As teachers we want our students to discover things, but we also like to be in control. Sometimes we feel like we don’t have enough time to let students discover. We just want to jump in and give them a list of steps. I believe there is certainly a place for teaching the list of steps, but I have also seen the power of discovery and I can’t go back to the days before I used it. Now, I start every unit with a discovery lesson. Some topics call for more involved discovery activities, like volume of cylinders, cones, and spheres. Other topics are less complex like simple probability. In this post I would like to share the discovery activity I used with my students for simple probability.

**The thinking process to get students to discovery simple probability**

Sometimes when I do a discovery lab with my students it is arduous and difficult. They struggle and stretch and reach for understanding. Other times the concept is reachable rather quickly and they get it fast. I think there is a place for both types of discovery labs or activities. I think that in this discovery lab students had a somewhat strong background in the topic which made it easier for them to see what was going on.

**Step 1-Building background**

We always build background before we start the discovery activity. In this lab the background we focused on was what fractions represent. Students were given a fraction and a bar cut into parts. They had to shade in the fraction. They were good at some of the fractions and not so good when it came to fractions that could be simplified. This was not a shock as I have seen them struggle with this before. This warm up activity gave us a chance to discuss this concept.

Students did this part of the discovery lab with a partner. They had a lot of good discussion and debate about what they should shade in. I always find it humorous when when students tells another student, “I told you so,” about a math concept. There was some of that going on during the activity. They definitely struggled to shade in a fractions that was equivalent to one half when the bar only had two parts.

This activity only took about 5 to 10 minutes and really set the stage for the discovery activity.

**Step 2-Making observations about simple probability**

I am a math specialist at my school and I teach a class of students who have historically struggled at math. One way that I build their confidence in math is through discovery labs. I want to promote an environment where students can make mistakes and fearlessly learn from them. Discovery labs are a great way to cultivate students’ willingness to try things out. Also, there is nothing like watching the light turn on when they figure it out on their own.

Actually, this discovery set-up the perfect situation for them to feel confident because it was not too much of struggle. The task they completed in the observations section consisted of two parts. First, they were given 6 fractions. Then they were given probability situations. They worked together to match the situation with the probability, or fraction. I did not give them any help or hints. Most of them figured out what to do rather quickly.

**Discovery in action**

After students had matched the situation with the fraction they had to explain why they chose that matching pair. This part of the activity pushed them to write down their thoughts and their mathematical reasoning. Some of their answers were spot on and showed their reasoning like, “Because there is only one 3 on a 6-sided dice.” Other answers made me chuckle like, “It was my last square left.” I emphasized that they needed to show their reasoning for all of them, including the last matching pair.

Next, I had students write their reasons why they chose that answer on posters around the room. You can see some examples below. This really seemed to get students to make sure their answers were good because they knew their peers were going to read them. (a little peer pressure can be a great thing).

**An added challenge**

I did make one modification from the original time I did this activity. I made some of the answers be simplified fractions from the the probability they would see on the spinner. For example, if the chances on the spinner was 2/4, then the correct answer that students would need to choose would be 1/2. This added a little bit of a challenge. They couldn’t just get away with saying it was the last answer left. They needed to figure out why that answer works for that situation and justify their answer.

**Step 3-Drawing conclusions and reflecting on learning**

We always have some sort of reflection in our discovery labs. Usually, students write a rule about the concepts and then reflect on their learning. When it comes to writing the rule, I like to give them a sentence stem to get them started because it doesn’t come easy to them. Therefore, in this discovery lab I gave them the sentence stem, “In simple probability you have to _______________, and then ____________. This really gave them the structure they needed to explain what they observed in the discovery process.

Finally, students answered questions about what they learned, how it related to real life, and what questions they still had. These all pushed them a little bit. Students can get used to there being one right answer. But with these types of questions they have to come up with answers. Here are some of their response to these questions:

- “I learned that you have to have two numbers for a probability. Also, it is like a fraction. Also, I learned it is about what the chances are.”
- “…if there are 8 pieces and it is one half that you have to break it in half and it will be an even number.”
- “It has a lot of fractions and counting pictures. It will have a bigger number on the bottom and a smaller number on the top.”
- “When I do experiments there are chances.”
- “…that there is chance in everything.”

As you can see, students’ responses can be insightful and sometimes they make you chuckle. I use their responses to help me see what they understand which helps me to know what I need to emphasize when we go to the learning part of the lesson. In this case I needed to emphasize simplifying fractions in probability.

**Putting it into the interactive notebook**

Discovery learning flips the way that we normally teach. Instead of giving students notes and then having them work with the concept we do it the other way around. So, when we finish the discovery lab we jump right in to writing notes in our interactive notebook.

Sometimes, with discovery labs, something magical happens. When we get to the notes part the students can fill-out the notes without a lot of help from the teacher. That’s when you know they internalized what they learned in the lab. That’s exactly what happened for some of the students in this situation. They were finishing my sentences and were very confident in their answers. It was an amazing teaching moment. I hope you have a similar experience because it is very validating as a teacher to see this happening.

The foldable that we glued into our interactive notebook organized the information by having students go through the process of finding the simple probability 4 times. First, they found how many times the event is possible. Then they found how many total outcomes are possible. Finally, they made it into a fraction. This works great as a reference for when they are completing practice activities later on.

**Try it out…**

You can try this activity out by purchasing our discovery lab here, or you can create a similar activity for your students. My challenge to you is to let students discover before you give them notes. It makes a huge difference in conceptual understanding and retention. If you have never done this before, I’m confident you will see something really cool happen.

Try it out and let us know how it went by commenting below. We love to see and hear about how these activities play out in other people’s classes. Also, be sure to check out this post with 9 other activities and resources for teaching simple probability.

Until next time!

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