Circles, Divisibility, and Wonder in Mathematics

Watercolor Meditations on the Divisibility of 12
Meditations on the Divisibility of 12

I’m working with a student who is having a hard time memorizing his times tables. Indeed, I suspect he does not have a concept of numbers as representing quantities–I think sometimes he sees them as separate numerals, some kind of squiggly code that doesn’t mean much to him. In performing operations, he does his best to memorize the algorithms, but of course they do not stick. I’m working on ways to get him to internalize some of the qualities of numbers, hoping that this will give him a scaffolding on which he can attach deeper learning.

The Page of 12s
The Page of 12s

We started working with a circle. I marked off 12 equally spaced marks on the circumference (similar to a clock face.) I then had him draw lines connecting marks when counting by different numbers. For example, when counting by 3s, the lines form a square. Counting by 4s you get a triangle. Then something crazy happens–, when you count by 5s, you get a star that crosses itself over and over again until you reach home. This is where the real math starts happening: Can you look at your data and predict what will happen for other numbers?
When I started working with this problem, I knew it had something to do with divisibility, and I thought it would be a cool way to use colored pencils and get away from numerals and algorithms. It turns out this problem is a lot richer than I anticipated.
I was surprised to notice that numbers that add up to the base create the same pattern. For example, 3 on a circle of 12 will make a square as will 9 on a circle of 12. 8 on a circle of 20 will make a 5-pointed star, as will 12 on a circle of 20.
I got excited to prove my observation that if n divides p and n divides q, then n divides p-q. Here’s my proof:

Definition of Divisibility: if n|p that means there is some number k for which kn=p (I remember this definition from an old college number theory class somehow . . . )

Given: n|p and n|q

Prove: n|(p-q)

n|p kn=p

n|q jn=q

Definition of Divisibility


Subtraction Property (of Equality)


Distributive Property

Let h=(k-j)

Then, nh=(p-q)


Therefore, n|(p-q)

Definition of Divisibility

OK, I know this is a pretty simple proof, but it made me inordinately happy to remember how to do it, and I truly was surprised to see the shapes on the circles match each other the way they do.

You can also use these circles to illustrate which fractions can be reduced and which cannot (the same idea as showing whether two numbers are relatively prime, or share a common factor.) If the number of marks around the edge of the circle corresponds to the denominator, and the number you count by corresponds to the numerator, then irreducible fractions will be the ones where the lines meet every mark. For fractions which can be reduced, the greatest common factor is the number of untouched marks between the lines.

I’m not saying this should be an algorithm for reducing fractions, but I’m excited because it is a visual way to understand a concept that baffles many of my students.

My hope is that no matter how much we do with this problem, I can demonstrate that numbers have qualities that can be observed and generalized. Observe, form a hypothesis, make a prediction, test your hypothesis . . . repeat as necessary. In my opinion, this is where the real learning is!


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