42 Folds to the Moon January 19, 2012

One of my students just can’t wrap his head around the power of exponents.  Can you blame him?  This week we learned that it would take just 27 folds of a piece of paper for the stack to reach the height of Mount Everest, and then just 15 more -a total of just 42 folds - to reach the moon.  As we started the lesson, students guessed “one million” and “47 billion!” folds to reach the moon, so you can imagine the shock (and disbelief) in the actual number 42!  Maybe the weirdest part is to think that it would take 41 of the folds to get just half-way to the moon and then just 1 more to make the second half of the journey. 

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But how can this be?  How is it possible that a thin sheet of paper easily ripped in half can reach the moon after a mere 42 folds?  Well, let’s see….

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The simple Algebra 1 exponential growth formula is:

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As the thickness of a piece of paper is roughly 0.01 centimeter, we’d fill in our equation as:

 

 

 

 

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This gives us a very large number of centimeters: (43,980,465,111).

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Dividing this number by 100 will give us the equivalent number of meters: (439,804,651),

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and dividing by 1,000 will give us the equivalent number of kilometers:  (439,804).

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For all us Americans stuck on the Imperial system, 439,804 kilometers is approximately 273,281 miles.  The moon is, on average, 238,855 miles from Earth at any given time, so 42 folds of a piece of paper will actually get us PAST the moon!

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So how small would the surface area of the top paper on the stack be?  How thin will be this paper tower to the moon?

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VERY thin!

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The exponential decay formula is almost exactly the same as the exponential growth formula except that there is a (1 – r) in place of the growth formula’s (1 + r).  To write the equation for how thin this stack of paper to the moon will be, we have to think about a funny occurrence in the stock market…

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To go from 1 to 2 is a 100% increase:  100% of 1 is added to itself to get 2.

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But to get back to 1 is a different story:  to go from 2 to 1 is a 50% decrease.  Just 50% of 2 is removed to get to 1.

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So when your stock goes down 30% on Monday, it’s not back to where it was if it goes back up 30% on Tuesday.  If your stock goes down 50% on Monday, it’s got to go up 100% on Tuesday to get back to where it was.

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Getting back on track (excuse me, not a fan of Wall Street), our decay equation would be written as:-

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This equation yields the incredibly small number: 6.37 x 10^-12 or .00000000000637 centimeters.  P was set to 28 to because a 9.5×11 sheet of paper is about 28 centimeters long.

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We didn’t do this last part in class, which is a good thing because all of a sudden I’m having a hard time wrapping my head around exponents!

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To the Zero! [power] June 15, 2010

“Any number to the zero power is equal to 1,” my teachers would say. “Why?” the students would ask. “Because.” the teachers would declare. And this would usually end it. Sometimes a persistent student would again ask, “Why?” to which he’d get the slightly more creative answer, “That’s just the way it is.” Because of the mystery surrounding the zero power as a kid, I always got the feeling that there was something supernatural about it; my teachers held the key and didn’t want to share.  

 

It wasn’t until graduate school that I finally found out why a number to the zero power always equaled one and how to explain this once mysterious phenomenon.

 

Now as a math teacher, I take the time to explain the zero power. It’s not mysterious, it’s just division. Whenever a question about the zero power comes up, I stop, go to a side board, and ask “What’s 2^3? How about 2^2?” When the kids answer “8” and “4”, I then ask, “Ok, what’s “2^1?” I create a table of their answers on the board:

 

    2^3             2^2              2^1

     8                 4                   2

 

We then step back and look at the pattern.

 

Soon the students see that the common difference between numbers is 2. “We divide by 2 to get to the next number,” they say. I then go on to ask, “Then what’s 2^0?”

 

Of course most students will answer 0 at first. I’ve realized that it’s a natural reaction to answer “zero” whenever hearing “zero” and “multiply” or “divide” within the same lesson. Other students will put 2^0 into the calculator and answer “1”, but that’s what I did as a student and it wasn’t good enough. I always wanted to know why.

 

I direct all students back to the table we created to find the next term “2^0”.

 

    2^3             2^2              2^1             2^0

      8                  4                 2                ??

 

“If we divide by 2 each time to get to the next term, what’s 2 divided by 2?” I ask. The students will answer, with an “oh, obviously!” tone, “1.”

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Another way to think of this concept is through exponent rules:

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To divide terms with exponets, we subtract the exponents, here giving us a 0 exponent.  If we then back up and look at the original fraction, we have a number over itself, which is equal to 1.  Therefore, any number to the 0 power is equal to 1.

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calculators KILL negatives! (uh, raised to even exponents, that is:) May 17, 2010

 

What’s negative 2 to the fourth power?  16?  -16?  If you put “-2^4″ into the TI-83, you get -16.  But we know that (-2)(-2) = 4 and (-2)(-2) = 4, and (4)(4) = 16.  So why does the calculator give us -16?

 

This post is no doubt for the high schooler and not for someone addicted to the )( buttons on the calculator like I am.  I parenthesize.  It comes from a fear that something will go negative that should be positive.  I have reminded my students more times than I can count to parenthesize, so many times, in fact, that I am more than sure that most tune me out as soon as they hear the first syllable.  But still the negative raised to an even number sneaks past the best of ‘em.

 

The evil negative base reared its ugly head again today when I graded papers on the geometric sequence an = a1 • r^(n-1) where:

an = the value of the nth term

a1 = first term’s value

r = ratio of change (ie “doubling” would be 2)

n = the terms placement (ie: 5th term would be n = 5)

 

“Find a7 if a1 = 5 and r = -2.”  The answer I or course got more than the correct answer was ” -320″.  What should the answer be?  “320″.  The problem should be written out first as: 5(-2)^(7-1) to make the process clear.

 

At least no one gave -1,000,000 as an answer.  There’s still hope!