## Discovering Pi: Understanding The Formula for Circumference and Area of a Circle

In this 3 act math task, we will explore the relationship between circumference and diameter of a circle to “bump into” Pi, then show how Pi can be used to help us find the unknown **circumference of a circle** and **area of a circle** by teaching through task.

Thanks to Alice Aspinall and Chez Cetra from Walkerville Collegiate for inviting me to be a part of your Pi-Day that inspired the creation of this task. Looking forward to learning with you all again real soon!

## Act 1: Spark Curiosity

In the first act of this **real world math problem**, we first ask students to create a chart with the words notice and wonder at the top.

Then, they write down all that comes to mind as they watch the following video:

Ask students to take a minute independently to finish writing down anything they notice and wonder.

Then, give them 2 minutes to discuss with their neighbours, before we share out as a group and I jot them down on the board.

Here’s some of what students noticed and wondered when I used this task for the first time at Walkerville:

- How big is the circle?
- Why is there a white spot on it?
- I noticed that there are square tiles around the outside.
- How many square tiles are there around the outer edge?

After having students share out their ideas and celebrating some of the creative thinking, we then narrow the question down to:

How many square tiles are wrapping this circle?

We will have students make a prediction based on what they see and share that out.

If students are reluctant to share initially, I might ask “who has a number higher than ____” or “lower than ____” and that almost always gets a few hands in the air. Students feel like you’re talking directly to them when their number is higher than the number I’ve tossed out there.

## Act 2: Revealing Information to Fuel Sense Making

After sharing out predictions and hearing what students feel they need in terms of given information, you can show them this video or the animated gif below:

Alternatively, you can show this image:

Now that students have some information to work with, they can get to it!

Sharing and celebrating different strategies to solve the problem can really help elevate the lesson to a dynamic and innovative learning experience. I try to find a really “messy” solution to celebrate the need for students to brainstorm by writing **anything** and **everything** they are thinking rather than worrying about their work being a perfectly organized process.

## Act 3: The Big Reveal

Once students have shared out, it is time to experience the solution of this real world problem, share this video:

Or the animated gif:

If you would prefer to show a still image, here it is:

The intention of this first task is to really explicitly pull out the relationship between the **circumference and the diameter of a circle to reveal Pi**. For so many students, they are uncertain as to where Pi comes from, even if they engaged in a lesson previously intended to help them make sense of it. It is important for us to come back to these ideas and ensure that they truly understand this relationship so they can make better estimates when dealing with **circumference** or **area of a circle**.

## Sequel: Act 1 – Spark Curiosity

While students might think the fun is over, that couldn’t be further from the truth.

Show them this image:

Then, ask them to make a prediction and share with their neighbours.

Be sure to have students share their strategies so we get an understanding of whether they are just making a gut shot guess, using a spatial approach or whether they have some prior knowledge in the area of a circle (i.e.: using procedural fluency with the formula) to come up with a number.

## Sequel Act 2 – Information to Fuel Sense Making

Show student this video:

Alternatively, you might consider using this animated gif:

Allow students time to discuss what is going on here. We want students to see that if we cut up the pieces of the circle smaller and smaller, we will eventually have a rectangle and finding the area of a rectangle is something we’re pretty good at.

Can students determine what the dimensions of this new rectangle is?

I’d hold off on showing them to see what they can come up with.

Using concrete manipulatives would be a great way to let them explore this.

Cut up a circle and using marker on the outer edges, where do those edges end up when you rearrange the pieces to make your rectangle?

## Sequel Act 3 – The Big Reveal

After students have shared out their strategies, let them see the visual to ensure that they truly understand where the area of a circle formula comes from:

Alternatively, an animated gif can be used:

## Grab The Resources!

Want to use this task in your classroom?

Grab the Animated Keynote Slide Deck, Powerpoint Slide Deck, and all of the downloadable videos and images by clicking below.

Did you use this task in your classroom?

Please share your results in the comments section below!

## Share With Your Tribe:

## About Kyle Pearce

I’m Kyle Pearce and I am a former high school math teacher. I’m now the K-12 Mathematics Consultant with the Greater Essex County District School Board, where I uncover creative ways to spark curiosity and fuel sense making in mathematics. Read more.

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