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 41 of the folds to get just half-way to the moon and then just 1 more to make the second half og 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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Top 10 Myths about Math Education December 29, 2011

The following article and its 10 myths about Math education were posted on the Gideon Learning Blog.  From inquiry being the catch-all to memorization being a dirty word, this article hits all the kinks in the way we teach, or at least how we’re told to teach (see Myth #10), math today.  You can see the full article by clicking on the picture or by following the link underneath.  p.s. It’s not just the reps of the Boston Teacher’s Union who roll their eyes at the mere mention of TERC (see Myth #2)….

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http://gideonlearning.wordpress.com/2011/12/28/ten-myths-about-math-education-and-why-you-shouldnt-believe-them/

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If Pigeons can do it, so can you December 24, 2011

Here is a fascinating article on the mathematical intelligence of…. pigeons. 

(to read full article, click on the pigeons)

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http://news.discovery.com/animals/pigeons-math-animals-111222.html

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The Distributive Property (“F.O.I.L.ing”)Through Pictures December 15, 2011

The transitive property was always my favorite as it could be applied to so many situations.  I like chocolate, there is chocolate in those cookies, so I like those cookies.  Totally useful.

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But a close runner up to this cookie property has got to be the Distributive Property.  With strange rules of “first, outer, inner, last”, I liked its mystery.  I could multiply two things together with no mention of a multiplication sign and somehow it meant something.  Something big.  I was doing real Algebra now.

It wasn’t until I became a teacher that I really had to think about what was being done.  My students would make mistakes when “F.O.I.L.ing” (I do not like this acronym.  What if one piece is a trinomial?) and I would attempt to explain what was happening.  It’s difficult to explain something that has been taken for granted for 15 years.  But as I made my way through my graduate program where being able to explain math was seen as the most important, I began to rethink this important property.

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The Example:

I always like to start with a concrete example.  Let’s take the problem “14 x 7”

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“14 x 7” is no easy problem for most of us as neither of these numbers is easy to work with.  To begin, let’s look at “14 x 7” as a geometric area in a picture:We can easily count up the small rectangles to find how many there are, though that would take time and leaves a lot of room for error.  Or, we could break the picture down into smaller pictures to make it easier to work with:

Here, we’ve broken “14 x 7” down into (10 + 4) x (5 + 2), or simply (10+4)(5+2).  Is this form familiar?

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Now we can see that “14 x 7” = (10 + 4)(5 + 2).  And now we can simply use multiplication find the areas of the different colored pieces and add them up:

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10 x 5 = 50

10 x 2 = 20

4 x 5 = 20

4 x 2 = 8

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50 + 20 + 20 + 8 = 98!  And in fact, 14 x 7 = 98.

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The Generalization:

Now let’s make a generalization that we can apply to other similar problems:

Here, we’ve replaced all of the numbers with letters and we can rewrite the problem as:

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(a + b)(c + d)

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Using the method we used before, we multiply each colored piece to find its area and then add up all the areas to find the total:

 

(a)  x (c) = ac

(a)  x (d) = ad

(b)  x (c) = bc

(c)   x (d) = bd

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The area is: ac + ad + bc + bd  !

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Not the prettiest of answers, but done correctly.  Using this model, can you multiply (3x + 4)(5x + 2)?

We’ll use the same picture because “x” can stand for any number at all.

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We have:

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(3x + 4)(5x + 2)

(3x)(5x) = 15x²

(3x)(2) = 6x

(4)(5x) = 20x

(4)(2) = 8

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Putting the pieces together, we have the trinomial:

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15x² + 26x + 8 !

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The Error:

The biggest error I have seen with the Distribute Property is forgetting to multiply a piece or two.  Students sometimes will answer:

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(3x + 4)(5x + 2) = 15x² + 20x + 8

Can you see what they forgot?  Can you imagine what other mistakes could be made?

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If you always remember the area of each piece, you will be The Best Distributor and Master of the Distributive Property!

 

 

 

 

 

The ZeroSum Ruler… free November 29, 2011

(Construction instructions can be found in the original ZeroSum ruler ebook (also free:))

How Observant Are You? November 28, 2011

How many times do the people in white pass the ball? 

The answer is somewhere between 13 and 19.

How About Better Parents? November 20, 2011

Read the entire article at http://www.nytimes.com/2011/11/20/opinion/sunday/friedman-how-about-better-parents.html?_r=1&ref=thomaslfriedman  or by clicking the article above.

