Illustrating Why a Negative Times a Negative is a Positive Using the Number Line!
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The number line is incredibly flexible in the different mathematical operations it can illustrate. Of course, students become familiar almost immediately with how it can be used for addition and subtract, and the way it represents those as left and right shifts. Scaling helps us represent multiplication and division, and understand the connection between the two.
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Inevitably, though, students also want to understand how to represent multiplication by negative numbers. Although there are other models that can help us understand this (repeated subtraction, for example, rather than repeated addition), my personal favorite is to take advantage of scaling on the number. When we let our scale get closer and closer to zero, we can see the number line contracting, until at a scalar of zero, we see the entire number line collapse to a single point. Pushing the scale past zero reveals that multiplication by negatives is actually flipping the entire number line around. This means that the items above 0 now flip below zero (so, a positive times a negative is negative), but even better, also shows that the items below 0 now map above it (so, the famous maxim "a negative times a negative makes a positive").
Not only can this reveal to students deeper structure than we might think the number line capable of, it also primes them for a geometric understanding later on of complex multiplication, and the role of the imaginary unit.
#negativetimesanegative #multiplication #numberlinemath #numberline #scaling
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