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.
Grade is directly related to the right triangles that we have been doing in this past week. It is measuring an angle that is on a right triangle. In order to measure a grade, you need to have a right triangle and the lengths of the two legs. At first I thought that I could solve for any tangent problems using rise over run. But when I calculated it, the rise over run angle and the tangent weren’t the same measure. This confused me. I asked a fellow student, and he told me that the rise over run is just the slope of the line, and not the measure of the angle. He did tell me however, that I could use rise over run to find the measure of a line if it was plugged into the right equation. He told me that if I plug in the inverse tangent of the measure of the slope, I could find the measure of the angle. I had to think this through my head for a while, and plug it into my calculator, but it did work. I then realized that this is just how we learned to find angles using inverse tangent. But instead of using rise over run, we use opposite side over adjacent side.
ReplyDeleteI think that this connection is what confused me before. There are many ways to complete a math problem, and many more ways to tell it. This was another way of telling it. I think that it is a cool way to do these problems, and I think that since we all have done rise over run at one point in our math classes, it could be useful to teach. I certainly think that it would help me if I were taught that as well as the opposite and adjacent sides. Did you know that you could plug in rise over run into tangent problems? I thought it was a pretty cool thing for me to find out.
A grade is a slope, or the rise over run. It basically measures how steep a canyon, road, or handicap ramp is. This relates to right triangles and tangents. Finding the tangent can find the grade of the slope. Trains have less gradeability (the ability to ascend grades) than cars. On railroads, a steep grade can be a problem. Even a 1% grade could cause the train to be able to pull half of what it could pull on a zero grade surface. So, for grades steeper than 7%, a rack railway is typically used. This helps keep the train on the tracks. When I say a 1% or 7% grade, its just the ratio of the grade multiplied by 100 into a percent. For example: a 0% grade is flat, and a 100% grade is straight up. A 1% grade has a slope of 1/100.
ReplyDeleteSteepness
ReplyDeleteA "grade" of a road is simply the slope, rise over run, also known as the vertical distance over the horizontal distance. It can be expressed in many different ways, such as percentage. Basically, the steeper the slope, the higher percentage it is and vise versa.
As humans, we face many slopes in our lifetime, it's a fact. Whether the slope is a handicap slope or a theoretical slope or "journey" to reach a goal. Maybe that's why we call our school results grades, because it is a slope, some slopes are steeper than others. The higher grade you receive, this shows you how much effort you put into the slope. It takes more energy to earn a higher grade, which can correspond to why higher slopes mathematically have higher percentages.
The grade of a slope is exactly what it sounds like-- a slope.
ReplyDeleteGrade is expressed in the following equation:
% grade = rise/run x 100
A slope with a grade of 9% is one in which 9 feet of altitude is gained for every 100 feet of distance travelled, which is actually steeper than it seems. A handicap ramp, which must rise 1 inch for every foot travelled, has a grade of 8.3%. The steepest road in San Francisco, Filbert Street, has a grade of 31.5%, meaning that when driving on this road, one gains 4 inches in elevation for every foot they travel.
This also means that the road climbs at an 18.4 degree angle, because the negative tangent of 4 over 12 is 18.4. The tangent, a trigonometric function, can tell us the angle based on rise over run, or the rise or run based on the angle and one of the value. The tangent function is opposite over adjacent. To elaborate, this means that the equation must be set up like this:
tan(angle) = (opposite side) / (adjacent side)
Reworded, the equation becomes:
tan(angle) = (rise) / (run)
And that is how the trigonometric function of tangent finds it's way into rise over run, and subsequently, grade.
The idea of a grade in the road is called the slope (known as rise over run, a bigger number means a higher or steeper slope of the surface compared to it being flat, or horizontal. Slopes are seen everyday in life; just recently our class studied trigonometric functions and finding the length of a side in a right triangle given an angle and the length of a side, or finding the angle of the slope and given the length of two sides, finding both solutions when plugged into the right equation.
ReplyDeleteSlopes can be found in our lives such as when we drive along hills, using less gas and simply letting the car go on its own when we have a steep hill, or using a lot when we go up a hill. The amount of slope or "grade" can also be seen directly at school as we occasionally have large thunderstorms, flooding our lower campus due to the gradual but large steepness of the school.
