Manny Ramirez adds his fractions… January 7, 2011

Usually when adding fractions, we never ever ever ever ever add the denominators together.  That is, except for in baseball.  In a season of baseball, a “whole” is the entire season of at bats, not any one individual game. We won’t know what that whole is until the end of the season, so we keep adding the at bats (denominator), and tallying the numerator (hits), to find how many hits per at bats Manny has at any point in time during the season.  Weird, right?  But true!

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Scenario 1: Manny Ramirez’s batting average is 5/7 (ie .714, “Batting a 714!”, WOW!  Go Manny!) after two games: one game of 3/4 (three hits out of 4 at bats), and another game of 2/3 (two hits out of 3 at bats).  In other words, Manny has hit 5 times in 7 at bats, which was realized by adding the numerators and adding the denominators. 

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But we’re told to never ever ever add denominators, so what happened?  What happened is simple: the “whole”, which is the basis of fractions, is defined here as the entire Baseball season at any point in time.  At this point in time, Manny’s whole season has consisted of 7 at bats.  The “whole” in baseball grows as each game progresses.  In fact, if we were to use the adding fractions algorithm to get a common denominator, we’d get 3/4 + 2/3 = 9/12 + 8/12 = 17/12!  Manny can’t possibly get 17 hits after 12 at bats!  That’s just nonsense! 

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Wait, I don’t get it.  I hardly do, either.  But let’s try…

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Let’s think a bit more about Manny’s at bats.  Maybe if we thought of every at bat as its own whole, that is, each at bat is like a coin flip – he’ll either hit or not – we’d begin to understand what is happening.  Ah, we do!  BUT, we also have to keep in mind when we’re looking at his batting average: after 7 hits.  There is a common denominator here, it’s 7!  7 is the, albeit temporary until the next game, sample space.  When we look at 3/4 + 2/3 = 5/7, what we’re really looking at is (1/7 + 1/7 + 1/7 + 0/7) + (1/7 + 1/7 + 0/7) = 5/7! 

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This can be seen with eggs, too…

Scenario 2: Here, we have 8/12 + 2/4.  If we add (8+2) and (12+4) we will get the fraction 10/16, and there are, in fact, 10 out of 16 spaces filled with eggs.  However, we completely disregarded the fact that the two containers are different sizes.  Let’s see what happens if we really ignore the discrepancy in container size:  If we first reduce 8/12 to “2/3” by chopping the numerators and denominators both by 4 (allowed!), and reduce the 2/4 to “1/2” by the old halfsies method (also allowed!), and then try adding the numerators and denominators together, we’ll end up with 3/5.  3/5 is definitely not the same as 5/8 (reduced from 10/16 by halfsies).  But why? 

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We have to remember what we were doing, just like Manny had to remember that outfield is just as important as batting.  When we added the original numerators together (8+2) and the original denominators together (12+4) we were working with raw data, just like in the case of Manny Ramirez’s batting average.  What we really did was add (1/16 + 1/16 + 0/16, … you get the idea.  We defined the sample space as 16 because there are 16 total spaces for eggs, and we disregarded the different sizes of the containers.  If we first take the time to reduce the fractions, we change the fractions from ones that represent real information (actual egg numbers) to one that represents the proportion of eggs in each container.  Herein lies the problem.  How big is our whole?  We need to clearly define it.  If it’s 16, that’s fine if we consider 2 containers to be one whole.  But if we consider each container its own whole, we need to do things differently…    

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If we are merely looking at how many eggs we have versus how many egg spaces, disregarding the discrepancy in egg carton size, we find that we have 10 eggs out of 16 total spaces.  16 is the whole.  This is useful information to have when baking a cake.  Or a few cakes and some French toast.  But if we first allow ourselves to reduce the egg carton fractions individually to 2/3 and 1/2, we change the problem from looking at one whole of 16 to two separate, differently-sized wholes of 3 and 2.  Once we do this, we enter into the realm of WHOLES.  And this is OK!  This is what fractions are all about!  There is a way to add wholes of different sizes; you just have to define how large you want your whole to be. 

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But first, we have to remember an old mantra I heard somewhere, or didn’t hear anywhere, that Math is a Language.  Math is a language just as Portuguese is a language.  In Portuguese, you can’t talk in straight verbs, people would think you cracked your egg!  There are rules to follow when speaking Portuguese, and the same is true in math.  If we want to add 2/3 + 1/2, we absolutely can, but we first have to remember that each of these two fractions has already been given a clearly defined whole: one is the denominator 3 and the other is the denominator 2.  To add these portions of wholes, we have to first decide how large we want our end whole to be, and it can be any number.  It can be 1, 2, 1.17, 2.14, anything.  But what number makes sense, and more importantly, what number is easy to work with?  How about 6?

