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.

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Brent, on May 13, 2009 at 1:08 pm said: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”. 😉

Mark Dominus, on July 27, 2009 at 10:32 pm said: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).

Math At Play Blog Carnival #7 - Onomatopoeia, on May 15, 2009 at 3:01 am said:[…] Dyer presents Plat Diviseur (Fractions on a plate) posted at The Number Warrior. He says, “Simple questions about a French plate lead to a […]