Grade and slope are basically two interchangeable words. If you want to find the grade of a hill, road, or railway, you can use the slope formula. The formula for slope is rise over run. Finding the grade or slope of something is related to what we have been doing since the beginning of the year. In a right triangle, you can find the slope of the angle by dividing the height by the length of the triangle. To get the % grade, put the denominator as 100.
ReplyDeleteSo what does it mean when you see signs with a percent grade on it? What if the sign says the slope of the hill, road, or railway, has a 9% grade? This means that for every 100 ft you travel horizontally, you will be 9 ft higher than when you started. The slope of this incline/decline would be 9/100. A 0% grade is horizontal and can easily be driven on. A 100% grade is vertical and is impossible to travel on.
Response to 9%
ReplyDeletePercents. Grades, Slopes- All have different meanings, but at the same time they can be alike. In school, the higher the grade/percent in a class is what we all aim for, and if we try hard enough we will achieve it. Getting a high grade takes a lot of effort. Going up a slope with a high grade takes a lot of effort. Yes, there will always be that person who can walk up a steep hill faster than you, or one who aces a test without trying, but those people- they are lucky. In reality, a low percent you get on a test is not something to be proud of, and getting up a on a hill with a 50% grade is something to be way more proud of compared to getting up a 2% grade.
When one becomes old and frail, it is imperative that they do what they can in order to maintain a happy, healthy, lifestyle, while still minimizing their physical effort. In order to get around, many people of the older generation use wheelchairs. To maneuver around, and in order to get places, instead of going up and down stairs, one must go up and down a slope. The slope for a handicap ramp is in the ratio of 1:12, meaning that for every foot horizontally, the slope has to decrease by one inch. If the slope was 2:12 it would be much more difficult for people of age to go on them. In order to accomplish tasks, such as climbing a mountain, getting a good grade, or even wheeling your wheelchair up a slope, you need to exert energy.
Technically a slope is "rise over run," which is used in mathematical procedures, and in the daily world. In the real world, slopes/grades should ultimately get you somewhere, but only if you put in the effort!
GA2
ReplyDeleteGrade is one name for a concept that has many. In mathematics it is generally refered to as "slope" which is rise/run, or change in x/change in y. Colloquially, one could think of grade as "steepness." This idea is used to indicate the steepness of roads, hills, and canyons among other things. To express grade in a percentage, one uses the equation 100 x rise/run. Albuquerque is a very flat city so people who live here might not be familiar with the application of grade to roads.
When you pour granular material onto a flat surface it forms a cone. The angle between the surface of the material and the horizontal surface it is poured onto is called the "angle of repose."
The primary trig function that is related to grade is tan(angle) = rise/run.
Grade, commonly referred to as slope, is simply rise/run. In order to find %grade we can use the equation 100(rise/run) and that would give us how much we fall or rise over a certain distance. For example, if you had a 10% grade that means that you would rise or fall 10ft for every 100ft of distance ran. The equation can also be expressed as tan a= triangle h/d
ReplyDeleteWe can now see why handling a 9% grade may be cumbersome, but a 100% grade is just plain "suicidal" because taking into my example above, if we had an 100% grade that would mean rising 10ft for every 10ft! If I am out biking or jogging I want a 0% grade from now on because that's just plain flat ground.
Quick story on how this concept may have helped my family in the past. One camping trip we decide it would be nice to go up into the peaceful mountains of Colorado and wanted a new campsite for a change instead of the one we always went to called "Redondo". So instead of taking a turn at the Redondo campsite we tried a new one called "Iron Gate". Now, the road to the campsite may have been perfectly fine for any old car carrying some light tents, but we were pulling a full size pop up camper! There were no %grade signs anywhere in sight and had we have seen them I think we would have thought differently before journeying up that hill. For the first 15 mins or so our Bronco handled itself just fine, but then the hill started to get steeper and steeper. We had no choice now but to continue on, we had already made it this far. Bad choice. The next 45 mins were filled with cursing, stopped cars and screeching tires, as we slowly made our way up the hill. We made it to the top with smoking brakes and exhausted campers. I don't think we will ever camp up there again and from now on I'll try to see if there is a %grade sign on the road BEFORE we spend an hour trying to get up it.
