Composite Functions and E.S.P.

An opener I have used in Algebra 1 encourages discussion of composite functions, but more importantly, allows me to show of my amazing ESP powers!

To start, you’ll need a wizard and/or magic hat.  A cape is also acceptable.  If all else fails, the cardboard crown from Burger King will suffice.

Have students choose any number, and write it down, out of view of you…the wizard!  Encourage students to choose a number which is easy to work with.  Now, step-by-step, have students perform the following operations, keeping track of their answers as they go along, and always hiding their paper from you.

1. Take your number, and double it.
2. Take your result, and add 4 to it.
3. Take your result, and divide it by 2.
4. Take your result, and add 4 to it.
5. Take your result, and double it.

Now, have students share with you their result.  Using your incredible predictive powers, you will be able divine their original number.  Utilize temple massaging and mysterious gestures to maximize the effect…..

If a student says “46”, their original number is 17.

If a student says “84”, their original number is 36.

If a student says “20”, their original number is 4.

I usually do this activity twice with a class.  The first time around, I suggest the students use a simple number.  In the second round, I’ll allow them to try to stump me, which usually leads to some awful mental math experiences for me.

So, what’s happening here?  In this activity, students are asked to do three different things to their results, which can be described by functions:

• let f(x) = 2x
• let g(x) = x + 4
• let h(x) = x / 2

The “secret” to this ESP puzzle is revealed in this 2-minute video, used making the great FREE iPad app, Doceri:

So, as students reveal the result, your job is to subtract 12 from it, then divide by 2.  This will always yield the original number, unless the student mis-applied any steps along the way (this happens!).  Also, in the second go-round, I will always have a student whose answer is 50.98, or something worse.  Have your mental thinking cap on!

This leads to a great discussion of the “inside-out” nature of composite functions, and the ability to recycle functions.  Challenge your students to come up with their own function puzzles, and amaze the world with their math ESP!