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.

Fun with Fractions: Day 2

I’m blogging our fractions work daily–read Day 1

Today kicked off with a quick review of yesterday’s homework. It was pretty simple, just comparing ten pairs of fractions using their fraction strips (no, I had no way to verify and no, I don’t care). Then there were two questions designed for Higher Order Thinking.

…and, close almost kills them. “Which is greater, 4/5 or 8/10? Using your strips, find the two fractions that are most near in value without being equal.” This was definitely a “but, what’s the answer?” moment. The instructions were perfectly clear, you need to fold the strips. But that concept of close really threw them for a loop. Estimation is a scary scary concept for them.

I did a quick overview of number lines, starting to introduce the idea of equal partitions, sequential v proportional reasoning (which number do you place first between 0 and 4? Most of my students place 1.)–but only for a quick couple minutes.

Moving on, we did a couple more comparison problems as a problem to get us warmed up and ready to go, and then they moved into groups and got a set of 10 comparisons. We didn’t want to get rid of context entirely, but we ultimately want them to classify these by type and didn’t want any groups mistakenly using things like, “these were all boys and these were all girls!” or, “these were about food!”

On the document above, they’re organized for teachers; I just made the questions only larger and printed one set per group of four. We gave them scissors to slice up the problems and then they solved these in their groups. I offered up the answers to anyone who was interested to emphasize how little I care about the answers and how much I care about their explanations. Most of the groups were trying really hard, which was awesome. (Side note: it should be missing piece–7/10 and 6/9 is super hard to explain!)

We ended up needing about half an hour to solve them, and then we gave them 10 minutes to sort them into groups of like problems. We brought it back together to generate categories–the students were having great conversations in their groups but were scared to speak up, but we got there eventually. After matching an example to each one, we assigned each group one category to write up for their peers. [This is where I say this was awesome, which I think it will be, but we have a shortened schedule for PSAT testing for sophomores and didn’t finish. Sadness.]

How do you prompt students to answer tough questions without giving them what you want to hear?

I developed this lesson with my team; the file was created by CC.

Fun with Fractions: Day 1

My students don’t understand fractions. They hate them, they don’t get them, etc. They know plenty of the “rules” but they don’t why they’re doing anything or how it actually works. We’re currently in the midst of Number Sense, and this is how we’re doing fractions. A huge thanks to my sister for her work on this individually and with WISME and CCLM in Wisconsin (for, um, fourth grade math). I’m not comfortable posting the powerpoints here, but am willing to share them.

Day 0:

Fraction Screener. 20 minutes to mostly model fractions in various models, representing things like sharing items equally, sharing items with a remainder, equivalence and partitioning.

Start fraction strips. We cut a rainbow of paper strips (9 colors), as well as spare “practice” strips from scrap paper (yay! Another use for all that blank-on-one-side mental math!) My sister said to cut waaaaaay more than you need, and it still wouldn’t be enough. True. I did and it wasn’t. I asked the students if they could work on folding for homework, and they gamely said yes. [If you do something like this: no writing on the strips, with the exception of blacking in the fold lines to make them easier to see/count. That’s how you find which one is which–counting, not looking at a number.]

Day 1:

While some students made a lot of progress over the weekend, many of the students came in Tuesday with several still-flat strips. We walked around and helped (and so did some of the already-finished students) and I had some very eye-opening conversations. “I need help with ninths.” “Ok. …any ideas which strip ninths might have a relationship with?” “….eighths? tenths?” and the like. Same thing with tenths. This is an on level ninth grade math class, and if anything my students have impressed me so far both with their ability and their willingness to try things. I don’t think this is unique. My students aren’t the only ones dying to blindly multiply by the other denominator ad nauseum because someone told them to.

We took a couple quick pauses in here for:

  • How many numbers is a fraction? (Consensus: probably two. Bizarre: the two kids who said “something else”)
  • How many folds did you make on each strip?

After that, we had the students fold each strip so that one unit was showing. We used the phrasing, “One share of size one-half” and so on, first with single shares, then moving into fractions like 3/4 (way harder to say–try it! Its tongue-twisty!). I tried to stay away from the technical language of numerator and denominator because I felt like students would be more likely to approach this in a new and different way (as opposed to multiply-all-the-things) if they were using different language.

