If you've learned about graphing lines, you've almost surely learned about the concept of slope. This concept becomes more and more important as math gets more and more advanced—one of the basic forms of calculus (called differential calculus) has two ideas at its core: the concept of slope, and the concept of limit (which you may not have studied yet). Yet many students learn a formula for slope without really understanding why that formula makes sense. This essay will try to explain why that formula absolutely must look the way it does if it is going to capture your basic intuitions about slope.
Let's imagine that you're riding a bike (a magic bike that can't crash!) on a hill. To keep things consistent, let's say you're riding from left to right like the cyclist below. When you look at the pictures, think about what your intuition is telling about which hill is the steepest.
Most people, I think, have no trouble seeing that cyclist C is riding on the steepest hill. You can even say that C's hill is steeper than B's hill, even though one is uphill and one is downhill (going from left to right, remember). Comparing the steepness of A's hill with B's, though, is a little trickier. They appear to have about the same steepness in opposite directions.
The concept of 'slope' was created to put a measurement to this intuitive concept. By 'measurement' I mean that we're trying to come up with a number that quantifies the intuition you already have. This number should increase as the steepness gets bigger (like measurements of distance, weight, temperature, etc. increase when the length, mass, hotness, etc. get bigger), and if we can manage it, the ideal formula would also reflect the idea that hills A and B have the same slope in opposite directions.
So to come up with a way to quantify this idea, let's look at some other examples of steepness—this time using stairs instead of hills. Compare the steepness of the staircases below. Note carefully that among staircases A, D, and E, the rise of each successive step of the staircases is exactly the same. Only the run of each step (the space the penguin has to stand on) changes. This is clearly one way to change the slope of a staircase; even though the penguin has to step up the same height in staircases A and E, we want to be able to say that staircase E is nonetheless steeper.
The other way to increase the steepness of the staircase, of course, is to increase the rise of each successive step while leaving the run alone. This is demonstrated by staircases A, B, and C.
These two notions give us our first hint about how to construct a calculation to measure slope (of a staircase, at least). Namely, the two things that determine the slope are the rise from each step to the next and the run (or depth) of any individual step. Furthermore the slope has to increase either when the rise increases (e.g. staircase B to staircase A to staircase C) or when the run decreases (e.g. staircase D to staircase A to staircase E).
But we can say more. Let's look more closely at the stairs above. Staircase A's rise and run are equal. To create staircases D and E, I doubled and halved the run (respectively). To create staircases B and C, I instead halved and doubled A's rise (respectively). And when I rearrange the staircases like I did below, you'll see that doubling the run and halving the rise give the same resulting slope. And, of course, vice-versa.
Since most of you have come here for further explanation of slopes, you probably already know the solution to the slope calculation problem. The trick is to divide those numbers. Notice how well division works here. If we put the rise in the numerator (top) of the division to get the familiar (to most) "rise over run", everything we want to be true about the number really is true! If the rise increases by doubling, but the run stays the same (from staircase A to C), then the fraction increases—in fact, it doubles. Likewise, if the run (the denominator) is cut in half instead (staircase A to E), the fraction also doubles. Just what we want. And of course, decreasing the numerator or increasing the denominator decreases the fraction—again what we want. Note further, that we decided to put the rise on top in order for the fraction to increase and decrease in exactly this manner. If we had put the run on top instead, the slope numbers would increase when we'd want them to decrease. Be sure you understand that decision before you continue reading.
But "rise over run" is an informal idea. I almost hate for my students to remember it that way, because it doesn't guarantee that they'll remember how to use it to find an actual number when they need to calculate a slope. So let's go further, and see how to calculate using that idea. We will need, of course, some points on a graph to do this.
First, let's calculate the slope between points Q and R. We need to know the lengths of the rise and the run from Q to R. That is, this calculation requires that we know the change in the vertical (y) coordinate and the change in the horizontal (x) coordinate if you walk from Q to R. That calculation is easy, right? The y-coordinate changes 3 (going from 2 to 5), and the x-coordinate changes 4 (going from 5 to 9). So the fraction we decided on above gives a slope of 3/4.
