While it's not written like that usually in Greek mathbooks (I most often see the coefficient of x in a linear function be "b" or even "a"), I would imagine that the "m" is there to help with calculations using that line as tangent to a curve? Since it would be unique.Why is a straight line y=mx+c?
A quadratic is y=ax^2+bx+c, a cubic is y=ax^3+bx^2+cx+d, why is a straight line not y=ax+b? Why c, and particularly why m?
I like how you put meaning to those equations .While it's not written like that usually in Greek mathbooks (I most often see the coefficient of x in a linear function be "b" or even "a"), I would imagine that the "m" is there to help with calculations using that line as tangent to a curve? Since it would be unique.
Of course it's counterintuitive to start with that (in middle school), when it will be years before you use it for the tangent to a non-linear function. Then again all sorts of secondary school math are presented heterochronically (I am sure a simpler term exists in English, but apparently this one is also there in English).
I didn't see meaning in them in highschool either - back then I was only looking for personal 'tricks' to do things automatically. And sometimes the "meaning" is not even discovered - but for many part of it has been. What I like is that obvious (but abstract) tautologies (or even not so obvious ones) in algebra can also acquire very non-abstract meaning elsewhere, such as in geometry. Eg the inequality ab<=[(a+b)/2]^2 implies (I suppose among other things) that the square is the orthogonal parallelogram with the smallest perimeter for a set surface area => the algebraic phrase can be written also as functions with their min/max.I like how you put meaning to those equations .
I found that to be the hardest thing to do coming out of high school and entering university with applied math as a degree. People tend to not explain every single thing in the equations and you could not find the meaning anywhere... That got me stunned for a long time.
Yes , yes I like your point. I spend 12 years in and out of the university and I finally manage to get the lowest possible grade in order to finish it haha but I was really happy that I could explain to myself one thing. That functions and interpolation can be used to find different "things" in time. So just if we have a random sets of points in a dimension let's say a normal coordinate system we can find a representation of those points as a drawn out function, a representation of time and that Pi and the golden ratio can be used as a perfect process in time.... Don't know if I'm making sense haha to you at least I was happy when I though of that got the lowest point and never gave it much though after...I didn't see meaning in them in highschool either - back then I was only looking for personal 'tricks' to do things automatically. And sometimes the "meaning" is not even discovered - but for many part of it has been. What I like is that obvious (but abstract) tautologies (or even not so obvious ones) in algebra can also acquire very non-abstract meaning elsewhere, such as in geometry. Eg the inequality ab<=[(a+b)/2]^2 implies (I suppose among other things) that the square is the orthogonal parallelogram with the smallest perimeter for a set surface area => the algebraic phrase can be written also as functions with their min/max.
I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.Yes, the form (x^2+2x)/2(x+1) is inconsistent (as you have shown with the two counter-examples where the same way was applied. But I can imagine why it was the preferred way:
If you try to establish, through algebra (ie not through visualizing the graph) where this function is positive and where it is negative, you would never leave it as three functions because then you would need to establish combinations of +.- in three things instead of 2. Of course you can do the numerator as two first, but then keep the result as one polynomial.
From discussions in Facebook by math teachers for those grades, it sadly seems to be inconsistent. Better tell the kid to write both/all in the examI see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.
As a high school math teacher my answer is :I see what you are getting at, but I am asking this from the point of view of tutoring a school child. If the question in the exam is "give this in the simplest form" what should the answer be to get all the marks? There must be a rule.