Quite by accident, which is how my life usually occurs these days, while tutoring at the community college, I went to an algebra site to check something. I don’t remember what it was, but when I tutor, I frequently encounter problems I remember but don’t recall exactly how to solve. I understand ellipses and hyperbolas, but I forget how to find the foci or the latus rectum. I have to look it up.
While on the site, I discovered the solutions were posted by volunteers, so when I had a quiet stretch, I gave myself a user name and logged in, solving a few problems that afternoon. I found it relaxing, which I am sure would surprise many for whom math is an odious chore. What I have learned, besides hyperbolas and ellipses, was more than math itself. Those who do not like math might read on, for you will be surprised. Those who do like math will likely shake their heads in agreement.
The first lesson comes early in Ringo’s and George’s lyrics: “You’ve got to pay your dues, if you want to sing the blues.” MATH TAKES PRACTICE, just like the piano. I practiced the piano an hour a day and took lessons for three years. I got better. Oh, I never got past a couple of recitals, where a dozen of us played solo to our parents and a few others. Wow, I was nervous. But I did fine. I played “By the Sea,” which I had memorized. I played it well and everybody clapped. I never thought I had musical talent, and to be sure, I don’t have much. But I could play the piano; I could read music and even change it into different keys.
I think latent math talent exists, too, but one has to follow the guidelines, of which practice is the most important. Practice allows one to solve problems, but it has a bigger advantage. When one needs such math in the future, while it may have been forgotten, it returns quickly. I never forgot the slope of a line. I did forget the point slope formula and quickly relearned it. I forgot how to integrate by parts, but I re-learned it enough to astound a few people in graduate school, 30 years later, when I blurted out the integral of log x one autumn afternoon in Las Cruces.
Sit with me as I tackle online a routine problem. Routine problems are ones I can do without pencil or paper. People submit them to get help. Watch my thinking, but more importantly, WATCH HOW I MAKE MISTAKES.
Joel and Nicole each together have 350 coins. When Joel gives away half of his and Nicole a third of hers, they now have the same number of coins. How many did they start with?
I love mixture problems; I’ve never had to review them. It’s sort of like a guitarist who learned “Don’t Think Twice” in the 60s, never played it since, and tries to play it at a gathering. He may not tune the instrument quite right, and he gets a few chords wrong, but he plays the song, and it is appreciated. Math is intertwined with music; an eighth note is held twice as long as a sixteenth.
I let Joel’s coins = x and Nicole’s = y. I could let Nicole’s equal 350-x, a trick I use, if I choose to use only one variable. Musicians have tricks when they play, too. They put a song in D major, rather than in D. They invent stuff. I have in math, too. Back to Joel and Nicole.
x + y =350. That is a fact. Translate: Joel and Nicole together have 350 coins.
Joel gives away half his coins. He has half left, (1/2)x. I put parentheses around the numbers, because 1/2x is not the same. Hey, you play in E minor on a piano, you may touch the D major key, same one, but it doesn’t sound the same. (1/2)x is not the same as 1/2x. We’re no different in math.
Nicole gives away 1/3 of her coins, so I first write she has (1/3)x. I am not correct, and when I later check the problem, it isn’t right. I return to the beginning, BECAUSE SOMETHING ISN’T RIGHT. I don’t convince myself it is right, I don’t dictate it is right. It is NOT RIGHT. I have an open mind and start over, asking WHERE DID I GO WRONG? A lot of politicians ought to ask themselves this question. The guitarist knows when it doesn’t sound right, too, asking himself where he went wrong. I and the guitarist are on the same wavelength. WE KNOW IT JUST ISN’T RIGHT. Oh, I discover, Nicole has (2/3)s of her coins left, not (1/3). What was I thinking?
Can you see that somebody like me, good in math, makes a simple mistake? If you aren’t good at math, did you ever realize how many mistakes mathematicians make? We make them all the time!!!
OK, so (1/2)x=(2/3)y . Now, there are at least three different ways to solve this, but I’m not going to play the song in 3 different keys, just one. I’m lazy, and I like my math simple. If I double (1/2)x, I get x. If I double (2/3)y, I get (4/3)y.
x=(4/3)y, I like this. It feels right, just like hitting the proper chord feels right. You sense it. We’re brothers here. The sense is well known in sports, where it is called “the zone”: Bill Bradley was once interviewed during practice. He made a 20 foot hook shot while talking: “You have a sense where you are.”
Now instead of x+y= 350, I have (4/3)y + y=350. One variable. But y = (3/3)y. You’d be amazed how often we math guys multiply by 1, which doesn’t change anything. Not only that, we multiply by really strange “1”. Here, it is (3/3). Sometimes, it is (√7 + 2/)(√7 +2). That is also 1. Even stranger, we may add 0, because it doesn’t change them. Crazy. Until we add 36-36 to an equation that has (x2+12x), allowing us to write (x+6)2 – 36. Both of those are equivalent, but we can do things with the second that we can’t with the first. I now add the y’s: (4/3)y + (3/3)y =(7/3)y, which equals 350. I flip the fraction over, because (1/2)(2/1)=1 and (7/3)(3/7)=1, and I want 1y or just y on the left. I must multiply the right by (3/7), too, and without boring anybody, y=150.
I make another mistake, a rookie one. I usually solve for x, but if Joel had 150 coins, Nicole had 200, and one can’t divide 200 evenly by 3. I WAS WRONG. Two minutes later, I realized I had solved for y, and when I went to the top, I had CLEARLY WRITTEN BUT FORGOTTEN that y was Nicole’s. She had 150; that checked just fine.
This is a simple example to a math guy. We make many mistakes. All of us. We copy the problem wrong, we forget a minus sign, we add wrong. Yeah, that too. I’m just a guy who plays with math to relax, hitting a lot of wrong notes along the way. Like the guitarist, I make good music, but “You Know, it Don’t Come Easy.”
I paid my dues.
Tags: MATH AND MIKE
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