GA2: Beauty and Geometry Response

GA2: Beauty and Geometry

 

I was just reading about the mathematician Tom Zhang and his fascination with "twin primes."

Please read this interview with this brilliant mathematician and consider at least one of the many things he's saying.  Choose something about his views of mathematics and write about it.

To get your creative juices flowing, a couple thoughts I had about the interview include (but are not restricted to):

1. The idea that mathematicians are born, not made.

2.  He views math as beautiful and interesting not at all for the application. He loves math for itself and for the way mathematics helps him use his mind.

3. I remembered something I read, written by Harold Jacobs in his Geometry text book from 1974:

Pythagoras was a Greek geometer who lived about 2500 years ago.  He wondered whether he could teach geometry even to a reluctant student.  After finding such a student, Pythagoras agreed to pay him an obel for each theorem he learned.  Because the student was very poor,  he worked diligently.  After a time, however, the student realized that he had become more interested in geometry than in the money he was accumulating.  In fact, he became so intrigued with his studies that he begged Pythagoras to go faster, now offering to pay him back an obel for each new theorem.  Eventually, Pythagoras got all of his money back. 

Now ok, a bunch of you admitted that you do some math in secret or in ways and times that you didn't think you were actually doing math. A bunch of you claimed that over use of technology contributes to math illiteracy.  Will any of you to admit that there's something in math --anywhere -- that you've found lovely, beautiful, cool, interesting, intriguing, puzzling, worth thinking about, or simply fun?  Oh, do share!

                                                                                                                                                                   

While I was reading the interview, I noticed that Tom Zhang isn't very talkative but he is a very focused person. He is so quiet because he is thinking. During the questions of the interview, he doesn't ramble on about the questions, he gives a concise answer; a simple yes or no if he can. I thought it was interesting how he didn't show his work to his colleagues because they hardly understood the problem and he wanted to wait until he was done and had finalized a result. This also leads onto where when he finally found an answer he showed no emotion. I found this interesting because I think he just felt relief sweep over him, not being excited or anything; just closure to what he has been working on for so long.

Because Tom Zhang is so quiet and shy, he becomes really focused about his problem. I found this to be a reason to why he would never give up, and just keep trying.It was also interesting that his breaks he had between the solving the problem were listening to classical music and reading novels.He didn't like to watch TV. It felt like Tom Zhang just kept calm during his whole experience and had his ways to stay calm. That is what helped him push through. I like how he says he is famous, but he seems so down-to-earth about it. His personality of being quiet, shy, focused, and calm routed him to success.

Overall, I got the feeling from this interview of Tom Zhang that if you stay focused, determined, and calm you can solve any problem.

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.

"Why are We So Illiterate in Mathematics?" Response

Why are We So Illiterate in Mathematics? August 22, 2013

And why does our common culture perpetuate the illiteracy?  As a math teacher, it makes me crazy to see our common culture supporting bizarre impressions of numbers and shapes and to see how students, made victims of these notions, can sometimes struggle with what used to be the most basic mathematical ideas.

Take, for example, the ice "cube." A cube is a regular hexahedron: a polyhedron that has six congruent faces, each of which is a square (think dice).  I don't know about you all, but the freezer of my youth held rectangular plastic pieces that froze water in the shape of roughly hexahedrons, with all edges that are line segments that are either parallel or perpendicular to each other.  Now, mind you, my own freezer mocks me in my passion for mathematics by producing a solid with two edges that are arcs and  two pairs of edges that are parallel line segments; these solids are definitely not cubes. Therefore it is no wonder that current students are confused by the word "cube."

Also take the coach, well meaning that she or he is, encouraging players to give 120% (or even a larger percentage) effort.  How does that happen?  When I fill my glass to the brim, it is 100% full. When I try to pour more water (or orange juice) into the glass, it overflows. I cannot add more than 100%.  Thus it is with athletics. How can you possibly give more than all you have?  No wonder students struggle with the concept of percentages in elementary school or middle school or in high school with the idea of probabilities summing to 1 or 100%.

My son, as a young child, participated in many sports, but particularly loved basketball.  He played in a pee-wee kind of league in Colorado Springs that played wonderful games, but clearly someone in the program wasn't a math teacher.  The teams played 3 quarters in a game.  Yes, instead of playing 4 quarters to make a whole game, they played 3 twenty minute sessions that they called "quarters" and the game was meant to be over.   When I asked the ref when we'd finish the game as we had only finished 3/4 of the game, he looked at ME as if I was the crazy one.  If they played 5 twenty minute sessions, I don't think they would have liked to play fifths.  That word has other implications.

Now ok, I'm contemplating these kind of challenges I face each day in the classroom as I walk my dog, Karma (as in "good Karma") around my neighborhood in Albuquerque and I start noticing the numbers on mailboxes.  I live at 1198.  To our left as we face the street live our wonderful neighbors at 1196.  Across the street from them is 1197; order is maintained with the folks across the street from us: 1199.  But much to my consternation, the wonderful folks to our right live at not 1200; they live at 11100.  Is this a problem unique to Albuquerque? 

