Mathematical Habits of Mind - Conjecturer

We had a nice emergence today in class.  We were practicing the habit of pattern sniffing.  Everybody has been assigned a 3D shape and they are trying to become experts in that shape.  Today, they started to count the vertices, edges and faces.  When we got around to the cone, a disagreement arose.

How many faces does a cone have?  

Some students said 1, others said 2.  Same problem with a sphere.  How can something have no faces?  It must have one, right?  We switched courses and started to conjecture.  A conjecture is a theory that you have before it has been tested.  It is like a hunch, it hasn't been proven.  We wrote three statements on the board and each child filled out their responses.

1) I think a sphere has _ faces because....

- 1 face because I feel something, so there must be a face

- 0 faces because faces are flat

- 0 faces because it rolls, and things with faces cannot roll

- 0 because a 2D circle has a face, but a 3D circle is not made out of 2D shapes

- 1 face because there can't be nothing

2) I think a cylinder has _ edges because...

- 2 edges because it is not smooth

- 2 edges because a circle needs an edge

- 0 because I can't see any points

3) I think a cone has _ vertices because...

- 1, because I can touch the sharp part

- 1, because there is only one pointy part

- 0, because there are no straight lines, and vertices are only on straight lines

- 6, because there are shapes inside that make the cone round

- 1, because a vertex is a point and it can't be round


Great conjectures.  We need to investigate further.  Where should we go next?  I was thinking of consulting an expert....