How do we move the typical “ray, line, and line segment” lesson beyond merely memorizing definitions? How do we get students *actually thinking* about this topic?

As always, it helps me to ask myself, **“What is interesting about this concept?”** ([Not just “challenging”(https://www.byrdseed.com/challenging/)).

I realized that the concept of **infinity** was staring me right in the face. **Infinity is a go-to interesting topic!** I remember being a kid and struggling to imagine a line that truly went on *forever*. As I thought some more, I realized that **both a ray and a line have infinite lengths**. But a ray stretches into infinity in one direction while a line goes to infinity in two directions.

I knew I had my interesting idea! I had found some *controversy!*

As we learn about these three similar shapes, I want you to wonder with me: which of them is

the longest?

Obviously, the line segment is out right away. It has a defined length.

So I want my students to go home pondering this baffling paradox: **a line and a ray are both infinitely long, even though the ray has an end-point.** Are they *really* the same length, even though one keeps going and one clearly stops? Are there *different sizes of infinity!?!?!?!?* 🤯

We’ve gone from “memorize-the-definitions” to “argue about infinity” by finding a bit of controversy.