GA2
ReplyDeleteGrade, slope, and incline, are all used to express the angle of inclinations of one surface to one that is horizontal. The steeper the hill or road, the higher the percentage of inclination is going to be. This relates to the tangent of a right triangle and what we have been learning in class. Environmental design is also a component that has to do with grade. Drainage, needs a certain slope to keep the city safe in case there happened to be a flash flood or heavy rain, which we have recently experienced in Albuquerque. The steeper the road the harder it's going to be to travel up it and going down with broken brakes can be just as dangerous. The formula of a grade is 100x rise/run which is how tests are graded, the rise is the number of questions you got right/ the run, the total points maximum. Grade is relevant to school because the harder it is to achieve something the harder you have to work.
GA2
ReplyDeleteGrade refers to the rise, slope, or incline of a horizontal surface; a higher number of grade indicates a steeper slope, rise over run. The grade of a horizontal surface relates to the tangent of a right angle, taught to us in our last lessons of class. Grades are important on roads, railways, and in environmental design. Gradients, used on railways, limit the load a train can haul; even, a 1% gradient affects the train's pull on a haul drastically. A train, with the same constant speed on horizontal and inclined surfaces, will only be able to pull half the load on a slope. Steeper railways will present a challenge for trains to travel up the slope and travel down with feeble brakes, creating a dangerous situation if gradients are not brought into consideration. A grade can be found with the equation tan (angle) is equal to the rise over run, and multiplying the rise over run by one-hundred, one can find the grade percentage. In school, our work is constantly being graded, but effort makes a difference. When a "slope," or test, is approaching, we must put in the effort to overcome that slope with a good grade, such as 100%. On railways with lower gradients, a train does not need to put in as much effort to haul a load compared to a railway with a higher gradient. Grades and effort correlate with each other. Higher grades require more effort.
GA2
ReplyDeleteGrade, 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.
GA2:
ReplyDeleteSlope, grade, percents, degree of tilt, they are all the same thing. If they are all the same thing than why is it that a 9% grade on a road is quite steep; whereas, as 9% on a test is failing? Slope can simply be found by finding the rise over run of a line. Secondly, the slope of the road, or anything for that matter, can be related to the sine function rather than tangent or cosine functions. The sine function is used to find the grade of the road, because it is still possible to find the slope, even when the horizontal distance id unknown. Take a regular right triangle for example. If you don't know what the base, or horizontal distance is, but you know what the angle of elevation is, you can use the sine function and divide the length of the opposite side (rise) over the length of the hypotenuse(run). This makes sense, because the formula for the sine function is sin= opposite/ adjacent, and the slope of a line is equal to rise/ run.
In everyday life, most people probably wouldn't know how to find the slope of a hill, or what it is for that matter. So how does this apply to everyday life? For people driving along a road, and see a sign that has the percent grade on it, in our case 9 %, they will know for that every 100 feet they travel, they will be 9 feet higher than they were at the last point. 9/100 is the same universally, no matter what the case is, but 9 % can vary drastically depending on the circumstance, but makes complete sense if you understand the logic behind it.
GA2
ReplyDeleteGrade is a word that has many different contexts depending on what situation it is used in. Slope is one synonym for grade, and is used when referring to the angle of inclination of a line found in a graph for example. When speaking about academics, grade is also used to indicate the level of progress one has made in a certain area. In the real world, like this blog post was concerning, grade is the physical amount of slope when applied to the real world.
When discussing grade, or slope when it is used in mathematics, the equation used to find it is m = y2 - y1/x2 - x1. In the real world, the grade, or the steepness of roads, tracks, and etc. is determined by the observation of how much change in altitude there is over the horizontal distance traveled in 100 units of measurement. This works out to result with the equation % grade = (rise/run) * 100. By knowing the grade, and the change of altitude over each foot or more, one can also find the angle of elevation of the road, ramp or etc. by finding the negative tangent of the opposite side/adjacent side. This simplified, if you visualize it is essentially the same as the negative tangent that can be found for the slope, or rise over run. It's cool that grade ends up working into trigonometry as well.
Concerning grades in academics, it also ties in with the relation that if something has a higher slope, then it becomes harder and harder to get up. In another sense, it's also interesting to see sometimes how wrong we can be, or how right when we take for example, say a test, and either assume that we know everything already and do poorly on it, versus studying hard, and just like the biker on the hill in Massachusetts, power through and as a result, do well on the test.
