I'm a sub, and today needed to determine the number of faces in a cone and a cylinder with the students. There was no answer key, and I could not find the teacher's manual, (though I tried to sort of "piece it together" through deduction while looking through a similar chapter in another manual.) The other 3rd grade teacher said it wasn't actually a standard for that level and she wasn't quite sure, so I'm consulting you 4th grade teachers!
I've "googled" this and keep coming up with contradictory answers - some sites consider the curved areas "faces" (differentiating between curved and flat faces), while others say that only the flat areas are faces. So what say you all? How does your elementary text define a face? (I counted the curved surfaces, so I said 2 for a cone, 3 for a cylinder.)
I did leave a note for the teacher stating I was not sure about some of these solids, so I guess she'll have to take care of the mistakes with them tomorrow! But this is just driving me nuts!
I've been teaching 4th grade math for years, and I know how confusing this can be. There are lots of contradictory resources out there. I agree with the other poster that face should be interpreted to mean "flat face". It's funny...when I searched to find a good definition for you, one that I came across said a face is "a polygon that is a flat surface of a solid figure". Interestingly enough, the definitions I use contradict this, because I would consider the cylinder to have 2 faces (the circles), but circles aren't polygons.
Thank you very much for your responses! Yes, it can be very confusing trying to label and classify the parts of some geometric shapes and solids. Hopefully, the next time I'm called upon to teach geometry, I'll have some clear resources in the classroom that I can consult.
There are conflicting definitions in the math world...so you need to go with the definitions presented by the district's curriculum and resources. For example, in my district, we use the definition that faces are flat surfaces (does NOT specify "polygons") of 3-d figures...therefore a cone has one and a cylinder has 2.
Another example:
Some mathematicians define a rhombus as a quadrilateral with 4 equal sides and 2 pair of parallel sides.
Others define a rhombus as a quadrilateral with 4 equal sides, 2 pair of parallel sides, AND specify that the angles are "oblique".
So where's the confusion?
Well, under the 1st definition, a square would be considered a rhombus, but under the 2nd definition, it wouldn't. http://mathcentral.uregina.ca/QQ/dat....02/beth1.html
Actually, a circle is the limiting figure of an n-sided polygon when n increases without bound, so it is technically not improper to consider a circle to be an infinite-sided polygon. Therefore, your definition of a 'face' holds for the base of a cone or bases of a cylinder.
A polygon is a closed plane figure bounded by three or more line segments. Three or more means that then number of sides can be as large as you like. A circle is actually the limiting figure for an n-sided polygon as n increases without bound. In other words, it is perfectly proper to say that a circle is an infinite-sided polygon.
So, we have established that faces must be flat and faces must be polygons and that a circle is, in fact, a polygon. Therefore a right circular cylinder has two faces. In fact, we can generalize even further to include oblique cylinders and cylinders where the base is elliptical rather than circular.
However ... An edge 'is' created on a cone and cylinder. An edge is where 2 faces (surfaces) meet and ...to determine the volume of a solid figure, you must determine its 'inside' capacity. Without a curved face to determine the area... the 'solid' doesn't exist. Cone is 1 flat and 1 curved face (2), cylinder is 2 flat 1 curved face (3) A sphere is one curved face!
Ironically I too subbed today and this came up and I tought it differently than the teacher I subbed for. The book discribed a cone as a flat surface conecting two line segments. Since there aren't any line segments in a cone I thought there were no faces on cones and therefore cylinders. Later I found out that the teachers found another place in the same book where thay said that a cone has one face therefore they decided as a team to teach it that cones have one and cylinders have two. Although they admited that they aren't sure of what the real answer is either.
I think it is wonderful that 4th grade teachers are discussing these concepts with each other!
However, in high school geometry we use the words faces, edges, and vertices for polyhedra only. Polyhedra are 3-dimensional shapes composed of polygonal regions, any two of which have at most a common side. (Just a fancy way to say only things like prisms and pyramids.)
I agree that a circle is the limit as the number of sides of a polygon increases to an infinitely large number. If you continue down this road...not only would the bases of a cylinder be "faces", but the cylinder would have inifinitely many rectangles in its lateral surface area, and the cone would have infinitely many triangles in its surface area.
It looks like we may be confusing some of our students. So, in HS geometry, we refer to the 2 bases of a cylinder and the 1 base of a cone. Neither the cone, nor the cylinder, (or for that matter, a sphere) is considered to have edges or faces (as they are defined in geometry).
As a second grade teacher, I teach the students that they are bases not faces because the surfaces that are put together to make the figures of cones and cylinders are not polygons.
I teach seventh grade math, and we also have had the debate in my classroom with my students on whether or not a cone has one face to two faces. The fact is that a cone has two faces. The definition of a face being a flat side comes from the net of a shape. When you look at the net of a cone, it has one flat circle and one flat sector (triangle with a curve). That being said, a cone has two faces and not just one. That should help a bit. When thinking about 3D shapes it is good to take a look at the net to find out how many faces a shape has. Check out this link: http://www.mathsteacher.com.au/year8...s/cylinder.htm
Personally, I believe the definition of a cone evolves with school level, just as other math concepts evolve with grade level and deepen as their developing minds can comprehend them.
At the elementary level, students are going to learn the basics about 3-D figures from a 2-D perspective, as young minds cannot process the complexities of shapes from a 3-D perspective. Textbooks are not going to get into the difference of a face and a base and the details of all the definitions behind 3-D shapes. But at the elementary level they should be calling the flat surface a base, not a face. From a 2-D perspective, it looks like 2 lines that meet at a point on a cone, which by definition, is a vertex. So that may be the reason why the point of a cone is a called a vertex sometimes at the elementary level.
As students progress through school, student's are taught about nets in order to help bridge their minds from the 2-D world to the 3-D world. The net of a cone is actually a circle and a sector of a circle, both of which are not polygons, therefore still would not have any faces.
By the time they reach High School, students' minds are capable of thinking in 3-D. They learn the complexities of 3-D shapes from a 3-D perspective, but in a finite context. In a finite context, a face of a polyhedron is any of the polygons that make up it's boundaries. A cone's base is a circle, not a polygon, therefore not a face. Also in this context, a vertex is defined as a point where multiple faces meet. Therefore, with no faces, a cone cannot have a vertex. Furthermore, an edge is a straight line where two faces meet- no faces, no edges. So the definition of a cone from a finite perspective has a base, no faces, no edges, and no vertex. Also students typically work with right circular cones, and oblique cones are introduced later on.
When students reach calculus, they learn the concepts of limits and infinity. A cone's definition then will change in the infinite context as others have mentioned. The limit of an n-gon as n reaches infinity is, by defintion, a circle. So people can argue that a circle is indeed a polygon and then a cone's base is also a face. Also, it's rounded surface is comprised of an infinite number of triangles extending from the axis of a cone. Since triangles are polygons, one can argue that a cone has a vertex because it is a point where an infinite number of polygons meet.
So I believe the confusion lies in the fact that people do not clearly state where they are in the evolution of a cone when stating their defintion. It creates a lot of confusion when definitions are being thrown out there on the internet without specifying their target audience. But it is clearly a great example of having student's think more deeply into mathematics and expand the possibilities! It becomes a great classroom discussion!
Math Geek
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02-26-2007, 07:13 PM
But this is just driving me nuts! 
