Why Perpendicular?

I realized part of the reason I haven’t blogged much this year (despite saying it was one of my main goals!) is because my team is so awesome. There are four regular ed teachers who teach double period Algebra 1, and we’ve done it in various combinations for several years now. This year, we’re newly combined and changing up the way we teach some things.

My contributions are often sharing awesome things I found (but that others wrote)–great for my students but not really anything to blog about. I’ve been loving Math Equals Love INB pages. I’ve implemented way more Desmos activities. I’m not standing still–just not very original I guess.

Anyhow, last week we did parallel and perpendicular lines. We started with parallel and a fairly routine look at parallel lines, focusing on identifying slopes and writing equations of lines that were parallel through a given point (largely to practice writing equations). I (badly) started introducing perpendicular at the end of the period, and homework focused on parallel lines. That day, C (one of my awesome team members) came in to talk through what he had just tried in his class: he gave his kids dot grid paper and had them determine where perpendicular slopes came from.

He tried writing up an activity sheet for me (which he is graciously letting me share below) BUT I didn’t use it either. I tried out what he discussed the next day in my class–and after using his sheet myself to practice decided to instead give dot grids in dry erase pockets. That way, students aren’t limited to a certain number of points (which may mean you have to start at only one of a couple places to be able to complete a square.

When students entered, the Do Now was posted on the board: Make as many squares as you can on a 3 by 3 dot grid.

3by3

And as the bell rang, each student got a dry erase dot grid, marker and eraser. My morning class got the idea; my afternoon class (which OF COURSE C was observing so we could talk through this) did not listen at all. Some kids were trying to make squares on an entire sheet of dot paper (whhhhhhy???). Finally, we got the idea and I had students submit in a poll how many squares they got. I should have been clearer that I would have students come up to show their squares, which hopefully would have made them count a bit better! For a 3 by 3 dot grid, the max is 6, but none of my students got more than 5. I showed them the 6th and then told them to use 4 by 4 dots to make as many squares as they could. Again, the maximum a student found was 18, but you can get as many as 20.

Warm up over, we took out our notebooks as well. I had students draw a segment with a slope of 1/3 on their dot grid–and then complete a square. Some students could not figure out how to do this, so I was glad for the dry erase. When we all had it, I had them copy the square into their notebook, and then make a square with a slope of 3/4 (which was tricky so I would maybe switch that for something easier?) on the dot paper. We followed the same process for -1/4 and -2/3 (I may have one of these be an integer slope for next year).

image2

Then I asked them what they noticed about the slopes. After they discussed with partners, we shared out and used those ideas to build our definition of opposite reciprocal slopes. Previously, I’ve usually just said that perpendicular slopes are opposite reciprocals and practiced finding them–but I didn’t feel like students understood WHY that happened. This took a little more time, although not as much as I would have thought, and I think students have a much clearer idea of what makes two slopes perpendicular to each other. I still did Quick Poll them on the calculators to practice finding the slopes, and I want to add more in about differentiating between the two–when I just asked for perpendicular students were generally fine, but asking for one OR the other led to way more wrong answers. Maybe something where they have to circle or underline or write the symbol would help? I thought about having them find both the parallel and perpendicular, but the problem seems to be deciding WHICH one to find.

From there, we practiced more writing equations with Math Mats.

Let me know if you try this out–it’s still very much a work in progress, so we would love any feedback you may have.

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