This also, as a bit of a tangent to this topic, reminds me of something we learned last year in science where, for instance, you swam 100 meters in a pool, you really wouldn't be doing any work because although you travel a distance, you end up in the same spot, resulting in you not gaining any real ground. Luckily this isn't the case for everything, and hard work always seems to pay off!
Grade has different definitions of course. The first is the percent you get in a certain class. However, that is not the one you have talked about here. The other type of grade that we are referring to is the % grade of a slope. It refers to the inclination of a surface. These two different meanings of Grade are inverse.
ReplyDeleteFor example,a 100% grade, or slope, would mean that the hieght increases 10 ft every 10 ft. It would be impossible to go up or down, because it's almost equivalent to a 45 degree angle. Try driving up that!
Alternatively, a 0% grade is just flat ground, which would most definitely be more comfortable.
The grade is the steepness of a slope. It is expressed mathematically by the formula m = y2 - y1/x2 - x1. It's the rise over the run, and can be seen clearly on mathematical graphs of lines. By knowing these things, one can figure out the angle of elevation of a surface. This also can link into trigonometry.
It is interesting how one word can have two meanings so diverse and so opposite mathematically. With one type of grade 100% is good, with the other, not so much. It's important that we know the difference between the two so we can apply them to life and not end up in unwieldy situations.
slope, which is Rise/Run, is commonly referred to as grade when you're talking about roads, hills, canyons, etc... In every day life, we see countless examples of grade. One example could be the road on campus, or another example could be Paseo Del Norte all the way down past the river. In order to find the grade of a road, you can take the vertical distance that the road extends upward and divide it by the length of the road, which will get you the slope. This slope will be the same as the tangent function, which is opposite/adjacent. However, the slope calculates the rise/over run while the tangent calculates the angle measure. Therefore, you could measure a grade with either the angle measure or with the slope. Finding a grade relates to trigonometry because right triangles and functions can be used to find the grade.
ReplyDeleteIn real life, if you were biking, you would want a very shallow grade, because as the angle or the grade gets steeper, it becomes much harder to get up. This is because gravity is working against you and because you have to move vertically up much quicker than you would if it was a shallow grade.
Grade, percent and slope, while all have dissimilar meanings, and should not be interchangeable. They all center around the same idea, but they are not the same thing. The grade in a subject at school is not the same as the grade discussed above. The grade or slope explored above is talking about the degree or angle of the inclination of a surface. In the real world this is talking about things like the steepness of a road or hill. In this case the higher the grade, the steeper the road. Both slope and grade can be found by a simple mathematical equations, 100X rise/run. The difference between the two is slope is measuring steepness (of a hill) while a grade, specifically one in a school context, shows how much effort was put into something like a test. In the case of a slope a lower grade is often better and takes less energy to make it to the top. In a grade at school a higher one is better because it means there was more correct answers. This is why they should not be used as synonyms. They are measuring two different areas that do not relate. Why compare two items that are not comparable? They are opposites. In different contexts a 9% is both failing and passing. It’s simply a matter of understanding in which areas a 9% is a pass or a fail.
ReplyDeleteGrade is a measure of how inclined something is. The percent grade tells how steep an incline is. The higher the grade, the more inclined it is. Grade is also referred to as slope. Slope (or grade) is measured by rise over run. To find the slope you measure the altitude and the horizontal distance it covers. You divide the altitude by the horizontal distance. www.spinlife.com (and Ms. Mariner’s blog) says that for a handicap ramp, every inch of vertical height should have twelve inches of horizontal length. If a handicap ramp is trying to reach a place somewhere three feet off the ground, the ramp must cover 36 feet vertically. The grade of this ramp is about 8.3% (you find that by dividing 1 by 12 and then multiplying by one hundred to find the percentage). This is where it gets interesting—the slope or grade of a triangle uses the same equation as the tangent. Imagine triangle ABC where angle A is the smallest angle. To find the tangent, you would divide the side opposite angle A (side BC) by the side adjacent to angle A (in this case, side AC). To find the slope, you would also divide side BC by side AC. However, when finding the tangent, you would get an angle measure. The slope however would not equal an angle. It would equal how much horizontal length there is to vertical length.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteThe grade of a road is directly related to the tangent function, as well as the slope of a line. I'd come across the term a few times, but never quite understood it. In math last year, slope was described to me as rise over run. To find the percentage grade of the road, you take the tangent of the angle of elevation (the angle the road makes with the ground) and multiply that by 100.
