One of my favorite things to teach is piecewise functions in Advanced Algebra. For my Honors classes, it comes pretty early in the year.
Last year, it was the first time I really convinced my class I could teach them something.
I started the year with a quick review of functions and inequalities. My students were unimpressed. This class was supposed to be a challenge, and instead it seemed easy.
I asked if the students knew domain & range, and of course everyone had it down. The x is domain, the y is range. Done.
Yep, sure, sounds good guys.
So lets get to some piecewise functions. And then I lost them. Mostly because they thought they could find me whenever, and could just catch on to whatever I was teaching. [Last year was the first and only section of any kind of Advanced Algebra offered at my high school, so they had no point of comparison. That won’t be the case this year.]
I did a brief lecture of the concept of using just parts of a piece of a function, which went ok. Then I went over how the domain and range worked for a piecewise function. I’ll spare you. It wasn’t very exciting. Also, they didn’t really get it.
The next day, the opener had students graphing one of two functions, depending on which side of the pair they were seated on. I told them to take one piece of paper from either partner and rip it in half (so that they had the same scale). So they each graphed their function, and then I had them fold their papers on the line x = 2 (or whatever, really). This helped a lot with the piece part of it, but the domain and range was still fuzzy.
In retrospect, these two activities should have been the first things I did.
The opener was the same, but then I had students write down the domain and range for ONLY their section of the graph, before they pieced them together. Separating it like this seemed to really help them understand that it was only one part of the equation, and that the composite determined the domain and range.
Then they pieced together the two graphs and wrote the piecewise function. Finally, each student rewrote their domain and range on a number line, and then placed the number lines together to see overlap and draw a new number line to see the domain and range for the piecewise function. For these functions, the range was continuous, but the range had a gap, which I think was helpful.
Then the fun began. Each student (I had 29) got one part of a function, which was labeled 1-10 and A-C—ie. 2B or 9A, and then a piece of graph paper. Each student oriented their graph paper horizontally and then graphed the function they were given.
Then the students found the others with the same # (letters didn’t matter yet). Once they had a group of 3 A-C, I gave each group a full size sheet of paper with a piecewise function laid out—the only thing missing were the parts of each, which the students still had. Those went into blanks labeled 3A, 3B, 3C (etc) on each sheet. I printed them out on a variety of fun colors but that’s totally optional.
Once the students had glued or taped their expression in the correct place, they knew the domain of their portion of the piecewise function. Using that information, they cut their graph according to their domain, then wrote out both domain and range (onto two number lines if needed—it really helped my students).
Finally, each of the graphs was put together on the bottom of the paper, and the domains and ranges were combined to find the domain and range of the piecewise function. I would give them one more piece of graph paper for this, but I didn’t last year. That caused some issues if there was a gap in the domain, since students physically cut the graphs into pieces—if no one had 2<x<3, there was a missing strip between two sides of the graph.
This activity really helped my students understand where piecewise functions were coming from and was really fun to teach as well. Everyone was engaged the whole time, and really into what came next.
What is your favorite subject to teach? Activity structure?
[I can’t post the document or images right now because I’m on vacation with very limited internet, but I have it still. You can email me for it or leave a comment.]