Give guidance and feedback to students' plans, encouraging, refining, and helping them to be responsible for their own learning. Maintain high expectations for completion of chosen options. Summarize by reviewing the whole, bringing students "full circle" to the experience with which the learning began.
Teacher and class lecture discussion on usefulness and students write essay.
Objective: To help students to see the usefulness of what they have learned.
Activity: The teacher conducts a lecture/discussion emphasizing the many uses of inverse functions in the world. (Note from Bernice McCarthy: I applaud this departure from 4MAT. Here this teacher is moving back to the combined methodology of Quadrants One and Two in order to take the students into an awareness of the usefulness of what they have just learned.) Clearly, since the students are in a class of precalculus, they will probably be taking many upper level math courses in the future, certainly at least two terms of calculus. Therefore, the future usefulness of inverse functions for these students is obvious. Also, in this age of high technology, it is becoming increasingly necessary for students to be familiar with mathematics and the natural sciences. So again, inverse functions will be very useful to them. In my lecture/discussion, I emphasize these points. But the students find it more interesting to discuss more specific uses of inverse functions that they can understand immediately, rather than some time "in the future." So we talk about the field of communication theory (telephones, telegraphs, radios, stereos and television). It is impossible to pick up a bare wire and talk into it and communicate with someone miles away. However, in a telephone, a transducer transforms sound waves into electrical impulses (the FUNCTION). These impulses can be sent to someone miles away. When these signals reach the other person, an inverse transducer (the INVERSE FUNCTION) inverse transforms the electrical signals into sound waves which can be understood by the other person. All forms of long distance communication use this principle. For those who are interested in codes, the same principle is used, although on a much more sophisticated level: a device is necessary to encode a message, and another device is necessary to inverse code the message at the other end. Cable TV companies also use this principle to ensure that everyone with a TV set does not see their programs without charge. They broadcast their programs on a UHF station; theoretically, it should be possible to tune in on their programs. So they use a coding device to scramble their signal. An inverse coding device is necessary to unscramble the signal, which they supply when one subscribes to their service. Without the inverse coder, if you try to illegally tune in on their programs, all you will see is "snow." In more advanced mathematics, many equations cannot be solved directly (especially in the field of differential equations). The Laplace transform is a method of transforming a differential equation into an algebraic expression. This expression can be easily manipulated algebraically, then the inverse transform gives you the answer to the original differential equation. Whichever examples the teacher decides to use to illustrate inverse functions, it is very important to show the students the usefulness of what they have learned, even if they cannot exactly understand all the details. They enjoy the feeling of having learned something useful. Some teachers tell their students they will find out "one day" how useful their learning will be. I believe this only decreases their willingness to learn. Have the students write a short essay on the usefulness they feel inverse functions will have in their own lives.