ReplyDeleteIn Albuquerque, I find that if I'm heading west (away from the mountains) I have to watch my speed more carefully than if I'm heading east. The mountains create a natural slope in the road, just enough so that I barely have to touch the gas as I head home from school on Osuna. Comanche's another road like that, if I'm not careful I'll wind up doing 45 after turning off Tramway.
In looking at the description of the road with a 9% grade, I wondered what angle that makes with the ground. The Wikipedia article says that percent grade figures are reached by multiplying the tangent by 100. So, by taking the inverse tangent of .09, I found the angle to be about 5.14 degrees. That doesn't sound like much, but definitely enough to make it hard to go up on a bike.
Grade is another kind of slope, just expressed as a percent rather than a fraction. It is rise over run times 100. Using the tangent function you can also learn a little about a slope, but in a different way. Using SohCahToa, we know that it uses opposite adjacent. That means on a right triangle, the hypotenuse is the road /slope, opposite is the rise, and adjacent is the run.
ReplyDeleteThese small things, slope and angles are very important in our everyday lives. They help us stay safe on the highway avoiding ups and downs, and they slow us down on our way to school or running in a cross country race. I also read a little about trains and how they are affected by slope. It is really interesting to learn about how big of an effect a 1% slope can have. It said a 1% slope cuts the amount of weight a train can carry in half. That would make a huge difference going thousands of miles because you would have to double your trips. This also shows you how much harder it would make biking. Since you are the one providing the power to propel you, you also feel physically all of ups and downs of where ever you are riding.
GA2
ReplyDeleteTo me, slope is the measure of the incline of something, which is measuring just the distance the "hypotenuse" travels lengthwise, as well as the height in the same amount of time (basically rise over run). The grade is more a a percentage, such as in tangents, sines, and cosines. In the real world, going up hill a smaller grade is better (in most opinions), because it results in less effort to get up the hill. But going down, a greater grade is less effort.
My dad used to love the Tour de France, and he was a really good biker and triathlete. He'd explain to me that the bikers in the Tour de France would bike up hills and mountains with 9% incline sometimes! I understood how steep that was, and the bikers in the race still astound me that they can bike at such steep paces. He also explained to me that the steepest parts of the race was set up in mountain stages and they are the most difficult. There are four categories in the mountain stages and some that are beyond the categories that are 7% or more! Category 4 is shorter and easier than Category 3 climbs that are 5 kilometers and have a grade of 5%. Category 2 climbs are the same length or longer with an 8% grade while Category 1 climbs last 20 kilometers 6 percent grade. Beyond category climbs include an altitude difference of at least 1,000 meters and have an average grade of at least 7%.
ReplyDeleteWe have some 9% inclines here in Albuquerque that are biked frequently such as Tramway to the Sandia Casino. On Saturdays I see people climbing up with their legs pumping, and I see others that coast down the street sometimes going upwards of 35 mph!
Grade, slopes and percents don't necessarily the same meaning, but definitely center around one main objective: to reach a goal. Even though grade slope and percents don't mean the same, they are still all connected. Slope can be defined as rise over run, or in the case of a road, vertical distance over the horizontal distance. The higher the grade, the steepness of the slope increases and the higher the percentage is for that slope. One can relate this to sine function of trigonometry. This is the best function to use because one can still find the slope of a road without having to know a horizontal distance. To relate this to what we are learning in math, you can find the horizontal run of a right triangle using sine if you know the angle and divide opposite side of the triangle by the hypotenuse (which is the rise).
ReplyDeleteThe grade can effect a slope, which can end up altering the percent, by the amount of effort put in to keeping the slope going. In real life, a grade depends on the amount of effort is put into the slope (or test). If it is clear that one doesn't care and didn't study, they will receive a lower grade (percent) vs. the person who spent countless hours the night before.
In the english language, we have many words that sound the when you here the individual word, but have completely different definitions including know and no, knew and new, are and our, ect. We also have a lot of words that are even spelled the same, but have different definitions, and one of those words is grade. For most people (at least students) the only "grade" that we care about are the grades we get in school. We use this term to express the quantity of how well we did on an assignment, quiz, or test. For most of these, calculating your grade is very easy. All you do is divide the number of points you received by the total points that you could have earned.
