I’m super behind on this so this might be a long post but here it goes.
During our curriculum meeting at the end of last year, a teacher from another high school in the district introduced us to how he likes to teach linear functions. It was quite different than how we had done linear functions that year and how I had seen linear functions set up in various textbooks or online resources. We all agreed that this year we would adopt that flow and see how we liked it.
I loved it. I really think that intuitively it makes significantly more sense. There are some things that I would still rearrange a little, but overall these two units had a good flow. Unit 4 focused on graphs of linear functions, and Unit 5 focused on other representations of linear functions.
The first topic was slope. I don’t think that we, as a district, focus enough on the idea that slope is a rate of change and I unfortunately didn’t realize that until after we had gotten through these two units. That’s definitely something I want to work on next year.
On the first day of slope we did a lot of discovery. I would give them two or three coordinates, and ask students to find the next coordinate in the pattern. Some students did better than others but I definitely think it helped them start thinking about “rise over run” before it was ever mentioned. The students then also worked with partners on a Gizmos activity involving slope.
On our first day of notes, we started discussing the fact that while it was possible to find the rate of change from parabolas and other curves, we would focus on linear functions during Algebra 1. We discussed “rise over run” and how it tells us how much the x- and y-variables are changing throughout whatever problem we’re working on. Then we began discussing how to identify positive, negative, zero, and undefined slopes and drew out a slope roller coaster as well as our names.
We spent a day practicing how to find slope from a graph and how to identify intercepts from a graph as well.
After we had spent 3 days on slope, we began discussing linear functions a little more in depth. We started by doing multiple representations of a parent function story problem.
We had already practiced doing multiple representations during the previous unit, but now we were able to really discuss what the slope and intercept meant in the context of a situation. We also talked about what it meant to be a solution to a linear function in the context of a situation.
Direct variation came next. This was meant to give us a good opportunity to talk about parent function transformations, since now our intercept is staying the same but we’re going to have different slopes for different situations.
The students did very well with this. We did some more practice problems on direct variation situations the next day, where I projected one of the four types of representation onto the board and they had to come up with the other three. This lead to some pretty creative verbal scenarios involving everything from puppies to aliens to stealing money… Toward the end of this lesson during the last class of the day, one of the students really understood where this was all going.
“But Miss Roberts, puppies don’t weigh zero pounds when they’re born.”
YES. I love when students do my set ups for me! So we took the exact same situation and discussed how all four of the representations would change if a puppy weighed 2 pounds when it was born. What happens to the table? The graph? The equation? We even were able to discuss how not everything is linear forever. Does a puppy continue to gain a pound every week, even once they’re an older dog? What would happen if we all continued to gain weight at the same rate that we did when we were younger? It was a really amazing conversation. I never said the words “parent function transformation” and yet the majority of the class conceptually understood and could analyze various situations that involved them.
This then obviously led really well into learning about slope-intercept form, so for the next few days we practiced how we can use slope-intercept form with equations and inequalities.
I didn’t force my students to do enough practice with this. Almost all of them still struggle to remember y=mx+b, what the m and b stand for, and/or the differences between the inequality graphs. I think that conceptually this unit flowed incredibly well, and that I was more concerned with that implementation than I was with how much they could do during this unit – something that I’ll have to change for next year.