Final Answers

Last week, my AP Statistics class took their final exam: 20 multiple choice questions and 4 free response, basically a half-AP.

I wanted to review the answers with them, especially noting some common mistakes, and also give them a chance to reflect on how it went. I’m really excited with how it went down.

Students went into their (new!) groups of 4 (new seating chart today) with a copy of each question, distributed one per person. My second class had half groups of 3 and half 4, which didn’t matter here (the ghost just got the fourth question and they passed through the ghost).

They got 2 minutes to silently answer as much as they could on the question in front of them, then passed and had two more minutes, and so on. Then they had 10 minutes as a group to come up with their best answers to all four questions. There were a ton of good discussions, both in writing before they were allowed to talk (“I got that too!” “I think its center not spread because…”) and verbally.

Once that wrapped up, I reviewed solutions and they scored their group solution. It also gave me a chance to remind them how rigorous the AP exam is–for students who are used to getting perfect scores fairly easily, the difficulty of the AP exam comes as a shock, and they have a hard time wrapping their heads around 75% correct on multiple choice being GOOD.

It was one of those periods in which I kind of felt like I wasn’t doing much, but I think it was so valuable for my students. This would even work well for a word problem (each person does one step) or explaining several different kinds of problems for a review. By the time they were allowed to talk, several students were really invested in justifying their perspective.

Have you ever tried a write-around strategy like this? Any tips for how it could go even better next time?


The Twelve Days of Statistics

A little late about this, but bookmark this one for next year!

Last year, I posted the Twelve Days of Algebra, which I got from ICTM several years ago. But my Stats students were a little hurt when I had to sing them the 12 Days of Algebra in Stats. So we fixed that right up.

My awesome resident took lyrics that some former students had worked on and refined them and now we have:

Behold, happy 10th day of Stats class!

[Lyrics by Miranda De Young, 2013]

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.


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?

Sharing is Caring: MTBoS

I’ve participated in a couple of the Missions put together by the fine folks running this fall’s Explore the MathTwitterBlogosphere.

And I’m cheating a bit, but thats ok, because its well within the spirit.

My school is four years old (or at least, will be once our seniors graduate in June), which means the majority of our staff is pretty young and inexperienced. At nine years, I have far more experience than many of my colleagues. I love sharing (its why I have a blog), and while I think it would be awesome if my colleagues all subscribed to a bunch of blogs…probably not happening.

So I started a distribution list. It began as emailing my residents cool things I saw that I thought we could use in the classes we taught together. Initially I just thought other residents might be interested, but it felt rude to only ask them, so I opened it up to the department–and a whole bunch of people were interested. (Again, almost all of the teachers in my department have less than five years experience and most are in their first or second year).

Google Readers passing this summer messed up my list for awhile, but I’m back on track with Feedly. If you’re interested in doing something similar, here’s my method:

  • Set up a googlegroup (Ok, get your more-technically-proficient-colleague to set up a googlegroup for you because you’re lame)
  • Email your department and see who wants to sign up. I explained that I usually send out a couple things a week, the text is included in the email so its very easy to read, and you don’t have to read it (I’ll never know). I aim to make it as low stress as possible because I think that encourages people to sign up. I’m also pretty open that I look for blogs that fit me, so I don’t send out a ton of geometry (I don’t teach it, so if thats all a teacher posts on I wouldn’t subscribe)
  • Subscribe to a whole bunch of blogs in Feedly.
  • Read them on your computer or your iPad. If you read it on your computer and want to share it, save it as unread because you don’t bother to set up a mail client.
  • Read something cool. Think others might be interested. Hit email on your Feedly app on your iPad and share it. Never write more than two sentences intro, and sometimes write nothing.
  • Share.

I have gotten SO MUCH positive feedback about this. I hear colleagues refer to things I sent out, people have tried these things–its awesome. And it takes very little effort on my part, a win all around.

Do you have a distribution list? How do you share?

