I can show you more than that

In a previous post, I talked about my grading system based on the mantra, "Show me what you can do." Here's a quick example of a student's response to a quiz item that shows why this proficiency-based approach feels so right:

One of my learning targets was:

___1.5. I can solve linear systems and represent the solution symbolically and graphically.

I stated that objective at the top of the quiz, along with several others, then gave the following task:

The solutions to the following system of equations are provided. Show that you can use the elimination and substitution methods (use each one once) to solve these problems.

{y = 3x+6
{2x + 4y = -4              solution is (-2,0)  

{7.5x - y = 10
{15x - 4y = 10            solution is (2,5)

What it Means to Solve (Again): Finding Wormholes

In two previous posts, I have explored what it means to solve linear inequalities and what it means to solve quadratic equations. The latter post describes a classroom moment in which my students and I contemplate what it means to find and visualize the (complex) solutions of a quadratic equation that has no real solutions. I wrote:

"A picture formed in my mind of an invisible, ethereal, wormhole-style thread binding the two curves. How could we represent the functions so that the "intersection" would be visible?"

This post documents my post-class exploration of that issue.