Probability for Babies

Ok, not babies, but not AP Stats students, which is what I’m usually rambling about.

I teach two sections of Algebra 1, and I like to use that last-day-before-Christmas for some fun with probability.

Once students end up in my AP Stats class, there’s a lot of fixing to do. They’re pretty sure they know probability. Its simple stuff really. Six blue marbles? 8 red? Done. Except that the “probability” they’ve had beaten into then (for how many years!) isn’t all that useful. Probability is not, be definition, neat and tidy. Its LONG-RUN. It’s THEORETICAL. It doesn’t tell you what “will” or “won’t” happen, it tells you what might happen.

I still remember talking to the nurse in college, and saying I wanted to stop taking the pill because it made me gain weight. I’d started it in March of my freshman year to help with my cramps and a few months later the improvement was minimal but I’d promptly gained 10 pounds. The nurse’s reply?

“This is a time in a lot of young women’s life when they gain weight.”

“What, March? I don’t think so.”

“Gaining weight on the pill is a myth. Only 2% of women actually do.”

Subtext: 2%= Impossible. I couldn’t actually gain weight because of the pill, because only 2% do, and 2% is close to 0, so no one does for real. She was totally serious. It did not occur to her that I could be in that 2% (I was. I ignored her, stopped taking, and the weight promptly disappeared.)

I want my students to explore situations like that, even if we can’t “answer” them.

Sure, I know some more sophisticated ways to calculate probability. But that doesn’t make simulating some halfway-there situations doesn’t have value too.

Here are some of the ways I’ve played with my kids, and awesome ones I’ve seen from others around the MTBoS.

  • Jackpot
  • Fire!
  • Roulette (get an iPad app, project it, and let them make bets, then simulate on the calculator)
  • Bumpy Flight: an excellent set up from Mathalicious. I’d have my students make a prediction, then simulate it using slips of paper, technology or whatever else they can come up with. Pooling our results is informative enough, and we can even make a recommendation at the end.
  • Greed: Everyone stand up. 5 loses, everything else gets points. Roll (let’s say…a 3) and tell they can keep their 3 points and sit down or they can risk it and keep standing–but as soon as you roll a 5 you lose it all. Play a couple rounds. See who won the most overall and what their strategy is.

Last year, we did Greed, Roulette and Jackpot, but I definitely want to throw Fire! in my rotation too. I love that these are fun, get kids thinking, and starting getting across the idea that the same probability doesn’t mean the same result.

Jackpot

I’ve talked some about Probability and I’m sharing a couple of simple ways to explore probability with your students. While you not be able to get exact answers, these are great ways to test out how probability really works, and whether what is “supposed” to happen actually does.

So you know Jackpot. It’s a Holiday classic on the Country station here, and super easy to simulate.

Listeners call in and the nth caller gets to guess how much is in the Jackpot. If you guess the exact amount, you win. Otherwise they tell you if you’re too high or low. There’s several times a day you can call in, and the idea is that listeners are keeping track at home.

Here’s how I run it in my class:

  • Everyone write down a number between 0 and 1000.
  • Choose someone to make a guess.
  • Say, “Sorry ____, thats too low/high” in your best cheesy gameshowhostvoice.
  • Get your Vanna White up to the board. All you actually need to know is how many guesses have been made, but its more fun to have them write the amount too (someone may even catch that it would be helpful to record if its too high or low, but I never point that one out).
  • Continue calling students for guesses.
  • Once you finally get it, have Vanna record the # guesses and winning total. Even the winner if you want.
  • Tell them all to write down a new guess, then repeat the above.
  • Hopefully by the next round they realize its totally pointless to write down an initial guess; they should be revising as they go.
  • Then split them into groups of 3-4, and show them how to use randint on their calculators.

There are a couple places you could go with this:

  • Have them go from 1000 to 10000 and see how many more guesses it will take
  • Make a class dotplot or other representation of how many guesses it took
  • Tell them you plan on doing this in an upcoming assembly but they can pay you off to get to guess at a certain time. Think about it on your own first, then talk it over with your partner. Discuss as a class why they picked that number of guesses. (Some kids will want to guess 5th to make sure they ALWAYS get a guess, others will want to go 10th because if it does get to them they feel they’re likely to get it.)
  • See if any groups can figure out how to make an optimal number of guesses.

This is pretty informal, but my kids love it and it gets them thinking about how the same situation doesn’t mean every detail is identical.

What are some fun ways you have to get kids thinking about probability?

Back to Basics

There are so.many.things out there on math blogs. I share my own, I forward others to my department, I try them in my class.

And I love it.

After this post, a comment sent me to Math With Bad Drawing’s Probability stories, and Ben Orlin graciously gave me permission to repost them on my (private) class blog. One of my students even asked when their next bedtime story was!

I love coming up with really interesting ways to teach things, and engage my students, and do inquiry.

But sometimes I need to take a step back and remind myself that isn’t always the answer. (I think.) As I mentioned, my AP classes have just started Probability. I’m not that great at teaching it. It’s my students weakest area, so its a safe bet its mine too (and my fault). Most of my class had to retake that test last year.

So I spent a lot of time and agony on this years schedule. I added a couple of days. I tried to come up with great things to do.

