Dear Mom, I know something you don't know

This is something I did in the very beginning of the year that I meant to blog about and just never did.

At our school, students are introduced to the graphing calculator in Algebra II, which means I've gotten to show kids lots of awesome techniques to problem-solving that they haven't been exposed to yet.

One of our first tasks of the year was solving systems of linear equations by graphing, specifically using the graphing calculator (TI-84).  I asked the kids if they knew when the graphing calculator was invented and when it became standard use in high school mathematics courses.  I asked if their parents would know how to use such a device.

The students assured me their parents would have no idea how to use a graphing calculator, so I asked them to write their parents a letter describing how to solve a system using the TI-84.  I was somewhat surprised--every student (I have about 100 young algebraists) sat quietly and wrote.  And wrote.  And wrote.

This was my favorite:

Yo momma, it’s yo boy ________ and I’m telling you how to solve for y.  Add x on both sides then you will end up with y=x+0.  On the other problem you are going to subtract 8x on both sides and then divide by 4 on both sides.  You should end up with y=-2x-6.  Then you want to put this in your graphing calculator and press graph.  BOOM!  There’s the graph.  Then will want to press 2^nd then calc, then press enter three times and you should have your coordinates.

Systems of Linear Equations Activity: 3 Cases

So, this isn't anything super fancy, but it worked quite well with my Algebra II kiddos (without much prep on my part, which doesn't happen often), so I wanted to archive the idea and hopefully get some feedback/ways to improve it.

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)

The given line (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.