# When measures of steepness disagree

I.

My students know a lot more about skiing than I do. I grew up in Skokie, IL — an exceptionally flat place, we went sledding down at a pile of garbage called ‘Mount Trashmore’ in Evanston — but a lot of my students go on vacations to resorts and stuff in the winter.

Once or twice, a Jewish youth group took me to Wisconsin to ski. Wisconsin sort of has hills. A midwestern ski resort is the sort of place where you can choose whether to slide down a hill on skis or an inflatable tube. It is also home to the tamest “Black Diamond” slopes in the country — colder but otherwise not much different than the slides my son plays on at the park.

Anyway, that’s what I know about skiing. Glad to get that off my chest.

II.

Towards the beginning of my trigonometry unit — after studying the tangent ratio for several days — I showed this picture to my geometry class. In whole-group, I asked my students to notice as much as they could, and after that I asked the class to try to figure out what all the numbers represented:

When I teach trigonometry, one of my first goals is to help kids see that angles and the tangent ratio both are measures of steepness. Trigonometry is the art of moving between these two different measures. With a trig table or a calculator you can take an angle and look up its associated ratios, and you can look things up the other way (ratios to angles) too. This is true for all the trig functions, and my students encounter it first in the context of the height-to-width ratio.

If you’re trying to describe the steepness of a ski slope — again, not a major concern growing up in Skokie — you could talk about the height:width ratio, or you could talk about the angle of inclination. That chart above rates the difficulty of ski slopes in terms of the angle, but it just as well could have done it in terms of ratios. (I asked my students to draw slopes with heights and widths in each zone.)

The American with Disabilities Act describes the appropriate steepness of a ramp in terms of both measures:

ADA Ramp Specifications Require a 1:12 ramp slope ratio which equals 4.8 degrees slope or one foot of wheelchair ramp for each inch of rise. For instance, a 30 inch rise requires a 30 foot handicap wheelchair ramp.

Every ramp, hill, slide or mountain has a steepness. To bring that physical concept into the realm of mathematics, we have to measure it. But there are many ways to measure steepness, and often we want to be able to move between them. That’s a big part of what trigonometry is.

III.

Before really launching into the trig unit, I task kids with a series of “Which Is Steeper?” problems.

Along with everything else, these problems also really help kids use the height:width ratio as a measure of steepness.

What I’m looking for is for kids to fluently use three little micro-skills:

• when two ramps are the same height (or the same width), the ramp with less width (or more height) will be steeper
• when heights are different, scale one ramp until its dimensions match the other’s, and then directly compare the other dimension
• in general, compare the steepness of two ramps by dividing the height by the width and comparing the ratios

The way I see it, these micro-skills are important background knowledge to support the procedures for finding missing sides of triangles using trig — especially if you come into this work without a lot of comfort with ratios and setting up and solving equations like $\frac{5}{x} = \frac{17}{19}$.

(I do a lot to help kids with ratios, but I don’t usually focus on setting up and solving the equations. Maybe I should.)

IV.

Once I think my kids are getting comfortable using the height:width ratios to find missing sides of right triangles, I show them the physical trig table. There is so much for kids to learn from the trig table — I think it’s a shame when students move straight to looking up values on the calculator.

The most amazing thing about the trig table — at least it’s my favorite thing, and kids often get excited by it — is what happens as we approach 90 degrees. The sine and cosine functions change a bit, of course, but the tangent values just explode:

Kids often are surprised by this, but it makes a lot of sense. Adding another degree of steepness always makes the height:width ratio larger, but not always by the same amount. If your ski slope is very, very flat, then going up by a degree doesn’t increase the ratio very much. If your slope is a double black diamond, though, upping the steepness by a degree leads to a radical change in the ratio, a huge increase.

I always try to use this as an opportunity to introduce some important language to my students: the relationship between steepness ratios and angles is non-linear; a small change in the angle doesn’t always have the same impact on the ratio’s size.

When I think of multiple ways of measuring things, I usually think of pairs of measures that stand in a linear relationship. The nurses measure my newborn daughter’s weight in terms of grams and pounds. When you lose a pound of weight you’re losing 453.92 grams — always. It doesn’t matter how much or how little you weigh. A pound and 453.92 grams are simply interchangeable.

But a lot of pairs of things in the world vary in non-linear ways. In a sense, an additional year of investment is worth more in the future than it’s worth now; a falling ball drops faster as time goes on. I don’t know how many opportunities there are to study this in terms of measurements, but it seems a fruitful arena for chipping away at the assumption that everything is linear.

V.

And, now, we get to the question that has been bugging me for the last few months: How much steeper is an 89 degree ramp than an 88 degree one? A lot or a little?

Remember: whether with ski-slopes or with ramps, there are two ways to measure steepness. You can measure it in terms of the angle or in terms of the ratio.

From the point of view of angles, the 89 degree ramp is just as different from the 88 degree ramp as a 21 degree ramp is from a 20. Which is to say, a bit steeper.

But look at the ratios! Maybe we should think in terms of height:width, in which case the 89 degree ramp is much steeper than the 88 degree ramp, especially compared to what happens when you add a degree of steepness lower down the trig chart.

One way out of this conundrum would be to assert that one of the measures of steepness is the actual, true measure of steepness. But any choice seems arbitrary. Both angles and ratios seem perfectly fine. Why choose one over the other?

