The Absolute Value Graphs Warm-Up is partly a review of yesterday’s lesson and partly a preview of today’s learning. It is an informal assessment of what students do and don’t understand about the absolute value function.
Problem (1) will likely require students to get out some notes or reference materials, so if they struggle to start, ask them: “What could help you solve this problem? What do you need to know or remember in order to tackle this question?” If students know that your response to their request for help will always be a question, they will start asking and answering more of their own questions.
A dialogue might proceed as follows:
Student: “I don’t get this!”
Teacher: “Okay. What could you do about that?”
Student: “You could show me how to do it.”
Teacher: “Or…? What is another option?”
For a while students will complain about this kind of response, but eventually, they start answering their own questions. At this point of the year I tell them, “If you want me to answer a specific question about the content, ask it in a way that shows me you already did some thinking and put some work in.”
Instructional Note: It can be helpful to have a reference poster showing a sample problem about continuity on the board, so if there is a student who is interested in making one of these posters, they can use this as a piece of portfolio evidence at the end of the unit.
Problem (2) will challenge students. When you offer help always connect absolute value back to distance. For instance, “When it says that the absolute value of x is the same as the absolute value of y that means that x and y are the same distance from 0.” These clues will enable students to think and make important connections as they plot points.
Problem (3) is a preview today's investigation. Again, I tell students, “If you can figure out this problem, you will be ahead of the game.” This motivates students who like to move at their own pace to dig in make a good start.
As students work I make observations to help me decide how to proceed with the investigation. I am particularly interested in whether students are more engaged in Problem (2) or Problem (3). I will use this information to decide which resource to use in the investigation.
Instructional Note: The way that I facilitate this investigation depends on how students engage with the Warm-up. The level of student of understanding of absolute value functions I observed in the opening determines the flow of the Investigation.
If students really struggled with problem (2), use the resource below called “Find points that…”. The purpose of this work is for students to think about the meaning of absolute value and how the absolute value function appears on a graph. The questions also show different transformations of the function. If you choose to use this resource, you can ask students to write function rules for each of the graphs they create using either absolute value functions or piecewise functions.
If students showed mastery on Problem (2), but struggled with Problem (3) during the warm-up, I will use Graphing Absolute Value Functions Example which scaffolds the process of graphing an absolute value function and provides more practice.
Either way, the focus of the lesson is on representing these relationships with piecewise functions, and this is what the main assessment Express Absolute Value Functions as Piecewise Functions is about. The assessment includes a lot of graphs so that students are motivated to consider shortcuts for creating graphs and writing piecewise functions to match the graphs. Eventually, I expect students to "invent" or use the idea of a vertex for the absolute value function. When they make this connection, they will be able to reason about and visualize absolute value functions more efficiently (MP7).
For students who don’t need to practice graphing the functions, they can start formulating a generalization about how to write a piecewise function directly from an absolute value function (see Absolute Value Piecewise Function Generalization). This pushes students to use mathematical practices to explore the structure of both functions and describe the relationship between them.