## Plat Diviseur (Fractions on a plate)

As Denise has pointed out regarding this plate …

… it’s better to leave as a mystery picture at let the students figure out what’s going on.

This plate is designed for cutting cake, pie, etc. so that a varying number of people can get equal slices. For example, if you have 7 people at your party, cut at the 0 and everywhere there’s a 7.

Before I go on, here’s what Problem Pictures has to say:

The plate lets you divide into 3, 5, 6, 7, or 9 pieces. Why do you think 2, 4, and 8 have been left out?

Make your own design for a dividing plate.

Both are worthy activities; keeping them in mind I believe the entirety of this lesson can flow from extremely simple questions.

1. What is this?

French ability, guessing from the picture in the center, and raw induction may all help here.

2. What’s missing?

This should prompt the students to spot the absent 2, 4, and 8 on their own, and lead them naturally to pondering why.

3. What else is unusual about the plate?

Hopefully the students will note the 369 sharing a position being rather odd. This can lead directly to sharing divisors, and questions along the lines of: if the plate could be cut for up to 12 people, which numbers would be in the same position? This may then segue into: how many points would have this coinciding going on? Is there a way to design the plate for 12 people so it doesn’t look muddled?

The students may also remark on the 0 being strange as the “universal cut” number. I’m still not sure what to make of this aspect.

4. How could we make one ourselves?

Rather than giving students straight directions, it’s more fun to provide them supplies (like string or measuring tape or a protractor) and let them figure out their own procedure.

### 4 Responses

1. Yeah, I think the 0 ought to be a 1. If no one’s going to eat the pie, you don’t need to cut it at all. If you’re going to eat it by yourself, you can make a single cut from the center to the place marked “1” and then eat the resulting “piece”. 😉

• I think this is right.

The correct generalization of the pie plate, I think, is that for each lowest-terms rational number a/b, the integer b appears on the circumference at position 2pi a/b. In particular, the integer 1 should appear at 2pi 0/1. The cutting rule is then that you cut the cake at all positions marked with d for all divisors d of the number of people. If you are cutting for 9 people, you cut not only at the 9s (corresponding to 1/9, 2/9, 4/9, 5/9, 7/9, and 8/9 of the way around the circumference) but also at the 3s (1/3 and 2/3 of the way around) and at the 1s (0/1 of the way around.)

If you accept this, then it is clear that 0 should never appear, since it is not the denominator of any lowest-terms fraction in [0, 1).

2. […] Dyer presents Plat Diviseur (Fractions on a plate) posted at The Number Warrior. He says, “Simple questions about a French plate lead to a […]

3. […] In #7, Jason Dyer (@jdyer) suggested an open-ended discussion about the Plat Diviseur. […]