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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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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For a picture tour on how to deal with fractions: Multiplying Fractions With Pictures!