Don't Memorize Formulas! Visualize

It's been a minute since I've gotten "mathy," so I felt it was time to dig back into it a little. Last time, I talked about using the structure of expressions as a means to make sense of the math rather than memorizing formulas. I kind of jumped a head and got into circles since it was right around Pi Day, so let's take a few steps back. This will be kind of a "part 2" to that entry. 

Continuing with the theme of not memorizing formulas, let's look at some math "common sense," for lack of a better way of saying it. If you draw things out, you can often visually see the math behind formulas. The formulas weren't magically plucked out of the air. The formulas you come across in math tell a story. They describe exactly what needs to be done. In the case of area formulas, they tell you how to put the pieces of the shape together to figure out how much space is inside of the shape. So let's look at an example of this "common sense," the area of a triangle.

Why is the area of a triangle 1/2bh (one half times base times height)? Well, let's look at a rectangle. The area of a rectangle is its length times its width (l ∙ w). If a rectangle is 4 units long and 3 units wide, drawing this array out will show that there are 12 square units inside the shape, or 4 ∙ 3 = 12. Most students have no problems with the area of a rectangle; it's a pretty easy concept to grasp. Before moving on, a slight note: mathematically we don't often use length and width for a rectangle. Instead of length, we'll refer to base, and instead of width, we'll refer to height. So a rectangle would be base times height, or b ∙ h.

If we were to draw a diagonal down the center of the rectangle, we would split it into two triangles. Each triangle would be half of the area of the original rectangle. This is why the area of a triangle is one half times the base times the height. You are literally cutting the rectangle in half. So the area of the triangle with the same base and height as the rectangle is half the area. This is a fairly easy notion for students to grasp visually; it's almost "common sense" when you look at it drawn out. So again, rather than just memorizing formulas, add meaning to it.

Once you understand the area of a triangle, you have a terrific tool at your disposal. You can visually break down so many other areas using triangles. From there, you can build the formulas for the areas using the triangles. I never had students memorize any perimeter, area, or volume formulas. I had them construct the formulas by "deconstructing" the figures visually. Two examples are shown below.

You can divide a trapezoid into two triangles. Each triangle has the same height and the bases are the "top" and "bottom" of the trapezoid. Add the areas of these two triangles together and you have the area of the trapezoid. I would have students do this and label each part with a variable and build the formula for the area of a trapezoid for themselves just like shown above. A similar process can be done for a rhombus. If you divide it into two triangles, you see that the two triangles share the same base, which is the horizontal diagonal of the rhombus. The two heights of the triangles together form the vertical diagonal of the rhombus. When you put it all together, you see that the area of a rhombus is half of the product of its two diagonals, which is actually the formula for the area of a rhombus. Students often find building this formula kind of fascinating. You can find the area of pretty much any polygon by dividing it up into a bunch of triangles and then adding up the areas of the triangles; doing this algebraically with variables will typically yield the formula for the area. 

Building the formulas this way is far more meaningful. Just memorizing a formula doesn't allow a student to gain any meaning to it. They have no understanding to attach the formula to. So, they often just forget it. It has no meaning. When a student constructs a formula by visually breaking it down like this, they see what each part of the formula means and make sense of it. I find that students often remember the formulas once they attach meaning to them like this. Or... if they forget the formula... since they have a concept of where the formula came from, they can rebuild it. Or... at the very least... they can divide the shape up into triangles and find all the areas and add those up. Visualizing like this is just another way of understanding or seeing structure, which is a very important part of mathematics. So again, don't memorize.... visualize... see the structure.... make sense of the math.