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# represent slope in a linear equation

Question: Initial question “Why are there letters in math?” I changed this to “Why are variables used in
math?” I finally decided to alter this to:

Why is the variable m used to represent slope in a linear equation?

Findings:
• Slope is used to describe the steepness of a line
• Slope is determined by calculating the rise over the run of a line
• Linear equation is y = mx + b where x and y are the unknown variables and b is fixed and a constant representing the y-intercept
• Reported that “m” comes from the French monter (to climb), but this is considered urban legend.

However, it was interesting to learn that Decartes (who was French) did not use m
• Some other thoughts I came across:
a.) m stands for move and b stands for begin (student, Robby Grant)
b.) The m is for “modulus” of slope (professor, John Conway)
c.) “It just happened” (Mathematical Circles Revisited 1971, mathematics historian, Howard W. Eves
d.) blog with over 2000 postings with the possible meaning

One post…
You guys are getting too caught up in semantics.
Y=mx + b
Y=cx+i
Y=wx+c
c=Y/w-x
x=(y-c)/w
etc.etc.

With y being output, x being input, and everything else being constants is still going to graph a linear line no matter what.

• Earliest known use of m for slope was in 1757 in Vincenzo Riccati’s memorir, De methodo Hermanni ad locos geometricos ressolvendos
• Contridicted by Jeff Miller’s (teacher at Gulf High School in Florida) research that stated earliest known use of m for slope is a British text by Matthew O’Brian, A Treatise on Plane Co-Ordinate Geometry, in 1844
• One account I read about was from a source who claimed that they were told by Mary Dolciani (teacher,
writer and researcher) that y = mx + b was decided by the French. It was interesting because this same
source reported while visiting a French high school the teacher used y=sx+b. When this source asked
• Swedish textbooks use y = kx + m
• Netherlands use the equations y = ax + b or y = px + q or y = mx + n
• Austria uses y = kx + d
• Russian textbooks use y = kx + b
• Webster’s New International Dictionary (1909) gives the slope from as y = sx + b
• Current trend is to use y = ax + b

Other questions that occurred:
Why has it been reported that Descartes did not use m for slope in his work?

Connections:
1. Students can come up with their own ideas for a variable to represent slope and try to convince the class why we should use their variable
2. Students can create their own “myth” to explain why m is used for slope

Why I picked this question:
Students have asked me this in the past and I have wanted to give a more detail and I realized I didn’t have much interesting to say about it.

I came across this when I was researching “why are letters used in math” and students ask me this all the time and I have no idea…so I thought it would be interesting…also on the Math Forum site it states: “This question has been researched by math historians but has not yet been answered definitively.” This made me a bit curious.

In Conclusion:
While I cannot give an answer for why m is used in the linear equation, at least I have some information for my students when I am asked this question each year.