Then we moved through some practice problems having students fold the strips and use them to represent and then talk about different comparisons. Some other discussion worked its way in there nicely, such as when one student said that more shares is bigger and was paused to develop some counterexamples.

I had already done this in my Honors class, and the show in Honors went something like this:

Do three comparisons using fraction strips. resist models. get annoyed. everyone’s frustrated.

This was a lot better. Students are still uncertain, but are trying their fraction strips, willing to trust us and hopefully ready to summarize comparison rules tomorrow (we’re calling it the Big 4).

A few notes on this and the posts that will follow: I’m not going for how-to-compare-every-last-fraction you can make up. I want my students to understand that thirds are larger than fifths because a unit is being split into equal pieces. I realize sometimes, you do need to multiply. But a lot of the time you don’t, and a basic comprehension of a number as being closer to 1 or 0 would go a long ways.

Do you teach fractions? Do you get annoyed when your students resist what you want them to do? Have you ever folded fraction strips?

Sounds like my school…

(See below for a full explanation behind this). We’re training teachers to work in some of the toughest schools in Chicago (the nation?),  so its helpful to have a shared vocabulary. And part of that shared vocabulary comes from this guy:



We roll out strategies throughout the year, and mentor teachers model them for their residents. Full disclosure: This was HARD for me. I have excellent classroom management, but a lot of it is fully my personality, and you can’t reproduce that.

Mentor: Then get them quiet.

Resident: But how?

Mentor: Stare at them.

Resident: I do stare. Nothing happens.

Mentor: Funny, works for me.

Not helpful. But its tough to retrain yourself to be more transparent, even if it does help me grow professionally–discomfort is not fun! Some of these can be cheesy or overdone, and I definitely don’t think teachers should be robots, but being able to say “strong voice” and have both parties know what that means and looks like is really helpful.

My favorite of these strategies, and one that makes my school distinct from other schools I’ve taught at is narration.

Narration is exactly what it sounds like: narrating what’s going on. It’s always positive, and can be either behavioral or academic in focus. It’s immediate, and does wonders to help guide class in the right direction.

Here’s an example:

Pass across your papers, one per pair, and work with your partner at a voice level one. You have three minutes. Go.

[slightly softer] the first row is all ready to work, I hear Samantha discussing problem one with Jose, Evelyn is writing while Samuel tells her what to do, the left side is at a voice level one.

It’s even better when students might be confused about what to do–instead of repeating the directions over and over, students can catch on to what should be happening by the narration.

Elizabeth has her notebook out, Amy is writing down the first problem. Alan is checking his work on number one.

Sometimes my students don’t do what I want them to do because they aren’t sure what that is. So confusion looks like noncompliance. But its such a little thing, they don’t want to ask, and sometimes you get yelled at for not knowing what you should be doing, so I’ll just sit here and hopefully I’ll figure it out… Students hearing the narration above get everything they’re supposed to do. I should get out my notebook and write down the first problem, and then I should check my work.

It works like magic on volume too. Ten minutes into your Around the Room and its getting a little loud? “Eli is working silenty. Esme and Diana are at voice level one”. And like magic, the volume just drops. No one got yelled at. No one did anything wrong. Just a quick reminder about what RIGHT looks like (never wrong) and we get back on track.

Try it out–if you feel awkward doing it in class, narrate your spouse or kids or roommate. Dave is clearing his plate and putting it in the dishwasher…Think of it as I spy. Narrate all of the good you see, and see how many students try to match it!

Have you ever narrated before? Are you going to try it?

Full Backstory:

My school is part of Chicago Public Schools, but also part of a network of schools within that. (That is not code for charter. We are not a charter school, the key difference is in our governance structure–we follow all CPS requirements AND all of our network requirements.)

Our mission is twofold, both training teachers and turning around failing schools. This is a teaching blog, not politics, so my whole point here is that I teach in one of our training academies and am in fact one of the trainers. Trainers–mentor teachers–are assigned resident teachers for the year. Typically each mentor teacher is assigned two residents, but this year I have only one (this is lovely, but I feel bad for her. I am a lot to take alone.) Our residents are selected by June, spend the summer in training and remain with us Monday-Thursday (Friday they have grad classes) for the school year.