Invite students to share some of the sets of side lengths that form triangles. Record these sets and display them for all to see. Then ask students to share some of their observations. The goal of this discussion is for students to notice that, given the lengths of two sides of a triangle, there are a range of possibilities for the third side. Students should be able to identify some side lengths that work and some that do not work. Ask students:

• “Given a triangle with side lengths 6 and 10 centimeters, can the third side be 50 centimeters? 1 centimeter? Why?”
• “What are some possible lengths for the third side of a triangle given two sides are 5 and 6 centimeters?”

If students struggle to determine if three side lengths will form a triangle, consider asking:

• “Name three sides lengths from the previous activity that form a triangle. How did you decide that those side lengths create a triangle?”
• “How might grid strips help you in your thinking?”

The goal of this discussion is to come to a consensus about lengths that will and will not form a triangle. Invite 2–3 students to share their reasoning about whether the side lengths 4, 6, and 100 form a triangle. Before moving on, make sure students have a clear argument for why these three side lengths don’t form a triangle—for example, “The side lengths 4 and 6 aren’t long enough to touch the endpoints of the side length 100.”

Ask students to raise their hands if they think 4, 6, 10 and 4, 6, 2 will create triangles. If students do not all agree, match up students with different answers and give students 1–2 minutes to convince each other. Select students who changed their initial answer to discuss their new answer and reasoning.

Tell students that some people would argue 4, 6, 10 and 4, 6, 2 form triangles because the sides all touch at their endpoints. This argument seems reasonable but is not correct due to how a triangle is defined: a polygon with 3 sides and 3 angles. Since a polygon is defined to be two-dimensional, it must have some area. Side lengths of 4, 6, 10 and 4, 6, 2 make a line segment, which is only one dimensional and contains no area. If mathematicians had defined a triangle differently, for example, “a figure made up of three line segments that touch at endpoints,” then 4, 6, 10 and 4, 6, 2 would fall into the definition.

Conclude the discussion by telling students that geometry is built on logical reasoning, but reasoning must start with some shared understanding, like precise definitions and proven theorems. Now that we have a definition of a triangle, it is part of our shared understanding and we can build logical conclusions about triangles.