Students will use spatial reasoning and proportional reasoning to first estimate and then investigate the relationship between the circumference and the diameter of a circle in order to determine whether the circumference or the height of a glass is longer.

The purpose of this Make Math Moments Task is to allow students to develop an understanding of the proportional relationship between the circumference and diameter of a circle.

This will be achieved by first having students use ratio reasoning to explore the multiplicative relationship between the circumference and diameter of a circle, followed by the discovery of π by revealing the rate through partitive division.

Some of the big ideas we hope students will walk away with include:

• There is a proportional relationship between the length of the Circumference and diameter of all circles.
• We can refer to this multiplicative relationship as a ratio of circumference to diameter lengths of π:1 (or approximately 3.14:1, or 314:100, or 157:50, etc.).
• If we divide the quantities in this ratio using partitive division, (π cm of circumference) ÷ (1 cm of diameter), we reveal the rate of “π cm of circumference per cm of diameter” or approximately 3.14 circumference cm per diameter cm.

If you are uncertain of some of the terms used in the previous sentence such as ratio, multiplicative relationship, rate, or partitive division, then you’ll definitely want to dive into the course The Concept Holding Your Students Back as we dive into these ideas in depth.

Show students this video:

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

You can optionally decide to show this video clip and image to give students a better spatial perspective of the glass:

Here’s some examples of what students might notice and wonder. Be sure to jot them down on the board or chart paper:

• I notice a clean kitchen.
• I notice a glass.
• I notice string.
• I wonder why there is a string on the counter?
• I wonder if you will be using the string to wrap around the glass??
• I wonder if the person is about to pour a drink?
• And many others…

At this point, you can answer any notices and wonders that you can cross off the list right away. Things like “I wonder whether the person is going to pour a drink?” can be addressed right away to show students that we are indeed listening to their noticing and wondering and that we value student voice.

Now you can reveal why the string is on the counter. It is because the question we are going to challenge students to think about is:

Which is longer: the circumference of the glass or the height of the glass?

Make an estimate.

To make this question as interesting as possible, we can specify that we are going to compare the circumference at the very TOP of the glass where it is most wide.

Be sure not to skip over asking students to make an estimate as this is a very important step in the Curiosity Path. This will force them to use spatial reasoning alone to try and come up with an initial estimate and to share their reasoning with their neighbours. 

Providing students an opportunity to make an estimate and try to articulate their thinking with their peers provides a very low floor opportunity for them to not only better understand the context, but to also begin nudging them to think about what will be important to make their estimate more precise as we continue through the lesson.

At this point, we want to give students the opportunity to improve their estimates by engaging in an exploration of the circumference and height of various drinking glasses and/or other cylindrical objects you might have available to you. 

Provide students with a string or paper strip to use as a measuring tool.

You can share the following or similar prompt with your students:

Explore the circumference and height of the different glasses and/or cylindrical objects around you. 

If you do not have a variety of glasses or other cylindrical objects with various diameters and heights, encourage students to create their own by rolling up pieces of paper and taping them. 

Have students engage in this exploration before updating their predictions based on the patterns they are noticing.

After students have compared the measurements of the various cylinders (i.e.: comparing height, circumference, diameter, etc.), you can give them some more information to allow them to make one final update to their estimate which should now be getting closer to a conjecture or a claim rather than just an estimate.

Your prompt might sound something like:

After exploring the various cylindrical objects, use this information to make a claim and convince your neighbours. 

Have students share their final updated estimates with their partners and give students an opportunity to share their strategies and conclusions based on the activity and this new information.

Monitor student thinking by circulating around the room and listening to the mathematical discourse.

After students have shared out their final estimates based on what they learned from the exploration, it is unlikely (but not impossible) that students would have noticed a relationship between the circumference and diameter of the circular opening of each glass since they are likely focusing more on whether there is a relationship between the circumference and the height of each glass. 

Without completely funnelling student thinking, attempt posing purposeful questions that may help draw some attention to the relationship between the Circumference and the diameter of each cylinder.

Consider prompts such as:

What do you notice about the length of the Circumference of each cylinder? Is longer or shorter than you expected?

And/or:

I notice that you're comparing the Circumference of each cylinder to the height, which is great! I wonder if there are any other attributes you could measure on the cylinder to see if you notice any other interesting relationships?

And/or:

Which cylinders (point towards the different cylinders the student is working with) do you think have the longest Circumference? 

(Wait for response)

Why do you suppose that is?

This particular prompt hopefully assists some students in focusing their attention towards what other attributes are getting larger/smaller as the Circumference of each cylinder gets larger/smaller.

Some students might notice that the area of the base gets larger as the Circumference gets larger. Celebrate this notice. Asking them what else gets larger may help some students land on the fact that the wider the cylinder, the longer the Circumference. 

