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.