Students will explore the relationship between a cylinder, cone and sphere when calculating volume. Through this investigation, students will have opportunities to begin to conceptualize that the volume of a sphere is \(\frac{2}{3}\) the volume of a cylinder, provided the cylinder’s height is equal to the diameter, or twice the radius of the sphere.

Students will explore the relationship between a cylinder, cone and sphere when calculating volume. Through this investigation, students will have opportunities to begin to conceptualize that the volume of a sphere is \(\frac{2}{3}\) the volume of a cylinder, provided the cylinder’s height is equal to the diameter, or twice the radius of the sphere.

This task will allow students to develop a deeper understanding of big ideas.

Some of the big ideas that will likely emerge in this task include:

• 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;
• The volume of a sphere is double the volume of a cone with an equivalent radius and equivalent height; and,
• The volume of a sphere is 2 thirds the volume of a cylinder with an equivalent radius and equivalent height.

The following sequence of problems provides an opportunity for students to conceptualize the ratio relationships between the volume of a cylinder, cone and sphere.

In particular, these visual math talk prompts will highlight that calculating the volume of any cone or sphere begins with finding the volume of a cylinder with the same radius and height.

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

About how many cubic centimetres of popcorn could you hold with this container?
Approximate without the use of a calculator.

With the image projected, give students a reasonable amount of time to reflect on and then work through the question, with strong consideration given to student collaboration as they reason and think their way through to what the volume of this cylinder might be.

After a reasonable amount of time, invite students to share their approximations/calculations.

The goal is to once again get students to think about how they can approximate the area of the base and use this quantity to then scale vertically to reveal a reasonable quantity for the volume.

At this point, students may be more inclined to simply use the formula for area of a circle rather than decomposing the circle into a rectangle as we had done earlier in the unit.

Since the radius is 6 cm and 6 cm squared is 36 cm squared, about 3 times this quantity will give an area of about 108 square centimetres. Students using a calculator will likely land somewhere around 113 square centimetres. Be sure to celebrate those who approximated and then adjusted up rather than simply punching into a calculator.

With the area of the base now known, we can conclude that we now have the volume of a 1 cm tall cylinder; approximately 113 cubic centimetres.

From here, students can scale the height of the cylinder upwards using a partial products approach until reaching the desired height of 12 centimetres.

Students who arrive at a total volume of between 1,290 and 1,380 cubic centimetres are demonstrating a pretty high level of precision without leveraging the use of a calculator.

Be sure to record student thinking in a space (eg., whiteboard) visible to all students. This honours the contributions of all students.

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

About how many cubic centimetres of popcorn could you hold with this container?
Approximate without the use of a calculator.

Since this cone has the same height and radius as the cylinder in Visual Math Talk Prompt #1, the intent is to promote students reasoning with the volume relationships that have been uncovered throughout this unit.

Since we know that the volume of 3 cones is equivalent to the volume of 1 cylinder with the same height and radius, simply taking 1 third of the volume of the cylinder is a highly efficient approach that some students might choose to use.

Since 1 third of 1,350 cubic centimetres is 450 centimetres cubed, an answer within a 50 to 100 cubic centimetre range would be reasonable for those working without a calculator.

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

About how many cubic centimetres of popcorn could you hold with this container?
Approximate without the use of a calculator.

Since this sphere has the same height and radius as the cylinder and cone in the previous Visual Math Talk Prompts, the intent is to continue promoting students’ reasoning with the volume relationships from this unit. In particular, noting that the volume of 2 cones is equivalent to the volume of 1 sphere with the same radius and a height equivalent to double the radius.

In the animation, students will see that it takes 1 cone to fill 1 hemisphere or 2 cones to fill a full sphere.

Prompt students with:

How many spherical popcorn containers can you fit in the cylindrical container? Make an estimate.

Give 30-60 seconds for individual think-time before offering students an opportunity to share their thinking with a classmate and then a whole group share.

To support learning during the investigation, students can use manipulatives from an available math kit, such as geometric solids. Alternatively, the visuals provided during the lesson can also be used.

Prompt students to:

Use your understanding of the volume relationships we have explored throughout this unit to refine your prediction.

Ask students to share with the group some of the different volume relationships they recall from earlier in the unit.

Once students have shared their thinking, you can show them the following struggle image:

For some students, they may notice that since the volume of 3 cones is equivalent to the volume of a cylinder with the same radius and height and the volume of 2 cones is equivalent to the volume of a sphere with the same radius and a height equivalent to double the radius, then there is a 3 to 2 ratio relationship between the volume of the cylinder and the volume of the sphere.

Giving students the opportunity to work with 3D geometric solids would be ideal at this point so they can fill and compare their results with neighbours.

Give students the opportunity to discuss what they notice with their elbow partners and share out to the larger group.

As students are investigating, consider what purposeful prompts and/or teacher moves you might leverage to guide students toward emerging the volume relationship between a cylinder and a sphere.

Pay close attention to the strategies that students are using.

Do students:

• Use Spatial Reasoning and/or 3D geometric solids to determine the relationship?
• Use volume relationships explored earlier in the unit to emerge this new volume relationship?

Facilitator Note
Reminding students of the work from previous lessons can be used to support the development of student thinking. For example, you might consider sharing an image or physically placing the cylinder, cone and sphere from your manipulative kit side-by-side on the table at the front of the room.

After allowing students to share what they discussed with their peers including any new learning that emerged from their investigation, we must explicitly consolidate to ensure that all students understand the volume relationship between a cylinder and a sphere with equivalent radius and a height equivalent to double the radius.

Leveraging our volume relationships between a cylinder and cone as well as a cone and a sphere, we seek to lean on algebraic reasoning to emerge the volume relationship between a cylinder and a sphere.

In particular, since students know that there is a 3 to 1 volume relationship between a cone and a cylinder and a 2 to 1 volume relationship between a sphere and a cone, we can leverage algebraic substitution to substitute the volume of 2 cones for the volume of a sphere.

The result transforms:

\(V_{cylinder} = 3(V_{cone})\)

to

\(V_{cylinder} = V_{sphere} + V_{cone}\)

Since the volume of a sphere is equivalent to the volume of 2 cones, we can also say that the volume of a hemisphere is equivalent to the volume of 1 cone.

Therefore,

\(V_{cylinder} = V_{sphere} + V_{cone}\)

Is equivalent to

\(V_{cylinder} = V_{sphere} + \frac{1}{2}(V_{sphere})\)

Simplifying, we can say:

\(V_{cylinder} = \frac{3}{2}(V_{sphere})\)

We can also look at the relationship by determining how many cylinders it would take to fill a sphere:

\(V_{sphere} = \frac{2}{3}(V_{cylinder})\)

This progression can be viewed via this silent solution animation video which also serves as our reveal video for this lesson:

Students will complete the following consolidation prompts independently.

Consolidation Prompt #1:

A cylinder and a sphere have the same height and radius.
If the volume of the cylinder is 1,200 \(cm^3\), what is the volume of the sphere?
Craft a convincing argument.

Reveal Silent Solution Animation Video:

Consolidation Prompt #2:

Without the use of a calculator, about how much popcorn can a cylindrical container with a radius of 7 cm and height of 14 cm hold?

Consolidation Prompt #3:

The cashier is using a cone shaped scoop to fill a cylindrical popcorn container.
How many scoops will it take to fill up the container?
How do you know?

Consolidation Prompt #4:

Your friend buys a cylindrical container of popcorn, while you choose the container shaped like a hemisphere with the same radius, but half of the height of the cylinder.
How much more popcorn did your friend receive?

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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