AWESOME post on the “divisibility by 3″ trick April 3, 2011

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Author Josh Rappaport had written a blog post on the divisibility by 3 trick.  If you’re not familiar with this trick, it states that by finding if the sum of the digits in a number is divisible by 3 then the number itself is divisible by 3.  For instance, the number 12,345 is divisible by 3 because 1+2+3+4+5 = 15, which is divisible by 3.  Taking the trick even further, the digits in 15 – 1+5 – add to 6, which is also divisible by 3!  Neat stuff!

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But why does this work?  I guess if I had to, and I had a bunch of time!, I may have been able to figure this out (I’d disappoint my professors if I couldn’t), but I thought I’d ask Josh for the cheat.  He wrote the best blog post I’ve read in a while on WHY this trick works… How to See Why the Divisibility Trick for 3 Works.  Check it out!

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My Harvard Math for Teaching Thesis: Complete! And ready to share… March 20, 2011

After many many years of jumping through many many hoops, I am finally graduating with my MA in Mathematics for Teaching in May.  My thesis, Negative Number Misconceptions in High School: An Intervention Using the ZeroSum Ruler is right now at the printers being printed and bound.  I don’t know about you, but that instantaneous feeling of relief after taking a final exam or passing in a final paper stopped hitting me sometime in college.  So now, I’m just feeling a bit burnt out.  OK, completely burnt out.  But I’m sure it will hit me soon since it kind of needs to; I need to now get in a post-Bach program to get my Initial teaching license.  I like to do things backwards.

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So here it is for download!  For all to read!  Or maybe to just glance.  In my study, the ZeroSum ruler proved effective in reducing eleventh grade error on integer addition and subtraction problems (especially with negative integers).  If I wasn’t so burnt out, I’d want to test it with younger kids.  Imagine how our world would be if my eleventh graders actually mastered integers when they learned them in, and only in, 7th grade.  But that’s in my thesis.]

 

ZeroSum ruler’s 62% success rate! March 9, 2011

 

The ZeroSum ruler improved my student’s understanding

of integers

by 62%

in a very short 2 weeks. 

Surpassed even my high expectations!

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HaPpY Calculating!

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Why we add fractions the way we do… a visual tour January 15, 2011

Why do we add fractions the way we do- by getting the common denominator?  A legitimate question!  The following is a visual explanation of why we need to do so…

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When we add two fractions with different denominators, we have been taught to “find the common denominator”, then add the numerators only.  But why not add the denominators too? 

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Let’s try to add 2/7 + 3/5.  We know from our standard algorithm that we would change both denominators to 35, then change the numerators by making sure to keep the ratio between the numerator and the denominator in tact.  We’d end up with 10/35 + 21/35 = 31/35. 

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But how can we see why this works?  Let’s first look at both as pictures: 

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The picture on the left represents “2 out of 7”, and the picture on the right, “3 out of 5”.

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To add means to combine, and fractions- with the exception of improper fractions- represent amounts less than one.  So, we want to combine these two shaded regions into one, or, if we can make more than one, we want to see how many “ones” we can make.  But the shaded bars aren’t the same size.  And how big is “one”?

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In the algorithm we’d “find the common denominator”, but what does this mean and look like?  It means we have to change the look of these two fractions so that their numerators represent portions of whole broken up into the same amount of pieces.  To do this, we break the picture on the left up into fifths and the picture on the right up in to sevenths. 

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Now our two fraction pictures are broken up into the same amount of pieces, and each piece is the same size. 

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To justify this, let’s look at the dimensions of each area…

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The fraction picture on the left has an area of (7×5).  The fraction picture on the right has an area of (5×7).  Because of the commutative property, we know that 7×5 = 5×7, so both fraction pictures have an equal total area (denominator), and that area is 35 spaces.  For this same reason, the shaded spaces in both pictures are also all the same size.   

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But now we have 21 (out of 35) shaded pieces in the fraction picture on the left and 10 (out of 35) shaded pieces in the fraction picture on the right.  Can we do this?  Is 21/35 the same as 3/5?  Is 10/35 the same as 2/7?

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Because we can break a “whole” up into as many pieces as we want, these fractions are equal.  For example, if two people both had a liter of soda each, one could give small cups of soda to 21 friends, and the other friend could give larger cups of soda to 10 friends.  If both empty their bottles, both gave out the same amount of soda despite giving it out to different numbers of friends. 

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It’s the same with these two fractions.  Once we have set shaded regions (3/5 and 2/7), we can break these regions up into an infinite number of pieces and still have 3/5 and 2/7.

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Now we can begin adding one to the other.

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A total of 31 spaces are filled in when we take the shaded spaces from the fraction picture on the left and add them to the picture on the right.  So, 31 out of 35 spaces are now filled in, or “31/35”.

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If we had used the standard algorithm, we would have added the numerators of the fractions (after we found the common denominators) to get this 31. 

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2/7 + 3/5 = 10/35 + 21/35 = 31/35, or 4/35 less than a whole.   