ReplyDeleteAnother type of grade is slope of a hill compared to 0% grade which is the flat ground. Slope is classified as the rise divided by the run. For example, if you were standing on a hill and wanted to figure out its slope, you would just measure how tall the hill is, then divide it by the potential length that it would have if it was just flat land (or half the length if the hill has a peak then goes back down). When you measure these two variables, you can find that the hight and length meet at a right angle. The hypotenuse of the "right triangle" that you just made is also the slope of the hill. Another variable that can help you determine the grade of the hill is the angle of elevation. The greater the angle, the steeper the slope will be. So, the higher the grade, the harder it is to achieve. Whether you want a high grade in a class, or if you are trying to conquer a steep hill on your bike (assuming you are going uphill).
Slope and Grade may seem like two different concepts, but they are the same: grade=slope. If you are using the wrong “grade” then they have no relation, but when you’re talking about how steep a ramp or hill is it makes sense. I asked my dad a little about this topic because he is an engineer, and you motioned they worked on this kind of stuff. He told me the “9%” means it’s 9 feet (for example) up for every 100 ft over. I looked up the topic and learned there is actually a Road Grader. It’s a machine with a blade that will grade the road correctly. Not only are roads graded, but railroads and ramps are too. Railroad grades must have very low grades. Unlike roads, which can have anywhere from 1% to the vehicles “gradeability”, trains can only go up to about 5% grade. Most commonly, they are only 1%-2% grade. This may be another reason why it takes longer to get somewhere in a (old) train vs. a car. Slopes can be found all around you, and each was calculated just right. Thus, another example of how math is all around us.
ReplyDeleteHello! I enjoy have enjoyed learning about functions and such lately because I am an avid golfer. Knowing how far the ball needs to go to carry onto the green is key to putting up a good score. Most often, the green is not on the same plane that you are on. you have to know how much higher or lower the green is than where your ball is. An example of what I'm trying to convey to everyone can be found on the 2nd hole at Tanoan Country Club. With a good tee shot, you play to a green that is elevated at least 15 yards above where you and your ball lie. If you looked at the hole from a good spot, you could turn this shot into a triangle, and decide how far the ball would need to fly to get to the hole (telling you to give yourself more yardage rather than less). If my ball sat next to the 120 yard marker on this hole, know that if the course was flat, I would hit what I call a "baby pitching wedge." But, guessing that the hole is really about 15 yards above where my ball is, I probably need to fly the ball over where the "flat shot" stopped. So, I would probably hit a "full pitching wedge." I also know that from my ball to the hole is a 7.13 degree incline or about a 12% grade. The idea is that if there is more grade percent, then you travel a longer distance to travel than if you journey was flat.
ReplyDeleteThough the concept is a bit confusing to me, I have always thought of grade and slope as being the same thing. But to be more specific, grade is used when talking about roads and slope is a more mathematical "rise over run." I read this blog a few days ago, but got a bit confused. Today was my brother's birthday and it's a tradition for us to hike to along the Sandia Crest to the tram on his birthday, so in the car today I brought the subject of the "9%" to my dad. As we zoomed through the winding road up the mountain he told me that a 9% slope is probably the steepest we would be on while driving up the mountain. He then continued to tell me that whenever he does triathlons a 9% is considered steep. At first when he said this I thought "Really?! That's the steepest?" But whenever you see a biker biking up that road to the crest they are always huffing and puffing, so 9% is fairly steep. But keep in mind, when measuring grade and slope you always have to take in to consideration the circumstances of the hill you face. Back to driving up the Sandia Mountains today, as we passed the ski area my dad added that Al's Run, a fairly well know black run at Taos Ski Valley, is about a 20% slope. This helped me put it all back to scale. So if 9% does not look that steep but is on a bike, and 20% is steep and not bike-able then what is 90%, that much be really steep! Railroad grades have to be significantly lower because the weight of the train would make is much harder to stop in an emergency.
