Piecewise functions have been something my students have always struggled with. They don't always get they are taking parts of the function and graphing them on the same graph. They'd like to graph the whole graph of all functions on the same graph or they don't have an idea where to get the points to graph. So, I am changing up piecewise functions this year based on some suggestions from the Math Twitterblogosphere.
The first thing I did was introduce piecewise functions via Mathalicious' Domino Effect lesson. (shameless ad - I've met Karim and had wonderful conversation with him. He and his company are doing some great things to help teachers. It is worth the money to subscribe and have access to the lessons. Go check it out. I'll wait. :-) ) I had used this at Hedge's suggestion - she had used it in her Algebra 2 classes to introduce piecewise functions.
On the second day, I adapted what Maggie has done with her PreCalculus students when introducing piecewise functions. She created an investigation where the students graphed the individual functions and cut out the pieces needed and put them together on a graph. What a great idea! Granted, right now we are working with piecewise linear functions, but the idea is fantastic. So, I took her idea and modified it to fit what we were doing with piecewise functions. Here is what I came up with:
I should add here that the graph is larger and has the printed axes on it - for some reason, I can't get it to show correctly when uploading to either box or scribd. If you want the file, send me an email at lmhenry9 at gmail dot com.
We only had time to work through the first two functions, We did the first one (mostly) together as a class, so that students would understand how to put them together. Once the first one was complete, we talked about what a piecewise function is and how it is created. We discussed the domain restrictions and why we cut out the individual pieces. On the second function, we talked about how to determine which x-values to use in the table and they walked through the process. When it came time to cut out the functions, students asked how to determine where to cut the graphs and we discussed the domain restrictions again. Here are some samples of what my students did:
When the kids returned on Monday, we went over the last question and discussed how to deal with the more open-ended domain. I then went over 2 piecewise examples with them without doing the formal cut-and-paste. With the remaining 20 minutes, I had students work on 5 problems in class. Most students got through all 5 problems and I was surprised that not only did they have very few questions on how to start, but that they were working successfully on the problems they attempted.
What a difference compared to previous times I have taught this concept! In the past, I have not even broached this topic with the regular Algebra 2 students, only the Advanced students, and even then, they have struggled with it. What a huge difference - students actually seemed to understand what parts to graph and to put it together on one graph. I am really pleased with how this turned out. Thanks Hedge, Mathalicious, and Maggie for all the great parts to this lesson!