If you’re interested in joining my list, let me know and I would be happy to sign you up. 

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!!

Fun With Fractions Day 4

Catch up here on earlier adventures, including Fraction Strips, Comparing Fractions and Equivalency.

Today we worked with ordering and locating fractions on the number line. We started off with a half sheet with 0-1 number lines on one side, and 0-10, 0-100 and 0-1000 on the other. We worked through five sets (we skipped the last one because it was taking forever), having students place the numbers on the line and then putting them on the board & debreifing their results. It went fairly well, although it felt a little draggy. It might be helpful to do one number, then another, then another to help keep the class on a more similar pace.

From there we moved on to a fraction line up. Nothing really revolutionary there, but we tried to be very intentional in our set up. Each group got a poster paper and ten fractions, so two for each student plus two left over.

  • First, look at your fractions and decide which one is bigger and which is smaller.
  • Then decide who has the biggest fraction in the group.
  • Draw your number line and decide what number to go up to.
  • Take one minute to decide where you’ll place your first fraction.
  • Go around and place on fraction at a time. First put your fraction down, then mark its location, then explain to your group why you placed it there.
  • Your group should give you any feedback (bonus: try to do it in the form of a question, instead of saying “wrong, it should go here.”
  • Continue until your first eight fractions are placed then place the last two as a group.

After they were finished, we had them do a gallery walk and write their comments on the other groups papers–I even got part of this on video and some of it was great! One thing I would do a bit differently is to have them move TWO groups away from their own–I saw a lot of looking over shoulders trying to see their paper to compare it.

One group was a little disappointed to see that all the groups had the same fractions–I think next time I might do 8 identical fractions and then change up the last two just a little (so use all fractions with a similar idea behind them like 13/25 and 7/15 and 9/19) so they can especially look at something new on the gallery walk, in addition to confirming their own thinking.

We’re moving on to number lines tomorrow, but with large numbers and then decimals, but I’ll continue this next week Tuesday when we pick back up with mixed numbers and adding fractions.

Fun With Fractions: Day 3

After we wrapped up our summary of how to compare fractions, we moved in to equivalency.

Everyone unfolded all their fraction strips and started finding where they had matching folds. By this time, everyone was pretty clear on how to use their strips and has a better understanding that all of our strips are the same size for a reason. After about thirty seconds, we choose one volunteer to tell us one match they had found. They picked /5 and /10 and the class helped list every fraction that matched on that list. From there we introduced equivalent and had the students spend about 15 minutes in their groups using their strips to list as many matching fractions as they could.

After all of the resistance to this kind of work in my Honors class, this class’ attitude was really refreshing. Neither of us saw any lists of multiplied numbers; they really worked with trying to find matching pairs visually*. We listed a few more on the board at the end of the time, and then go into defining equivalence and multiplying by a whole with any factor.

Then we introduced some other models. I really struggled with when was the appropriate time to do this–I didn’t want to tie students to the strips, but I also didn’t want to have so many things going on that students couldn’t focus on the important conceptual understanding. Ultimately, I decided to put it here, after comparison and wrapping up equivalency. We first did tape diagrams/fraction bars, and then moved on to number lines. We did a little the day before around partitioning number lines, but I saved most of it for today. Its definitely harder to work with eyeballing your own partitions, so we actually first flipped our notebooks sideways so we could make an 8″ + long line that worked with the fraction strips. It was a nice bridge between using something that was already the correct size (the strips) and emphasizing that the placement on the number line needed to be accurate.

I’m out for Day 4 at a training, but hopefully they’re doing their best at ordering fractions on a number line. (Although I already am thinking we could have done better by very intentionally choosing fractions that all use the same partitions for more problems to really get into that idea before making them need several different kinds of partitions (/5 and /8 on the same line). There’s always next year!

How do you help your students order fractions? Any ideas beyond “multiply all of the things”?

*I don’t have any problem with multiplying by a whole, but most of my students don’t have any idea why they’re doing it or what it means.