Posts tagged ‘square root’

Hiking Routes, Square Roots, and Trail Ratings

Not all those who wander are lost.

You may have read that line in Gandalf’s letter to Frodo Baggins in The Fellowship of the Ring, but nowadays you’re more likely to see it on the t-shirts and bumper stickers of hikers.

Hiking is a popular sport, and the 47 million Americans who reported that they’ve taken a hike in the past 12 months (Statista) had a lot of different trails to choose from: American Trails maintains a database of over 1,100 trails, and Backpacker‘s list of America’s Best Long Trails offers an impressive 39,000 combined miles. Plus, there are thousands of miles of trail not on either of those lists. With so many options, how are you supposed to choose?

A wealth of information is provided for most hiking trails. But while some information — like distance and elevation gain — is absolute, other information leaves room for interpretation. What does it mean when the Craggy Pinnacle Trail just outside Asheville, NC, is described as a “moderate” hike? The Explore Asheville website says,

Moderate hikes could range anywhere from a few to ten miles with an elevation gain up to 2,000 feet.

By those standards, a three-mile hike with a 10% grade would be considered moderate. No, thank you.

Unfortunately, there is no standardized system for determining trail difficulty. Most of the time, the trail rating is a nebulous qualitative combination based on an examination of the terrain, trail conditions, length, elevation gain, and the rater’s disposition.

But I tip my hat to the good folks in Shenandoah National Park who have attempted to quantify this process. Their solution? The simple formula

$r = \sqrt{2gd}$

where g is the elevation gain (in feet) and d is the distance (in miles). The value of r then corresponds to a trail rating from the following table:

 Numerical Rating Level of Difficulty Estimated Average Pace (miles per hour) < 50 Easiest 1.5 50-100 Moderate 1.4 100-150 Moderately Strenuous 1.3 150-200 Strenuous 1.2 > 200 Very Strenuous 1.2

Elevation gain is defined as the cumulative elevation gain over the entire hike. So if the hike climbs 300 feet over the first mile, then descends 500 feet over the next 2 miles, then goes back up 200 feet to return to the start, the elevation gain is reported as 300 + 200 = 500 feet.

Old Rag is one of the most popular hikes in northern Virginia. Known for the half-mile rock scramble near the top, this trail boasts an impressive 2,415 feet of elevation gain over 9.1 miles. Applying the formula,

$r = \sqrt{2 \cdot 2415 \cdot 9.1} \approx 209.6$

which means Old Rag’s level of difficulty would be “very strenuous.”

This formula could lead to several activities for a middle or high school classroom:

• Draw an elevation map depicting a trail on which any type of hike (from easiest to very strenuous) would be possible, depending on how far a person hiked.
• With distance on the horizontal axis and elevation gain on the vertical axis, create a graph that shows the functions for easiest to very strenuous hikes. (See Figure 1.)
• If you were on a trail with an average elevation gain of 300 feet per mile, how long would you have to hike for it to be considered a moderately strenuous hike?
• If one 5-mile hike is rated “easiest” and another 5-mile hike is rated “strenuous,” what’s the minimum possible difference in elevation gains for the two trails?

Students could also do a comparison between this trail rating formula and the geometric mean, if you wanted to go really crazy.

Figure 1. The graphs for various difficulty ratings, using the Shenandoah trail rating formula.

Just as every good hike comes to an end, so must this blog post. But not before we laugh a little.

As it turns out, there’s a math joke about hiking…

An actuary has been walking for several hours when the trail ends at the edge of a river. Having no idea how to cross, she sees another hiker on the opposite bank, and she yells, “Hey, how do I get to the other side?”

The man across the river — a math professor — looks upstream, then downstream, then thinks a bit and finally says, “But you are on the other side!”

It’s a math joke about hiking as much as any joke about any topic is a math joke, if you insert the correct professions.

There’s a great non-math joke about hiking, too…

A fish is hiking through a reservoir when he walks into a wall. “Dam!” he says.