ReplyDeleteIn terms of basic ideas, grade, slope, angle of repose etc., all lead to the same idea. Slope is measured by dividing Rise by Run, or the altitude divided by how long a distance it covers horizontally. Grade is the measure of how inclined something is. Angle of repose is the maximum incline the slope can be at where material will stand without sliding. The trigonometric function that ties all these terms together is the tangent function. This helps us find the grade using the Rise over Run equation. You can still find the slope of a road without knowing how long it goes on for. With your example of the road outside Otis, Massachusetts, we know that since the grade is 9%, the you will rise 9 feet every 100 feet or just over and inch in elevation for every foot the you go horizontally. Here in Albuquerque, we could find the grade of Paseo Del Norte from where it starts on Tramway and ends down past the river. Generally, railroads will have to have a lower grade than regular roads. This is because trains weigh so much more and require extra distance to stop in case of an emergency. If there was a high grade, it would be virtually impossible to stop the train.
ReplyDeleteGA2
ReplyDeleteA grade, more commonly known in mathematics as a slope, is the rise/run of a line, expressed as a percentage, which can be done by simply multiplying the rise/run by 100. If you draw a line on a sheet of graph paper, then draw and connect the lines that are the rise and the run, a triangle is formed. Then you can do something very interesting. Since you know the distance of both the rise and the run, you can find the angle of elevation by this equation:
Angle of elevation= tan-1 (rise/run)
By finding the angle of elevation, you have just found, in simpler terms, how steep a certain path or road is.
But slope and percent grade don’t just apply for math class. Take running for example. Everyone knows that running down a hill or on a flat stretch of grass is 100 times easier than running up a steep hill. In a race you have to conserve all of your energy for every uphill stretch, knowing that every uphill will be the hardest parts of your run. In running, the idea of the grade is everything.
When coming across a road that was going downhill or up hill I actually never knew that it was called a grade. I always just referred to it as the slope of the hill and the higher the percent the steeper the road. 0% is a flat surface and 100% is straight down and would indeed be suicide for anyone to drive on (or completely over). It is used to give drivers a heads up that they're coming to a really steep hill and allows them to either prepare to break if they're going downhill or prepare to step on the gas pedal to make it up the hill. Even though 9% doesn't seem like a lot I think a runner would object. Running up a 9% grade(slope) would definitely be a whole lot harder than running on a flat surface. Talking about the grade of a hill reminds me of when I go camping with my family. We go to a place called Morphy lake near Mora. Anyway, to get there you have to go up basically a mountain for about 30 minutes and it has a constant steep road. When my dad drives the mini van he is constantly having to step hard on the pedal for fear of going backward, but has to go extremely slow so he can watch out for oncoming cars (since the road is so narrow). It's always a scary drive, but knowing theres a steep hill makes it easier so its not such a big surprise.
ReplyDeleteGrade can also be defined as slope, or the ratio rise over run. The percent grade of the slope can be determined by calculating the rise over run and then multiplying by 100. The method of finding the slope relates strongly to trigonometry because of the formation of the right triangle. If the grade of something is seen as a triangle, you could use the Pythagorean theorem in order to find the rise or run. Sine, cosine, and tangent can also be used to determine the sides or angles.
ReplyDeleteI had never thought of grade describing the hills and rises that I, as a native of our beloved and ridiculously hilly Albuquerque, have gone over time and time again. I often run in my neighborhood, and I always dread the incredibly steep inclines. I can now see these hated hills as grades; I learned that one particularly brutal hill has a grade something near 20%. The number seems small, but not when I'm running up the hill like it's a flight of stairs. It is very interesting to see a way in which the trigonometry we learn in the classroom relates directly to my daily life.
Grade and slope are basically the same thing. Either of these words can be used when describing the % of a road. The incline of a road is also the same thing as the slope or grade of a road. The grade or slope of a road can be found using the formula for slope, which is rise over run, and then this multiplied by 100. This formula is the tangent of the angle of inclination. So if you take a right triangle, then you take the opposite side and then divide it by the adjacent side you have slope, then whatever that happens to be multiplied by 100 will give the % of grade. For example a road that has a 12% grade goes twelve feet vertical for every hundred feet horizontally.
