Section 30.4 Closing Ideas
One way of understanding multiplication that was not discussed here was "multiplication is repeated addition." We side-stepped that one because it's a very limiting perspective of multiplication. The limitation of repeated addition is that it is limited to only integer values. It does not make sense to add something a half of a time. But it does make sense to talk about half groups of objects, such as a half bags of chips.
This doesn't mean that the concept of repeated addition is wrong. If you go back through the diagrams, you can see that it's sitting there in plain sight. But it simply does not have the flexibility that these other images do. Even though we did not explore them in this section, the concepts of area and groups of all extend into fractions and decimals.
In fact, we already saw groups of when we were working with fraction multiplication. The product \(\frac{2}{3} \cdot \frac{5}{4}\) was two-thirds of a collection of wedges of size And we were able to obtain that value by taking the wedges, cutting them into pieces, and taking out of each of those new collections.
It turns out that area also works with fractions, though you do need to be a bit more careful with your drawings to know where your integers are. Here is the representation of the product \(\frac{2}{3} \cdot \frac{5}{4}\) as an area.
Notice that inside of each unit square there are pieces, and that a total of 10 pieces have been shaded in. This means that the final result is (which reduces to If you were to multiply straight across, you would get the exact same result.
These are connections that are both obvious and non-obvious at the same time. If you understand the idea of multiplication being represented by areas, and if you understand parts of a whole, this picture makes complete sense. But if you struggle with one concept or the other, this can seem extremely mysterious and unnecessarily complicated. But that's the tension of mathematical thinking. As you grow in your intellectual sophistication, you can start to see connections arise in a very natural way. But if you don't have the complete toolbox, it's easy to simply be overwhelmed and frustrated. As we approach the last part of this book, we hope that you are becoming more and more like the former than the latter.