My brain is so fuzzy today. I’m struggling to type this because I just can’t think straight and I’m not sure why. Just one of those days I guess.

I introduced the concept of slope of linear equations today.

I began by pairing my students and giving them the silly drawing below.

I totally made up the elevation of Mt. Doom this morning and I realize it’s unrealistic so save it, LOTR and earth science pedants.

As you can see, my Mt. Doom as the Eye of Sauron at the top of his tower, Frodo and Sam at the top, and a few eucatastrophe eagles on the way. Yes it is terrible. My middle block let me know as much.

I then asked the students to identify which portion of the path up the mountain, notated by the dots and lines, they thought was the steepest and then be prepared to defend their choice. I did not give them any tools or definitions to do this.

Some students goofed off instead of trying but many of them circled a part of the path they thought was the steepest. However, no one was able to defend their choice other than to say “This part looks the steepest”. Some coaxing did get phrases such as “most vertical” or “least horizontal” but they were missing the vocabulary to really say anything other than that. And honestly I was really happy with their answers. I wanted some more discussion and I definitely need to improve my skills in leading a guided discussion, but overall it went well in all of my classes and I felt it was a good way to introduce the idea of comparing relative steepness of lines. Of course, I am borrowing heavily from Dan Meyer again here with the example he mentions in that famous video “Math Class Needs a Makeover” where he has the ski slopes. However, he does a much better of job of showing the layers of abstraction than I did today. BUT, tomorrow we will look at the same picture in Desmos with some grid overlay so find the true answer.

After we did this, I drew two lines on the board and had the students compare the steepness to continue the discussion. I purposefully drew the first line to be very shallow but rise higher than the second line which was very short and very steep. I asked them to compare the steepness of the two lines. I told my students to pretend they were lazy like me and asked them which one they would rather walk up. This worked well in all of my blocks, but it worked especially well in my final block where some groups of students got very heated about the definitions of steepness and which one was steeper. But it still wasn’t the way I wanted it to be in terms of student conversations.

My biggest struggle I think with student conversations is that if I break them into smaller groups, I cannot be in all of the groups at once, pushing them to start arguing. (Once they get going they don’t need me) and they get off task. But if I do a whole class discussion like I did today, most of the students don’t get to talk or argue about the math. I need to get better at designing tasks that need less guidance from me so that the students can get started without me and classroom management so that everyone is working and also setting expectations early and modeling how small group discussion is supposed to work. I think there is a place for whole class discussion but my lesson today would have been served better if I could have had small groups working together. This has been my challenge all year.

After this discussion we moved to notes where I gave formal definitions of slope, talking about the ratio of change in y to change in x, teaching them the greek letter delta for change, and rise over run. I really emphasized that rise over run is just a way to remember the definition of slope and that change in y over change in x is more important to know. I’ve seen many students in later classes remember rise over run but have no idea what that means for calculating slope and I would rather not perpetuate this. I really wanted to help them get a more intuitive grasp of slope, but I realized today that I don’t know what kind of tasks and lines of questioning really help build this intuition. This is a very glaring gap in my content knowledge. I’m not saying I don’t know how to explain slope, I do, and well I think, but I don’t really want to explain it. I want to help them build understanding. My brain is too fuzzy right now to think on this. The words are coming to me faster than I can type and it kind of hurts. Blargh. The best I could do for building intuition today was to keep relating slope back to our idea of steepness and saying “How far does the line go up or down and how long does it take the line to go up or down? That’s slope. That’s change in y divided by change in x.”

The rest of the notes today after giving the definition of slope and talking positive, negative, zero, and undefined was working examples as a class. Then I set them on their own work. The notes took longer than I thought because of the discussion (and presentations at the beginning of class) so they did not have enough time to finish practicing in class. I will allow them to work on it again tomorrow and try some more ideas for building intuition.

I need to stop typing though, I keep making mistakes and my head is too fuzzy to think straight it is starting to give me a headache.

Thanks for reading.

With regard to your remark about the small group/whole class conversation divide, I have had great success this year with whiteboards for small groups to record their thoughts before presenting these thoughts/arguments to their classmates. This has worked really well with my older students and pretty well for my Geometry kiddos.

I appreciate your resistance to the ‘rise over run’ business or any similar memorization technique!