Back To The Future

by Sr. Alice Hess
K-12 Archbishop Ryan High School
Philadelphia, PA

Although I have tried many approaches throughout 32 years of teaching math, nothing seems to work as well as this when it comes to getting beginning algebra students to discover for themselves the rules for multiplication of signed numbers.

By way of introduction at the overhead projector, I announce these conventions:

Next, I pose the following questions:

1. If you were in the habit of gaining $5 a week, two weeks from now would you be richer or poorer than you are today? (+5)(+2) = +10
2. If you were in the habit of losing $5 a week, two weeks ago were you richer or poorer than you are today? (-5)(-2) = +10
3. If you were in the habit of gaining $5 a week, two weeks ago were you richer or poorer than you are today? (+5)(-2) = -10
4. If you were in the habit of losing $5 a week, two weeks from now would you be richer or poorer than you are today? (-5) (+2) = -10

With a little thought and common sense, students correctly respond to each question and, in the process, develop "their own" rules for multiplying signed numbers.

Another concept that Algebra students seem to have difficulty grasping is that division by zero is not defined. I lead them to discover the dilemma caused by trying to do so by posing and acting out the situation that follows:

1. If I have 12 cents in my pocket (and I jingle them around), to how many people can I give 6 cents? 12 / 6 = 2
2. If I have 12 cents in my pocket, to how many people can I give 4 cents? 12 / 4 = 3
3. If I have 12 cents in my pocket, to how many people can I give 1 cent? 12 / 1 = 12
4. If I have 12 cents in my pocket, to how many people can I give 0 cents? 12 / 0 = ??

As various incorrect answers are prematurely called out, I proceed around the classroom dramatically giving nothing to each student, continually jingling the 12 pennies in my pocket as I go along. Sometimes to add to the drama, I go across the hall and ceremoniously give zero cents to the principal and the school secretaries and anyone else I happen to see. Eventually the students arrive at the conclusion that division by zero is not defined.

As a follow-up activity to this scenario, my class acts out Sally Ferry's The Mystery of Madame X which was published in the Clubs Section of the May 1981 issue of Mathematics Teacher. The setting for this short effective skit is a fraction and Madame X-2 is the denominator. Every time the X is replaced by 2 a murder takes place until a "professor" from the audience solves the mystery by correctly explaining that in a rational number, a/b, b cannot equal zero.

As simple as these ideas might seem, the students, for all their sophistication, are amused and instructed.

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