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Both 3 and 2 go into 6, so we can make the common baseball season, er, we can make the common denominator 6.  We do this by un-reducing the fractions:

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2/3 = 4/6 by multiplying the top and bottom both by 2.

1/2 = 3/6 by multiplying the top and bottom both by 3.

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Now we have our new common sample space, our new common whole, our new Common Denominator!  It’s 6!  Now we can add the numerators and come to 4/6 + 3/6 = 7/6.  The new common egg carton has 6 spaces for eggs and 7 eggs, or 1 carton and 1/6 of a carton.  We could make our sample space 12 and add 8/12 + 6/12 = 14/12, or one full carton of 12 with 2 eggs left over. 

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But wait!  Why do we have one egg left over in the first addition and 2 eggs left over in the second addition?  Remember, we’re no longer talking real eggs here; we left real eggs behind when we decided to look at each carton individually and throw sample space 16 [rightfully] out the window.  We are talking “proportion of the whole”, and with fractions, we can decide however big we want our whole to be.  How many at bats will Manny have?

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The Old Schoolhouse Magazine July 10, 2010

 

With any luck, and hopefully within the next six months, homeschoolers will open their copy of the The Old Schoolhouse Magazine to a nice review of the ZeroSum ruler.  I wrapped one up in its nice packaging and sent it off earlier in the week to be reviewed by the magazine, as dvertisements aren’t super expensive, but more than I can spend right now.   So we’ll see!  

 

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 “doublling” 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 gthe 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!

 

overkilling negatives? May 8, 2010

 

I know the ruler seems a bit overkill for a simple subject like adding positives and negatives, but I teach 11th grade in Boston and it’s the biggest stumbling block for even my students taking my advanced algebra class.

 

The problem is that kids are taught a “noun-verb” way of solving problems like “-12 + 7″. They are told to find -12 (noun, static number) and count up 7 spaces (verb, movement) to the right to see what number they land on. This is fine in a classroom with a number line taped to the desk, but it doesn’t teach the kids how to think about the numbers and a lot of kids will get this problem, and ones like it, wrong. It only gets worse with “x + 12 = 7 (solve for x)” or “y + 12x = 7x + 3 (solve for y)”. It’s the same problem over and over again, just disguised.

 

The problem with the number line and the “noun-verb” way of solving is that it’s not the way we think. It’s not even the way we are taught in school to solve these problems. In the Boston 7th grade curriculum is a book called “Accentuate the Negative” where the very first page of text has a caption over a kid’s head that reads something along the lines of “I owe my dad $4. I have -$4″. So this business of “owing” comes into play very early.

 

If I owed you $12 (-12) and I only paid you back 7 (+7), how much would I still owe you? Asked like this, it’s a simple problem. You’d count up from 7 until you got to 12, knowing that the answer would be in “owe”, or negative. In school however, the kids are told to start at -12 and count up 7 spaces. This is completely backwards from how we think.

 

So to get to my ruler…. The ZeroSum ruler allows a kid to find -12, find 7, fold the ruler in half and count the space between the two numbers’ absolute values. This is what we do when we are finding out how much someone owes us, and this is really the way we think. In time, and to answer your question about what a kid would do with numbers beyond -25 and +25, a kid would start to see the relationship between positives and negatives and that if you “owe” more than you “pay” (if the negative is further away from break even (zero) than the positive) then the answer will take a negative sign. But it’s really the space between the absolute values we are counting.

 

 

So, how much do I owe you? April 19, 2010

 

You friend borrows $22 from you.  He pays you back $15 the next day.  How much does he still owe you?  Asked this way, it’s obvious he owes you $7.  But give a kid the problem -22 + 15, and the answer mysteriously becomes, well, mysterious.  

 

WHY?

  

My students can certainly tell me how much I would still owe if I borrowed $22 and paid just $15 back. Like us, they’d probably count up from 15 to get to 22. But give a student the problem “-22 + 15″, and all bets are off.

  

For this number sentence, we are taught in school to find “-22″ on a number line and count to the right 15 spaces to find the number we land on. But this is not what we do in real life to find out how much someone still owes. There is a huge disconnect here.  In real life, we count up from 15 to 22, keeping a tally on our fingers of how many numbers we pass by.  We would never count up 15 from -22 to find how much someone owes us!  It’s no wonder students have difficulty with negative numbers with the way we are taught!

  

To plug my product, the ZeroSum ruler allows a student to count the spaces from 15 to -22 by folding the ruler in half at the pivot and counting from 15 to +22. When the positives are aligned with their negatives, they’re essentially finding the difference between the absolute values of -22 and 15.  This is the way we think and therefore a more natural way to learn.