Traditional Lesson:

(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?

Stealing from my favorite stats teaching book: http://books.google.com/books?id=FMX2FPHkG5gC&lpg=PA108&ots=5-xFs0B779&&pg=PA108#v=onepage On 9 September 1981, the winning number in the MA lottery and the NH lottery were the same!

ReplyDeleteWhat are the odds of (insert name of favorite hockey player here) scoring a hat trick?

ReplyDeleteThere was an episode of Numb3rs where a supposed psychic was doing the guess-the-hidden-card thing, and got them "all wrong". At which point the math dude went "um, ALL wrong?" and pointed out that that's as unlikely as getting them all right.

ReplyDeleteI dunno if it's a lot more motivating than your example, but it's got a video clip with a dramatic cop-show soundtrack?

(I think it was 1st season or early 2nd, that's all I remember right now.)

If you've got any gamers in your class, it's almost guaranteed that at some point they've played a game where they were waiting for some random event and eventually had a string of bad luck that got them frustrated. (especially if they're MMO players, you know, camping for raid gear or whatever) That might have some emotional buy-in - finding out EXACTLY how mathematically unfair it was.

Come to think of it, even better might be to simply play a (non-digital) game where this happens. (I can think of a board game example, "Lord of the Rings", but that one takes a couple hours to play.)

That said, that's just me brainstorming and I think your idea is cool. I'll probably steal it. :)

If you have world of warcraft players (as Josh G. said) then ask them how likely it is to get Baron Rivendare's horse and Wildheart Kilt at the same time.

ReplyDeleteData: http://www.wowhead.com/npc=10440#drops:0-15+1

A graph I made a while ago (answers): http://www.scottfarrar.com/geogebra/documents/Baron_Mount_Chance.html

This is mostly tongue in cheek. But if you know you have that one kid who is always asleep because they are playing World of Warcraft 50 hours a week... maybe this will get them interested.

I've always liked versions of the birthday problem. Determine if any two people in your class have the same birthday. Start breaking the problem down. What is a trial? What is success? What is the probability of success? What are the chances in a group of three people? Four?

ReplyDeleteJust a thought. Fiestaware is great, but I also like appealing to adolescents' self-centered natures. This is also a flexible problem that can get rather complex if students know logarithms and how to solve polynomials. It doesn't have to be that hard, but it can work at various levels.

--paul

k12.com blog

Shouldn't the probability be 1/72 for choosing a matched pair? WLOG, one person has color A for a plate and the other person has color B for a plate. Prob. of first person getting matching bowl A is 1/9 and prob. of second person getting matching bowl B is 1/8.

ReplyDeleteMmm, well, the problem with MMORPGs and drop rate on raid gear is a couple kids will be way into it and the rest will feel obligated to check out because they can't be perceived as paying attention something so dorky. I mean, I embrace the dorky, you know I do, but WoW is seen as decidedly uncool at my school.

ReplyDeleteStill and all, I'd be happy to find something useful that would be more interesting for them. I still favor this though because the situation arose naturally and p isn't artificially predetermined.

Hao you are doing 1/P(9,2). It should be 1/C(9,2). The first person doesn't have to get their color first.

No, but the first person has to get the right color. (i.e. that person has to get the bowl that matches the plate. it is insufficient for the two people to get the bowls whose colors match the plates, as the ordering matters when determining whether they have appropriate matches or if they have the other person's bowl, so to speak.)

ReplyDeleteIf you just want to be holding in your hands four dishes of only two colors, you need combinations. (Or calculate 2/9 * 1/8) That's what I'm going for. In my imagining of the problem statement, it would be equally acceptable to set the blue bowl on the green plate and vice versa.

ReplyDeleteBut that's what clarifying questions are for. Thanks for tipping me off to another way of understanding the problem, which I'm sure will be common given that the photo shows the matched colors set together.

I guess I am particularly picky about matching colors. :)

ReplyDeleteI know some people who are convinced that "Shuffle" on an iPod is somehow non-random, because it often plays several songs by the same band in a row.

ReplyDeleteThat ends up being a much tougher problem than the fiestaware, but would still get them thinking.

I think the fiestaware problem is really good, and that you will get more buy-in than you might think. (c.f. Dan Meyer's filling an octagonal water tank video - not something the average teenager already cares about). The key thing is not that it is relevant to their lives, but that (i) it is a concrete problem that they feel they can fully understand, and (ii) they don't feel like they are being led by the nose to a predetermined answer.

ReplyDeleteI know some people who are convinced that "Shuffle" on an iPod is somehow non-random, because it often plays several songs by the same band in a row.This is one of those old ironies about how people are bad at understanding randomness. In fact, Apple's shuffle algorithm

isn'trandom -- they modified it to be less likely by default to repeat an artist, in order to seem "more random". But evidently they didn't go far enough to convince your friends.

ReplyDelete"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."I let my class get sucked in to this for a 25-minute discussion about what probability really is. Big time investment, but it's paying off now because no questions have to be simplified - we don't ignore that one side of the quarter is heavier or that people try to avoid picking the same number twice, etc. Some calculations get harder and some don't. The biggest success is that the students think about which probabilities they can trust and which have them unconvinced.

Kate,

ReplyDeleteAs I was searching for more SBG help, I came across this post and thought it was my duty to preserve a future "SBG Hall of Famer." The redish-orange Fiestaware plates in your mom's cupboard should be donated to your physics teacher (http://www.orau.org/ptp/collection/consumer%20products/fiesta.htm).

The harm comes when tiny chips of paint are ingested as you scrape up food with a metal utensil.

We used these plates as part of our radioactivity lessons at UMO.

I liked the math lesson you came up with though!

Matt

Matt - Thanks. I'm aware the red Fiestaware used to be toxic, but these were just bought within the couple years.

ReplyDelete