The Choosing Inputs Warm-Up introduces a new type of problem. Students need to determine whether or not a data table represents a radical function. Sometimes even though questions seem repetitive to us (the teachers, being experts in the content) they are non-routine for students. Different presentations of concepts push students to think in different ways.
Problem (1) asks students to develop a general explanation of what makes a data table radical, and they may or may not write their answer. Likely their written answer will be less articulate than what they can explain orally, so as I circulate I always ask them: “What do you mean? Is that always true? Can you use the data tables provided as examples to justify your claim?” These kinds of questions help show students that no matter what they write, whether it is correct or not, they will still need to justify their answer. It also helps them to learn to use the examples given to support their claims (MP3).
Problem (2) is asking students to generate an example of a pair of inverse functions and to observe the relationship between the graphs. This is another important opportunity to ask students questions, like:
As students start to work on problem (3), I ask them to look at the transformations of the graph, rather than just turning this into a data table. This is a good example of a problem that can be tackled two ways, both abstractly and concretely (MP2). Students can identify the transformations visually: shifted up, reflected over the x-axis, and possibly stretched, or they can find the coordinates of the points and use those to find the function rule as they did in problem (1). As always, with any problem that can be approached in different ways, as I circulate I ask students to think about the problem in the other way. If they make a data table, I ask them how they could have used transformations. If they use transformations, I ask them how they could have used a data table. This helps show them that the goal is not simply solving the problem, but rather understanding the ideas behind it (MP1).
Use the closing to ask students to be more precise in their explanations of how to choose inputs. We need to choose values for x that cause the radical to be a perfect square number. To do this, it is sometimes easiest to see the expression in the radical equal to a perfect square and solve this equation for x to get the input. Students may not put all of this together, so it is important to push them to explain it thoroughly (MP6.)
The remaining questions on the Exit Ticket deal with the graphs and transformations. At this point, students should make some generalizations, but they may struggle to be precise. I ask them to use the examples from the Matching activity as evidence to support their answers (MP3). Again, a word list is a helpful scaffold:
Because upcoming lessons will continue to deal with graph transformations, these closing questions are more of a preview, so students will have more opportunities to flesh out their answers.