Surface Area to Volume Animation September 26, 2011

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The following [totally awesome] video comes courtesy of Dr. Brian Biswell.

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The Language of Math Poster August 19, 2011

Below is a poster I hang in my classroom every fall.  Each year it grows longer as more and more terms come up for the different operations of math.  When I was a kid, no one told me to look out for these words, or that math was even a language at all, which made word problems pretty tough.  By clicking on the poster you will be sent to the original Excel file on Google Docs.  Do you have any words to add?

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Link to Google doc:

https://spreadsheets.google.com/spreadsheet/ccc?key=0Asra4GjkRBNidGhoZlZYcjk4dmhISDlSNHJDbjBPTXc&hl=en_US

Audit of the Fed reveals $16 trillion in secret bailouts August 10, 2011

http://www.unelected.org/audit-of-the-federal-reserve-reveals-16-trillion-in-secret-bailouts

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Windows 7 users: What to do if Windows won’t update… July 31, 2011

Windows hadn’t updated for me in a while.  My computer would try, but would fail each time with Error Code 80070005.  It’s now finally working… installing 21 updates.  The very first update was a new virus scanner something or other, which I am guessing was the reason behind the glitch.  I know this is a bit off topic – though some math can’t be done these days without a computer - but I thought it was important to let everyone know.  I hope I am lucky enough to have avoided any major damage to my system, and I hope this helps you too if you need it.  -

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Here are the simple directions that I found online and worked for me.  I was a bit skeptical about the “renaming the folder” part, but my Windows is now finally updating so I am a believer.

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1: From your start menu, type services.msc in the search bar.

2: Find “Windows Update”, right click and choose STOP.

3: From your desktop ”My Computer” icon, search within C:// to find a folder named SoftwareDistribution.  (This file may be hidden.  If it is, go to folder options  and choose “see hidden files”).

4: Rename this folder SoftwareDistributionOLD. 

5: Create a new folder named SoftwareDistribution within the same directory.

6: Back in your Start menu, retype services.msc.

7: Right click on Windows Update and select START.

8: Restart your computer.  Windows updates should start automatically.  If not, type Windows Update into your Start menu search bar, type Windows Updates and change the settings to start automatically (recommended settings).

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I hope this saves at least one computer! 

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Can these World Statistics be real? July 27, 2011

Below is a screenshot from the website http://www.worldometers.info/.  I’m not yet sure what to make of its flood of numbers.

Geometric Transformations (video) July 18, 2011

Reducing fractions: One cookie = two cookies? July 11, 2011

Any kid will tell you that eating one of two cookies is not the same as eating two of four cookies.  In the first case, you only get to eat one cookie and in the second case, you get to eat two!  Yet in math, we are told that 1/2 is equal to 2/4.  How can this be?

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First, we have to be able to read fractions to understand them.  In other words, we have to remember that fractions are a sort of shorthand for longer phrases.  For instance, let’s take 1/2.

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1/2 can mean:

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“one out of two”

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“one divided by two”

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“one out of every two”

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“one for every two“

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Writing “1/2″ is so much faster than writing any of the above phrases.  And when we understand this, and that mathematicians often use abbreviations, we can begin to think about what “1/2” really is:

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Here’s two cookies:

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And here’s one out of two cookies:

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We took “one out of two cookies, or “1/2″ and showed the fraction “1/2″ with cookies!  This seems obvious, but may be a little misleading.  In our above example, it seems as though the numerator (1) represents the number of cookies we take and the denominator (2) represents the total number of cookies.  And in a way this is true!  But let’s look at one more example…

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Here we have four cookies…-

And here we take two of them…

We’ve taken two out of four cookies, or “2/4″.  We’re told that 2/4 is the same as “1/2″, but how?  Let’s remember our phrases.  “1/2″ can also be read as “one out of every 2“, and in fact we have taken one cookie out of every two on the table.  We can begin to see how 1/2 = 2/4. 

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Income and Debt with ZeroSum Ruler July 10, 2011

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One of the ZeroSum ruler’s main purposes is to calculate debt/income problems.   In the problem “I owe you $12 and pay you back just $7. How much do I still owe you?” how do you come to your answer?   Do you count backwards from $12 to $7?   Or do you count forwards from $7 to $12?   No really, how much do I owe you?   How did you figure this out?    The ZeroSum ruler allows the student to count forwards instead of backwards just like we do in real life!    So why do we make our kids count backwards in school?

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