TPC and GA2: Human Calculator Skills Response

TPC and GA2: Human Calculator Skills

 

Yesterday in class, Elijah Ray made a swift calculation in his head.  Occasionally, someone in class blurts out the product of a couple of two-digit numbers with lovely speed.

When can YOU do this?  Under what circumstances can you multiply swiftly?  Do you ever make a (let us all pause here) geometric illustration of your calculation in your head?  Can you picture something with, say, algebra tiles?  Let me illustrate.

 Let the following illustration of a rectangle represent the product of 43 and 42. Let the horizontal represent 40+3 and the vertical represent 40+2.   The blue square represents 40x40; the green bars each have dimension 1 x 40; the yellow squares each represent a 1 x 1 unit. 

 

 One (of three) options for your blog this time is to relate this idea to areas.  Demonstrate how these tiles support or illustrate the traditional way to multiply these two numbers together:

 

 

...or how the tiles support FOILing:




But there's more. I'd like to draw your attention to two modern-day mathematical geniuses  from India. 

One woman, Shakuntala Devi is known as "The Human Calculator" for her ability to calculate products of large numbers in her head very swiftly.  From the on-line version of the India Times:
Shakuntala Devi once competed against a computer to see who could come up with the cube root of a 9 digit number first and she defeated the computer at this challenge. The same year, in 1977, Shakuntala Devi was asked to give the 23rd root of a 201-digit number; she answered in 50 seconds! On June 18, 1980, she demonstrated the multiplication of two 13-digit numbers 7,686,369,774,870 × 2,465,099,745,779 picked at random by the Computer Department of Imperial College. The correct answer was presented by her in just 28 seconds!

I've always loved stories of Srinivasa Ramanujan. One of my favorite stories is the following, fromDurango Bill's site:
    If you mention the number “1729” or the phrase “Taxicab Problem” to any mathematician, it will immediately bring up the subject of the self-taught Indian mathematical genius Srinivasa Ramanujan. When Ramanujan was dying of tuberculosis in a hospital, G. H. Hardy would frequently visit him. It was on one of these visits that the following occurred according to C. P. Snow.

   “Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’”

A second option for your blog this time is to search the web for yet another "human calculator." Have your mathematician be from relatively modern times (the last 100 years) and write something about him or her.  Include your internet sources in your blog.

A third option for your blog this time is to find an interesting story about one of these two folks that I identified above that are NOT on the sites I've provided for you.  Also be sure to include your internet sources in your blog.

                                                                                                                                                                  

Scott Flansburg hold the Guinness World Record for the "Human Calculator". Flansburg has been teaching for more than 20 years. He is able to subtract, add, multiply, take the square and cube roots of numbers in a short amount of time. He said the only that holds him back is that he cannot speak as fast as he gets the answer. I think it is really interesting that he can do math in his head faster than professionals on calculators. He has appeared on many talk shows such as Oprah, The Ellen DeGeneres Show and ABC’s Good Morning America.

The Human Calculator reminds me of when I was watching Live! with Kelly and Michael when a teen came onto the show. He was wearing a math Olympiad shirt and he solved mathematical questions against Kelly and Michael. he answers the questions way faster than than the two hosts. The questions aren't as complicated and complex as the ones Scott Flansburg does, but he is still impressive. I am unable to find a link or video on him. I just remember sitting on the couch with my mom trying to see how fast I could solve the problems.

The Human Calculator also reminds me of when I was in about 4th grade we did a race against another student on a calculator. It was simple multiplication to 12 x 12. Almost all the students were faster than the calculator. Once you commit the math to memory it becomes easier and faster to yourself.

Sources:
http://scottflansburg.com/

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.