Solving a math problem can be as easy as
adding two plus five equals seven or as difficult as describing the geometry
of a shape. So is there any method that applies to any problem in general?
Here are some basic tips that will lead to some mathematical solutions.
Reading a math book is sort of like reading a drawing book, because you
should read it with a pencil. You should read the theory and then examine
the proofs of how it is formed. If the book is divided into chapters then
read the entire chapter. Then if the book has a problem section (as math
books often do) do as many problems as possible until you fill comfortable
with your understanding of the subject.
This is simple advice, but now is the important part: the challenging
word problems. These usually deal with the application of the math theory.
This part can actually be fun. Yes that it: fun. It is fun because you
don’t know the answer to the problem and have to use your mind to
come up with a solution. It is a test of imagination and knowledge. Can
the knowledge of the theories you learned be applied to a real word application?
Now read the question a couple of times. Try to see what kind of variables
are given and determine what the problem is that needs solved. Do not
look at any examples or math references yet. Try to find your own solution
to the problem in your head. Then look back at the examples where you
reached a problem that your own theories didn’t seem to solve. This
is a simple method, but it helps you to better understand the problem,
test your own ideas, and better understand the solutions that solve the
problem.
After the first inspection of the problem, determine if it is purely mathematical
or contains some graphical elements. It is very common for a math problem
to have a graph or a shape. Which is seen in trigonometry and geometry
and even calculus. If the problem is graphical, draw a picture. If it
isn’t you are going to have to master “the art of the equation.”
Restated, you may have an abstract math concept. Changing the equation
might cause changes that your mind can’t exactly follow.
But dealing with the graphical first, it helps to draw a picture. It may
be useful to use a Cad program. But just because you are using the computer
doesn’t mean that you don’t have to plan your work. In the
cad program draw the information that the problem gives. Then try to draw
the answer or the representation of what is occurring in the problem.
This gives you something tangible to picture in your mind. Pictures tell
thousands of words, and you just drew your math problem. You are on the
steps it takes to solve the problem.
Computers may also be used to crunch numbers. In general, there has to
be a general understanding of the equation. Computers can crunch numbers,
but the user has to have the proper input. The computer is useful for
testing the results of several equations or graphing input, but the computer
operator must solve the equation.
Now for the most rewarding part. The reason your mathematical quest began.
It is the solution. But just as important as the solution is the way it
is achieved. Are the steps logical and easy to follow? And is the solution
correct? Test it. See if the solution is in the same range the answer
is expected to be. Review the steps and check your solution.
Now is the moment when you test your knowledge and hard work. The solution
either works or it doesn’t. But just because the answer is wrong
doesn’t mean nothing was learned. And if this is a problem that
is asking for a new solution. That is an answer or solution that hasn’t
been tried before. If it is a new method or a problem that will take time
or a life time to solve, such as a “Theory of Everything”
the effort spent in trying to reach a solution may be a step towards or
a “building block” to continue research.
In conclusion there is just those problems that intrigue the mind. And
they appear when studying math or uncovered through real world situations.
Solving these problems is similar to solving any problem rather it be
on a test or in our studies. These problems can be fun and something that
was undiscovered may be found. Math explains and is defined by the world
we live in. Let’s just have some fun and see what discoveries we
can make.
In math there are all kind of little discoveries to be found. You just
need to find something interesting or see something that can be done in
a different way. Maybe it is a challenging problem or that section of
application of the theory. But don’t misjudge the value of your
solution. Little discoveries add up to more and more discoveries. Maybe
even a major discovery lies hidden in plain sight.
Here is an example of a problem
that finally worked for me.
It took some time working off and on. Finally before I solved it was
a period of trial and error. I knew the problem could be solved, but I
was unsure you could solve it just using elementary mathematics. I learned
a lot and think more work can be done with this problem.
Here is an example
of another problem that I can’t solve without using the instructors
and books solution.
I thought that you could just have one integral to solve the entire problem.
This problem has led me to review calculus. I rediscovered “e”
and now better understand the books solution. But I still believe there
is a discovery to made here. I believe there is a simple integral that
describes the problem. I have spent hours trying to find the simplest
or even a different solution. Not all my work is shown here. Half of that
work turned out to be mathematically untrue. But as mentioned before I
didn’t stop the problem before I learned something. You will always
learn something while trying to solve a challenging problem or any math
problem you face.
