To make a conjecture, or educated guess, about an idea in mathematics, we can start by experimenting with different values to find out what is true in a given situation. By trying out different side lengths, we were able to figure out that the length of one side of a triangle must be less than the sum of the lengths of the other two sides. This conjecture turns out to be true in all situations, but we will convince ourselves of this fact in another lesson.

Consider this progression of conjectures about the side lengths of a triangle:

• One side can’t be too long.
• Any one side must be shorter than the other two sides together.
• Any one side’s length must be less than the sum of the other two sides’ lengths.

Which conjecture is the most clear? These statements may look like a progression of ideas used when developing a conjecture. It is helpful to have a partner to share our thoughts with so that we can refine our ideas into stronger and clearer statements.