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.

Number Sense: Whatever That Means…

Next up in Algebra 1 is “Number Sense”–I pushed hard for this one, and we have around three weeks. As I talked with a coworker today, he pointed out that much of number sense can’t exactly be taught–a point I only partially agree with.

Wikipedia says, The term “number sense” involves several concepts of magnitude, ranking, comparison, measurement, rounding, percents, and estimation, including: [10]
  • estimating with large numbers to provide reasonable approximations;
  • judging the degree of precision appropriate to a situation;
  • solving real-life problems involving percentages and decimal portions;
  • rounding (understanding reasons for rounding large numbers and limitations in comparisons);
  • choosing measurement units to make sense for a given situation;
  • comparing physical measurements within and between the U.S. and metric systems; and
  • comparing degrees Fahrenheit and Celsius in real-life situations.[10]

All good points, if technically stated.

I’d call number sense being able to understand relative size and equivalency of numbers, estimate both individual quantities and combinations (like products and sums) and locate numbers on a number line. I know I’m missing a lot here, and I plan on refining that definition a lot (notice something? Let me know in the comments!).

That’s all good, but I need to determine how that translates to the classroom. I talk a lot about fractions below, and have a lot more to say about them, but have been really pressed for time. I worked with my elementary school teacher/math specialist sister to develop a lot of what I did, including rethinking my OWN understanding of fractions, and I’m really excited for where my other class goes with this.

One thing for sure: I want to use Estimation 180 throughout this unit, hopefully daily (discovered through Infinite Sums).

Here’s my very work-in-progress plan (these are topics; some might be several in a day, others might take multiple days):

  • Prime Factorization
  • Place value (naming place values, building numbers, identifying place value of a specific digit and vv, naming numbers correctly)
  • Powers of 10 and equivalent forms (x 10, x 0.001 etc)–we will use the Pyramids again here
  • The Fraction Screener (to give me an idea where my students are)
  • Folding Fraction Strips & Defining Unit Fractions
  • Modeling fractions using fractions strips and number lines
  • Comparing fractions & summarizing comparison rules
  • Locating fractions on a number line and representing fractions several ways
  • Fraction equivalency (Stuck on: how to prove two fractions are equivalent)
  • Ordering fractions, then locating them on a number line, including a gallery walk style group placement activity
  • Adding, Subtracting, Multiplying and Dividing fractions (maybe. I teach 9th grade. My kids have seen this over and over and over–I don’t want to repeat the stuff they’ve already seen/heard/done, so I’m only doing this if I really think I can add value)
  • The ONE (using pattern blocks to determine the whole for different size partitions
  • Converting between fractions and decimals (use a calculator? I’m not sure I see too much value in having them convert to /10, /100, /1000 but I’m open to the idea)
  • Equivalent numbers and expressions (both 3/6 & 2/4 and 4+7 = 2*6-1)
  • Basic probability (relating it to fractions too, since who doesn’t like beating a dead horse)
  • Ordering decimals and large numbers on a # line (including a great world populations number line activity)
  • Distance on a number line (and scale? Will this help prepare for graphing?)
  • Order objects (the online game I found here via Infinite Sums)

This is where the order falls apart a little bit…

  • Classifying Numbers
  • Properties of real numbers
  • Perfect squares
  • Benchmark values (along the lines of estimation 180, the average could reasonably be which value,
  • Other number systems, like Mayan and Egyptian (honestly, because this will be on a test I have to give at the end of the quarter)
  • Perimeter and area; frequency tables (not even number sense. Just on dumb test.)

Overall ideas: Does this make sense? About how much would I expect this to be? That’s really the point of number sense, isn’t it? To understand how much it is, without going through a bunch of rules, especially rules that we understand only as rules and not as concepts. The one thing I don’t want to do is rehash the same old same old they’ve already seen year after year and never quite understood.

What does Number Sense mean to you? What am I missing? Any ideas?