I enjoy a good math joke as well as any other football player; people recognize this as they link comics, cartoons, and what-not to me on facebook; I do delight in the humorous additions to my day.  But I squirm at the one that had one triangle talking to another triangle: "You are so obtuse, you wouldn't know an isosceles triangle if it bit you in the hypotenuse."  Now stop right there.  The cartoon has a cute one-liner and anthropomorphizes triangles, but only a right triangle (and not an obtuse triangle) has a hypotenuse.

So perhaps I need more to do with my life so I'm not so concerned with such trivia, but I already fill 100% of my time between the triangle of 1198, math classes, and a tonic with ice cubes. Real ones, mind you.

For your blogs, folks, how about you have a choice.  1. you write on your theory about why U.S. Citizens aren't very savvy mathematically or 2.  you find some mathy idea to write about that interests you. Remember to submit it to canvas. Remember to make it your own; you are not to copy something from somewhere else. 

                                                                                                                                              

I definitely think that U.S. citizens are not the best at math. I think this because the technology we have been greeted with makes us start to become lazy. We have become too dependent on the technology and use the calculator on our smartphone for simple addition and subtraction problem. This makes our minds start to become less engaged in math because we think "why do something that takes energy and more time when something else can do it for you in a lesser amount of time". This is how most of America is starting to think with the higher technology we are being introduced to so we have become lesser found of math. 

In the street numbers I also don't understand how they work. I get confused and it makes me scared that if I want to drive to a friend's house I've never been to how will I find it without using a gps or maps on a smartphone? This also shows how technology just makes math more confusing and people start to lack on the ability. We depend on the smartphone to do things for us again. 

I also find it strange that the "quarters" of the basketball game was in the reality of math thirds. I think the word "quarters" has adapted to being just "parts" of the game because that is usually how a sport game is timed. Quarter was just used to make it sound "official" because who would want to call it thirds? This shows how some Americans just aren't that into the real math of things. It makes us become less aware of the math of what is happening around us.

Ms. Mariner's Blog on JAMM'n Peaches Response (Q1, Homework1)

JAMM'n Peaches: May 2013

We lived in Grand Junction, Colorado, at the juncture of the Grand and Colorado Rivers.  Just East of town is the town of Palisade; just west of town is Fruita.

Our rental house, about 1500 square feet of damp stucco and thin pine floorboards from the 1920's, sported a flower garden.  When I moved in, I didn't pay much attention to plants; after all, I killed most of the houseplants that I had ever owned.  First bloomed the Peonies.  Each large lacy fluffy light pink billow was so fragrant that the scent of one in a simple spherical bowl would fill the house.  Different flowers bloomed in succession until the roses, the many roses, bloomed. The wirey and pokey stalks of plumes spread across the yard in wild wonder, blissfully taking over all the play area.  Our neighbor, Barbara, a proud nonagenarian, leaned over the fence; "I have some clippers you could borrow for those rose bushes," she said.  I smiled, knowing that if I did anything to these plants then surely they would die.

One morning, I found the clippers and some gloves on a table in our back yard.  I began using the clippers more like a sythe on the rose bushes so the size of our yard would be increased, allowing for more play space. Later that day, I returned the clippers to Barbara and thanked her. She smiled and gently said, "You know, if you cut the roses at an angle five leaves below the last bloom, then your roses would blossom again."  I smiled back, knowing that I could not have cared less about the roses.

It was only 2 weeks later when I found myself cutting the roses as she had recommended, and not even a month later when our world became filled with Elizabeth, Sunrise, Blood Orange, and Pink Beauty roses.

And so it went with peaches.  Bushels appeared on my porch from well-meaning friends.  "Surplus," they said, "From the orchards in Palisade and Fruita. Can them."  I smiled, knowing that I didn't like peaches and liked cooking even less. Clearly, the peaches would rot.

It was barely 10 days before the lids of my first set of Jammin' Peaches began making friendly popping noises as the peaches cooled inside the pint glass jars.

But it is the smell of the roses and the thick air of boiling peaches that keeps me returning, in my mind, to our damp stucco home in Grand Junction. It is the smell of surprise, of change, of adaptation and the smell of comfort.

GA2
I really like how the dynamic of the cutting the roses at an angle five leaves after the last bloom and how it made a difference in the rose bushes. It made them look more elegant and fuller. I like how this idea of roses lead to the peaches. This blog post shows a good idea of how one idea can lead to another even when you don't really enjoy doing what it is, but it still comes out to a great product just like in math class.

I approach mathematics in a simple way. I try not to think to hard about it or I become frustrated. I use my notes to help me also. I like math, it is sometimes challenging and easy. I like math because I like the puzzles it brings forth to my everyday life. I learn math the best by doing the beginning of the chapter/sub-chapter on the board in class while taking notes. My best math experience is whenever I receive a good grade on my math tests, An effective learning tactic for me is taking notes while a teacher is writing down key parts of the chapter/sub-chapter in the classroom.