Students will explore the relationship between the area of the base and the height of a cylinder when determining volume. Using this relationship, students will investigate volume in order to determine which figure will hold the most popcorn.

Special thanks goes to Dan Meyer whose Popcorn Picker context inspired the investigation explored during this lesson and the extensions shared throughout the unit.

In this task, students will explore the relationships between a cylinder, cone and sphere. Using these relationships, students will determine the volume of each figure. This lesson will allow students to develop a deeper understanding of the following big ideas.

• The volume of a 3-dimensional figure can be found by determining the number of cubic units that can be contained within the figure;
• The volume of a prism can be determined by finding the number of cubic units required to cover the base and multiply by the number of layers (i.e.: the height);
• The volume of a pyramid is one-third the volume of a prism with a congruent base and equivalent height. Therefore, the volume of a cone is also one-third the volume of a cylinder with a congruent base and equivalent height;
• Varying the width of the base of a cylinder (i.e.: the diameter) has a greater impact on the volume of a cylinder than varying the height of the cylinder; and,
• The lateral area of two cylinders (i.e.: the surface area of the cylinder excluding the base and top) can be equivalent while producing inequivalent volumes.

Show students the video of a cylinder being made from an 8”x11” piece of paper rolled widthwise and a second cylinder made of the same size paper rolled lengthwise.

Then, ask students:

What do you notice?

What do you wonder?

Give students a reasonable amount of time (eg., 1-2 mins) to do a rapid write, keeping the video paused on the two cylinders side by side

Now invite a whole group conversation centred on student noticings and wonderings. Be sure to record student thinking in a space (eg., whiteboard) visible to all students. This honours the contributions of all students.

Some noticings and wonderings that might emerge are:

• I notice two pieces of paper
• I notice two cylinders
• I notice the cylinders are different sizes
• I notice one cylinder is taller than the other
• I wonder how big the paper is
• I wonder if we’re going to see other shapes
• I wonder if anything is going to happen next

At this point, you can respond to questions that you can cross off the list right away. For example:

• Each piece of paper is the same size
• You could explain who the person is
• You could explain what the person was trying to do
• And so on…

Now that students have had an opportunity to share their thinking, we can find a landing spot on the first challenge students will focus on:

Which cylinder can hold more popcorn?
Make an estimate.

We can now ask students to make an estimate using reasoning and trying to be as strategic as possible. Providing students with 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.

Ask students to share their estimates, but only their estimates without their thinking or reasoning yet. Invite all estimates into the conversation, and depending on the nature of what is shared and language used, could be an opportunity to discuss concepts, such as understanding quantity relationships and change.

With the image projected, students can be asked what information might be helpful for them to work through the problem. This invites students into the conversation as co-constructors of their own learning. Allow for individual think time, students can then turn and talk to share their thinking in small groups, and finally a whole group discussion. Being strategic in how we respond will help lead students down the curiosity path towards conceptualizing the formula for the volume of a cylinder. By inviting and recording (eg., post-it notes, whiteboard etc,.) all students’ thinking is validated and honoured.

Facilitator Note

Should students need support conceptualizing the formula for area of a circle 

be sure to check out Going In Circles! | Exploring Pi: Formula for Circumference of a Circle before working through this unit they, should first check out Going in Circles

Provide students (or ask students to gather) two 8.5” by 11” pieces of paper for them with the intent of updating their original estimate with a more precise estimate.

Prompt students to:

Use two pieces of 8.5” by 11” paper to create the two cylinders we saw in the video; one created by holding the paper lengthwise and the second by holding the paper widthwise.
Update your estimate. 

Consider the following prompts to help guide next moves:

How might you use these cylinders to help you determine which will hold more?

Are there any tools available in the classroom that you might use to help you measure?

Do you have any other tools or strategies you might consider using to help update your estimate to a more precise one?

You might consider having popcorn, unifix cubes, or other non-standard volume units that could be used to help students shift from an estimate to a measured approximation.

Facilitator Notes: Some students may be surprised by the results when building their cylinders from paper, for example, the top and bottom are in the shape of a circle and are the same size. 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 see which two-dimensional shapes are used to compose a cylinder? Are students making a connection to previous learning on creating and using nets to build other three-dimensional figures, such as a triangular-based prism.

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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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