# Exploring Covariation with Bottles of Water

Author:
Angela
Suppose we have two different types of bottles we can use to fill with water. The first bottle is cylindrical shaped and the second bottle is spherical shaped. Let's explore the cylindrical bottle first:
1. Move the point labeled 'Move' and determine the relationship.
2. As we increase the water's height and volume, what type of graph do we get?
3. Should this make sense?
**cylindrical model based off of ideas from Dr Phil Clark (The Bottle Problem - Cylindrical Bottle)
Now we want to fill up a spherical bottle. Suppose we have the bottle shown below:
Now we want to graph the relationship between the water's height and volume (same as before). First, get out a piece of paper and see if you can graph the relationship before we explore with Geogebra. After you have tried the problem, follow the steps below:
1. Move point A from STOP 1 to STOP 2. What do you notice? What relationships are formed? How does this relate to the water being poured in? Should this make sense?
2. Move point B from STOP 2 to STOP 3. What do you notice? What relationships are formed? How does this relate to the water being poured in? Should this make sense?
3. Overall, describe the relationship between the water's height and volume. In other words, as we pour more water in our spherical bottle, what happens to the water's height and volume?

## Wait! There's one more bottle!

Go the following link (made by Andy Greasley) and play around with the various sliders. How does this cone/pyramid bottle differ from the spherical? Describe its graph. www.geogebra.org/m/tFQC2sM7