When I go #$2$, I like to fold the toilet paper into layers so that I don’t get my hand dirty.
My process is that I take a piece of paper with $8$ rectangles. Folding it in half, I get $4$ rectangles which results in a piece with two layers. Next, I fold that piece in half resulting in a piece with $2$ rectangles which results in three layers. Then I fold that piece which results in one rectangle having four layers. Then I do my business.
My question is, what can describe this process (an operation, a function, etc.)? Can it be generalized? Is it reversible? Is the domain continuous or discrete? Can this be applied as a math lesson in school and at what grade? Can this lesson have connections with different areas of mathematics or other subjects?
Let $x$ be the number of rectangles in the toilet paper ($x \in (\Bbb{N} \cup{{0}})$). The pattern would indicate that $log_{2}(x)$ is a good function if I fold the paper in half evenly every time. This function would work because the paper with 8 rectangles folded in half three times collapse to only one rectangle with 4 layers. I see it as 8 rectangles collapsing to one rectangle after folding three times in half.
I see undoing the folding as the inverse function of $log_{2}(x)$ with its corresponding domain, namely $2^{x}$.
If I fold the paper evenly in thirds, I would get $log_{3}(x)$ and its corresponding inverse $3^{x}$.
Let $a$ be the number of foldings. Then $log_{a}(x)$ would be the function and it’s inverse would be $a^{x}$ for $a \in \Bbb{N}$ and $x \in (\Bbb{N} \cup{{0}})$.
Now, this would be a good lesson for algebra 2 class or pre-calculus class. Some of the concepts to talk about could be function, domain, range, codomain, inverse function, continuity in domain, etc.
But $x$ and $a$ could be extended to a more abundant number system, say the nonnegative real numbers. I would say that since $x$ is the number of rectangles in paper and the rectangles can be folded unevenly and at any point in the length of paper (as long as the foldings are the same size), the domain of $log_{a}(x)$ is continuous. That is, $x$ and $a$ are in the nonnegative real numbers as long as each folding is the same. For example, fold 8.125 rectangles into equal parts of 1.75 rectangles.
In this sense, both $x$ and $a$ are continuous and discrete.