   SEARCH HOME Math Central Quandaries & Queries  Question from Makayla, a student: I've just started the 4th grade and I am having trouble with a question on my homework. The questions is: Carrie is arranging sticks to make shapes. she has made a closed figure, with 10 sides How many vertices are there? I believe the answer is ten but I'm not sure. We have two responses for you

Makayla,

Since the title of your problem is 'problem solving', I think the key is to look at your reasoning.

I suspect that you drew an example, and found that your example had 10 vertices.
This is certainly a good start. If you imagine shifting them around a bit, or making some longer and shorter - does the number of vertices change? This is also part of your reasoning.

You can back up the reasoning in a couple of ways: is 10 a magic number?
Is there a pattern that works for three sides, four sides, five sides? This pattern gives some back-up for your reasoning. (Good problem solving works for more than one problem!)

If you imagine walking around the shape: edge vertex edge vertex .... because it is CLOSED (have to use that somewhere) you can write out a 'cycle' with 10 edges which is general enough that you can just 'count' the vertices.

Hope this helps.

Walter Whiteley

You are right, Makayla!

In a closed figure, each side attaches to two vertices and each vertex joins two sides, so it makes sense that the number of sides and vertices is the same.

Stephen.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.