Multiplication is scaling

We’re often introduced to multiplication as repeated addition. However, this is an incorrect model that leads to confusion when you start tackling more advanced mathematical topics. A better mental model is to think of multiplication as scaling. In simple words, when you multiply, you’re stretching a quantity $x$ by a factor $k$—even when you’re using fractions.

$$y = kx$$

For some reason, when we learn algebra, we’re not exposed to the idea of arithmetic (and algebraic) operations as smooshing, sliding, and stretching numbers. However, that’s what these methods do; they “transform” quantities. Let’s dissect this.

As we said, multiplying is aking to scaling a quantity by a factor to obtain a different amount. To find the scaling factor, you do $k = \frac{y}{x}$. If you know the amount and the stretching factor, you can retrieve the original number with $x = \frac{y}{k}$. These operations come up in the discussion of direct and inverse variation too, but that’s a topic for another time.

If multiplying by a factor, $k \geq 0$ is akin to scaling in the positive direction, then doing it by a factor $k < 0$ is like pulling in the opposite (negative) direction.

Multiplication by a fraction is the same idea, but we’re scaling by a factor that is a fraction, therefore, we’re scaling by a factor that’s lower than a whole number.

If we multiply $x$ by a fraction with $1$ in the numerator, $x \frac{1}{k}$, because of how fraction multiplication works, we’re performing $\frac{x}{k}$, or scaling down the number by the amount in the denominator. It follows that we can interpret division as “mushing” our number down.

Now, we’re left with multiplication by $1$ and by $0$ which are a bit trickier to conceptualize in this manner (they can be explained empirically or using thought exercises). Unfortunately, I don’t have a good intuition for these two cases. This thread is helpful though.

Let’s look at a couple examples where we “scale” the number $2$.

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