Many Lines Through One Point

Today's lesson is all about the standard REI.5:

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

I want students to understand a few things: first of all, that it's a non-trivial task to find three or more lines that all intersect at the same point, and it's amazing when we can get that to happen.  I also want them to understand that it's a "legal" algebraic move to multiply an entire equation by some factor and then add it to another.  Not only that, but doing so will produce another line through the same intersection point as the other two, and that's pretty cool!

I describe today's opener in this video narrative.  You can access the graph on Desmos by clicking this link.

A few further notes about this opener: 

• Kids will want to get up and talk about it.  In full disclosure, the photo I'm sharing here was staged after the fact.  But during class, there are always a bunch of kids who go up to take a close look, and some great conversations always happen up there.
• I love that this is a self-checking activity.  If a student tries something and it doesn't work, that's clear to the student.  Conversely, when something does work, it's super satisfying, and the live-updating nature of Desmos makes it neat to watch lines "snap into place".

After students work on the opening challenge for 10 minutes, I distribute this handout: Computer Lab Graphing Lines.  As I describe in the narrative videos here, I try not to say too much about the connections between this document and the opener.  I think it's best to talk through the handout while we both take a look at it, so I've prepared a few video narratives:

• Here's a video description of the front of the activity.
• Here's the back.
• And here are some extensions.

One point to reiterate is that a big point of today's lesson is to understand that this is a legal algebraic move.  I make these notes on the board to make it clear to all students what I'm talking about.

With a few minutes left in class, I ask for everyone's help.  I say that we're going to make up a new system of equations.  I get each of the coefficients in the equations you see here by asking volunteers to name numbers.  Then I say that tonights homework is to find four more lines that intersect at the same point as these two.

Note that it doesn't really matter what that intersection point is.  In fact, the "messier" its coordinates are, the better!

I've found that I can gauge the level of understanding at the end of a class by seeing how excited kids are to give this a try.