## Lesson Plan

# Devils Tower Math - Volume of a Cylinder

- Grade Level:
- Middle School: Sixth Grade through Eighth Grade
- Subject:
- Math
- Lesson Duration:
- 30 Minutes
- State Standards:
- 8.G.C.9
- Thinking Skills:
- Applying: Apply an abstract idea in a concrete situation to solve a problem or relate it to a prior experience. Analyzing: Break down a concept or idea into parts and show the relationships among the parts. Creating: Bring together parts (elements, compounds) of knowledge to form a whole and build relationships for NEW situations.

### Objective

How big is Devils Tower? How much rock makes up Devils Tower? Use this lesson to help your students practice their geometric equations by calculating the volume of the Tower.

### Background

Devils Tower is a geologic formation in northeastern Wyoming that is roughly the shape of a cylinder. Formed from magma, Devils Tower is an igneous rock called phonolite porphyry. As the magma pushed its way up through layers of sedimentary rock, it cooled and formed hexagonal columns. Erosion moved away the sedimentary layers, leaving the harder igneous formation exposed for us to see today.

Students will practice finding the area of a circle and the volume of a cylinder using the measurements of Devils Tower. Although the Tower is not a perfect cylinder, it is a great example of how using formulas can help us find answers to real-world questions or problems. An extension can be used for advanced students by finding a closer estimation of the volume of Devils Tower by thinking of it as a truncated cone instead of a cylinder.

### Preparation

Each student/group will need three pieces of paper of equal size (standard 8.5" x 11" works fine). For effect, it is recommended to use three different colors.

Using a snack food (e.g. popcorn, potato chips) as a filler for your cylinders can increase the number of learning styles this lesson includes.

Review the formulas for finding the area of a circle and the volume of a sphere.

### Materials

Materials explanation and formulas for using the Tower to find the volume of a cylinder.

### Lesson Hook/Preview

If you were stuck in a tube (cylinder) filled with popcorn, could you eat your way out? How much popcorn could a cylinder the size of Devils Tower hold?

### Procedure

#1: Familiarize your students with Devils Tower. Use images from the park's website or videos from our YouTube page (link at bottom) to help them visualize the formation.

#2: Ask how big they think the Tower is. We can find quick facts about how tall it is (867 feet) or how big the base is (1 mile in circumference), but what if we wanted to know how much space it takes up? We will need to calculate the **volume** of Devils Tower.

#3: To calculate volume, we have to think about what shape the Tower resembles. What geometric shape does it look like? A **cylinder**!

#4: Review the volume of a cylinder, then use the three pieces of paper to create three different cylinders.

- Cylinder A - tape the long sides of the paper together
- Cylinder B - tape the short sides of the paper together
- Cylinder C - cut the paper in half horizontally (the short side), then tape the shorter ends together to form a short, wide cylinder

#5: Have students predict which of the three cylinders has the largest volume. *Most suspect the volumes are equal, but this is a common misconception, confusing volume with surface area.*

#6: How can we test which has the largest volume? *Use small objects (e.g. beans, popcorn, rice, etc.) to fill the cylinders to see which holds more. Tip: stack the cylinders inside of each other (tall and skinny in the center) and fill the tallest first; remove that cylinder to visually demonstrate that the center one holds more. Remove the center cylinder to show how much more volume that short, wide cylinder can hold.*

#7: Was anyone surprised at the answer? Review the formula for volume of a cylinder (V = π*h*r^2) - what has more impact on the volume, the height or the **radius**? Why?

#8: If we know that Devils Tower is 867 feet tall and has a circumference of one mile, can we calculate the volume of the Tower?

#9: Allow students to work the problem, providing help if necessary. Remind them that they can find the radius by converting the circumference into feet (5,280 feet = 1 mile).

#10: Review answers and discuss how/why people judge the size of objects. What reason(s) would we have for wanting to know the volume of the Tower? *At one time, people thought they could use the Tower rock as gravel for roads.* What other ways can we apply this formula to the real world? *Volume of a: can of soda; drinking glass; bottle of shampoo.*

### Vocabulary

**Cylinder**: A solid with two congruent parallel faces called *bases*. Lines joining corresponding points on the two bases are parallel.

**Volume**: The amount of space an object occupies; or, how much space is enclosed within a container.

**Radius**: The length from the center of a circle to its perimeter.

### Enrichment Activities

As an extension, ask students to consider that Devils Tower is not a perfect cylinder - the base has a larger area than the top. Our volume calculation is an over-estimate. How could we calculate the volume of the Tower more accurately? *Think of Devils Tower as a truncated cone, or a cone with the top chopped off. See materials sheet for details on extension.*

### Additional Resources

More on the geology of Devils Tower:

https://nature.nps.gov/views/Classic/Index_DETO.htm

### Related Lessons or Education Materials

This lesson was originally created by Wyoming Agriculture in the Classroom. Please visit their site - www.wyaitc.org - for more information about the organization and finding other teaching materials related to Wyoming resources.