And there is a very mathematical list about hiking, which might be considered a joke if so many of the observations weren’t true…

Eight Mathematical Lessons from the Trail

1. A pebble in a hiking boot will migrate to the point of maximum irritation.
2. The distance to the trailhead where you parked remains constant as twilight approaches.
3. The sun sets at two-and-a-half times its normal rate when you’ re trying to reach the trailhead before dark.
4. The mosquito population at any given location is inversely proportional to the effectiveness of your repellent.
5. Waterproof rainwear isn’t. But, it is 100% effective at containing sweat.
6. The width of backpack straps decreases with the distance hiked. To compensate, the weight of the backpack increases.
7. The ambient temperature increases proportionally to the amount of extra clothing in your backpack.
8. The weight in a backpack can never remain uniformly distributed.

Go take a hike!

Mental Math and the MCWC

How long would it take you to find the sum of 2 two-digit numbers?

Or 4 four-digit numbers?

Okay, let’s get really crazy… how long would it take you to find the sum of 10 ten-digit numbers?

You can decide whether you’ll do the calculation in your head, on a calculator, or with paper and pencil. Your choice.

With a calculator, it took me 91 seconds to find the sum of 10 ten-digit numbers.

Without a calculator, the winner of the Mental Calculation World Cup 2014 needed only 242 seconds to complete 10 problems in which participants were asked to add 10 ten-digit numbers. On average, that’s just 24 seconds to do in his head what took me a minute-and-a-half with technology.

Holy smokes!

Competitors at the MCWC do a number of mental calculation tasks. The following exercises will give you an idea of the computations that they do.

Exercise 1. Sum of 10 ten-digit numbers.

$\begin{array}{r} \mathbf{4190187220}\\ \mathbf{3967093178}\\ \mathbf{8567125486}\\ \mathbf{1005683165}\\ \mathbf{3635944647}\\ \mathbf{7645865467}\\ \mathbf{3506970235}\\ \mathbf{6710259450}\\ \mathbf{2 347 894 647}\\ \mathbf{\underline{ + 4 995 420 559}}\\ \end{array}$

Exercise 2. Multiplication of 2 eight-digit numbers.

$\begin{array}{r} \mathbf{18467941}\\ \mathbf{\underline{ \times 73465135}} \end{array}$

Exercise 3. Square root of a six-digit number.

$\mathbf{\sqrt{530179}}$

Exercise 4. Day of the week for a calendar date.

$\textbf{8 December 1721}$

Exercise 5. Multiplication of 3 three-digit numbers.

$\mathbf{184 \times 734 \times 635}$

These sample exercises are taken from the examples in the MCWC 2014 Official Rules.

If you were able to complete all five of those exercises in, say, less than 3 minutes, then you might be ready for MCWC 2016. (Note that for Exercise 3, you needed to be accurate to within 5 × 10-6.)

But if you’re like me, you’ll probably want to skip the competition and keep your calculator close at hand.

Rational Thoughts

I saw the word irrational a lot on Pi Day, and my friend Patrick Flynn recently invited me to join the Global Anti-Rationalization Foundation (GARF) on Facebook. Both of these things reminded me of the following story, which comes from a four-decade veteran of the classroom, John Benson of Evanston Township High School.

A student of John’s solved a problem and obtained a fractional answer that contained the square root of pi in the denominator. But then the student did a curious thing. He started to multiply the numerator and denominator by the square root of pi.

“What are you doing?” John asked.

“Rationalizing the denominator,” the student responded.

(If you don’t understand why that’s funny, you may be too young.)

Thanks to calculators, the process of rationalizing denominators is no longer necessary. One could make the argument that it wasn’t really necessary before the days of calculators, either, but there was at least some reason for doing it back then — converting a fractional result to a decimal was difficult if the denominator was irrational. If the denominator could be converted to a rational number, then the division would be less arduous.

When I worked for MathCounts, I was fortunate to spend two memorable years with John Benson. He served as a member of the MathCounts Question Writing Committee. John wrote phenomenal problems, had keen mathematical insight, and could put you on the path to a solution for every problem. But most importantly, John was inspirational. I once attended a presentation by John, and I loved the quote he displayed at the end of his talk: “You can’t get burnt out unless you were once on fire.” When he talked about his classroom, you realized what’s possible in math education. His philosophy has kept both he and his students inspired for nearly 40 years, and the following statement from John, which he claims to repeat often at parent-teacher conferences, will help you understand why:

I promise that I will try hard to make sure that something happens every day in class that cannot be replicated anywhere else. If a student is not in class, that student will never be able to recapture that teachable moment. They will have missed something.

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.