(Maybe we’d try to further plant things on a human foundation; how much more effort would it take to climb up each of these ramps? Let’s run experiments that measure physical exertion; maybe we could use physics to model this. Steepness would just then be an expression of human exertion. This is a weird idea.)

Another way out could be to deny that there is any single thing that we’re measuring at all. Maybe steepness isn’t one single thing — it has an angle dimension and a ratio dimension. But what does that mean?

I really have no idea what to think. As we near 90 degrees it seems that the two measures of steepness disagree on how much of a difference a small change makes. Which means that we’re measuring the same quantity (steepness) with tools that are fundamentally incompatible.

What does it mean for two measures to be incompatible? What other measures are like this?

In trying to sort this all out — and I hope it’s clear that I’m awfully confused — I’ve been also thinking about something Freddie deBoer wrote about educational testing:

Incidentally, it’s a very common feature of various types of educational, intelligence, and language testing that scores become less meaningful as the move towards the extremes. That is, a 10 point difference on a well-validated IQ test means a lot when it comes to the difference between a 95 and a 105, but it means much less when it comes to a difference between 25 and 35 or 165 and 175. Why? In part because outliers are rare, by their nature, which means we have less data to validate that range of our scale.

Could that help us think about what’s going on with steepness? Clearly there is no such validation problem when it comes to the steepness of right triangles — we can always draw more! — but maybe there is something analogous going on. We might say: it just doesn’t mean very much to get precise about how steep a very steep ski slope is. Numbers break down, our measures of steepness fall apart, and all we can say about very steep things is just the tautological thing — they’re pretty damn steep.

That is, there just is no way to precisely talk about the steepness of a very steep ramp, as the measures disagree.

But that seems weird too, and I’m lead to the conclusion that I don’t understand this very well at all.

## 5 thoughts on “When measures of steepness disagree”

1. Zach says:

It seems to me that tangent ratio is the natural measure of “steepness” (or “slope”), while degrees is a more fundamentally a measure of “angular distance”.

To see why, just consider a vertical wall. It’s infinitely steep, right? You can’t make it any steeper. So it’s only natural for the value of its steepness to be “infinity”. The tangent ratio also has the pleasing feature that it changes sign whenever the “direction” of steepness changes.

Compare that to the steepness of a wall measured in degrees. Saying it’s “90 degrees steep” would mean nothing to somebody who didn’t already know that humans had arbitrarily decided that there are 360 degrees in a circle. (This is true, even if you use radians, the more “natural” unit of anglar distance.) They would assume that a 91 degree wall would be even steeper, but that’s nonsense.

On the other hand, degrees or radians are a much more natural way of describing the abstract concept of “angular distance”, divorced from any particular reference frame. If you use degrees, you can add or subtract multiple angles to get the total net angle. If you rotate something about an axis without changing it’s radius, a rotation of X degrees always moves you Y distance, no matter where it started.

Come to think of it, this dichotomy falls naturally out of the basic principals of calculus. Taking the derivative of a function always gives you the slope in ratio form, not as an angle, because you have a defined reference frame (the x and y axes, representing the dependent and independent variables of the function). It doesn’t make sense to rotate a slope by another slope. But if you’re working in polar coordinates, your angular ordinate has to be in radians (or degrees), not as a tangent ratio, because in a polar coordinate system has complete rotational symmetry: no angle is privileged over any other.

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2. Peter says:

Suppose you asked instead “Which is shallower ?” and not “Which is steeper?”. For example, how long does it take a ball to roll along the incline from the top to the bottom? (It’s not a perfect analogy, as the answer is the square root of cosecant). The opposite happens: it’s at small angles that any nice, apparently linear relationship breaks down. The gaps between 5 and 4 and 3 and 2 and 1 degrees become larger, until on a perfectly flat surface the ball will not budge from its initial position.

Small angles are where you have a wonderful linear (in fact, practically one-for-one) relationship between angle and height:width ratio and so expect to be able to measure steepness well. But your corresponding measure for shallowness in that range is wildly nonlinear in degrees. Surely “steepness” and “shallowness” are the same phenomena, no?

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3. revuluri says:

This is interesting! I distinctly remember my “aha!” feeling when I got that the tangent ratio was just the slope (closely followed by, “why did no one ever mention this?”)

I was interested by your analogy to measures that get iffier away from the middle. I think the reason is the same; there are two “natural” denominations for “how different” (score and share of the population) and there’s a big difference in the slope (elasticity, first derivative, etc.) between the two at the ends compared to the middle.

The comparisons I was thinking of were actually with examples of inverse variation: gas mileage is one (that’s been studied in psychology or behavioral economics, but likely less familiar to students, especially in NYC); SPF for sunscreen is another. (How different is SPF 35 from SPF 50 from SPF 100?)

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4. M Caswell says:

You can see ahead of time that slope and angle are going to be “mismatched” scales in some sense, just because slope increases infinitely, where angle does not.

Your interesting post made me think about what is really “the middle” of the slope range. Is it 45 degrees, where the tangent ratio is 1:1? But at this point, slope is already changing a good bit faster than angle. Looking at a full trig chart, it looks like the tangent ratio increases linearly with the angle somewhere between 7 and 8 degrees. That’s not very intuitive!

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5. I’m curious about the skiing classifications chart. Because the tangent ratio is increasing more slowly at small slope angles, I would think that the rate of skiing difficulty also increases slowly. Thus, I would expect a wider band for the beginner slope, rather than a narrower band.

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