2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

We can't all be Rembrandt. Heck, we can't even all be Bill Watterson. Drawing anything fancier than a seriously messed-up stick figure is, frankly, beyond most of us. That's why geometrical drawings are so nice—there's no fancy perspectives, colors, or hands to worry about. Just slap some straight lines and angles on there and call it a day.

For this standard, students will be using really specific parameters to draw various polygons—it's more about following directions than having any artistic skill. And as usual, triangles are front and center here. Sheesh. Quit hogging all the attention, triangles.

Given three angles or sides, students should be able to figure out whether they can draw a unique triangle, a bunch of different triangles, or nothing at all. If the problem asks for a triangle with 5-inch sides all around, there's only one type triangle they can draw. But if it's a triangle with one right angle, one 8-inch side, and one 4-inch side, they've got a couple of options.

They'll also use drawings to solve problems visually, like, "Can a triangle have three obtuse angles?" Trying to connect three obtuse angles into a single shape will pretty quickly show 'em how impossible this is. (And if you've got a student who can construct a triangle out of three obtuse angles, they might be an actual wizard.)

This is a good chance for students to informally "prove" various rules about triangles, like the fact that all three angles add up to 180° and the sum of two sides is always bigger than the third side. The whole standard is essentially a giant visual aid. Instead of fiddling with a bunch of abstractions, your students will instinctively know that a triangle's angles can't measure more or less than 180°, because they've literally tried to do it with a protractor and ruler.

By the way, students won't need to know or use the Pythagorean theorem yet; this standard is all about using trial and error to explore how different shapes are constructed. All those fun, math-y bits will come later.