What Do Students Need Help With?

Algebra
Generalizing from y = mx + b so that the result is an equation like y = 2x + 5.
Students benefit from being reminded that they are finding the equation that describes the entire line not just a point.
I like to ask them "How many points are on the line?" -- the answer I like the most is "Infinite" but "Too many to count" or "A real lot" are very acceptable answers.

It takes a while to see that (x-5)/(5-x) = -1!!


Calculus and Number Lines

The idea of limits and that inscribing an n-sided polygon in a circle, the more sides you have, the closer the area is to the area of the circle. So as n approaches infinity using a circle with a radius of 1, the area approaches pi (3.14159265....).

Secant and tangent lines have nothing to do with trig functions secant and tangent!!

A really cool problem was f(x) = [x] + [-x] which involves the greatest integer function.
The TI-84 function for greatest integer function is Int. The greatest integer function is best demonstrated with a number line and asking the student what was the last number that you passed. If you are exactly at an integer, then the answer is that number.
For example, [2.5] = 2 we passed 2 on the way to 2.5.
and [4] = 4
But [-2.5] = -3 we passed -3 on the way to -.25 (but we have not yet passed -2!!)
The cool thing about this problem is that for most x's, such as x = 2.5,
f(x) = [2.5] + [-2.5] which comes out to
f(x) = 2 + -3 = -1
The exception is at exactly an integer -- f(4) = [4] + [-4] = 4 + -4 = 0.
This graph looks like a straight line like y = -1, however at each integer value, there is a discontinuity, as y jumps to 0 at -3, -2, -1, 0, 1, 2, 3, etc.
This is best seen by setting xmin = 0.99 and xmax = 1.01 on the window menu -- at x = 1 there is a dot on the x axis!!

Trigonometry

The Law of Cosines is really the Pythagorean Theorem with a little extra.
We have always been using it except that if we can use the Pythagorean Theorem then we must have a right triangle!
If we rewrite a^2 + b^2 = c^2 as
c^2 = a^2 + b^2 - 2ab cos C (cos 90 = 0)
so we are used to seeing it as
c^2 = a^2 + b^2


General Math (Percents)

I love to work with students on 'trick questions' .

For example, you buy a stock at $100 -- it goes up 10% and then down 10%, what is its final price?

You buy a stock for $40 -- it goes up 50% and then down 50%, what is its final price?

And, you buy a stock for $60 -- it goes up 100% and down 100%, what is its final price?