But you know what I think we need right now? Some practice. We need to do some problems. We need the time and the space and the permission to draw 20 Venn diagrams and fill them in correctly, til it isn’t at all scary anymore. We need to find conditional probability of six different situations, one at a time. And we need to make sure we have our vocab and probability rules down.

Ironically, this is so, so easy to plan. I wrote 5 problems. I’m done teaching for two days. It feels lazy, but that doesn’t mean it is–and it doesn’t mean that’s bad for my students.

I wish I was better at really teaching this so they got it…but until I am, giving them to time and space to really practice (and review each problem, one by one) is the most beneficial thing I can think of.

Do you feel guilty when you do “boring” things in class? Any more ideas on how to help my kids with probability?

Lions and Tigers and Bears (Rambling about probability again)

If you’re any good at teaching probability, you may want to just stop reading for at least a week. I’m on a probability kick as I head into introducing probability in AP Statistics.

I wrote earlier in the week puzzling through what to do about probability in my AP Statistics reasons. There are a variety of problems, but my biggest frustration is that I can’t figure out the problem. That doesn’t usually happen to me. When a lesson goes badly, I usually know which part. When my students aren’t getting it, I can usually narrow down where the disconnect is and then focus in on that. And with probability I just…don’t know. And it kills me.

I mentioned my drama to a coworker, who jokingly asked if I’d heard of formative assessment (novel!). I have, obviously, but my issue seems to come that they are able to do it until we mix things together, they take a test, and it all goes to hell.

So there are two major issues at play:

  • I need to figure out where they are not understanding so that I can make a plan to fix it
  • I need to give them ample time to practice.

I struggle with the last one. Sometimes I feel like I’m “wasting” class time if they just have a work day for their survey project, and yet I think they’ve gotten more out of really doing their surveys (I have a couple groups taking a true SRS of the school, despite my caution not to, and rocking it). So I need to give them probability they can do (you know, because I taught them so they can actually understand it), and then I need to shut up, step back and let them practice doing it.

I think that might look like two things:

  1. Letting them play with some pretty open-ended probability (like Fire!) and then talk about it, without strings attached or to prove a point. I did this last year somewhat when we would play Jackpot and just see what we got. No calculations, just let’s see. I also think Ben Orlin’s stories are awesome–go check them out now, and thanks Planting Ideas for the tip! I’m posting one on my class blog Friday.
  2. Giving them AP and pre-AP style problems on probability and letting them work those out too, maybe first in groups or pairs and then individually.

Oh, and back off my schedule. My pacing is good. I have enough time to review. Matching my own stupid calendar, which no one but me cares about, does not get me bonus points. Or even a cookie. If my goal is for my students to learn Statistics, and ultimately pass a college level exam, that’s where I need to go, not matching a timetable I made up.

What’s your best tip when you’re struggling with a topic? Any brilliant (or even decent) ideas about how I can help my students understand the basics of probability?

Losing (My) Marbles

My AP Statistics class starts Probability sometime next week (I’ve scheduled a couple days of how-to-answer-multiple-choice in there, so I’m not precisely sure when). I’m dreading it.

I think I’m less-than-great at teaching it, but I also think I come up against a lot of baggage when we hit probability. No one thinks they know z-scores. Students readily accept that I will teach them new things about scatterplots. But probability? I’ve got that down.

Probability is easy, right? We started those problems in third grade, and for whatever reason it shows up every.single.year. Marbles? Jelly beans? Candy? Socks? Shoes? Elvis’ jumpsuits? Done ’em all.

And so I take a bunch of students who “know” probability and…it all falls apart.

I suspect the breakdown happens in between the students feeling like they’ve totally got probability (marbles) and getting at the idea of a long-run frequency of something occuring. Instead of probability as “over many tosses, about half will be heads”, it becomes “you get heads half the time, because that’s just the way it is. And since I just got tails, heads is next.”

So how do I undo that? How do I help them see probability as long-run chance, and never ever a sure thing (well, unless its 0 or 1)?

Part of me thinks I should try to teach as much probability as possible without using the word “probability”. Or maybe do some exploration of different probabilities before I start formally teaching the actual content.

Part 2 is how tricky vocabulary is around probability (especially since English isn’t my students first language). Last year I did a foldable with vocab but I’m not sure how helpful that was (if at all).

Should I have the students model the same situation several ways? Generate our own data and use that?

How do you teach probability? Any ideas of where the disconnect might be? Help!!

Bingo and Expected Value

I’ve been trying to find quick little probability related teasers to talk about in my non-Stats classes, and I think this is a great example of a quick problem for AP Stats and a quick thing to talk through in my other classes.

A few screenshots of the Bingo game that somehow ended up on my iPad and I play just because it’s there. They plan these things you see…every day I log in, I get 15 free tickets, and then you can get rewards for things and you get more free tickets. Tickets are good, because you buy bingo cards with them. The other thing you need is coins, to buy power-ups…

Bingo Coin Wheel

You get one free spin a day! And then you win that many coins! But what if I want more? You can always buy another spin…

Bingo Buy Spin

Worth it?

(Just one more…I’m sure I’ll win this time…)