Students will engage in another exploration highlighting the relationship between circumference and diameter of a circle called Pi (π), then consolidate this new learning to emerge a formula for circumference of a circle.

In this task, students will explore the concept of circumference through estimation followed by measurement. This task will allow students to develop a deeper understanding of big ideas, including the following:

• When exploring measurement relationships of any circle, there are three (3) standard measures that can be used to reveal more information about the circle:
  • Circumference: the perimeter of a circle.
  • Diameter: any straight line segment beginning and ending on the outer edge of a circle and passes through its centre.
  • Radius: any straight line segment beginning at the centre of a circle and ending on the outer edge of the circle.
• Knowing one of the standard measures of a circle provides information about all other measures;
• There is a multiplicative relationship between the length of the diameter and the circumference of a circle. In other words, as the diameter of a circle increases, the circumference increases multiplicatively (not additively);
• For all circles, the ratio of the circumference to the diameter is the same (approximately 22:7); and,
• For all circles, the rate of each unit of circumference length per unit of diameter length is the same (approximately 22/7 or 3.14) and is known as Pi, π.

Note:

There are some very complex mathematical understandings happening “under the hood” when we explore the differences between ratios, rates, and other nuances along the journey through proportional relationships. If you want to build your mathematical content knowledge to assist in better understanding the progression through comparison, additive thinking, multiplicative thinking, ratios, rates, and proportional relationships, consider joining us in our 9-module self-paced course called The Concept Holding Your Students Back.

Watch the following video that gives you a sneak peek into a virtual classroom where this lesson is being delivered via online learning:

Note that for privacy reasons, you cannot see student names or hear their voices, but you can get a sense as to how you might facilitate this lesson in a virtual environment and extrapolate to how this could look/sound in a face-to-face classroom environment.

Show students the video below showing someone walking into a building a seeing a circle on the floor surrounded by white tiles.

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

Write down all of the student noticing and wondering. For example:

• Is that a school?
• Who was holding the door?
• 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?
• And many others…

Take time to acknowledge the noticing and wondering your students have engaged in and try to answer any that you can address right away.

Then, land on the following question:

How many square tiles are there around the outer edge of the circle?

Give students an opportunity to make an estimate by first thinking independently and then having them share their estimate with a neighbour.

Ask students to estimate a number of square tiles that they think is too low and a number of square tiles that they think is too high before making their “best estimate”.

After sharing out estimates and recording them on a number line, ask students what information they could use to make their original estimates more precise.

Facilitator Note:

Be sure to reiterate that you want to demote one-to-one counting here. This might mean that you do not leave the image of the entire circle up on the screen, but rather leave the image showing the upper portion of the circle to promote additive (skip counting, repeated addition, etc.) or multiplicative thinking (doubling, tripling, etc.)

After they brainstorm and share out, show the video below:

Follow up with:

Update your estimate.

How might we convince someone that the quantity you come up with is reasonable without the use of a calculator?

Students have now been given enough information to now make calculations using mental math to improve their estimates based on spatial reasoning and/or the prior knowledge they acquired from the investigation that took place comparing circumference to the diameter from the previous day.

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Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.

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After students are finished sharing what they noticed from their exploration, share what really happened by showing the following video:

Here is a screenshot from the final frame of the reveal video:

Answer: 164 square tiles.

Provide students an opportunity to investigate whether this relationship is true for all circles and to reflect on their learning by providing the Printable PDF Handout [ DOWNLOAD ]:

The Printable PDF Handout gives students the opportunity to explore the following:

Using string, cubes, and other non-standard measurement tools, measure the distance around several circular objects (circumference), as well as the distance across the middle of those objects (diameter).

Record your measurements below.

Be sure to make note of the difference between the ratio of Circumference to diameter (approximately 3.14:1) and rate of Circumference length units per diameter length unit (approximately 3.14 Circumference length units per diameter length unit).

To learn more about the subtleties of a ratio and rate, consider exploring this lesson from The Concept Holding Your Students Back course.

On the backside of the printable PDF handout, you’ll find these questions as well:

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