I am constantly astounded by the direction 9 year old students can go in their thinking. This morning we were discussing 3D shapes, clarifying and defining. What makes a shape a pyramid and what makes a shape a prism? What are the distinct differences between pyramids and prisms? We had a wonderful array of 3D shapes to sort and explore with. Suddenly one student asked, "Is a cone a pyramid?"

"Good question, who wants to theorize?" I asked.

"No'" exclaimed one chap, "because it only has one face at the side and it is not triangular in shape."

There was a consensus of nods, but one divergent miss shouted out. "Not necessarily, maybe the sides are actually made of many very, very small triangles."

"How many would we be talking about?" I asked.

"Well lots and lots, because the more triangles you have the more the base will look like a circle," she explained.

"Ah, you have hit upon one of the great mathematical discussions. If a line is defined as straight, how many lines make up a circle?" I challenged.

"Lots and lots and lots of them," the students were starting to get animated now with the ideas, but as the bell was almost on us I asked them to take a quick vote on whether a circle has zero straight lines or infinite number of straight lines. We then voted on the fate of the cone. Did it qualify as a pyramid? We decided that these were both debatable. Finally, I asked, "What about the prisms. Using the same logic and arguments, is there a shape that could debatably be a prism?"

"Oh, the cylinder, " a group stated,

"Because, the curved face could actually be millions of very small rectangles joined together," it was explained by one.

"Good question, who wants to theorize?" I asked.

"No'" exclaimed one chap, "because it only has one face at the side and it is not triangular in shape."

There was a consensus of nods, but one divergent miss shouted out. "Not necessarily, maybe the sides are actually made of many very, very small triangles."

"How many would we be talking about?" I asked.

"Well lots and lots, because the more triangles you have the more the base will look like a circle," she explained.

"Ah, you have hit upon one of the great mathematical discussions. If a line is defined as straight, how many lines make up a circle?" I challenged.

"Lots and lots and lots of them," the students were starting to get animated now with the ideas, but as the bell was almost on us I asked them to take a quick vote on whether a circle has zero straight lines or infinite number of straight lines. We then voted on the fate of the cone. Did it qualify as a pyramid? We decided that these were both debatable. Finally, I asked, "What about the prisms. Using the same logic and arguments, is there a shape that could debatably be a prism?"

"Oh, the cylinder, " a group stated,

"Because, the curved face could actually be millions of very small rectangles joined together," it was explained by one.