Shifting Functions: How do they move?

Narrative: I am going to start class today with a brief discussion about Mathematical Practice 7: look for and make use of structure. This is the mathematical practice that I would like my students to think about and focus on today and tomorrow. I want students to try and find the structure and patterns in the numbers in the equation and how this relates to the graph. And then use these patterns to make general rules for shifting. To help students understand the practice better, I would give them a very elementary example: we know that 1+3=4 and 3+1=4, if we know that 2+5=7 can we generalize what 5+2 will be? I predict that my students will have difficulty really understanding what it means to generalize so I think this basic example will help that.

Next, I will present page 2 of the flipchart and have students add the definition of Argument to their math dictionaries. Again, emphasizing the importance of students rewriting this definition in their own words. By talking about the argument of a function, we can lead students into this activity by asking them to figure out what effect changes of this ‘part’ of the equation can have on our graph. Students should now work in their teams or individually to complete the shifting investigation (part II: problems 1-5). With about 10-15 minutes remaining of this section time, push students to start part III: problems 6-9 on the worksheet. It is important that students get time to work with both the shifting and stretching investigation.

Environment: I am going to encourage my students to work individually for this activity, but to use their team as a resource if needed. They will also be allowed to use a graphing calculator in this section and the next to make better use of the time allotted in class. This is probably a good time to emphasize to students the difference between a sketch of a graph and an accurate graph. These graphs are expected to be accurate.  I will present the team rules again, emphasizing the rule that helping a teammate does not mean giving them the answer! Depending on how needy my students have been the last few days, I may also start implementing Question Cubes at this time. I do a really great job at over-scaffolding and not making my students think for themselves. However, I do know that they need to think for themselves to really learn. So these question cubes are my way of making sure kids really do have a question on something before they ask me and it keeps me focused on making the kids figure stuff out on their own!

Differentiation: For struggling students, help them to see the structure in the equation. For example, if a student is struggling on question 2, I would ask them questions like: “If we want to shift a graph up, which values would be changing? The argument or the function values (y-values)? Would they be getting larger or smaller? What can we do to the equation to make all y values larger by 5?”

For more advanced students who zip through this activity, have them complete their graphic organizer the rest of the way (the original one, not the revised one). Some of these terms such as asymptotes and line of symmetry may need to be researched by students. You may want to have textbooks or internet access available for that.

On pages 3-6 of the flipchart there are 4 Clicker Questions to check students current understandings of parent functions and shifting. I am going to be sure to emphasize to my students that this is what they were supposed to learn today. These are the types of questions they should now be able to answer WITHOUT a calculator after completing today’s activity.

It is important that students are able to answer these questions without using a graphing calculator to really be able to demonstrate mastery of today's learning objective. It is essential that students are able to identify the key shifts of these functions by just analyzing the equation. This concept of shifting functions is spiraled throughout the first semester's curriculum so it is helpful if students really understand this early in the course.

Assign homework 2 from this unit. In this homework assignment students will review some past concepts from other courses such as solving an equation and evaluating functions in function notation. Today’s new learning of shifting basic functions will also be practiced.