ReplyDeleteWhen looking at the grade of a road, it becomes apparent that it resembles a trig function. The Grade of a road is exactly like the slope, just put in different proportions. if a road has a 10% grade, you might not think much of it especially when in a car but in reality, a 10% grade is a rather steep road. To find the grade of a road, you have to take the rise over the run and multiply it by 100. So this means that a 10% grade would be rising 1 foot for every 10 feet traveled. Still doesn't seem like much right? Well most hills around Albuquerque do not exceed a 6% grade. So what does this mean for the roads themselves? Is getting a 100% no longer what is strived for? If a road had a grade of 100% it would be at a 45 degree angle to the ground. This would be nearly impossible to drive up. This can tie back into our last discussion about uninformed students. This can cause for confusion especially with kids. They might ask their parents, "Mommy, why does this road have such a bad grade? a 6% is terrible, what did it do?" The explanation of this to a little kid would be very difficult, which would cause anger in the parent, thus creating a tumbling ball of frustration! haha This is yet another example of when math can be applied to the real world.
ReplyDeleteSlope is measured as rise over run, and grade is then measure as 100x rise over run. When driving along a road, you never see a sign letting you know the road you are about to drive on has a slope of y=2/3x+14. Instead, you see a sign letting you know the road you are about to drive on has a grade of 12%. Grade is the steepness of a hill, and this directly correlates to the tangent function-opposite over adjacent. For instance, if you picture Carlisle between Lead and Central, the grade of that road is very different than that of an open stretch of highway in northern New Mexico. To picture one of these roads as the hypotenuse to a right triangle, you can find the grade and use the trig function. The 9% grade has a very different meaning when comparing school and a hill. In school, a 9% grade can be the difference between passing a class and failing a class, but a hill with a 9% grade is not nearly as dramatic an issue.
ReplyDeleteGA2
ReplyDeleteSlope, grade, and percents. They all have different meanings with context, but in one way they all mean the same thing. Grade can be a percent, such as in a class, is the number of questions correct divided by the number of questions total. A slope is rise over run, multiplied by 100 to get a percentage. A measure of incline. And the higher the measurement, the more steep something is going to be. If a ramp inclined vertically 2 inches for every 18 inches horizontally, it has an 11% incline. In real life, however while biking or driving a car, a lower percentage/grade/slope would be better conditions for driving.
The idea of slope is directly related to tangents. Since they are angles (slopes), Tan(slope)=Opposite(rise)/adjacent(run).
I thought this idea was very interesting that slopes relate so directly to tangents.
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ReplyDeleteGrade can also be called slope, incline, gradient, pitch, or rise. Last year in Geo/Algebra 2, we became familiar with the term slope and rise over run when studying graphs and such. To find the grade of something like a hill of ramp, we find the rise/run, then multiply by one hundred to get a percent value. When we use grade in a trigonometric function, it is tangent. If we imagine a triangle, the height of the triangle is the opposite side of theta, and the bottom of the triangle is the adjacent side of theta, thus giving us rise over run or opposite over adjacent. Its also kind of funny how the term grade can apply to something such as a handicap ramp of the grade we earn on a math test. It takes a lot of effort to push something heavy up a hill with a steep slope just like it can take a lot of effort to get a good grade you want on a math test, but if you work hard, you can accomplish many great things.
ReplyDeleteI think that a 9% grade is significant. Imagine having to bike up say Paseo del Norte. This is a road that most people drive on. Its not very steep but it is also not flat. It is a hard ride up it just because you have to keep a consistent speed inorder to got up the hill. Paseo probably has a grade around 5%. So if it is hard going up paseo at 5%, then going up a 9% hill would be a bear.
ReplyDeleteThis idea of a grade reminded me of when we drive up or down to a ski resort such as Taos. The grade on the road is fairly great so you have to be prepared. I think that having those signs on the side of the road that warns the drivers really helps because you can be read for the grade of the hill.
Tangent is sine over cosine, and slope is rise over run. You can say that (if you are graphing the horizontal distance on the x-axis and the vertical change of a slope on the y-axis) rise is sine and run is cosine, which makes the grade, or slope, tangent.
ReplyDeleteI was interested to see on Wikipedia that the steepest railroad had a grade of 13.5%, and that a grade of .77% was last on the list, but still on the list. It made me wonder what the steepest railroad I've ever been on is. The thing about "Compensation for Curvature" confused me.
Also, I agree that although a 9% grade may not sound like much, it's kind of steep. I don't know what the grade of the Crest Road is, but I have tried to bike the very short distance up to Tinkertown from my house, thinking it would be effortless, and ended up going more slowly than I would've had I been walking. Coming down, though, you can go the whole way without pedaling once.