Circles have several properties: diameter, radius, area, and circumference. The diameter, radius and area are often described in expressions and equations that are related to π (“pi”). However, circumference is almost never discussed with as much detail.
There are a wide range of math topics covered in the classroom, but not as much depth and exploration of these new ideas. Adding personal thoughts and experience to a math concept is a great way to help student engagement and promote meaningful learning.
When learning is more meaningful, students retain the information in long-term memory and recall those concepts with less difficulty.
Paper towel rolls are great way to use prior knowledge from students to build new ideas about math topics. Daily interactions with common household objects, such as paper towel rolls, can strengthen the relevance of math in everyday life.
Note on Teaching Supplies: Craft paper rolls can be purchased online from Amazon. Or you can use an empty paper towel roll.
From observation, we see that a circumference is the perimeter around a circle. If you walk around the entire perimeter of a circle, we find that the total distance around the circle is equal to the length of a segment that has been “un-rolled” from a round arc into a straight line.
Let us take a cardboard paper towel roll. The cross section is a circle.
If we make a single cut along its longest side, we can “un-roll” the paper towel roll into a flat sheet of cardboard. (The straightness of the cut does not need to be perfect.)
We know that a circumference is the length around a circle. We also know that the length remains unchanged. Also, the surface area of the paper towel roll along its longest length is also unchanged. The only change was the path of the length, which was transformed from a round arc into a straight line.
This activity demonstrates the powerful the idea of π (“pi”), which allows us to relate the diameter of an object to its circumference without the need for modification. We also see that circumference is simply the arc length that travels completely around a circle.
Key Vocabulary: circumference, length, area
CCSS.MATH.CONTENT.7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.