In March 2000, I wrote an Email message to a young student who had asked me about memorizing mathematical formulas. With his message he included some example formulas for perimeter, area, etc. Here was my reply:
“There are lots of different ways to memorize mathematical formulas. I have always found that the best way is to know how the formula was created in the first place. Then, if I forget the formula on a test, I can just go through the steps of re-creating the formula. It gives me a great feeling to know exactly how a formula works rather than just taking the letters and numbers and trusting the “magic.”
Here is an example, using the formulas you gave me:
>Perimeter: Rectangle: P=2 (L + W), Square: P= 4s
To derive the formula for a rectangle, first picture the rectangle,
W +-----------------------+ | | L | | L | | +-----------------------+ W
The perimeter, as you probably know, is the distance around the
rectangle. To do that, we simply need to add up the four sides.
The answer is L + W + L + W. This is equivalent to the condensed
formula that you gave: P = 2(L+W). Is this pretty easy?
The only problem you may run into is that you may use different
letters than were used in the “official” formula. For example, you
might use “a” and “b” instead of “L” and “W” and get a formula
like P = 2(a+b). There is nothing wrong with this alternate
formula. If you use it to calculate a perimeter, it will work fine.
So the only question is whether your teacher expects you to quote
the exact formula with the right letters or if he/she wants you to use the formula to actually solve a math problem. If you are solving a math problem, then the teacher will never see the formula you use, so you can use whatever formula you want (as long as it works). If he/she wants you to write down the formula on a quiz, you might have to use the same letters. But hopefully the teacher is reasonable. You could do something like this on a pop quiz:
Perimeter of a rectangle: c = 2(a+b) where c = perimeter a = length b = width
Since you are specifying what the letters mean, the teacher should
accept this answer as correct, unless he/she’s really strict. (If
you don’t say what you use for your letters, the answer will not be
correct, because someone else who looks at your formula has to be
able to understand how to use it.)
Here’s another example of deriving formulas: What about the area
of a triangle? Let’s say you already know the easy formula for the
area of a rectangle: A=LW. A triangle looks like half of a
______ | W / | / L| / | / | / |/
In that case, the area is half that of the imaginary rectangle:
A = 1/2 L*W. Now, of course triangles have different shapes, so
does this formula work for all triangles? The answer is yes. If you
doodle on the back page of your quiz, you can draw various triangles and fit rectangles around them. Some of the triangles may cut the big rectangle into three pieces instead of two, but, when you look at it, it still will look like half the area. Also, hopefully you’ve previously made an attempt to memorize the formula, and you know that it’s a real easy formula, so it has to be right! The only difference is just the letters we used. The “real” formula uses B and H for base and height instead of L and W. But we already talked about this.
“Base” and “Height” are easier to understand than “put an imaginary rectangle around a triangle as tightly as possible and take H for the height and W for the width of that rectangle and use that in my formula A = 1/2 L*W” … so you’d want to use the preferred “base” and “height” on your pop quiz unless you really were having a memory crisis!
The techniques I mentioned above can be applied to almost all of the formulas you gave me. But sometimes you may have to do something different because 1) the formula is too complicated, or 2) you don’t have enough time on the pop quiz to derive the formula because it’s a really short pop quiz and you’re expected to know the answer immediately.
The most complicated formula you gave me was the area of a circle,
which is A = 3.14*R2 (R squared). Actually, I hope you were told
that the number 3.14 is a special number in mathematics called PI
(and represented by a Greek letter). It is special because it appears
in a whole bunch of formulas, so once you’ve memorized 3.14159, you can really take advantage of it. If you know that 3.14 is PI, then that makes for a very interesting memory “picture”… you can think of a PIE for PI. Then you can think of two rats on top of the pie, eating it. You watch them as they gobble down the whole pie. The pie represents PI. The two rats represent R and 2 (2 for squared). The full stomachs represent the entire “area” of the pie. Actually, we have to be a little careful because “area” is a two-dimensional concept and the pie in our imagination is three-dimensional and we might think of it as VOLUME… we don’t want to confuse it with a volume formula we might memorize in the future. So… just modify the picture a little bit… pretend we have a very FLAT pie… one that is perhaps only half an inch thick. Perhaps this was a rejected pie by a baker, so he threw it out the door into the alley, where the two rats found it. WOW!! What a picture! You’ll be sure to remember the formula now!
The only other problem is memorizing what PI is in the first place.
There are different ways to do this. One way is to convert the numbers
to letters using the mnemonic alphabet as described in my
“How to Improve Your Memory” tutorial
(you might think of meat, run, lip, then
use that with “pie” to make a short little story). Another way, if you
have the luxury of time, is to cleverly create a sentence like:
“Hey, I feel a lumpy cantelope.” The trick is that if you write down
the number of letters in each word, you get PI! A third idea is simply
to write 3.14159 on a card and tape it to your bedroom ceiling, front
door, maybe even your Playstation console. It may take longer to
memorize, but soon you’ll have it memorized by “brute force,” and you’ll
be able to recall it very quickly without even a story. Since it’s such
an important number in math, it may be worth it.
Here is an example from another Email message:
“I found website very informative. Do you know of any information out there on memory techniques for remembering detailed technical calculations … ?”
Most of the good, published books out there on improving your memory include
a section or chapter about memorizing mathematical formulas. The basic idea
is that you convert the formula into a picture. For example:
_()_ = V --- A
This formula says that resistance is the ratio of voltage to current. For
the horizontal line, you might think of a table. V is perhaps a V-shaped
vase, and it’s on top of the table, whereas under the table is an Ant.
The Greek Omega symbol on the left side sort of looks like a horseshoe, and
you could pretend someone throws the horseshoe over to the other side of the
equation and hits the vase and it crashes. Some of the falling pieces
disturb the ant.
There, that is a beautiful and memorable picture for a formula!
For more complicated formulas, you just
creatively expand on the idea.