Have you made a Venn diagram before? What do you know about them? How can they be used to evaluate conjectures? Let's say my conjecture is that there are more multiples of two than there are multiples of three. Can you use a Venn diagram to support or refute that conjecture? By the way, a Venn diagram does not constitute a proof, but it will give us better insight to the problem. Also, did you know that a Venn diagram does not need to be made up of circles? It could be made up of any shape that can group objects – in fact, some Venn diagram groups do not overlap at all. Also, it is a good idea to draw a box around the Venn diagram. This will clearly identify those items that were considered, but did not fit in any group.

Let's investigate our original conjecture, that there are more multiples of two than there are of three by looking at the set of integers between zero and ten. I will start with a box to capture all numbers from this set, then I will need two overlapping shapes – one for multiples of two, the other for multiples of three, and a common section for multiples of both. Since I'll be counting the number of multiples of two and multiples of three, those that fall in the common section, and those that fall outside of the diagram sort of cancel each other out – they are neither multiples of two nor three, or they are multiples of two and three.

Pause the video now and work through the list. Be sure to place all elements of the set somewhere inside the box. Resume playback to check your work… Here is what I found. We had four elements fall in the "multiples of two" section, two elements fell in the "multiples of three" section, one fell in the shared section, and four didn't fall in either. Does this evidence support or refute our conjecture? What happens if we extend our set to include numbers up to twenty? Give it a try on your own and decide whether the conjecture is true or not true for all integers. Good luck!