One of the main complaints that students have when struggling with their math homework is that they don't understand why they need to learn this in the first place. After all, how often do we actually use calculus or trigonometry in our daily lives?
I always make an effort to correct this false assumption in my students. Everything that we learn in math connects to reality in often unexpected ways. For this reason, I like to find out what it is that interests my student, or what their career goals are, so that I may show them how the math connects.
Take the example of logarithms. For the student with an ear for music, I can explain how logarithmic scales describe the relationships between musical tones, and true understanding of musical theory requires an understanding of this field of math. For the student who plans to go into the medical field, logarithms can be used to help model the levels of medications in a patient's...
There's no such thing as the square root of a negative number. Right?
Since squaring a number is defined as multiplying it by itself, and multiplying a negative times a negative gives a positive, all squares should be positive. Right?
So any number you want to take the square root of should be positive to begin with. Right?
So what if it's not?
What do you do if you're chugging through a problem and suddenly find yourself confronted with
x = √(-9)
It seems like to finish this problem we'll need to take the square root of a negative number – but we can't, so what do we do? Drop the sign and hope nobody notices? Mark it as 'undefined' like dividing by zero? Give up? Cry?
Well, actually, we don't have to do any of that, because we've got an imaginary friend to help us.
i is a mathematical constant, whose sole definition is that i2 = -1. Or, in other words,
i = √(-1). i is an imaginary number...
I was very fortunate to have been taking electronics at the same time I was learning high school trigonometry. Like most folks, I was never good at abstract math, but being able to see physical demonstrations of math principles was a huge help.
One of the real "bears" for most math students is the concept of the imaginary number...and yet it's crucial to working with all alternating current electrical and electronics problems.
When I teach electronics, I have students ranging from postgraduate math majors to middle school students who want to take a few shop classes before high school. (Nice thing about working at a technical arts extension of the local University!)
I introduce practical trigonometry without even using the term, using nothing more than a straightedge, some graph paper, and a protractor. I avoid trig tables entirely UNTIL they understand how to manipulate the Pythagorean theorem inside out and upside down.
I then demonstrate...