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

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.

• Various cylinders (glasses, water bottles, etc.) for measuring
• on-standard measuring tools (i.e.: string, paper strips, the cord from earbuds, etc.)

Show students the following video:

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Replaying the video and/or leaving a screenshot from the video up can be helpful here.

Then, ask students to share with their neighbours for another 60 seconds.

Consider sharing the following video to give them a closer look at the glass itself:

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up includes:

• 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?

At this point, you can answer any wonders that you can cross off the list right away and of course, you can create your own answers for some of these wonders as you leverage your storytelling skills. For example:

• You could explain who the person is (is it you? a friend? a family member?)
• You could explain what kind of drink they were planning to have (orange juice? water?)
• You could start explaining what was going through your mind as the glass was placed on the countertop and mention that someone once asked you about which was longer: the distance around the top of the glass (circumference) or the height.
• And so on…

Spending times to acknowledge and address specific thoughts that students shared whether a notice or a wonder is crucial to building a culture in your classroom where students know that their voice is being valued and thus encourages them to continue sharing their thoughts and opinions later in this lesson and in future lessons.

After we have heard students and demonstrated that we value their voice, we can land on the first question we will challenge them with:

Which is longer: the distance around the top of the glass or the height of the glass?

Make an estimate.

Be sure to clarify that we will be exploring the distance (circumference) around the top of the glass where it appears to be widest.

Be sure not to skip over asking students to make an estimate using only their spatial reasoning skills as this is a very important step in the Curiosity Path. 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.

Provide students (or ask students to gather) cylindrical objects for them to explore with the intent of updating their original estimate with a more precise estimate.

Prompt students to:

Explore various glasses and/or cylindrical objects and compare the circumference of a base from these cylinders to its height using a non-standard measuring tool (i.e.: string, paper strip, etc.).

Record your results.

Are the results what you expected? Were there any surprises?

Record these reflections as well.

Facilitator Notes:

Some students may be surprised by the results they observe when comparing the circumference of a base of a cylinder to its height. For cylinders that appear to be much taller than its width, in some cases the circumference is still longer.

Other students may not notice much of anything, which is an important observation the facilitator can use to help assess how far along the continuum of comparison they might be.

Are students struggling to make a non-standard measured comparison to see that one length is longer or shorter than the other? Are students making additive comparisons (the circumference is a little longer than) or multiplicative comparisons (the circumference is about 1 and 1 fourth times as long as)?

Give students the opportunity to share out their findings from their investigation with their neighbours.

Walk amongst the groups to listen and observe their discussions so you can be intentional about how you select and sequence those who share and in what order. For example, giving students who haven’t necessarily landed on a firm conclusion first to share their current thinking is helpful to allow them to have a voice in this discussion.

Work your way to other students who may have noticed the surprise that in many cases, the circumference of the base of some cylinders was longer than the height even when the cylinder was much taller than wider.

After students have shared their thinking, give them time to revise their estimates based on what they observed and recorded from the investigation compared to the glass they see in the video.

Show students this short silent animation clip to reveal more information about the glass:

Prompt students by stating:

Which is longer: the circumference or the height?

Update your estimate.

You can also extend this prompt by asking:

About how much longer?

Convince your math community.

The intent here is to promote the idea of students estimating more precisely through a measured comparison using squares as a unit or count.

Facilitator Note:

Without providing students with a few purposeful questions, it is unlikely that many students will hypothesize that there is a relationship between the 8 square long diameter of the glass and the circumference. Thus, you will want to prompt students to go back to the cylinders they had already investigated to determine if there is some sort of relationship between the diameter and circumference.

A prompt such as:

The distance across the circle through the centre (the diameter) is 8 squares long.

Do you think this might be able to help us determine the circumference around the circle?

Revisit some of the cylinders you investigated to see if you notice a relationship that might be helpful.

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Consider sharing the following reveal video with your students:

Consider leaving the following screenshot of the final frame up for students to reflect on.

Students will complete the following consolidation prompts independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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We suggest collecting this reflection 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.

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