Often when students learn a particular skill or concept towards the end of a unit, I notice that they don’t have as much time to deeply understand that concept. My conjecture is that the lack of sufficient days of review or practice affects retention. We completed the Piecewise Functions unit with lessons that asked students to make a piecewise function continuous. The thinking and problem solving required to do this is mathematically rich. It is possible to use multiple representations to solve problems through graphical and/or algebraic approaches. Successful solution processes require enough steps that students can develop their own strategies through successful mathematical practice (MP1, MP2, MP3, or MP5). For these reasons, I decided to extend work on these types of problems into the the Absolute Value Functions.
The Piecewise Functions and Absolute Value Functions warm-up includes three sections which are each divided into 3 levels. So this warm-up gives students the opportunity to identify the right level of challenge for them. This is a good chance for you to provide some coaching, such as:
Asking these kinds of questions gives students the chance to reflect on their learning during class and over the week to make better choices about how to use their class time.
The big idea behind making piecewise functions continuous is that we want to make sure that each function that composes the piecewise function has the same outputs at the transition points. Rather than teaching students an algebraic algorithm to solve these problems, I keep asking them to repeat this big idea back to me, and then I ask them how to use this big idea to turn the problem into some equations that they can solve.
Over the course of the week, students will be able to develop many of their own ideas and strategies to solve these problems. Even though it seems to take forever, asking them to come up with equations and methods on their own enables them to really understand what they are doing. I focus on MP1 by not telling students whether they are correct or not. When opportunities arise I ask them to listen to each other’s ideas for solving the problems associated with writing the piecewise functions (MP3).
I like to give students a chance to talk to new people about their work, especially when a new problem or idea is being introduced. I assign partners by choosing people at different tables who have made different amounts of progress, but who are generally similar ability levels, and I ask students to find this new partner. I ask them to compare what they figured out on both sections of the warm-up, and to teach each other what the other one doesn't understand so far. This type of activity helps show students that there is a lot of knowledge in the room, and that the can learn from each other rather than waiting for the teacher to tell them how to solve a problem.
After they briefly discuss the problems with each other, I ask them to write a quick check out:
1) What does it mean to "make a piecewise function continuous"? How can you solve these problems algebraically and graphically?
2) What do you understand so far about distance functions? What do you still want to understand better?
By taking time to ask students to think about these ideas, I can help show them what the big ideas of the lesson are, and I can also get a quick sense of their metacognitive understanding of what they are doing. Usually based on their classwork, I know how well the are able to do the problems and usually I can get a sense of what they understand as I circulate and ask them questions, but I am often surprised by all of the things that do not translate into their written self-assessments of their understanding.