Why Is Slope Rise Over Run and Not Run Over Rise?Date: 03/05/2005 at 21:27:04 From: kristin Subject: Why rise over run??? Someone asked our math teacher yesterday, why is it rise over run and not run over rise... I was wondering since our teacher didn't know the answer, if you might. If you could answer this that would be so great!!! I think that maybe it could be both ways bcause like for instance: (I'm going to show my work here) (4,2)(3,-7) 2-(-7) / 4-3 = 9/1 y=mx+b 1/9 + 2 y=9/1x + 2 and when you graph it you put the dot at the 2 place and go up 9 lines and go out one line and put a dot (then connect those dots) BUT you can also put a dot at the 2 go out one, and up 9 and put a dot there (then connect) and it's the same thing. Do you think rise over run can also be run over rise??? Date: 03/05/2005 at 23:04:40 From: Doctor Peterson Subject: Re: Why rise over run??? Hi, Kristin. There are at least two questions here: why do we define the word "slope" to mean the ratio of "rise" to "run", and why is that the right number to use in the slope-intercept form of a line? The answer to the first question is that any number we assign to a "slope" ought to be bigger when the line is sloped more steeply. We want the "slope" to tell us how much the line is sloped. A steeper line goes up more in the same distance: o / / / o / / / / o o steep less steep If we used the "run" over the "rise", then the less steep line would have a greater slope, which wouldn't make sense: o /| / | / | o / | / | / | / | o-----+ o-----+ 6/6 3/6 <-- actual slope: steeper has greater slope 6/6 6/3 <-- run/rise: steeper has smaller slope! How about the question of what goes in the "m" spot in the equation? Let's look at the intercept and one other point on the line: | / | o (x,y) | / | | / | (0,b) o-----+ | | +--------------- What is the slope of that line? y-b y-b m = --- = --- x-0 x If we multiply both sides of the equation by x, we get mx = y - b and adding b to both sides gives mx + b = y So if we define slope as rise/run, then this is the equation that any point on the line has to fit. If you defined slope differently, you would get a different equation. So here's how it works: we define slope in a way that makes sense based on what the word "slope" means; then we find that we can use that slope value in an equation that describes any point on the line. One could use a different formula for something like "slope", and get a different equation; but this one gives us a reasonable definition AND a nice little equation to use it in. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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