Teaching probability gets kids interested because it so naturally involves dice, playing cards, and spinner. What 7th grade doesn’t love to spin things? Usually, they get in trouble for spinning things, so when we pull out the spinners and tell them to spin it, they look at us in disbelief. They wonder if it’s trick. We reassure them that they won’t get in trouble, and quickly they are hands on, having a great time experimenting with probability. I love how incorporating these manipulatives has a way of getting and keeping students’ attention. In this post I’ll break down for you how I chunk out and teach compound probability.

**What I want my students to be able to do with compound probability**

Compound probability has a variety of skills for students to learn. Often when teaching a topic, you can see a natural progression of building within that concept. However, with compound probability I want my students to be able to find the probability of compound events, make a tree diagram, and find how many combinations are possible. These skills are related but don’t necessarily build on each other.

When we start compound probability we have just finished simple probability, so students bring that knowledge with them. By the end of this unit, I want them to be able to answer questions like the one shown before. I want them to be able build on what they already know about probability, and apply it to more complicated situations.

To get started, I introduce students to all of our learning goals for this topic by adding the compound probability I Can statements to their math interactive notebook. I’ve been using interactive notebooks for a few years, but I recently started adding the I can statements to help students see the objectives and have a clear picture of where we’re going. If you want to know more specifics about using I can statements in the math interactive notebook, check out this blog post.

**Compound probability I can statements**

In a nutshell, we have a list of I can statements that we use through out the unit. We read them as a class and talk about them at the beginning or end of every class. I’ve used them both as part of an anticipatory set or as a closure activity. By using them in students’ notebooks and referring to them often, students know what they are learning. It gives them a place to reflect daily and evaluate how well they learn each objective.

Here is a list of the I can statements we used for compound probability:

- I can use a tree diagram to model compound probability.
- I can find the theoretical probability of a compound event.
- I can determine how many possible menu combinations and other compound events.
- I can determine the probability of the same event happening multiple times in a row.

Remember that this list is not the law :-). You might have more or fewer goals than we use. You may phrase them differently. There is nothing perfect about this list. But I hope it’s a great starting point for you. The important part comes from having a way for students to reflect on their learning and articulate what they have learned. I’ll dig into each specific goal, or I Can statement, in the sections below. You can get your own free copy of this exact I can statement insert for the interactive notebook, or a blank template for you to make your own, in the link below:

**I can use a tree diagram to model compound probability.**

We start with using tree diagrams to model exactly what happens in compound probability. Actually, tree diagrams can be great for organization. But unfortunately some kids just don’t think that way. Tree diagrams definitely have their pros and cons. It would be nice if we could just offer them as a strategy for students to use, but my students are tested with them. They have to be able to interpret and understand information presented in a tree diagram, so I have to make sure that as many students as possible understand them. To introduce this piece to students, I start with a discovery lab on this topic. Students get the chance to discover tree diagrams and what they represent before I formally explain them.

We also made real life tree diagrams from manipulatives. This activity showed me that many students didn’t understand the symbolism of the tree diagram. Physically creating tree diagrams gave students a chance to make a tree diagram. It also gave me a chance to really dissect their understanding and give them specific pointers.

We also worked through some typical problems using tree diagrams. This way of representing probability was completely new to every student. Because of that, students needed a lot of support and examples. Even with all that practice, a few of them never got to a place where they could see what the different parts of the tree diagram represented. For those kiddos, they just need a lot more work understanding symbolic representations in general.

**I can find the theoretical probability of a compound event.**

Next, we worked on theoretical probability of compound events. This went well because students used what they learned from simple probability, multiplied the fractions together, and then articulated what that means. We used a lot of different situations with compound events. We looked at different combinations of rolling dice, flipping coins, spinners, playing cards, and others.

I tried to increase complexity by giving them a variety of types of events like landing on odd numbers, or a number higher than 4, or a combination like red, green, or blue. Once students truly get probability, most of those additional complexities don’t really seem to cause a problem.

I work with students who struggle with math and I also have a fair number of English language learners. Sometimes they get a confused about words like even and odd. It was a reminder to me that not all 7th graders have a grasp on every concept that seems obvious from previous grade levels. So, impromptu vocabulary lessons happen often.

**I can determine how many possible menu combinations and other compound events.**

This topic can be shown in a very visible way. But it surprised me that some of my students couldn’t see why there were so many combinations. I think I have the curse of knowledge on this one because it makes so much sense to me (if you’re not familiar with the curse of knowledge, this Edutopia article is fascinating). To explore this side of compound probability, we watched a short video that showed menu options. Watching this video, I felt like I could actually see some of their minds being blown.

The video actually shows all of the possible combinations with five appetizers and eight menu options. After watching, we added three drinks to the situation and tried to find how many combinations were now possible. This threw them for a loop. Several students started off by blurting out guesses. So, we slowed down and went through the process of choosing a meal from the menu. Then, we talked about just how many paths were possible. I wanted students to see that by adding three drinks, the total combinations didn’t just go up by 3. It actually led to so many more combinations than they initially thought.

To get more practice with this, you can have students come up with their own examples of things that have combinations. They can then do the math and show how many combinations exist. The more chances they get to go through the process themselves and not just hear about it, the better their chances of remembering the concept later.

**I can determine the probability of the same event happening multiple times in a row.**

This objective consists of something like rolling a 3 on a dice 4 times in a row. Students have to figure out that they will first find the probability of one event occurring and then multiply that probability times itself 4 times. This seems intuitive, but when you are doing it for the first time, it isn’t. After a few examples and practice runs, students quickly caught on.

In fact, with this concept you can let students see if they can figure it out before you just show them how to do it. Each time there’s someone, or even a couple of someones, who will figure it out. When they have that moment of discovery, the learning is so much richer than just taking notes and practicing.

**Common misconceptions with Compound Probability**

While there are misconceptions students struggle with in compound probability, even more of an issue is the fact that students need to be good at simplifying fractions when they work with probability. We spent a lot of time practicing this skill leading up to our probability unit. Many students had forgotten how to do it and the practice was essential to their success in the unit.

Also, a misconception that they have with the tree diagrams is that they don’t see what it shows. They look at it literally. Then they think they are rolling the dice multiple time or drawing from 12 cards when in actuality it shows the possibilities. A tree diagram shows theoretical probability. It’s not recording the actual results of an experiment. This concept comes as something new for them, and it really helps if you just embrace that as a teacher. Otherwise, it can be easy to make assumptions and get frustrated when students don’t get it. For my students that struggled, I realized I needed to step back and work to build up their background, or schema, to help them understand what was happening.

**Take-aways and putting it into the classroom**

One of my biggest take-aways from this unit was to remember to go in a logical progression. Also, students need a lot of practice before they really own this concept. Luckily, there are a lot of great practice activities out there for this topic. This topic lends itself to being very engaging and hands-on.

Also, students get excited when they experience success. Then, you can use that success to build their confidence in other areas. For my students, using this series of I can statements to teach this topic helped them recognize just how much they were learning. They felt so accomplished when they moved from, “I don’t even know what (fill in the blank) means,” to, “Of course I can do that. It’s easy!” I hope that if you’re not using a series of I can statements to organize your unit, that you try it out. I promise you’ll see your students understand and get more ownership over what they’re learning.

For more activity ideas and resources, check out the post: 12 Easy Engaging Compound Probability Activities.

If you needs some task cards with lots of different types of questions on compound probability, check out our set of Compound Probability Task Cards. Thanks for reading! Until next time!

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