9%.
Fun to bike down but a workout to bike up, a 9% grade earns a failing grade in my gradebook.
Outside of Otis, Massachusetts is a road with a very steep hill. Put in neutral, our standard transmission car just cruised down the hill. On the way up, first gear was the way to go, so to speak. The rhomboid sign reported a 9% grade (“Test your brakes,” it warned). A biker was huffing her way up the slope while a second simply sailed down the hill. What’s the 9% mean? If 60% is passing and 90% is an “A,” what’s 9%? Doesn't seem like much; why the big deal on that hill outside Otis? Folks seem to always aim for 100%, but that would be suicidal in an automobile or bicycle and certainly not preferable. We, as humans, do our best to categorize (think: Kingdom, Phylum, Class, Order, Genus, Species or better yet square, rhombus, rectangle, parallelogram, trapezoid, quadrilateral); it seems we have categorized slopes (or grades) of hills as well.
Wales has a road with a 25% slope; I-70 into Denver from the west has a cool 6% grade. A handicap ramp has to be an inch vertically for every foot horizontally. Are these ideas related?
Your mission is to understand what these numbers mean and how engineers have come to categorize the grade of a road, ramp, or slope. Nice word there, by the way, “Slope.”
Yep, good ol’ Wikipedia actually has a description that works for us. It may seem a little dense and might take some slower reading than, say, Ted Geisel’s stuff, but it’s got all the ideas you need. In the wiki, there are triangles, a protractor shape, a trigonometric function, and some other very familiar words. Put the pieces together in your blog and you’re set for the week’s blog assignment. (Be sure you take out the irrelevant ideas for "grade" in my post -- this is meant to have nothing to do with the grade you get in class. That's a joke.)
More specifically, the assignment for both TPC and GA2: the grade of the road has everything in the world to do with a trig function. Which one? Why? Explain. Use roads that you've seen or know about or find on line. There's a couple different standards for handicap ramps (businesses vs private homes); find those if you'd like. Go bananas on this one -- where else do you hear about grades? What about the "angle of repose"? What's that? What about "railroad grades"? Choose something that interests you; don't feel as though you need to cover absolutely everything, but DO cover the idea of what a "grade" is. If you are one of those folks in GA2 fascinated by the number theory topic we touched on (Pythagorean Generators), you can choose to write on that instead of this whole idea.
More specifically, the assignment for both TPC and GA2: the grade of the road has everything in the world to do with a trig function. Which one? Why? Explain. Use roads that you've seen or know about or find on line. There's a couple different standards for handicap ramps (businesses vs private homes); find those if you'd like. Go bananas on this one -- where else do you hear about grades? What about the "angle of repose"? What's that? What about "railroad grades"? Choose something that interests you; don't feel as though you need to cover absolutely everything, but DO cover the idea of what a "grade" is. If you are one of those folks in GA2 fascinated by the number theory topic we touched on (Pythagorean Generators), you can choose to write on that instead of this whole idea.
GA2
Grade, percents, and slopes are all different words but can lead up to the same idea. In school we aim for the highest grade/percent we are able to achieve. In fraction form this is shown with how many you got right/ how many questions there are total. This correlates with the idea of a slope. The meaning of a slope is rise over run . this shows how far you are rising in your education. The higher the goal the harder it is to achieve. Like if you are running up a hill that is pretty steep then you'll get tired, but if you have been working and training up that hill it'll be easy. Same concept is being dealt with grades and percents, if you keep trying and studying as hard as you can it'll be easy. Good grades don't come from doing nothing. Even to the kids it comes easy to they too study maybe not as much but they do.
With the percent in slope. You would probably want a lower percent in incline. You would want this because if you have a 9% incline on the steepness while your riding a bike it'd be a nice little trip. Now imagine biking on that same pathway but with a 90% (what most kids strive for in school) incline. It'd be a lot tougher. The idea with percents and slope here are different then what you want in school. I think of it this way, slopes in steepness are like in golf. The lower the number, the better. The higher the number, the harder you must try to succeed.
After a lengthy google search, I found out that a 9% grade is around a five degree slope, which sure is steep for a road, but while looking at it on graph, is not. The grade of a hill is rise/run x 100. But, knowing enough Trig, I was able to quickly realize that this is just tangent (opposite/adj) of the angle of inclination, or as its called in class - theta. I didn't expect to read on Wikipedia that a train can only pull half of it's load on a 1% grade as easily as it would on a 0% grade. Reading this I realized that slope and grade have a enormous importance on our world, especially in construction, transportation, and landscaping.
ReplyDeleteThe height of a right triangle is the rise, the base is the run, and the slope is the hypotenuse. This triangle immediately pops off the page, of unit circle for me, into real life; usually nothing comes off the page in math. I see the grade now as a part of math and real life. This is one of those few moments were math actually seems useful to me.
I can't help but wonder if anyone is finding the grade or tangent of a road somewhere and using the formulas were using in class. I wonder if one day I'll be using tangent to build a house, or choose a bike trail.