Equations of Parallel and Perpendicular Lines Linear, Quadratic, and Exponential Models* slope

Holy cow, there’s a lot to dig into here. Read this post carefully.

geom26 question


geom 26 4 geom 26 geom 26 3 (640x422) geom 26 2 (742x800)


There’s a ton to comment on here. I doubt you’ll need much in the way of a prompt, but here goes: what mistakes are missing? You grade this test on Sunday; what does Monday’s class look like?

Thanks to Tina Cardone, who is not-so-slowly taking over this blog, for the submission.

6 replies on “Holy cow, there’s a lot to dig into here. Read this post carefully.”

Phew, there is a lot going on.

I think Monday looks something like:

Every student gets a sheet that has four big, clean graphs on the front. #1 has two lines that are neither parallel or perpendicular. #2 has two parallel lines. #3 has two perpendicular lines. #4 is a challenge one: two lines that LOOK parallel (or perpendicular if you choose), but are not. The idea being by the end of the period, a student should be able to prove whether or not those two lines in #4 are parallel (or perp).

Given that sheet, I’d start with a discussion on slope – I might do a fishbowl discussion. I’ve done that before where everyone has a mini-whiteboard and can share. Something to get them defining slope again – not me defining it for them. Once we’ve gotten slope back in our brains, we can have similar discussions about graphs #2 and #3. Finally, we talk about #4.

Now, that doesn’t get into point-slope and all that. But, like you said, it seems like too much to hit for one class period.

(Perhaps we split into groups where those that struggled have the discussions above and those who did better get into the point-slope problems and maybe create their own challenging problems).

My favorite, “… because it’s even, as long as it’s even.”

I like Tom’s suggestion a lot.

I’d start from scratch on Monday, something like this:

1. Ask students to draw the SAME line that I draw.
2. Kids find the slope of this line. We make sure everyone has done this correctly.
3. I then ask them to draw a line parallel to this line. (No need to find the slope of it yet. Just draw.)
4. Call on kids to share their parallel line [to the original] to the whole class.
5. Extend suspicious looking lines that may meet somewhere…
6. Now, everyone finds the slope of this line.

Hopefully this drives home the point that there are infinitely many parallel lines to a given line, and they have the same slope.

Repeat 1-6 above with another new line (maybe one with negative slope if first one was positive).

Then, do the same with perpendicular lines.

Or I don’t know.

Not on to Monday in my mind yet – but I am intrigued by a few mistakes that would have to be part of the conversation. Two students make the easy mistake that 2x is the slope, not just 2. I think this is hard to shake but easy to discuss. Do lines have the same slope everywhere? Does it matter what the x coordinates are of the points I pick to find the slope? Then it cannot be that x is part of the slope statement. Two students refer to -2x as the only other two. Have to unpack that one with the whole group unless they are the only students who wrote that odd comment.
Need words – from the students, not from me – about what they think slope IS. Try to stay away from numbers and try to slide the word rate in there somewhere.
Monday will be a busy day in this class

I am fascinated by the beautiful, blank, empty grids next to their answers. I might start by asking them if there is anything they could have done with the grids that might have helped them understand the problem better.

I don’t think anyone used the grid, which is a shame! This is a geometry class who had an algebra review question on their midterm. It’s clear to me we need to do some more Algebra review.

Interesting discussion – Speaking of which , if you wants to merge PDF files , my colleagues used a tool here

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