Take Something Apart ... Toilet Paper Rolls

As a pretty large family during a time of social distancing, we have a true abundance of toilet paper rolls these days! Instead of tossing them directly in the recycling, we took some apart to learn more about their structure. Then I took the deconstructed roll and photographed it because I thought it was quite beautiful. Imagine that: there is real beauty in the structure of a toilet paper roll!

If you can help us understand more about how the geometry of the tube is related to the geometry of the parallelogram(s) that result from unrolling it, please email everywheremath2020@gmail.com. Thanks!

What do you notice about this cardboard roll?
It looks like a short but very wide straw.
It's kind of a cylinder but it's hollow. Maybe it's better to call it a tube.
It has a diagonal kind of swirl on it.

What do you think it will look like when we unroll it? Why do you think that?
We've rolled rectangles (of paper) and sheets (of wax paper) to make tubes before. So maybe it'll be a rectangle when we unroll it? Or a long skinny sheet of cardboard?

Now that it's unrolled, what do you notice? What shape(s) do you see?
It's a quadrilateral (4 sides).
I see a parallelogram.
I see kind of two overlapping parallelograms.

What do the two parallelograms you see have in common? How are they different?
All their interior angles are the same.
One of each of their pairs of sides is equal but not the other.
I think the smaller one might be half the size of the other. (How could you figure that out?)

How do the area and perimeter of the two parallelograms relate to each other?
If the smaller one is indeed half the size of the other (because the top and bottom sides are half the length of the top and bottom sides in the larger one), then the area of the smaller one is half the area of the larger one.
We need to know more about the relationship between the side lengths to know for sure about the perimeters.

How do you think this roll was manufactured?
You can speculate about it based on what you notice and what you know. You can also find all kinds of cool videos online.

Why do you think someone decided to make the roll this way instead of another way?

What would be another way to make a roll? Want to make one? What are the benefits and drawbacks of different ways?

Have you ever seen any other objects that look similar to this toilet paper roll? Do you think they were made the same way? Why or why not?