This Warm-Up builds on the main skill students already have from this unit, graphing a piecewise function. The problems focus on the idea of continuity. This concept may be a bit of a leap for students, but the point of this is to give them the chance to figure out the day’s lesson for themselves.
The first problem includes two piecewise functions that use the same function rules but have different endpoints. They were deliberately written so that one of the functions is continuous and one is not, as a result of the endpoint change. As students discover this, I ask them: “Is there a way you could have figured out which function was continuous just by using the function rule, without creating the graphs?” They may or may not see it right away, but the more they think about it, the more likely they are to figure out the big idea of the lesson on their own.
When it comes to Problem 2, you can have graphing technology (calculators or computers) available so that students who want to look at graphs quickly can use those to check their answer (MP5.) If they struggle to start, I ask them, “Where might continuity break down? Where do we have to check to see if there is a problem?” At this point, if a few students come up with an idea, I ask students who still haven’t figured it out whether or not they want to hear some of the ideas that their classmates have come up with. This encourages students to reflect on their own learning process and make a decision about how they want to figure the problem out.
Problem 3 is a big challenge, so for today’s warm-up, I tell students that it would be awesome if they figured it out, but that they don’t need to worry if it seems confusing. Once all students have figured out Problem 2, I ask any students if they are willing to explain to other students how they figured out problem (3).
Finally, as a transition to the investigation of the lesson, I use the projector to show how I would used Desmos to solve the second problem using sliders. This helps make more clear what is meant by “parameters” because students can physically slide the sliders to change the blue line in the graph until it matches up on both sides with the other lines (see below.) Now, they are ready to start the investigation. They will likely start with the computers, because is what they have been shown, but I tell them that their job is to figure out an algebraic way to find the solution as well.
This lesson is full of big ideas and challenges for the students, so first question on the Exit Ticket is more straightforward: I want them to explain how to set up the problem in Desmos, using sliders, because I think this is a very intuitive way to start thinking about these problems.
The second question asks students to provide their own definition of continuity. I am hopping that after today’s class they can be more precise (MP6) than just saying, “It means the function has no gaps.” I want them to be able to say something more like, "the outputs of each of the function rules are the same at each of the endpoints." I am looking for a sentence that understands the structure well enough to show a person how to set up an equation in order to solve a problem. I plan to keep pushing students to dig deeper as they share ideas and shape their definitions.
The third question asks students to make a connection to prior knowledge using vocabulary: a system of equations arises when we are looking for two parameters and so we set up two equations that need to be somehow put together to find the values of the variables.
My goal for the closing is to help students learn to make their knowledge more public, for sure to get their understanding of the big ideas out there. Ultimately, I want the knowledge shared in these closings become a part of student's portfolios (via the problem sets that they include in their portfolios). The more they see these conversations as essential to their learning, the more engaged they will be in these conversations and the mathematical practices that arise when mathematics is discussed publicly.
If students ever struggle to answer a closing question thoroughly, I put a sentence frame with many blanks up on the board. For instance, to answer question (B), I might right: “A piecewise function is continuous if you substitute ______________________________ into ______________________ and end up with ______________________________.” This scaffolding helps students engage and often pushes their thinking more even than the open-ended question.