Regarding Average Rates of Change: teacher says (y2-y1)/(x2-x1) != (y1-y2)/(x1-x2)

[ViolentViolets]

Hello,

My advanced functions teacher believes that (y2-y1)/(x2-x1) is not equivalent to (y1-y2)/(x1-x2).

I am 100% certain believes this to be untrue as we went over two questions. He claims it's a "math miracle" it works and that the sign of the slope will be changed if you go through enough problems. Unfortunately he was unable to supply such a question, and his attempt to do so only proved me right further.

I did attempt to prove it by factoring out the negative one from top and bottom and then rearranging it to show that (y2-y1)/(x2-x1) is indeed equivalent to (y1-y2)/(x1-x2).

My proof:

= (y2-y1)/(x2-x1)

= -(-y2+y1)/[-(-x2+x1)]

= (y1-y2)/(x1-x2)

Therefore (y2-y1)/(x2-x1) =(y1-y2)/(x1-x2)


He had unfortunately dismissed all I had too say, stating that he will give an example when the class does instantaneous rates of change.

Have I gone mad, or is there another way I can attempt to prove this to my math teacher? He claims he has taught high school math for 25 years and I would hate if he kept telling his future students that this method is wrong.

 

[Dr.Peterson]

My advanced functions teacher believes that (y2-y1)/(x2-x1) is not equivalent to (y1-y2)/(x1-x2).

I am 100% certain believes this to be untrue as we went over two questions. He claims it's a "math miracle" it works and that the sign of the slope will be changed if you go through enough problems. Unfortunately he was unable to supply such a question, and his attempt to do so only proved me right further.

I did attempt to prove it by factoring out the negative one from top and bottom and then rearranging it to show that (y2-y1)/(x2-x1) is indeed equivalent to (y1-y2)/(x1-x2).

My proof:
=(y2-y1)/(x2-x1)
= -(-y2+y1)/-(-x2+x1)
=(y1-y2)/(x1-x2)

Therefore (y2-y1)/(x2-x1) =(y1-y2)/(x1-x2)

He had unfortunately dismissed all I had too say, stating that he will give an example when the class does instantaneous rates of change.

Have I gone mad, or is there another way I can attempt to prove this to my math teacher? He claims he has taught high school math for 25 years and I would hate if he kept telling his future students that this method is wrong.

You are entirely right.

In fact, it is usual to point out to students that you get the same slope for a line regardless of what pair of points you take, including reversing the order of the same two points, which is what you are doing here. So if he was right, the whole idea of slope would fall apart! I suspect that he really means something else entirely (or is trying to goad you into doing what you did, which does look miraculous to some students!).

Instantaneous rates of change, also called derivatives, are limits of these "difference quotients"; I can't picture what he might have in mind where they are not reversible, but whatever it is, it would not work backward to make your statement incorrect.

How about asking him for his counterexample, even though you are not ready to fully understand it, so you can look into it? Tell us what he says, and maybe we'll have more to say.

 

[tkhunny]

Hello,

My advanced functions teacher believes that (y2-y1)/(x2-x1) is not equivalent to (y1-y2)/(x1-x2).

I am 100% certain believes this to be untrue as we went over two questions. He claims it's a "math miracle" it works and that the sign of the slope will be changed if you go through enough problems. Unfortunately he was unable to supply such a question, and his attempt to do so only proved me right further.

I did attempt to prove it by factoring out the negative one from top and bottom and then rearranging it to show that (y2-y1)/(x2-x1) is indeed equivalent to (y1-y2)/(x1-x2).

My proof:
=(y2-y1)/(x2-x1)
= -(-y2+y1)/-(-x2+x1)
=(y1-y2)/(x1-x2)

Therefore (y2-y1)/(x2-x1) =(y1-y2)/(x1-x2)


He had unfortunately dismissed all I had too say, stating that he will give an example when the class does instantaneous rates of change.

Have I gone mad, or is there another way I can attempt to prove this to my math teacher? He claims he has taught high school math for 25 years and I would hate if he kept telling his future students that this method is wrong.

Try (y2-y1)/(x2-x1) for (x1,y1) = (5,7) and (x2,y2)=(3,4)

Then

Try (y1-y2)/(x1-x2) for (x1,y1) = (3,4) and (x2,y2)=(5,7)

The choice of points is quite arbitrary. If it were my child facing this in a primary or secondary grade school, someone would hear about it from me!

 

[Deleted member 4993]

Would it possible to get in trouble if there is a point of discontinuity between x1 and x2? Does the definition of "average slope" break down, if it encompasses a discontinuity?