Students will estimate how much popcorn you can pour from a cone shaped container into a hemisphere shaped bowl to lead to the emergence that the volume of a sphere can be calculated by doubling the volume of a cone with the same radius and height.

Students will explore the relationship between the volume of a cylinder, a cone and a sphere to emerge the formula for determining the volume of a sphere:

Volume of a Sphere, V=\(\frac{4πr^3}{3}\)

Some of the big ideas that will likely emerge in this lesson 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; and,
• The volume of a sphere is double the volume of a cone with an equivalent radius and equivalent height.

Show students the following video of the cone shaped popcorn container and a hemisphere shaped bowl.

Then, ask students:

How many kernels of popcorn will fit in the bowl?
Make an estimate.

By asking students to make an estimate, they will be forced to leverage their spatial reasoning which provides a 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 further, 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 sphere. By inviting and recording (eg., post-it notes, whiteboard etc,.) all students’ thinking is validated and honoured.

Provide students with a cone and a sphere that can be split into two equal sized hemispheres with the intent of updating their original estimate with a more precise estimate.

While you can also provide popcorn, using water, sand or rice will allow students to more clearly recognize the one to one volume relationship that exists between a cone and a hemisphere.

Note that the hemisphere must have the same radius as the base of the cone. The height of the cone and sphere (two identical hemispheres placed on top of one another to form a sphere) must be the same as double the radius OR equivalent to the diameter because a sphere’s height is the same as its diameter.

Prompt students to:

Explore the cone and hemisphere provided to update your estimate.

Describe the relationship between the volume of a cone and the volume of a hemisphere.

How might you determine the volume of a hemisphere? Convince your neighbour.

While working through this task, you might expect students to also draw on their previous learning as they begin to make sense of the relationship between all three figures explored (cylinder, cone and sphere) in this unit.

Facilitator Notes: Some students may be surprised that it takes 1 cone to fill up 1 hemisphere or 2 cones to fill up the sphere with popcorn. Other students may be ready to generalize that the volume of a sphere is \(\frac{2}{3}\) the volume of a cylinder. These, amongst other observations that will surface, can be used by the facilitator to help assess how far along their learning continuum students might be. For instance, are students struggling to see how filling up the sphere with popcorn using a cone relates to filling up a cylinder with popcorn using the same cone? Are students making a connection to previous learning on the relationship between a cylinder and a cone when calculating volume?

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Show students the following video showing each kernel of popcorn being moved and counted from the cone shaped popcorn container to the hemisphere shaped bowl:

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

Centering understanding on student thinking, reasoning and approaches should drive the direction when consolidating learning with students. This provides students with a meaningful context for making connections to their own learning or strategies used, and those shared by their classmates and peers.

Since the intent of this task was to reveal the big idea that the volume of a sphere is double the volume of a cone with an equivalent radius and equivalent height, we must ensure that we take the time to consolidate learning and ensure this idea emerges. From this new understanding, we can easily extend the formula for finding volume of a cone to simply double for finding the volume of a sphere.

The following silent solution animation video may be useful as you prepare for this consolidation. You might utilize some or all of the visuals to assist as these connections are made:

Before sharing any of the consolidation animation with students, use this opportunity to ask them to recall how we find the volume of a cylinder. By this point in the unit and given students have already explored finding the volume of other prisms in the past, there should be a common understanding that we multiply the area of the base by the height of the cylinder, V = \(πr^2h\).

This is also another opportunity to give students additional practice recalling that the relationship between the volume of a cylinder and the volume of a cone is 3 cones to fill 1 cylinder or 1 third of a cylinder to fill 1 cone.

Pause the silent solution animation to prompt students to once again share with their neighbours a formula (or rule) for finding the volume of a sphere with the same radius and height.

Our goal is for students to understand that it takes 1 cone to fill 1 hemisphere, or 2 cones to fill 1 sphere.

While this is certainly enough for students to calculate any sphere with conceptual understanding and procedural fluency, take this opportunity to help students see that many mathematicians simplify this notation to a more compact and concise formula leveraging substitution of variables given that the height, h can be represented with double the radius, 2r.

This process can also be helpful to provide students with an opportunity to simplify algebraic expressions involving fractions.

Upon simplification, one simplified representation of this formula is \((\frac{4}{3}) πr^3\).  while another is V = \(\frac{(4πr^3)}{3}\) .

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