with copies of some practice problems at the ready to hand out, once we got this one.
Suggestions? Better ideas? I readily admit this question kind of sucks, because there's no intrinsic buy-in. Nobody really cares about the dishes at my mom's house. Or maybe it is the best I can do in my classroom, because I thought of it, so I can do it justice. What do you think?