# Ceiling fan arithmetic

Summer has arrived in Minnesota, and that means we alternate between warm days where we open the windows and run the ceiling fan, and hot days where we close everything up and run the air conditioning (a luxury, btw, that our 1928-built home only got about five years back).

Not our ceiling fan.
Image credit: Brian Snelson (CC-BY 2.0)

Tabitha is naturally curious about how the ceiling fan works. In case you don’t have experience with them, or yours works differently from ours, here are the basics: There is a switch on the wall—just like a light switch—that powers the fan. Then there is a chain hanging from the fan itself that affects the speed. There are four settings controlled by the chain—High, Medium, Low, and Off.

By the time this conversation takes place, Tabitha and I have already explored a variety of ceiling fan questions, such as If the fan is off, should you pull the chain to turn it on, or head over to the light switch? and How many times can I pull the chain before my parents tell me to stop playing with the fan?

On this day, I ask Tabitha to flip the wall switch to turn on the fan, which she does. She then starts to stand up on the couch to reach the chain. I ask why.

Tabitha (9 years old): I want to see if it’s on high.

Me: But how will you know? If you pull the chain it will slow down, but that’s what it always does. So how will you know whether it was on high to begin with?

T: Well, it doesn’t always slow down, otherwise how would it ever be on high?

## So What Do We Learn?

There is some very deep math going on here.

Tabitha and I are playing with properties of modular arithmetic, but she (and you) don’t need to know the specifics. Things that go in cycles are all examples of this kind of math.

The classic example is time. You could say that later times have bigger numbers. 4 is later than 3; 12 is later than 9. This is just like my claim that every pull of the chain slows down the fan. Both of these claims are only sort of true. Three in the afternoon is later than 11 in the morning, despite having a smaller number. If the fan is on low and you pull the chain twice, it’ll be on high.

People study these things in great depth in the field of Modern Algebra, and the ideas are useful in all sorts of places.

## Starting the Conversation

Play with a ceiling fan. Talk about staying up all night. Notice together that weird things happen when the fan is in the off position, and at midnight and noon. Wonder aloud whether 12 o’clock is like zero (and if not, what is?)

Play around with basic facts in this ceiling fan environment. If it’s on high, how many pulls to turn it off? If it’s on low, how many to get it to medium? I just pulled the chain three times and now it’s on low—where was it before? Etc. Challenge the child; have the child challenge you.

Stay cool!

## 2 thoughts on “Ceiling fan arithmetic”

1. Tienchin Ho says:

This is so terrific! Thank you! I had tons of games for my girls when they were younger but now I have one learning linear algebra and one learning quadratics and it seems less fun than when we were playing with numbers. Playing the function machine game only goes so far. We do love geometry – tessellations, polyhedrons, rotations, symmetry, etc. Do you have any resources for math fun beyond arithmetic?

2. Miriam says:

Hey Christopher,
what an interesting method to unterstand the logic of arithmetic! My daughter is now 12 years old and at the moment she’s struggling with math. I think this method is a good way to make it easier to understand. Because of the fact, that she’s already 12, how can I adapt the task to a higher level? Thanks in advance!

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