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"royal road to geometry" defines modern fuzzy math. Slow the pace down, do fun, discovery learning, spiral the curriculum, and don't expect mastery of the basics (that's drill and kill). Then, whatever is left, you call math.

Perhaps their motto should be: "If there is no royal road to math, then redefine math."

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"I could do the easy, obvious problems, but the graph where 'one step to the right' seemed to be followed by '1/3 step up' stopped me cold."

[slope = the change in Y divided by the change in X. The change in Y = 1/3. The change in X = 1. One third divided by 1 is 1/3. Find the y-intercept and you're all done.]

Lines and graphs and their equations are easy compared to many things you have done. Don't make it worse than it is. All you need to know is:

1. Any two points (X,Y) on a graph define a line. Or, given a line on graph paper, pick any two convenient points on the line. You don't have to do just 'one step to the right'.

2. Given any two (X,Y) points on the line, use the two-point form to define the line equation.

3. The standard form for writing an equation of a line is the slope-intercept form.

Y = mX + b

Where m = slope and b = the y-intercept.

Once you plug in the two (X,Y) values into the two-point form, you can rearrange the equation into the slope-intercept form. You can also create the equation by finding the Y-value of where the line crosses (if it does) the Y-axis. This is called the Y-intercept or just intercept. For slope, it is the change in Y divided by the change in X. You don't have to use just one step.

4. Given any old equation, it is a line only if you can rearrange it into the standard slope-intercept equation form.

5. For vertical lines, well, I just remember that my teachers glossed over that. Slope 'm' goes to infinity; divide by zero; no y-intercept; just treat it as a special case. Messy. X = 4 is a vertical line that passes through X = 4, or X = 4 for any value of Y.

A better way is to treat vertical lines just like every other line. This is done using the implicit form of the equation of a line. (You put everything on the left of the '=' sign and leave zero on the right.)

Y = mX + b

is the explicit or functional form of a line, often written as:

F(x) = mX + b

The problem with functions is that only one value of Y is allowed for each value of X. This has a bigger impact when you want to define the equations for things like circles.

To get around this, you have to use either the implicit or parametric definition of a line. The implicit form would be:

aX + bY + c = 0

Where X,Y are the variables, and a,b,c are the constants that define the line. All slope-intercept forms of a line can be rearranged into the implicit form of a line. With the implicit form of the line, Y can equal zero and you don't get any divides by zero. This form may not seem very useful, but the constants a,b, and c have real geometrical meaning.

Well, I better stop now. Otherwise, I will start talking about the parametric form of a line (and everything else) and why it is so wonderful.

One of my biggest complaints about the math that I had was the very poor discussion of the reasons and uses for explicit, implicit, and parametric equation forms for geometry.

-- SteveH - 24 Jun 2006

Leading to my first Math Revelation of the day: it's not algebra!

It's coordinate geometry!

Or it's analytic geometry - the study of geometry using the principles of algebra.

I really don't get how American high schools manage to break maths down into separate subjects like they do.

-- TracyW - 24 Jun 2006