So, what arithmetic calculation did we do to get those numbers? Clearly, we subtracted "5 minus 2" to get 3 and "9 minus 5" to get 4. In fact, subtraction is just about always what the word "change" means in math. If the price of gas changes from $3.50 to $3.90 per gallon, we subtract "final price minus initial price" to get a change of +40¢. If it changes from $3.90 to $3.50 per gallon, we still subtract "final price minus initial price" to get a change of –40¢. So it should come as no surprise that we should be subtracting "final y-coordinate minus initial y-coordinate" to get the change in y-coordinate, which is the rise.
If you can remember that "change" requires a subtraction, then a more reliable way to memorize a word formula is:
And once you automatically associate "change" and "subtraction", you can memorize the full symbolic formula:
I include the formula that uses the little triangle (the Greek letter 'delta') because that's the way many people learn it. The 'delta' is supposed to be translated as "change in", so that it turns directly into the word formula above it. But for people just learning the formula, the delta can be confusing, so I recommend using the version with plain subtraction. It means "subtract the y-coordinate of point number 2 minus the y-coordinate of point number 1, and the x-coordinate of point number 2 minus the x-coordinate of point number 1, and then divide those differences". And at this point, each part of that should make sense: subtract because you're finding changes, and divide to make the numbers work in a way that matches our intuition.
But what if you use a different point in your calculation? Won't it turn out different?
No, it won't; not while all the points are on the same line. Remember that if your rise and run are each (say) three times as long, the slope should stay the same (remember conjoining the different staircases?). That's just what happens if we use point P instead of point Q in our calculation. From point P to point R, the change in y is 3 times as large (9; and notice that subtracting "5 minus –4" works well to get the 9, even though one of the numbers is negative) as it was from point Q. But the change in x is also 3 times as large (12) as it was before. The fraction works perfectly! 9/12 simply reduces to 3/4 again, and thus using P instead didn't make any difference. If the calculation had turned out a different result, the three points could not have been on the same line. This demonstrates an important geometrical interpretation of a line: a set of points all fall on the same line exactly when all pairs of points you might choose from that set give the same result in a slope calculation.
How did you know to make Q point number 1 and R point number 2. If I did it the other way, would I get it wrong?
No, you wouldn't get it wrong. It shouldn't matter which you call point number 1 and point number 2. The only difference in the calculation would be that you and I would subtract each of our numbers in the reverse order. Notice that it won't change the result: reversing a subtraction just gives the opposite result, right? (10 minus 17 is the opposite of 17 minus 10.) But that will happen in both the top and the bottom of the fraction. So I will be dividing a positive number by a positive number, and you will be dividing a negative number by a negative number, and our results will be the same. As long as you're consistent (don't call Q point number 1 on top, and R point number 1 on the bottom), the naming of the points makes no difference. In terms of your intuition about hills, walking from point R to point Q does lead you downhill, but you're walking backwards (since left-to-right is forwards, remember), so the slope hasn't changed.
What about downhill slopes? How are they different?
Well, we have a good example of that going from point G to point H. The change in y is 7 minus –8, or 15. The change in x is –2 minus 1, or –3. Note that I used G as point 2 in my subtraction—the subtraction looked easier that way for the y-values, and if it makes no difference, I might as well make it easy on myself. When I divide, I get –5 for the slope. The downhill slope showed up in the algebra as a negative number. Note that in the case of a downhill slope, either the rise or the run (but crucially, not both) must be negative, giving a negative result when you divide. This is, of course, exactly the last criterion we wanted our slope calculation to meet. If two slopes are identical, except that one is uphill and the other downhill, that will indeed show up in the measurements of the slopes. The two slopes will be exact opposites of each other: a perfect match for our intuition.