Systems Exploration and Ditch Diggers

I really enjoyed my Algebra class today...the pieces of the lesson went together well...

First, I passed out pieces of paper with an ordered pair to each student...asked students to determine if their ordered pair was a solution to either equation shown.  After students checked, I had all students whose points were solutions to -2x+3y=1 stand up.  I used Desmos to create a table and show that the points formed a line (well, some of them, didn't want to show all of the points for time purposes).   Then I had students whose points were solutions to -3x+2y=-4 stand, and repeated on Desmos.  Several students noticed that one of their classmates got to stand twice, and others pointed out that the lines intersected at that point, and that is must be a common solution.  The list of the points I used is here...came from a Teach21 West Virginia website.

Next, we did the exploration activity I described in my last post .  I had a random representative from each group come up to share their question and the conclusions their groups came up with.  Had some really interesting discussion with the extension that asked students to graph a system that had more than one solution.  One group thought that if they graphed three lines in the system they could have three solutions- each where two of the lines intersected.  Another came up with the idea of using two parabolas- the actual question just asked for two equations, and didn't specify linear.  One did mention that if it was the same line there would be more than one...but they didn't think that would make a system if it was the same.  I wrote two equations on the board (equivalent equations...one in standard and one in slope intercept form).  I asked a student to choose a random x value, and find the point on the line in slope intercept form.  I said "Wouldn't it be great if you managed to pick the PERFECT point from that line and it is also a solution to the other equation?"  Students checked it, and IT WAS! :)  So we tried it again with another point and IT WAS TOO!  So then I asked a student to just give me a random ordered pair, and students were disappointed when it didn't work as a solution to the system.  That started a whole conversation about infinity.

After all the groups had presented we moved on...I really wanted to use Dan Meyer's Ditch Diggers 3act problem.  I loved the comment someone made about tying it into the "Chunnel" so we started with watching a little clip about how the "Chunnel" was made, and how the engineers weren't 100% sure that the English and French sides would truly meet up as planned.  Then I played the Ditch Digger act one...asked my students what they wanted to know, and then what might help them figure it out.  Several students requested a grid on top of the image, so they could find some ordered pairs.  Enter Act Two.  We had to wrap up the lesson after recording the ordered pairs, but will start with it tomorrow.

I like to assign about 5 problems for homework each night...so I thought I'd use this...a little practice and extension at the same time.