Selling Math To Students – number tricks, cell phones and fractals October 22, 2012

Do you have a number trick you like?  I’d love if you posted it in the comments below!  Also see Super Cool Math Tricks.

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To begin each new school year, I like to sell Math a little.  So each fall I like start the year with my favorite number trick:

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1: Have a student choose an item: pen(3), book (4), board (5), or eraser (6).  You can also do this with fruit of different word lengths or anything else, really.  The important thing is that the words are all different lengths.

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2: Have the student:

a: Turn the word into its number

b: Multiply by 5

c: Add 3

d: Double the number

e: Add in favorite single digit

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3: Now ask for the number. 

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To “undo” the trick, simply subtract 6 (in your head is most impressive).  The first digit in the number you now have corresponds to the item the student chose.  The second digit is the student’s favorite number.

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The debunk:

Let’s say we chose “eraser”.  We’ll use E for “eraser”:

5E

Let’s say my favorite number is 3:

5E + 3

Now I double it:

2(5E + 3) = 10E + 6

Add in my favorite digit “n”:

10E + 6 + n

The 10 shifts the “eraser” (E) number to the tens placement and the favorite digit is in the ones spot.

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If this doesn’t work (it usually does), then I move to more drastic measures.  This Prezi has embedded clips from Nova: Hunting the Hidden Dimension because it is the most awesomest Nova special (or any type of special) ever.  The kids enjoyed this intro to their school year:

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The Math Book by Clifford Pickover is awesome! July 11, 2012

“Have you read The Math Book?”

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“Which math book?”

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“The Math Book?”

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“Umm, ??”

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“There’s a The Science Book, too!”

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#*%^&!!!!!

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No really!  There IS a book called The Math Book  and it’s by the fascinating and mind-boggling book-producing author Clifford A. Pickover, who you can even follow on Twitter! 

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Before I was introduced to this book, I would have been the red converser in the above dialogue.  I mean, surely the title The Math Book was taken back in the stone ages.  How could there possibly also be a The Science Book (and a The Physics Book and a The Medical Book, etc.) in modern times?

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I’m not sure how it happened, just that it happened and that it’s awesome.  The Math Book was published in 2009 and has a cooler format than just about any other math book I have ever cracked open (possibly tied with Glencoe’s high school math textbooks, but those are textbooks so are in a different category altogether) and certainly cooler than any other book about math I have ever read.  As you turn each page, you are surprised by yet another full-page picture relating to the previous page’s concise summary of some even stranger mathematical topic.   Here we can see the page on Zeno’s Paradox, a mathematical problem that has plagued mathematicians for centuries.  Did you know that Math states you’ll never make it out of your bedroom door?  If you get up to leave (dinner’s ready!), at some point you’ll be halfway between your computer chair and the door.  At some point later you will be halfway between the halfway point and the door.  You journey to the door is marked by an endless series of halfway points, and you can always take half of whatever distance you have left to travel (think the molecular level… and beyond).  Hope you stashed snacks!

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Ant stride lengths, cicada life cycles, spirals, knots and Murphy’s Law (doh!), fractals as used in animation, Cryptography, magic squares, slide rules, imaginary numbers, Möbius strips, Zero, monkeys typing on keyboards, monsters!, and dimensions far past our comfortable 3 are just some of the many topics, broken down by year, surveyed in this wonderful book.  If you’re a math teacher of students who often ask for extra credit, this book is a great jump-off.  The kids can look through the book for something that catches their fancy and do additional research until their heart’s – and page number requirement’s - content.   This is a truly awesome book for all ages.

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contact blog author Shana Donohue: shanadonohue@gmail.com

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Archimedes says “cone + sphere = cylinder”? April 14, 2012

Archimedes of Syracuse (87 BC – 212 BC), before the eternal time sucks of Facebook and DrawSomething, sat and imagined a cone and a sphere balancing one side of a scale, and on the other side balancing against these two objects was a cylinder.  The sphere he imagined has the same diameter as height of the cylinder and the cone and he reasoned that the scale would remain perfectly balanced.

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But would it?  Using just their volume formulas, which Archimedes did not yet have, can we see that this works? 

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Does sphere + cone = cylinder?

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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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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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Super Cool Math Tricks: Be a Human Computer! June 29, 2011

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Also see Selling Math to Students – number tricks, cell phones and fractals

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