## A small question on the slope formula

Why is the slope formula normally presented as $m=\frac{y_2-y_1}{x_2-x_1}$ $m=\frac{y_1-y_2}{x_1-x_2}$?

The latter sure seems to confuse students less.

### 19 Responses

1. Not really sure…it irked me when I first ran into the formula 2. Trying to represent that x2 >= x1?

3. If you’re just presenting the formula without context, it might be confusing. If you explain rise and run as the distances travelled to get from point one to point two, then it makes way more sense to subtract the initial value from the final value.

• Thanks for the comment!

Given $x_2 \geq x_1$ is not a requirement and the formula gets used by students quite a bit without (and also often without or at least forgetting the original context), I’m not sure that’s sufficient reason.

• Who cares whether x2 > x1? The change in x is still xfinal – xinitial, even if there was a negative change in x. Makes more sense when the slope has a meaning. Like the slope on a position-vs-time graph rather than a y vs x graph (is that a position-vs-position graph? Or is the graph meaningless? In which case, who the heck cares what the slope is on a meaningless graph??)… the slope is the change in position over the change in time. So delta-x over delta-t. So final position – initial position, and final time – initial time. But the slope formula as presented above I do find to be pretty meaningless. Both of your versions of it.

4. I like getting students to see that the two are equivalent, but I agree with anonymous about the reason.

5. I imagine it’s that when the slope concept is first introduced, one typically starts with the positive slope case. It’s natural to move from left to right when introducing the elements, even on the coordinate plane, so I think that the second point presented, naturally referred to with the coordinates (x_2, y_2), tends to automatically be above and to the right of the first point, (x_1, y_1).

From there it seems very sensible to present the slope formula as being m=(x_2 – x_1)/(y_2 – y_1), in order to make all the values positive — this way, there’s no dealing with messy negative-canceling and whatnot. Of course it doesn’t make a difference mathematically, but in early stages of pedagogy it makes sense that one would want to reduce (or at least focus) the student’s cognitive load to the most contextually important components of the concept.

I think from that point on out, the use of (x_2 – x_1) and (y_2 – y_1) becomes inherited wisdom.

• This is also supported by the common graphical trope of drawing the slope triangle. If you’re in the positive slope case and the points are as described above, since the sides of the triangle should have positive length, the natural inclination is to use (x_2 – x_1) and (y_2 – y_1).

6. I wonder what the implications would be of not using numbers at all? How about “(y-sub-triangle – y-sub-circle)/(x-sub-triangle – x-sub-circle)”? Or red and blue? It would have to be something where it was very clear that the order has to stay the same between variables though.

7. There are two realizations going on here, slope as rise over run (assuming that {x2,y2} is to the right and above {x1,y1}, and the algebraic realization that the two expressions are the same. The confusion is that the student hasn’t really materialized either realization very well. I say either because they are pretty much equal in sophistication, even if they are two realizations, and I don’t think a student can be (authentically) successful with one and not the other. The realization of rise over run, in the mathematical sense, is half way there. I mean, once you have established in your head that slope, or rise over run, is a quotient of differences, then the simple algebraic notion of equivalent expression (that if you negate the numerator and the denominator the result is the same) should kick right in.

If you break it down into parts like this then it is hard to see a student truly able to understand the math of slope in the first form (a quotient of differences) and not understand the equivalence of the second form. I would say the student isn’t getting either very well.

8. I don’t use that formula. I write $m = \frac {\delta y}{\delta x}$ and then I define $\delta y$ and $\delta x$ separately. This way I break down the formula into smaller pieces for students.

• That should be $\Delta y$ not $\delta y$.

• I always do the same. It also helps to relate the distance formula: $$d=\sqrt{\delta x^2 + \delta y^2}$$.

9. I always mix it up and write both ways down. When the learners ask me if it has to be one way or the other, we discuss the fact we are finding the distance or “rise” between the two points in the y direction and the distance or “run” between the two points in the x direction.

At that point, they understand it doesn’t matter, as long as they are consistent in putting the values from the same point first.

Kind of goes to dwees example of of delta y over delta x. I use that notation as well.

10. I don’t even bother with the subscript numbers… I just teach them to pick one of the points – doesn’t matter which – and subtract the other point from it.

(I say – dance with the one who brought you – to help remind them to use the same point first for both x and y.)

I have them do it both ways for a week or so, to drive home that they will always find the same answer both ways.

11. Would labeling the points as point A and point B and using letter subscripts help avoid the confusion?

As in m=(y_A – y_B)/(x_A – x_B)

Does the confusion stem from numerical subscripts being mixed with the numerical values they represent or is it more the order of the numerical subscripts? I imagine keeping good track of y_1 and y_2, and x_1 and x_2 of (1,2) and (2,1) can be difficult for students.

• If they do the “official” formula they will sometimes reverse the order and sometimes not (so for example flipping the y coordinates but NOT flipping the xs), and are especially more tempted to flip at their convenience so that there are no negative numbers.

If they just subtract in order — and there is absolutely no reason mathematically not to — they nearly always get it right (barring the 1,000,000 problems they have with integers, of course).

I imagine doing both formulas (if taught well) would work fine too and would let them evade some integer problems for just a little bit longer.

Also, I distinctly remember (I guess I was taught wrong in 7th grade or whatever it was) being paranoid when I was a kid about making sure x_1 =< x_2. On one occasion I even remembering forgetting if it had to be y_1 < = y_2 or x_1 < = x_2 and deciding to draw graph paper by hand to solve it that way instead.

12. I think it’s because you are dealing with position of the points, you want to differentiate between the initial and final position. When we move point it always travels up:-) Of course, it could also be that textbooks/teachers don’t want to scare the wits out of students with y1- y2 or x1-x2 who might associate it with taking away a bigger number from a smaller number which remains unthinkable for some even after Year 10:-)

13. usually x2 > x1 and y2 > y1, so m is ussually positif or going up, eq stock charts, sale charts, growth charts, etc..

if m negative it means going down

perhaps because people likes something going up then going down