What’s Up With The Rectangle?
A discussion arose in the Facebook group about "why the rectangle". Mortensen Math, Math-U-See and even Gattegno use the rectangle for teaching math concepts. Why? What is so special about the rectangle that it makes visualizing math so easy? The last question that was asked was, "When would I use this in real life?"
Polynomials and factoring them have value in all kinds of places in science, technology, statistics, and economics. The rectangle is an abstraction. You want to think abstractly. Why? The more abstract we think, the more relationships we notice between seemingly unrelated problems. Your ability to solve the abstract problems is going to get easier, which means that solving the actual problem, in front of you, in real life, is also going to be easier -- even if it has nothing to do with math. But understanding polynomial factoring helps us with rapid mental calculation, and that should be an interest to everyone today.
I'll be looking at the rectangle, the relationships it represents and, as it fits in, I'll work in that rapid mental calculation stuff I talked about.
The rectangle is just a grid or table for displaying information. We could use all kinds of shapes or graphs to display information, but the the shape we want to use should be the best shape to help us understand the information. The shape that happens to be most effective is the rectangle. So lets look at the statement 13 x 15 = 195. If we set this up as a basic grid we get the following:
The reason, in the video, I suggest you add diagonally is that we are keeping 100's, 10's and 1's separate for easier understanding.
If you are so inclined, add them across. It doesn't matter, just add.
Let's Take A Look At Another Rectangle:
If we had the problem ab x cdef, the grid would help us to keep all the information straight.
Did we actually multiply everything by everything else?
We can easily see from the grid that we did, indeed, multiply each row by each column.
This is called the Distributive Property or Law, in my next post we'll be looking at this a little more in depth.
This representation of math is called a geometric representation or the area model. It should be visually obvious as to why. If one side of the rectangle is 12 and the other side is 13, answering 12 x 13 = also answers the question, "What is the area of a rectangle that is 12 by 13?"
Did Caleb Gattegno or Jerry Mortensen come up with this great idea? Not even close. The area model for understanding math has been around a long time. You can find it in elementary mathematics textbooks, like Ray's, in the early 1800's.
What Caleb Gattegno and Jerry Mortensen did is recognize is that the combination of the area model with base-ten blocks would make math education a hands-on and visually obvious experience. Even though the men applied the area model differently, it remains the core of both curricula. As this series progresses, we will look at why that is.