All last week we had Pi on the brain as we learned the formula for calculating the volume of 3D shapes with circles. We got to see Pi in action! In preparing for the week, I really thought about how to help students 1) understand the relationship between the formulas for cones, spheres, and cylinders 2) get really clear on the vocabulary and individual components of the three formulas.

First, I did a Volume of Cylinders, Cones, and Spheres Discovery Lab with students. As I’ve written about before, I really enjoy these Discovery Labs, or guided inquiry lessons, to get students thinking about the concept and making connections and observations before jumping in to the formal instruction- it really just helps students get a deeper understanding of what it is we’re studying. We started by looking at the formulas to find the volume of spheres, cones, and cylinders. I wanted them to really understand what the formulas themselves contained before using them. Students were asked to look at the individual components of each formula, and make observations about the relationship between the three formulas:

Then, after looking at each of the formulas, students completed this graphic organizer to again really understand the individual components that made up each formula. We talked about what each formula has. For example, why do they all have pi? Why doesn’t the sphere have height? By going through this process, I wanted to make sure that students really understood what the formula itself means- kind of like translating a foreign language:

Finally, it was time to see the formulas in action. As a class we worked to find the volume of a cone, cylinder, and sphere which all contained the same radius. Before solving, students were asked to make a prediction about which volume would be greatest and we recorded those predictions on the board. My students were super in to seeing how their predictions matched up with the reality after solving.

Finally, students reflected on their learning. My favorite part was reading their reflections on what surprised them when calculating the volume. For example, the student below said “I was surprised that the cone was so much smaller than the rest.”

It was really interesting to hear what students identified after briefly working with the formulas for volume for these three shapes. I sorted students into 3 groups after reviewing their responses: the “Got It” group, the “On the Right Track, but Missing a Piece” group, and the “Pretty Sure That’s Not Really What We Were Going For There” group. This process was super helpful to know just what to key in on, and with whom, when doing the formal instruction and first practice problems.

##### Formal Instruction on Volume Formulas- Interactive Math Notebook

Then, we did the formal instruction with notes in our Interactive Math Notebooks. I made this foldable for students that helps them look at all three formulas. It also reviewed the key math vocabulary terms of diameter, radius, pi, and height. They need to own these terms in order to talk about what they are doing when finding the volume of these 3D solids.

I love using foldable graphic organizers for notes because they really help students organize what they are learning. And I love the creativity of making shapes that fit the topic. So of course this graphic organizer needed to be a circle itself. Because obviously, with all of the shapes being related to circles, it would just be silly to have a foldable organizer that is a plain Jane rectangle! Here’s what the foldable looks like closed and ready to go into an interactive math notebook:

##### Getting Proficient with the Formulas to Find the Volume

Last year, I underestimated how much practice students needed to be able to use a formula. I think my teammates and I assumed that it would be pretty straightforward and easy- just explain the formulas to students and then let them do some practice and voila! students would be able to find the volume of these 3D shapes. But, when we reflected on how they did at the end of the unit, we realized that they still didn’t own this concept. And we realized that a major reason why is that they weren’t terribly comfortable with the concept of using formulas- it was still relatively new for them. So just saying- here’s the formula let’s apply it- didn’t stick with them the way we had hoped.

To get students a deeper understanding or applying formulas to find the volume of cones, cylinders, and spheres , we spent one day with each shape, really digging in to each formula. During that day we practiced applying the formula to a variety of problems. We followed an “I Do, We Do, You Do” structure to each day, focusing on giving students a gradual release of responsibility.

In addition to more traditional practice questions, students were also asked to grapple with an unusual, or novel problem to apply not only the formula, but also develop their problem solving skills. Here is one example:

Students had gotten pretty automatic using the formula to solve- but they really had to grapple with this. They came to a bit of a halt and had to look at solving this problem differently. It was great to see them work to make a strategy of how they could figure this out, and how they could use the information they knew to find the unknown. My students definitely need more experiences doing this kind of thinking and work. While some of them got it, others didn’t, and after struggling with it for a bit wanted to give up and call it impossible.

When we talked through approaches together, it was fun to see the light bulb moment when those who had been struggling realized this was yet another situation where they could set up an equation and solve to find the unknown. I’m hopeful that by the end of the year they’ll all make that connection themselves, but they still need more practice with setting up and solving problems.

##### Getting Practice with Volume- and Having Some Fun

We had fun practicing the formulas to find the volume of cones, spheres, and cylinders. On our third day we worked with the volume of spheres. After two days of practicing with the volumes of cylinders and cones, they were proficient at using the volume formulas, so by the time we dug into the formula for spheres, they flew through the practice and we got to do some coloring 🙂 We enjoyed getting to the fun activity- the Volume of Spheres Color by Answer. It was a blast to watch students work through it and I got to just keep telling them, yep, you got this! My students absolutely rocked this activity.

What I really enjoyed about this year is that they got much more experience with the formulas themselves. They really got a deeper understanding of what the formulas meant and what the formulas consisted of through the discovery lab. Then, we layered on top of that better understanding more focused practice. Spending a day practicing each of the formulas by themselves really helped them own it. I felt like the whole week just had a really good flow to it and the concepts were chunked out just right for students. Everything we did can be found in this bundle at TPT, including a few games that we didn’t quite get to. Do I really have to wait a whole year to teach this topic again?

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