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# Cones and Spheres

## How Many Cones Does It Take To Fill a Sphere?

In this 3 act math task, the teacher will show short video clips to help students understand where the Volume of a Sphere formula comes from. Similar to the last Volume 3 Act Math Task: Prisms and Pyramids, the intention has been to leave Act 1 of each set very vague to allow for students to take the problem in more than one direction. The teacher can bring light to the learning goal of showing where the formula for volume of a sphere comes from after watching Act 1.

## Act 1: What’s the question?

My students usually explore the relationship between the volume of a prism and a pyramid, so this activity is typically considered an extension.

To begin, I show students this image:

Students typically make a connection between a sphere, cylinder and cone and I ask them to justify why they think this would seem logical (i.e.: all three have a radius/diameter, seem “round,” no vertices, etc.). While we could show a relationship between the sphere, cylinder, and cone, I show them this video to focus in on just one relationship:

I gave my students some time to chat with a partner and come up with some possible questions for this video. Since they had completed the Prisms and Pyramids exploration the previous day, most jump straight to volume. However, other options are possible.

As most expect, we narrow the question down to:

How many cones would it take to fill the sphere?

Students then have a moment to come up with their best guess and we share out and record the guesses in class.

Many students will say “3” since they noticed a pattern in the relationship between all of the prisms and pyramids explored the previous day.

Students then watch this video:

I then allow students to chat with a partner and consider “updating” their guess based on the new information given to them.

## Act 3: Experience the Answer

Students will then watch Act 3 in order to determine how close they were to the actual number.

# Visual Derivation of the Volume of a Sphere Formula

## Making Connections Between Volume of a Cone and Sphere

I then have a discussion with students and even let them try finding the volume of a sphere using the volume of a cone formula they used the previous day. This gives some additional practice with the volume of a cone formula and builds confidence that they already know how to find the volume of a sphere based on prior knowledge.

We then try to make a connection between the formula for volume of a cone and simplify the equation to give us a more efficient route to find the volume of a sphere:

## Step-By-Step Slide Deck

See the step-by-step slide deck that I built in Apple Keynote:

## New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

## Share With Your Learning Community:

I’m Kyle Pearce and I am a former high school math teacher. I’m now the K-12 Mathematics Consultant with the Greater Essex County District School Board, where I uncover creative ways to spark curiosity and fuel sense making in mathematics. Read more.

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