Saturday, May 5, 2012

Graph of a Function and its Inverse (VIDEO)

Using a technique from art class, this activity helps students visualize the reflection of the graph of a function about the line y=x when sketching a function and its inverse. Students who went on to calculus tell me this was one of the activities they remembered from my Pre-Calc class.

Be sure to turn on the annotations (if it is off) to see the directions and comments. This time, I decided to spare you from my voice and some random Creative Commons licensed background music I would usually add to videos.



I suppose you can also use this strategy in Geometry to help students visualize reflections. The novelty factor from doing this in a Pre-Calc class really helped students remember how a function and its inverse are connected graphically. This is especially useful when learning about logarithmic functions.

PS: This is my first time playing with annotations on youtube.

6 comments:

  1. Love this. About how long did you spend with it in class?

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    1. @Rebecka: It takes about 10-15 minutes, depends on student ability and maturity. I speeded up the video so it plays faster. The pace in the video is probably a little fast for students.

      It's a substantial chunk of class time, but I think it pays dividends in the long run. This is one activity former students who've gone on to AP Calc or who took regular calc still tell me that they remember from class. I think it's the novelty of it that helps them remember. I can see how this wouldn't have the impact it had, had the students done similar activities in previous classes (say Geometry or Algebra) before they did it in mine. But from this activity, many students remembered the relationship between exponential and logarithmic functions, at least graphically.

      The activity works better when the students partner up. If they miss a little detail, their partners can fill them in.

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  2. Had your students already seen inverse functions? How did you introduce this? I love love it and was thinking of using it as a discovery-based lesson for students to get some exposure to inverse functions before seeing a formal definition (maybe even deriving their own definition by looking at relationships between various linear functions and their inverses). Thoughts?

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    1. @Melina: I did this in Pre-Calc when I taught it. Inverse functions is in the Algebra 2 standards in California, so students *should* know them by the time they get to Pre-Calc. The sheer number of standards in Algebra 2 means it may not be taught or covered as thoroughly as it should be. Many don't really understand it or even remember it. I review the topic by going over the definition of functions and inverses along with the relationship between the graph of a function and its inverse. So it's not a surprise when students actually do this activity.

      I can see how this could be a discovery-based lesson. Perhaps, students plot points for a function in one color and points for its inverse in a different color, then follow the steps in the video by sketching the first function, folding, and then transferring to reflect over the line y=x. If they plotted the points and sketched the graphs accurately, the reflection should connect all the points of the inverse. They can then describe the relationship between the graphs.

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  3. Absolutely brilliant. Thank you for a great teaching and learning idea! Will be using this with my class in Sydney in a few weeks.

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    1. You never know when skills you learn will come in handy. I think I learned this technique for transfer in 5th grade. Would love to hear your experience with it.

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