If we can help students discover this relationship by bringing it to the surface during the exploration, this will make for a much easier consolidation.

The tools and representations you might see students using to explore and to make conjectures with their peers and/or the teacher include:

• Concrete non-standard measuring tools like string, paper measuring strip, etc.;
• Connecting cubes iterated around the circumference and then "stacked" to compare to the height; 
• Charts and tables; and,
• many others...

Select and sequence some of the student solution strategies and ask a student from the selected groups to share with the class from:

• most accessible to least accessible solution strategies and representations;
• most common/frequent to least common/frequent strategies and representations; or,
• choose another approach to selecting and sequencing student work.

Facilitator Note:

Even with carefully crafting purposeful questions such as those listed above, this exploration may not lead to students discovering the relationship between circumference and diameter.

Do not panic if students do not come to this conclusion, but rather celebrate their strategies for how they arrive at their conclusion of whether the Circumference or the height of the glass in the video will be longer.

Have students share their strategies and reasoning for determining whether the glass will have a longer circumference or longer height. Ask them to convince you and their peers that their answer is correct by sharing mathematical reasoning.

The circumference is going to be a lot longer than the height because even on this tall and thin cup (holding the tall and thin cup), the circumference was about the same as the height.

The cup in the video looks shorter and wider than this cup.

After you (the teacher) asked me about what other attributes I could measure to try and discover more relationships I noticed that the Circumference was definitely longer on wider cups than on thinner cups (holding a wide cup and thin cup and comparing their bases).

When I used my measuring strip to measure the length of the Circumference and diameter and compare them, I noticed that the Circumference was about 3 times as long as the diameter on both.  

The height of the cup in the video doesn't look longer than 3 times of the diameter, so I think the Circumference will be longer.

When sharing the height of glass (14 cm) and diameter (8 cm), then expect some students who have already been exposed to the ratio relationship between the length of the Circumference and the diameter may come to a conclusion such as:

“Since it takes a bit more than the length of the diameter 3 times to get around the circumference of the glass and double 8 cm is 16 cm, we know the circumference will be longer.”

Other students who have experience working with Circumference of a circle may be able to apply the formula:

C = πd

    = (3.14)(8 cm)

    = 25.12 cm approximately

Since the height is only 14 cm, then the Circumference is much longer than the height.

During the consolidation, be sure to gather student work samples or make note of student approaches while you were monitoring as students work productively struggling.

Facilitate a discussion by referencing some of the student approaches including their strategies, tools and representations. 

If a/some students noticed that there was a relationship between the length of the Circumference and the length of the diameter (whether it was on their own or from your purposeful questioning), be sure to ask one of those students to share their discovery with the group. 

If students have not noticed something about the circumference and diameter is “interesting” or developing some sort of pattern, then you should use two (or more) of the cylinders with different diameters to now directly prompt students to consider the length of the Circumference and diameter of each. Something like:

Look at the base of both of these cylinders. Visualize the length of each diameter and the length of each Circumference. What do you notice? Can you make any claims or conjectures?

Many students will notice that the Circumference and the diameter is longer on the "wider" cylinder and both measures on the "thinner" cylinder are shorter.

Ask students if they notice anything consistent about the relationship between the circumference and the diameter. Possibly even have a volunteer measure both the diameter and Circumference of both cylinders to compare them.

Some students might get stuck thinking additively where they see that a circle with a diameter of say 5 cm has a circumference of about 15 cm and thus the circumference is about 10 cm bigger.

Be sure to help students think multiplicatively by noticing that a circle with a diameter of 5 cm has a circumference that is just about “3 times as big as”.

Have students check this multiplicative relationship for all the openings of each glass and they should come to the conclusion that the circumference of each opening is about 3 diameters in length or 3 times as big as the diameter.

We can now name this relationship that we've noticed as the ratio of the length of the Circumference to diameter as π.

After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what really happened with this video:

Show the video below to allow students to see an approximate relationship between the diameter of the opening of the glass and the circumference of the glass. Ensuring students understand that this proportional relationship reveals that the ratio of the circumference to the diameter (is a multiplicative comparison (i.e.: 3.14 units of circumference to 1 unit of diameter) that can reveal a rate through partitive division (i.e.: 3.14 units of circumference per unit of diameter).

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

How much taller must the glass be to have a height the same length as the circumference?

How do you know?

Consolidation Prompt #2:

Design two glasses and challenge others in the class to determine whether the height or circumference is longer.

Have them convince you.

We suggest collecting these reflections as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

Consider extending this lesson with the following extension prompts such as the following:

Extend #1:

How much taller must the glass be to have a height the same length as the circumference?

How do you know?

Extend #2:

Design two glasses and challenge others in the class to determine whether the height or circumference is longer.

Have them convince you.