Fractions in the Fast Lane Response

GA2: Fractions in the Fast Lane: July 17, 2013, published August 19, 2013

"How did you happen to be so good with fractions," friends used to ask when I was in middle school.  Everybody knows everybody universally dislikes fractions.  For me, it was all about distance swimming.  I knew I could solve the world's problems during a long workout (though I'd forget the solutions to the world's most serious problems as I climbed out of the water); what I didn't know was how I was using swimming to solidify my working facility with fractions.  It was simple: as I swam 1,000 meters, I was constantly figuring out what fractions -- and what ratios were identical to the reduced fractions -- could represent how far I had swum and how much further I had to swim before I finished.  It started simply: if I swam 40 lengths in a 25 meter pool, then after 7 lengths, I was 7/40th done and had 33/40 to go.

Sometimes, however, I swam in the 20 meter YMCA pool and the numbers became different. I needed to focus and not just rely on memory.  I now had to swim 50 lengths to complete 1,000 meters.

My thinking soon became more complicated and required swifter calculations -- I moved to measuring what fraction of the swim I had completed for each stroke -- or even each partial stroke.

Bored with that, I began watching my teammates swimming in the neighboring lanes.  What were their ratios and how were their numbers different from mine?  At what points would we pass each other?

I long since moved away from my home town, stopped swimming, and became a math teacher. I forgot about fractions in the fast lane.

Then deep into middle age, I started swimming again.  And calculating fractions.  I kept this secret lest my lane-mates think me insane.

I don't always swim in pools.  There are lakes with cool fresh water, sunbeams that cut through the waves, and no visible bottoms. Plants grow through the water towards the source of the sunbeams, branching in infinitely smaller "Y" shapes at the same angles.  Bubbles surface and break into more and smaller bubbles from the depths; there are no numbers. Only fractals.  And chaos. And new things to think about.

Your blog: where do you use math in secret?  Or if you don't use math in secret, where might you start using math in secret or not in secret so you can increase your skills in math?

                                                                                                                        

I use math when I am doing my homework. I use fractions just as you do, to see how much work I have completed and how much I still have left to work on. I like to calculate these fractions into percentages because I feel like I get a better understanding of how much I have done and what is left. This helps me get a better view on how I can reach 100% rather than a number out of a number. I think this helps me push myself to get things done because I think to myself, I can do it just this more percentage to go until I reach 100%.
A place I use math in secret is when I am helping my mom put away grocery bags. I count the bags quickly when bringing them in from the car and when I start to put away the bags I get a fraction of how many bags I have put away and how many are left to go. When I first start putting away bags it is tedious because I know this will take a while until I finish. Once I start getting a higher fraction or percentage, I feel relived because I will soon be finishing.
When I use math in secret, I feel like the idea of fractions and percentages help me because they set a goal for me. It helps me get through tedious work. This goal is kind of like a way to keep on track too.I think when my brain is trying to figure out the fractions and percentages it distracts me from work and I will always try to work to a 100%.

9% Response

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.” 

http://en.wikipedia.org/wiki/Grade_(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.

                                                                                                                                                                                                                         

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.