A highlight in my math class during this past month was teaching area and circumference of a circle. I’m excited to share with you a twist on the traditional teaching of this topic that made a huge difference for my students. One thing that has thrown my students for a loop in previous years was getting the area and circumference of a circle mixed up. Over the unit I often heard this chorus of questions: What was that formula again? 2 pi r squared? Which one is this again- circumference or area? When do I have to square the unit?

So, this year I wanted to start off by preemptively addressing those common misconceptions. How did I do that? By building students’ conceptual knowledge before we ever talked about formulas and calculations. I started with a discovery lab approach to get them to explore circles. (I’ve talked about the discovery lab approach before here). They would draw conclusions and build their understanding before using any formulas to solve problems. And as we near the end of the unit, I haven’t heard the same questions that were so common last year. Hooray!

##### Getting started

We began by refreshing our memory of area and perimeter from previous grade levels. I gave students rectangles and squares on grids. I asked them to find the area and perimeter of each shape. My goal was to link to what they already knew so this didn’t seem like such a foreign concept.

Even though this was below grade level work, it turned out great. They already had a background in this topic, but they really noticed during our review the difference between counting the squares (finding area) and finding the distance around the outside (perimeter).

##### Exploring area and circumference

Then, I asked students to estimate. I gave them a circle printed on grid paper (with square inch grids) and asked them to color the circle blue. Next, students were asked to estimate the area of the shape. They could count the squares and come up with an educated idea of about what the area was. Coloring first helped the students focus on what they were measuring. We could easily talk about the area as “everything in blue”. The square inch grids were such a great addition. It helped students analyze the circle and have a visual tool to talk about its area.

Next, I asked students to trace around the circle with red. Once they had the outside of the shape color coded, I asked them to estimate the circumference, or the distance around the outside of the circle. I gave them some wiki stix I had laying around the classroom (but you could also use string or pipe cleaners) to trace around the outside of the circle. The wiki stix worked great because they were easy to shape to fit the circle. Then, students removed the wiki stix and measured. They recorded their approximation below their approximation for the area of the circle.

After all students made their estimates, I showed them the formulas and we calculated the actual solutions. They compared their estimates with the final answers. It was fun to hear students talking about how close they were to the actual answer, and talk about how their approaches worked.

One adjustment I’m going to make for next time is that I won’t talk about the formulas just yet. I’ll just give them the correct answers at this point, and we’ll talk about what those mean. I felt like hitting them with the formulas here was just a bit before they were ready for it.

##### Student Reflections on this Discovery Lab

Finally, I asked students to reflect in their own words. I asked them to explain what they thought the difference is between area and circumference. Then, students identified their most important takeaways from this activity. They wrote things like, “Oh, so the circumference is like distance. It’s the distance around the circle.” Yes! Happy teacher moment right there!

After completing this discovery lab, we got into the traditional lesson sequence. Students took notes in their interactive notebooks on the vocabulary they need when working with circles. I introduced the formulas to them. They practiced finding the area and circumference of a variety of circles before branching out into semi circles and other circular applications.

Throughout all of this work, this discovery lab activity served as an anchor of sorts. I saw students refer back to it often. What I didn’t hear, though, was the same level of confusion between area and circumference from students. By starting with a hands-on, inquiry activity, students built a conceptual framework that helped them make sense of the whole unit. I can’t wait to teach this unit again next year! If you want to check out the student resources I created to go along with this lesson, click on the picture below 🙂

Curious about other discovery labs I’ve used with my students? Check out these other blog posts:

- Teaching Systems of Equation with Discovery Labs
- Exploring the Volume of Cylinders, Cones, and Spheres
- Teaching Effects of Transformations
- Helping Students to Discover More Math with Discovery Labs

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