Ban Slope-intercept

Ban y = mx + b

Really I mean it!

Given a point (x1, y1) and the slope, m, you first write the equation y = mx + b with the slope substituted in for m. Then you substitute the coordinates of the point for x and y and solve for b. Now that you know all the parameters you can write the equation of the line. Sure it works. Sure it's correct.

Someone a long time ago thought this would be a great way to have kids write the equation of a line. Long ago: I learned it in high school and probably most of you did to. It's there in the textbooks to this day. But it is not the best way to write the equation of a line.

Why not go right to y = y1 + m(x - x1), the point-slope equation? Substitute in the 3 numbers and you're done? Fast, simple, no chance of an algebra or arithmetic mistake.

In addition to the efficiency, it makes more sense. Here's why. Let's start with a "real" example.

Five towns, Alpha, Beta, Gamma, Delta and Epsilon, in that order, are located on a straight road that starts at Alpha and goes uphill rising at the rate of 26.4 feet per mile. Beta is 3 miles (horizontally, not along the road) from Alpha. From Alpha is is 7 miles to Gamma, 10.5 miles to Delta and 14.75 miles to Epsilon. Beta is 827 feet above sea level.

How many feet above sea level are each of the other towns?

How far from Alpha should the "1000 Feet Above Sea Level" sign be placed?

I suggest you let your algebra 1 students work on this for a while. I'm pretty sure that together thy will figure it out without know about the equation of a line; without modeling the road with an equation.

Once they are familiar with the situation, you can lead them to writing the equation that gives the distance above sea level, y, in terms of the horizontal distance from Alpha, x, and use that to answer the questions.

Thy reasoning like this: Since we know the most about Beta we'll start there. We are 827 feet above sea level and 3 miles from Alpha.

To this we add the number of feet the road rises per mile multiplied by the number of miles from Beta, which is 3 less than the number of miles from Alpha: 26.4(x - 3)

(This may be the hard part -- using (x - 3) = the distance from Alpha. so spend some time making sure everyone gets this part.)

The height above sea level is y = 827 + 26.4(x - 3).

And there it is: our elevation is Beta's 827 feet plus 26.4 feet for each mile from Beta. And since Alpha is downhill from Beta, when x = 0, the (26.4)(0 - 3) will give a lower elevation for Alpha.

Striping the "real" parts away we have written the equation of a line through (3, 927) with a slope of 26.4.

When you have the equation of a line you want the y-coordinates in terms of the x-coordinates. So y starts somewhere say , y1, and then changes by an amount m for each unit change in x from x's starting value x1. The point-slope form gives you all that.

The slope-intercept form is really a special case of the point-slope form with x1 = 0. Did you ever consider that it may be easier to teach the general form first and the special case later?

You may want to somehow "prove" this (or maybe not). Here is a way to do that. Given the point (x1, y1) and the slope m, pick a general point on the line and call it (x, y). What makes a line a line is that the slope between any two points is always the same. So write:

SLOPE = SLOPE

Then multiply by (x - x1) and solve for y:


There are reasons beside ease of writing the equation to use this form.

1. The form makes sense: start at y1, increase by the slope times the "change in x."

2. The equation can be "simplified" into any other form if necessary.

3. The equation can be entered in a graphing calculator or other graphing program as is, without simplifying.

4. When you get to calculus the equation of the tangent line to a function, f, at (a, f(a)) has a slope equal to the derivative of f at a. The equation is the same.


and the reasoning is the same (start at f(a) and change this value by f ' (a) multiplied by the change from a to x).

5. Later, when you get to power series, the equation above is always the first two terms of a function's Taylor series centered at a.

The real reason is that it is so much faster and simpler.