In the first image on the left, you can see the large square has length and width of a, which would have an area of a*a, or a^2. There is then a little square inside with length and width b, for an area of b^2. Essentially, the little square is getting removed from the big one (a^2 - b^2). In the last image on the right, you can see that the length of one side is (a-b) and the top side is (a+b), which would mean the area is equal to the product of (a-b)(a+b). This means that a^2 – b^2 = (a + b)(a – b). The intermediate steps just show how to move the area around visually
I think this helps the most. The piece that doesn't "just click" for me is that area = product of multiplication. Manipulating symbols generally just feels more natural to me.
It’s worth rehearsing that instinct. The insight that areas are products is a really useful one for deepening your intuition of what integration is doing, as well as for enhancing your ability to interpret graphs and charts.
Like, if you have a graph showing power consumption over time, it’s great to be able to mentally recognize that, say, if the time units are hours and the power units are Watts, that the area under the graph will be counted in Watt hours; that a rectangle one hour wide by one Watt tall is one Watt hour, and so on.
Not sure if it helps, but trying: In a sense area isn't anything else than multiplication. It isn't like you have a) multiplication and b) area and then prove that a=b.
Rather, area IS multiplication.
The unit of "square meter" quite literally means "meter multiplied by meter".
I agree and I'd say that area is almost, in some intuitive way, the more basic thing and multiplication follows from that (although I know that's not mathematically true). The definition of multiplication for natural numbers is repeated addition (e.g. 3 x 5 is defined to be 5 + 5 + 5). Many people would see that as the count of a 3 by 5 grid of objects, and that's certainly how we'd explain the commutativity of multiplication in school. If those individual objects happen to be unit squares then you have area.
I would say area is integration, and that, for the simple case of integrating a constant function, equals multiplication of that constant by the length of the interval being integrated over (measure of the set, if you’re doing Lebesgue integration)
Imagine you have 100 square tiles that are each 1cm x 1cm. You can make 20 groups of 5 tiles each with them — this is what it means to say 20x5=100.
Now take each of your groups and arrange its 5 tiles into a vertical line. Each line is now 5cm long and 1cm wide.
Arrange the lines side by side. You now have a rectangle whose height is 5cm and whose width is 20cm. You already know its area 100 cm^2, because you made it out of 100 tiles that were 1 cm^2 each. And now you can see that its area also corresponds to the multiple of its side lengths.
