#MathArtChallenge Day 52: Crossing Lines

The Challenge: Drawing only straight lines where all lines must cross all other lines, how many can you draw?

Materials Needed: Paper, pencil, maybe a ruler.
Math concepts you could explore with this challenge: angles, calculus, lines, slope

(As I allude to with my warning -mostly for twitter folk- there is an optimal solution for this, and it’s very satisfying if you can find it on your own. If you want it spoiled for you, look below.)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

The optimal solution to this ends up forming a parabola.

So middle school teachers, this is a great way to introduce slope, because a successful image has all different slopes.

Early high school teachers, you can use this to encourage students to practice generalization skills in Desmos.

Calc teachers: here’s your intro to derivatives as slope. The derivatives are drawing the function for you.

And if you’re nervous about materials, feel free to ask students to fold this from any old piece of paper.

Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments! 

2.) Reason abstractly and quantitatively. This is a different challenge if you allow for more than 2 lines to share a single intersection. How does that change the challenge?

3.) Construct viable arguments and critique the reasoning of others. Make an argument that every line in this construction must have a different slope. Prove that this is true.