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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The Coolest Math Class Ever. Seriously! June 6, 2011

I was lucky to get into Oliver Knill’s class Teaching Math With a Historical Perspective, one of the choices within the Harvard Extension School’s ALM in Mathematics for Teaching program.  It changed the way I think and the way I teach.  He explained complex topics, such as code breaking and non-verbal proofs, with such ease.  He was inspiring and made me look at math in a way I had never before – from a historical perspective!  His site is worth checking out, especially if you can not get into his class!  (Click on the red circle to go to Oliver’s site)

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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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“A good curriculum is the best classroom management” February 7, 2011

I have worked for a lot of people, but the most inspiring boss I have ever had was a principal who was strict, forgiving of human flaws, hard-working and who lived by the motto “A good curriculum is the best classroom management”.

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I’ve never been a disciplinarian and never will be.  I value learning too much to stifle a kid’s personality in favor of keeping my classroom quiet.  Learning is loud, it’s fun, it’s rich, it’s not a library.  If learning was supposed to be done in complete sterility we’d all be able to teach ourselves in the quiet of our own homes.  Learning is the push and pull between student and student, student and curriculum, and student and teacher.  It should be a fun process.  Do you normally do things that are not fun?  The best classroom management is a good curriculum.

 

The article by Pamela Kripke “And You’re Out!” in the Huffington Post is very much in line with what I encountered during my years of teaching.  Kids would get kicked out of their classes for doing something against the teacher’s status quo and end up in my classroom.  I was constantly torn between letting them stay and being part of the “united front” against the student.  I never wanted to be a part of that front and never really was.  In my mind the right thing to do was keeping the kid happy and wanting to come back to school the next day.  With the dropout rate as high as it is and high school degree jobs steadily dropping, it was my job to keep learning fun and school an enjoyable place to go.  And that’s what I did.

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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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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!