Today is the first day of a unit about probability. Students have seen probability before, and to the extent that they've seen it, it's intuitive. We have the rest of this unit to debunk some of that intuition, but to get started I'll build on what they know, and review some key concepts. Each day will start with some questions about probability that increase in complexity over time. For today and tomorrow, our sample space will be a standard deck of playing cards.
As students arrive the second slide of the lesson notes is projected on the front board. It shows a picture of a deck of cards laid out on a table. I want students to be able to answer these questions about probability, but I don't assume that all kids have seen a deck of cards. Even if they have, this photo is a useful resource for our upcoming conversations.
One copy of this Opener Handout is on each table. I tell students to take a few minutes to try to answer each question. I encourage conversation, and for each table to reach consensus on all solutions. Kids like this. It's fun, and they say it "feels easy." That's because of the intuition built into probability. Now my task is turn that into deeper engagement as I show them something new.
I give each group five minutes or so to answer each question, then we go over the answers. Each answer is on a slide of the lesson notes. Each answer is expressed as a fraction, a fraction in lowest terms, a decimal, and a percentage. The "approximately equals" sign is used when decimal values are rounded from their fraction form. This is what I mean by taking advantage of student engagement when something feels easy. I have the opportunity to remind students about the different ways of expressing numbers. I tell them about the approximately equals sign. I say that in formal probability, it's most common to express probability in decimal form, as a number between 0 and 1, but that fractions and percentages are useful ways for people to talk about probabilities. In the media, we'll often see percentages. In conversations, we might say that we have a "one-in-four" chance of pulling a spade from a deck of cards.
A few of these problems are open to interpretation. For example, how do we define "odd numbers"? Is an Ace an odd number? The solution included here assumes that it's not, but if we change that definition, the result changes accordingly.
One hook for this Probability unit is that probability is about counting. I love telling students this, and seeing where we go with it. Today we'll start with an activity that frames the work of counting, allows us to apply probability to counting, and reframes the idea of sampling (which we started to explore in the previous unit) in terms of probability and running multiple trials to get the most accurate results.
This first investigation is called "Capture-Recapture." Dried beans are used to simulate the counting method used by ecologists, epidemiologists, and anyone else who might want to use statistics to estimate an otherwise "uncountable" quantity, when they try to determine the size of a population.
Version 1: Capture-Recapture from NCTM
Here is an NCTM Illuminations activity called Capture-Recapture that takes students through the process on a smaller scale. There is a four-page student handout that can be used. I recommend reading through this lesson to get an idea for what students do today.
This activity prompts students to count beans in a cup by "capturing" a small handful, marking them, and then "recapturing" before using proportions to estimate the number in the cup. We use a similar process on a much bigger population in the whole-class activity I describe below.
Version 2: How Many Beans in a Four-Pound Bag?
The big idea behind any capture-recapture process is that probability is a tool that can be used to estimate the size of much larger populations, or in other words, to count. So by using the process outlined in the NCTM activity, we can figure out how many beans are in a bag much bigger than the cups students use in the first version.
I hold up a four-pound bag of dry black beans, and ask, "How many beans are in this bag?" As I describe in this video, there are a few ways to answer this question. One is to count. We don't want to do that. One is to figure out how many beans are in a cup (or half-cup, or quarter cup), then multiply by how many total cups are in the bag. I tell students that I've tried it this way, and that I have my own estimate for what's in the bag, but that I'd also like to try the capture-recapture method on this "population" of beans.
So we pour all the beans in a bowl, throw in 300 white beans, shake it up, then use this technique. With students working in groups of three or four, students take samples. One member of each group comes up to the front of the room to collect a "sample." I use the same cups that were used in the NCTM version of the activity, filling the cup about half-way, and telling students to use what they know to estimate the total number of black beans in the big bowl. For this, students are required to use proportions. We know the number of white beans and total beans in each sample, and we know that we added 300 white beans to the mix. The total number of beans is the unknown.
I try to get each group to run at least two trials, and then to compare their results with other groups. Then we gather all results, and we see that the more trials we've run, the better an estimate we can make.
Here are the results of taking ten samples, and this is where we really get to dig into the idea that taking multiple samples is so important. If we only ran the experiment once, we might get an estimate as low as 4,833 beans or as high as 18,900. By taking these ten samples and averaging the results, we get an estimate of 8,539 beans. That's still a little south of the estimate I came up with by counting 450 beans in a half-cup and 10 cups of beans in the bag, but it's perfect for getting the conversation going.
These are the sorts of questions that statistics and probability seek to answer, and even if there's not enough time to dig rigorously into all of these questions, I'm excited to expose students to these ideas.
Hybrid - Using Both Activities
I think this can be a rich activity if you do both versions. I don't always have time to do both. In any given school year, I'll look at how many days are on the calendar and what it seems like my students need. The key is that my students get a feel for the big ideas: that probability can help us count, when we don't know a total, and it's necessary to conduct multiple trials to come to a conclusion, because otherwise the results might vary wildly.
I'll leave it up to you to decide what works best for your students - I hope that what you see here helps you to get ideas!
A guiding theme of this course, and of this unit in particular, is that we can review and remediate essential high school math concepts while exploring new, deeper, exciting ideas. As I describe in this video about this course and problem set, the course is designed for juniors and seniors who have previously had a hard time in math. Most of my students must still pass exit exams in order to graduate. We're nearing the end of the first semester. Just covering content in the same old way isn't going to do it, but denying the importance of passing the tests is also unproductive.
So: we have a new experience. Then, we hit some old-fashioned problems. The message to students is that math exists for incredible and useful reasons, and oh, by the way, there's this test you've got to pass, so let's get ready!
This lesson and the The Proportions Problem Set that I distribute at the end it keep that theme in mind. During the opener and the Capture-Recapture activity, it was natural for students to review some of what they know about percentages and proportions. To solidify that knowledge, I provide the problem set, which consists primarily of questions pulled from state exams. There are a few percentage questions that require students to read a box score from a basketball game; these are included to help bump up engagement and get students reading data. I find that these questions are equally useful for kids who love basketball and follow the NBA (engagement) and for those who don't (reading data with which they are unfamiliar).
I note above that my approach to the Capture-Recapture activity is flexible. This problem set might be a significant part of class, or it might be assigned as homework. It depends on how much time we spend on the main event.