I gave four lines in slope-intercept form and had my students get out a clean sheet of paper, fold it twice to create four quadrants, and write one of the lines at the top of each quadrant.

Then they were to write four categories (in each and every quadrant--oh my!):

- Given line (Y1)
- No solution line (Y2)
- Infinitely many solutions line (Y3)
- One solution line (Y4)

**Y1**) is the line I gave them.

**For Y2:**we talked about what would need to be true about the second line in order for it to never touch the given line. The kids were pretty quick to tell me that the lines would have to be parallel, and for that to be the case, the lines would need to have the same slope (and different y-intercepts, btw, cherubs). A-ha! We do remember some things from Algebra I! So, we decided on a line that was parallel to Y1 and wrote it in the category of "No solution line."

**For Y3:**we discussed what would need to be true about a line in order for it touch the given line at each and every point on that line. Well...it's gotta be the same line! Write that in the category of "Infinitely many solutions line."

**For Y4:**the typical case, but I love that this activity made them think a little deeper about this case. "So...what has to be true for a line to touch the given line once and ONLY once?" Pause. Pause. Pause.

Still, small voice: "Different slopes?"

Oooo...

"So, a line with ANY slope other than that of the given line will intersect with the given line somewhere?"

Pause. Pause. Pause.

Unanimously: "YEAH!"

Wohoo! So, we made up a line with different slope and wrote it in the category of "One solution line."[1]

Graphing calculator time...

For the given line of y=2x+1, our y= screen may have looked something like this:

We changed the features for Y1 and Y3 so we could distinguish between the lines and actually see the calculator graph the given line again for the special case of infinitely many solutions. I had them sketch these lines at the bottom in addition to stating the point of intersection for Y1 and Y4 (they could use their calculator).

I did one of these exercises with them and then had them do the same thing for the remaining three given lines on their own/with their partner.

About half-way through the period, I had them turn their papers over. Using the same quadrants and the same lines they created, we solved each of the three cases algebraically. That's twelve systems they solved in half a lesson. The goal was to get them to see that algebraically a false statement is related graphically to two lines that never intersect (no solution); that a true statement is related to two lines that always intersect (infinitely many solutions); and that a conditional statement is related to two lines that intersect once (one solution).

Again, it didn't take lots of prep and I think it really brought together the geometry with the algebra. I hope you're proud, Descartes.

[1] Lots of students would just change the slope of the given line but keep the y-intercept. Then, when they solved the system, they noticed that x always turned out to be zero. "Mrs. Peterson! I keep getting x=0! What's going on?" "What did you keep the same?" "The y-inter...oooo..." Light bulb. One kid was so excited about this I truly thought he was going to pee his pants. It's the little things in life.

This is a great idea. I love it because it forces them to think about all the possibilities at the same time and chose one that fits the requirements. Also, this activity has foldable written all over it. I'm totally borrowing it for when we start solving systems of equations.

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