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For why we need to first find the common denominator, see two or three posts down…

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All too common math mistakes! June 16, 2010

 

One of my grad school professors called the mistake (a+b)^2 = a^2 + b^2 the “freshman dream”.  I guess he meant freshmen in college, but freshmen in high school make the same mistake.  Come to think of it, a lot of people make the same mistake.  What other mistakes are common in Algebra?

 

Here are just a few I can think of:

 

x = 1.     NO.  Well, sometimes, but definitely not always.

  

-7 + 5 = -12.  NO.  This one’s a never and the reason I developed the ZeroSum ruler.

  

(a+b)^2 = a^2 + b^2.  NOPE.  There’s a middle term in there.  Write it out.  Don’t be lazy.  Find it. 

  

1/2 = 2.  NO.  “1 out of 2″ is not 2.

  

(x + 2) + (x - 5) = x^2 – 3x - 10.  NO.  This is an addition problem, not a multiplication problem.

  

(x+2) – (x - 5) = -3.  NOPE again.  That little – sign is a -1 in disguise and needs to be distributed.

  

(2/5)(7/3) = ….<blank>….  NO.  Multiply the numerators together and the denominators together to get 14/15.  Easy!

  

“Thirty percent of 140 = 140/.3″.  NO.  You’re going to get a huge number.  “OF” means to multiply, not divide.

  

|-5| = 5, so |5| = -5.  NO.  Absolute value is always a positive number.  It represents a distance.  Even if you walk backwards, you’re still moving some distance.

  

“find x^2 if x = -5″.  Answer: -25.  NO.  Jut because the TI-83 says -25 doesn’t mean it’s right.  Calculators use PEMDAS, which states multiplication comes after exponents.  Remember a few mistakes back that the little – sign is a -1 in disguise.  It’s the same here.  If you put “-5^2″ into the calculator, the calculator will square 5 and then multiply by -1.  To square the entire -5, use ( ).  Or just remember that a negative number raised to any even power will always be positive. 

 

(x^4)(x^5) = x^20.  NO!  This is a common mistake kids make.  But if they start by thinking about what “x^4″and “x^5″ mean, it’s easy to see that (x^4)(x^5) expands to (xxxx)(xxxxx).  Parenthesis right next to each other tell us to multiply, so you end up with (xxxxxxxxx) or x^9 or x^(4+5).

 

(x^4)^5 = x^9.  No, too.  x^4 = (xxxx) and the “^5″ means you have 5 (xxxx)’s.  So (x^4)^5 expands to (xxxx)(xxxx)(xxxx)(xxxx)(xxxx).  Parenthesis right next to each other still say “multiply” so we have 20 x’s in a row, or x^20.

 

I know there are more.  What did I miss?

 

 

Dividing by Zero Blows up the Universe! June 15, 2010

“Because the universe will blow up,” was the usual answer I got when my teachers tried to explain why we couldn’t divide by zero.  From a young age, I was a sort of anti-Pythagorean in that I believed people created numbers, not that the universe was ruled by them.  So why then did we create the divide-by-zero bomb? 

 

The best way I’ve found to describe why dividing by zero will destroy everything is to go back to translating fractions.  What does “1/2” really mean?  “1/2” translates to “1 out of 2” or “I have one piece of candy out of the two pieces on the table, so I have half of what is on the table.  My sister is a good sharer.”

 

Now try this with “0/2”.  This translates to “zero out of 2” or “I have zero pieces of the two that are on the table.  My sister’s cheap!”

 

Both of these situations are real.  You can have one piece of candy out of two.  You can have none of the pieces of candy.  Even if the fraction is an improper fraction, like “3/2”, certainly you can’t have three out of two pieces of candy; this makes no sense at all.  But then we remember that improper fractions can be written into mixed fractions, so “3/2” becomes “1 and ½”, and we sure can have 1 and a half of the pieces of candy on the table [leaving our cheap sister with just ½!  Haha!]!

 

So then comes “2/0”, which would translate to “2 out of zero” or “I have two pieces of candy out of the zero that are on the table.”  HUH??  This obviously doesn’t make sense! Despite what Little Orphan Annie and Jay-Z may lead us to believe, you can’t make something out of nothing.  It’s just basic physics. 

 

Once a student begins learning about slope and functions, the impossibility of “2/0” becomes even more obvious.  Let’s think of a graph that measures your height against your age.  “2/0” represents a rise (y-value or “height”) of 2 and a run (x-value or “time”) of 0.  This is to say that, for example, at time 0 you are 2 feet tall.  Ok, so maybe you were born 2 feet tall.  That’s possible.  Now let’s move up from coordinate (0, 2).  The slope of “2/0” tells us to move up 2 and over 0.  We move up two spaces to 4 feet tall and over to… over to nothing!  We stay at zero!  So a slope of “2/0” says that you can be 2 feet and 4 feet tall at the same point in time.  This is impossible!

To the Zero! [power]

“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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