Parallel and Perpendicular Lines

One of the main focuses during my first year of grad school was the idea of constructivist teaching.   If you’re unfamiliar with the idea of constructivism, in short, it’s teaching in a way that gets the students to discover (construct) on their own what you want them to learn.  At first, I really struggled to wrap my head around how that would work in a math class.  My goal my first couple years teaching was to explain the math so well that students  didn’t have any questions.  My goals are now very different than those first few years! The more I started using constructivist activities in my teaching, the easier I found it was to implement more of those types of activities into my classroom.

When I started implementing more activities that led students to discover the math last year, I was not very familiar with Desmos activity builder.  I’m slowly becoming more familiar with how to use it and am loving how it makes these discovery-type activities run much smoother for both me and my students.  There was little to no direct instruction when my 8th graders were learning about parallel and perpendicular lines.  For the most part, students discovered it all through a few Desmos activities I created (with the help of some Twitter friends).  🙂


Here are the links to the Desmos activities.

Parallel Lines

Perpendicular Lines (Thanks to Ilona for the help with this one!)

Equations of Polygon Sides (Thanks to @GrainBrowth for the help with this one!)


As I think back on how the lessons went, a few things stand out to me.

1.  I probably shouldn’t have been, but I was surprised that nearly every student used “guess and check” to find the answer to the question below.  That meant that we had to spend some time talking about ways that students could come up with a solution without using Desmos since I have yet to have students use Desmos for tests.  However, I think because of this I had far fewer students write “y = 4x + 10) for their answer because they saw what happened in Desmos when they tried that.

Screen Shot 2017-03-19 at 6.10.43 PM

2.  When I give an exit ticket at the end of class after doing one of these activities, I’m still somewhat amazed at how well students do.  It’s encouraging to see that this type of instruction works.

3.  In the past when I’ve taught this concept, my lessons looked something like this:

  • Get students to understand parallel lines have the same slope
  • Get students to understand perpendicular lines have opposite reciprocal slopes
  • Spend time using those ideas to solve problems

This year I decided to spend about 2 days on just parallel lines and solving problems with parallel lines.  Then about 2 days just on perpendicular lines and solving problems with that.  Then, I combined the two before having students look into the equations of polygon sides.  This seemed to go better for students.  The biggest mistake I saw students make was when they were asked to write a line perpendicular to another line through a specific point.  Many students who made a mistake on this type of problem, knew to change the slope because the lines are perpendicular.  However, when they would find the y-intercept, they would leave the slope the same, and then at the end when they wrote their final equation, they changed the slope.  However, in the past when I’ve taught this, students have made the same mistake.  I don’t have the data to back this up, but it seemed that fewer students made this mistake this year, and I taught this concept to more students this year than in the past.

4.  As students were working through the activities, I wish I had a better way to quickly check which students were on the right track.  The Desmos teacher dashboard is awesome, I’m just not that great at using it efficiently yet.  I mentioned to my co-teacher that I wish I had a better way to know if students were on track or not, but that I also had the same problem when students were doing this types of activities without Desmos activity builder -it’s actually worse without it.



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