Begin the class discussion by saying, "Today we are going to explore both positive and negative rates. Imagine a rate that is positive when a person moves from left to right in front of the number line and negative when he or she moves right to left." Allow time for student comments to indicate that they understand the idea. Continue by saying, "Next, imagine continuing the rate for one minute, two minutes, and so forth."
Do some examples at the chalkboard of rates, such as:
+10 yards/minute × 1 minute
+10 yards/minute × 2 minutes
+10 yards/minute × 3 minutes
...
Repeat the same examples for:
-10 yards per minute × 1 minute
-10 yards per minute × 2 minutes
-10 yards per minute × 3 minutes
...
Left-to-right movement in front of the number line is the model for forward rates. Right-to-left movement denotes negative rates.
Initially, the following situations are demonstrated:
positive time x positive rate = positive distance
and
positive time x negative
rate = negative distance.
Present the other two situations as opposites of the former two:
negative time x negative rate = positive distance
and
negative time x positive rate = negative distance.
Students may ask, "How could you have a negative time? Time moves only forward." Clearly, the students may be discomfited by the use of a model that accounted for only half the multiplication products. Allow students the chance to discuss this problem in small groups of two or three. Students may suggest (and if they don't, you may need to guide them into suggesting), "Maybe it's like a film that is rewinding."
Next, select two student volunteers. One student may operate the video camera while another student is walking approximately fifteen yards forward carrying a green "I am walking forward" sign. After that, the student will backward carrying a pink "backward" sign.
Before watching the videotape, discussed the motions the students think they will be seeing on the video monitor. Distribute copies of the student activity sheet and ask students to predict the apparent motion.
Students should have little difficulty correctly and confidently predicting the outcomes of the first three situations. When the tape of the students walking forward is rewound, she appears to be walking backward, just as the tape of a student walking backward operated in the play-forward mode appears to be walking backward. Despite the logical symmetry, the class may not be able to commit to an outcome if the tape of a student walking backward is played in "rewind" mode. Eventually, students should see that a negative rate times a negative time yields a positive distance! You may need to play the tape over and over to reassure students, double-checking that you see the pink backward sign in the rewind mode.
