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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Multiplying Fractions with Pictures! June 15, 2011

Fraction Multiplication: Of what?

Fractions are probably the most troublesome topic in Math.  As soon as a problem involves a fraction, kids freeze up.  In Math, of tells us to multiply.  How many shrimp are in five pounds of shrimp?  We multiply the number of shrimp in a pound by five.  Once we know this, fraction multiplication becomes a bit easier to understand.

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The How of fraction multiplication is easy – multiply the numerators and multiply the denominators.  When we show fraction multiplication with pictures, we need to remember of.

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Now to the Why.  To start, we’ll look at a relatively easy problem so that we can develop a pattern to follow with more difficult fraction multiplication problems:

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(1/2)(1/2)

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Translated into English, this problem reads “one-half of one-half”.  Here’s a picture of  1/2:

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And the area below in red is “one-half of one-half”:

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It’s easy to see that one-half of one-half is (1/4).  And in fact:

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(1/2)(1/2)   =   1/4

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Before moving on, let’s look more closely at one aspect of the problem above: the denominator 4.  Where did this come from?  To get that denominator, we needed to keep the entire circle (whole) in mind.  In other words, we needed to say that the red piece was “1 out of something”.  (Confusingly, out of means to divide in Math!)  The denominator is 4 because the red pie piece is 1 out of 4 total pie pieces in the circle.  Always remembering the entire original area is key in fraction multiplication.  Later, we’ll see the same is true with fraction division.

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To make the denominator easier to see, we can divide the circle twice: first vertically for the first fraction, then horizontally for the second fraction:

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It’s then easy to see that the overlapped area (numerator) and the entire number of pie pieces in the circle (denominator) create our answer.  This will always be the case.  It wasn’t a coincidence that the denominator was naturally created as we divided the circle twice.  Let’s use this pattern to solve a more complicated fraction multiplication problem:

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(2/7)(3/5)

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Because these two fractions do not have a common denominator, it would be hard to divide a circle into 7 (and take 2), then into 5 (and take 3), and analyze the overlapped area.  So instead, we’ll use nice, easy rectangles.  First,  2/7:

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In the rectangle above, two of the rectangle’s 7 horizontal bars are colored green to represent 2/7.  Now, keeping the whole rectangle in mind, let’s take 3/5:

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In the rectangle above, three of the 5 vertical columns are colored blue to represent 3/5.  Where to the two colored areas overlap?

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In the above picture, we can see that the overlapped area consists of 6 purple boxes.  But 6 of what?  Remembering our easy example (1/2)(1/2), where our denominator was the total number of pie pieces in the circle after our two rounds of dividing, let’s count the total number of boxes in the above rectangle.  The total number of boxes is 35.

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And in fact:   (2/7)(3/5)   =   6/35

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OLD FASHIONED CHECK: We know that (2/7)(3/5) = 6/35 from the algorithm “multiplying across”.

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To see this more clearly, we can look at the below picture and see that the area the fractions share, or the overlapped region, is 6 boxes, and the area of the entire region is 35 boxes.  “6 out of 35 boxes are double shaded”.

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In this last picture we can see that the area of the double shaded region is 6, or (2×3), and the area of the entire region is 35, or (7×5), which is why we multiply the numerators (2×3) and the denominators (7×5) when we find the product of two fractions.

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Also see Dividing Fractions With Pictures!

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If this was helpful, here is a free poster download for your classroom:

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Also see Dividing Fractions With Pictures! and Differences of Squares with Pictures!

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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!

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contact blog author Shana Donohue: shanadonohue@gmail.com

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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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