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
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
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
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
Videotaping Activity Sheet
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.
Cooke, Marcia B. "A Videotaping Project to Explore the Multiplication of Integers." Arithmetic Teacher, 41 (November 1993): 170-171.