Perhaps the Best Resource Is....
For the past seven years on this website, I have been suggesting resources of all types. The one resource I have forgotten or ignored is the close presence of a colleague or peer. Very often, they can shed a helpful prospective on your question or concern.
Too illustrate, consider the following e-mail request from a local mathematics teacher...
I have a math question that I hope you can help me with. I have researched this topic but have not found an answer I am satisfied with in the textbooks or online.
Well, I guess it is two questions:
I found that the standard form is helpful for graphing the x-and y-intercepts of an equation, for solving systems of equations, and is necessary to write the equation of a vertical line, but I still can't figure out why there are no negative A values or fractions allowed. The best I can come up with is that the equations look nicer, and are easier to deal with when finding the x-and y-intercepts, but that can't be correct because that's just aesthetics.
- Why is the coefficient of the A term positive in the standard form of a linear equation?
- Why can't we have fractions in the standard form of a linear equation?
Do you have any ideas or places I could look to find this answer?
How would you respond to this question? Part of the importance of the question lies in a need to understand (and possibly support) a textbook convention, while also perhaps being consistent with how other math teachers (at least in your school/district) teach forms of linear equations....let alone performance tests that match state requirements.
And finally, the example continues below with the e-mail response from a colleague to the posed questions/concerns. Do you agree or disagree...or have something else to add to the situation?
As to your question, your hunch is basically right. There are no mathematical reasons for A being positive or lack of fractions as parameter coefficients. It is nothing more than convention....and if not followed, nothing dire happens (i.e. the math cops do not appear!)...except.... Some thoughts:
My suggestion....ignore the rule and nothing will happen, other than perhaps a discussion about arbitrary conventions.
- Too many high school texts go further and state that A,B, C must be integers (see problem below with this restriction).
- Given form Ax+By=C, if A was negative, all one needs to do is multiply both sides by -1 to make A positive.
- If fractions occurred as coefficients, all one needs to do is multiply by LCM of all divisors and situation is corrected.
- BUT...what is the standard form of a horizontal line....such as y = 4. Then, A = 0, which is not positive....so, one needs to go back and say that A must be a positive integer OR zero.
- All of this sounds good, but the "true" definition has none of these requirements: "Ax+By=C where A and B are not both zero." [taken from James & James Mathematics Dictionary]
- Note that this "true" definition allows for case of horizontal line y = 4.
- Historically, the rules you mention were not in place....and algebra developed just fine.
- And lastly, the rules you mention can lead to complications. Suppose you start with a point slope form: y = (pi)x, which is a linear equation, producing a line with slope pi and y-intercept of 0. Under your rule restrictions, it is impossible to produce a Standard Form for it....as dividing by pi creates a bigger mess, yet pi is not really a fraction, yet it is not an integer.
- But, one might say...the previous example is a weird case that can be avoided...and I say, no way....as it happens to be the famous C=(pi)D, which is the linear equation relating circumference and diameter.