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-What is it?
Definition: The rise or vertical (up/down) change divided by the run or horizontal (left right) change, when
moving from one point to another on a line.
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-What does that mean?
Explanation: If the line goes up the slope is positive, and if it goes down the slop is negative. If the line is flat
with no rise the slope is 0 and if the line is vertical with no horizontal motion the slope is considered
undefined. Slope is also known as the ratio of rise to the run, or the quotient of the rise and the run.
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-How is it used?
Every Day Uses: Slope plays a part in driving, nursing, economics, construction, chemistry, playing golf, riding bikes, and hiking. Mathematically it is used
when working with the Cartesian coordinate system, which is a rectangular system divided into four quadrants. These quadrants are split by a horizontal xaxis and a vertical y-axis. Points in this system are represented in an ordered pair by two numbers, an x value and a y value (x, y), which describe a distance
from the origin or (0, 0). The y value gives the rise factor and the x value gives the run factor. Specifically, slope is used in two ways. Both ways need a linear
set of data. The first way is the slope-intercept form (y = mx + b) of the linear equation. For this way you need at least two data points. This form of the
equation is called slope-intercept form because both the slope (m) and the y intercept (b) are used to compare x to y. The y-intercept is the point when the
value of x is 0. There is also the point-slope form, which is
. For this one you only need one data point. Both equations are useful, it is
just a matter or what information you have or need that determines which equation will be the best for the situation.
-How can I find the slope of a line?
Calculation: Slope can be calculated by dividing the difference, or distance if you will, between the x coordinates and the y coordinates for two points which
come from a linear set of data. For example if we take two points (0, 4) and (5, 9). Remember the first digit is the x coordinate and the second digit is the y
coordinate. You calculate and divide the differences between 9 and 4, and 5 and 0. (9-4=5 and 5-0=5 and 5/5=1/1=1). This means the graph will go up 1 unit
for every 1 unit the line moves to the right. As mentioned earlier slope is represented by the letter (m) and if you plug in one of your data points for x and y,
use the number 4 as b, and the slope we just calculated, you will see that both sides of the equation are equal. (i.e. 9 = (1)5 + 4).
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-Why is the letter m used to represent slope in the linear equation?
Answer: Not known for sure!
Proposed Theories for why m is used to represent slope:
Because the French word for climb is monter and monter starts with an m. (Refuted as an urban legend, because Rene Descartes the creator of Cartesian
coordinate system and linear equations was French and did not use m for slope. The French in fact use the letter a for slope)
Because mountains have slopes and mountain starts with the letter m.
Because slope is sometimes referred to as the “modulus of slope” and modulus is defined as the essential parameter determining, or a number used to
measure, or the parameter which determines and modulus starts with the letter m.
Because the first letters (a, b, c) of the 24 letter English alphabet are used to represent constant values in equations, the last letters (x, y, z) represent
unknown values in equations, and the middle letters (m, n, p) represent the parameters in equations. This is thought to be a good way to explain the
variables, but not the believed reason for the letter m. The author of this theory was not the first person to use m for slope.
Because slope is the movement of the line and movement starts with the letter m.
It just happened!
Earliest know use of m: 1757 by Vincenzo Riccati in a memoir referencing Jakob Hermann, who was a friend and colleague of Leonhard Euler.
Believed creators of m for slope: Leonhard Euler, Rene Descartes, Vincenzo Riccati, and Jakob Hermann
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References
<www.algebrawithsteve.com/combine.pdf> (6/16/10)
<http://www.numericana.com/answer/trivia.htm#slope>
(6/16/10)
<http://www.unl.edu/tcweb/fowler/2006MathPapers/mForSl
opeAnthony.doc> (6/16/10)
<http://jeff560.tripod.com/geometry.html> (6/16/10)
<http://answers.yahoo.com/question/index?qid=200809241
70250AANrkBd> (6/16/10)
<http://mathforum.org/dr.math/faq/faq.terms.html>
(6/16/10)
<http://answers.yahoo.com/question/index?qid=200801021
43449AA9ctzg> (6/16/10)
Mathematics for Elementary Teachers: A Conceptual
Approach. Albert B. Bennet, Jr. (2004)
Fun Fact! Slope is not always represented with the letter m. Here’s
how it is represented in the slope-intercept form of the linear equation
in other countries! As you can see the variable for slope isn’t the only
difference.
y=mx +b
US, Canada
y=mx +c
UK
y=ax +b
France
Netherlands, Uruguay
y=kx +b
Russia
y=kx +m
Sweden
y=kx +d
Austria
y=px +q
Netherlands