Math, Eloquently

The Logic of Slope (Part 1)

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.

Any questions?

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.

a Plus b, Squared

Go ahead, try this:  ask a high school math teacher for the single student error across all grade levels that's the most infuriating.  I'd bet that the most common answer you get is some version of how students incorrectly square the sum of two numbers (as in (a + b)²) to get the sum of the squares of the individual numbers (as in a² + b²).  In fact, though, those expressions are not equal;  that's not how squaring works.  Indeed, in my experience, students who can absorb the intuition about why they are not equal are much more likely to succeed in future.  This lesson will try to make sense out of this inequality:

Before explaining why it isn't true, I think it's important to understand why so many people think it is true.  There are several reasons a student might think it's true, and each of those reasons has its own logic.

(1)  I know it's true for multiplication:

so why isn't it true if the symbol between them is addition instead?

This is a classic case of a student paying more attention to the notation than to what the symbols actually mean.  Squaring means multiplying an expression times itself:  (ab)² means (ab)(ab).  But crucially, that's a whole bunch of multiplications.  And we all know that you can regroup and rearrange items being multiplied at will (formally, we say that multiplication is associative and commutative).  And when you do regroup and rearrange, you see there are two a's and two b's all multiplied together;  thus a²b².  But look at what (a + b)² means: (a + b)(a + b).  Where's the possibility for rearranging so simply?  It's not there.  The mixture of plus and times makes this expression harder to work with than if it's all multiplication.

(2)  I know it's legal to do this thing called 'distributing':

so why can't I distribute the exponent just like I can distribute that funny symbol?  (What is that thing, anyway?)

That thing is a capital Greek letter 'psi', and I used it to demonstrate how anything can be distributed, even if it looks complicated (we'll need that idea in a minute).  But many students don't know that the word 'distribute' is actually a nickname for a much longer description of the process shown above:  it's the 'distributive property of multiplication over addition'.  Distributing leaflets over a parking lot means that every person in the parking lot gets a leaflet, and distributing multiplication over addition means that every element of the addition (the a and the b) gets multiplied.  It's just a fact about our number system that you can choose to add two numbers before you multiply the sum by a third number (the left side of the above equation), or you can multiply each of the two numbers by the third number individually first (the right side), then add the products, and you'll get the same result.  In fact, the equation in (1) above is a demonstration of a different distributive property: 'the distributive property of exponents over multiplication'.  And that one is true because multiplication is associative and commutative.

But the 'distributive property of exponents over addition' is simply not valid.  Various distributive properties are different mathematical processes (a subtlety you might miss if you don't know their complete names), and they don't all have to be valid.  In fact, most are not;  any one that is valid is special and important.

(3)  But...it just seems like they should be equal!

Well...only at first maybe.  It's a common misconception that mathematical processes that look right on the page or seem right in your head must be true.  Mathematicians approach it in the completely opposite direction—in a way, nothing should be true until you can prove it's true.  And equality in this case just...isn't true!  Too bad!  Out of luck!  Oh well, we'll learn to deal with it!

Once you look at this situation very carefully, using examples and various different mathematical interpretations, you'll find that you probably don't really even believe that the two sides could be equal.  You already have the knowledge to convince yourself of that.  Let's look at some of those examples.

If you're in a candy store scooping candy into a bag, and you buy one-and-a-half pounds of candy that costs $1.50 per pound, how much will you get charged?  You might not know the answer off the top of your head, but I'll bet you know that $1.25 is a ridiculous answer.  You're buying more than a pound of candy, so the price would just have to be more than $1.50.  Yet if you think that the square of a sum is the sum of the individual squares, you'd have to believe that $1.25 is right:

One-and-a-half pounds times one-and-a-half dollars per pound doesn't equal one-and-a-quarter dollars.  Note the crucial placement of the not-equals-sign.  Be sure you understand the reason for every step of that line of math.

I'll bet you can probably square the number 20 in your head.  2 times 20 is 40, so 20 times 20 is 400.  You might not be as quick with squaring 21.  The actual answer isn't important, but do you have the intuition that it's not 401?  Well that intuition comes from a deep-down intuition that the square of a sum is not the sum of the individual squares:

Again, some part of your brain already knows that (20 + 1)² isn't the same as 20² + 1².

What does (a + b)² equal, then?  Or, put another way, is there an expression without parentheses that always has the exact same value?  The formal algebra is below.  It uses the distributive property of multiplication over addition that we saw above.  The popular term for the process that you might know is FOILing:

Again, be sure you understand every step in that process.

Now,  a² + 2ab + b² isn't exactly an obvious way to rewrite (a + b)², but it does have the advantage of being correct!  If you try to interpret that in a different way, though (geometrically, for example),  it can make much more sense.  The natural way to understand the concept of squaring is through looking at the area of a square—which is calculated by squaring.  So below is a picture of a square whose sides are each a + b long.  To make that more clear, those sides are broken up into their separate a and b parts.

Asking about (a + b)², then, is just like asking about the area of that whole square.  But the whole square is broken up into smaller squares and rectangles, and we know enough information to calculate each of those smaller parts separately.  The areas of the two smaller squares are calculated below.

Notice that the areas of the two smaller squares together come nowhere close to totaling the area of the large square.  In algebra terms, we'd have to say that (a + b)² must simply be greater than a² + b².  Of course that means they can't be equal, which is exactly what we've been trying to understand!  This picture actually tells us even more, though.  It tells us how much greater.  Each of the blue rectangles has a length of a and a width of b, so they each have an area of a times b.  And there's two of them.  Which means precisely that (a + b)² = a² + 2ab + b², just as we saw in the algebra.

Finally, I'll show one more way to understand the original inequality.  This last way requires that you know what the Pythagorean theorem says, and I'll assume here that you do (if not, you can skip this part).  Note the square built on the hypotenuse of the triangle below:

The square has an area of c², which the Pythagorean theorem says is equal to a² + b².  But that is exactly the right hand side of our original inequality.  So it makes sense to ask about the square represented by the left hand side.  A square with that area would have to have a side length of a  + b.  But it's clear from looking at the triangle that a + b has to be bigger than c (walking along the hypotenuse must require fewer steps than walking along the legs of the triangle—technically that's called 'the triangle inequality').  That is, the two sides of the inequality each geometrically represent the area of a square,  but those squares can't be the same size, so the two expressions can't be equal.

That is, I hope, enough to convince you.