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The following [totally awesome] video comes courtesy of Dr. Brian Biswell.
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Dividing Fractions With Pictures! June 8, 2011
Of all my posts, this one gets the most hits. I think I know why. Fraction division seems like it should be simple. Afterall, ”flip the second fraction and multiply across” is a complete cake walk. But when we have to explain the process to a kid (or an overly-inflated interviewer), things can go very wrong. Why is it so hard? Recently, I met a new friend, Chris Fink, through my blog. Chris teaches Math in the California penal system. Through a series of emails back and forth, we both came to a better understanding of this tricky process. She was able to explain fraction division to her inmates (they all clapped and thanked her - yes, her - afterwards!) and I came to understand how to show the process through pictures a lot better thanks to her. I left my old post underneath the new stuff because, though wordy, it does give a bit more explanation. The following three screenshots (you can download the pdf here or by clicking on one of the three screenshots) are a decent start to How we show fraction division thorough pictures. Thanks Chris!
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Fraction Division: Not Just a How
Dividing fractions has got to be the algorithm we most often take at face value. The How – flip the second fraction and multiply across – is easy, while the Why can fill an entire chapter.
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Whenever we divide, we’re asking “How many groups of this will fit into that?” With, for example,
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10 ÷ 2, we’re asking “How many groups of 2 will fit into 10?” This is easy:
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Here’s 10:
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Here’s a group of 2:
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We can easily see that 5 groups of 2 will fit into 10:
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Unlike multiplication, division is not commutative. We cannot divide backward and forwards and expect to get the same result. For example:
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10 ÷ 2 ≠ 2 ÷ 10
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We always “flip the second fraction” in the fraction division algorithm as, contrary to logic, flipping the first fraction instead will not yield the same result. For example:
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The first number in a division problem is simply more important than the second number. The first number sets the stage while the second number asks, “How many groups of me will fit into your first number?” In 10 ÷ 2, we weren’t putting groups of 2 into any old number; we were putting groups of 2 into 10. We needed to keep the 10 in mind as we bundled our groups of 2. Division with fractions operates in the exact same way. Whenever a fraction is divided by another fraction, one of two possible outcomes occurs: a fraction less than 1 or a fraction greater than 1. Of the fractions greater than one, answers can be either whole numbers or mixed numbers. Whenever we deal with parts of wholes, things get interesting.
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We’ll start with a simple example where the result is a nice, easy whole number:
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If we ask “How many of the green pie piece will fit into the blue half of the circle?” we can see pretty easily that 3 will fit in perfectly. If we were to superimpose the green pie over the blue one, the centerlines on both pies would line up nicely, creating a common denominator of 6. Unfortunately, not all fractions superimpose over each other so nicely. To develop a pattern that we can use with more difficult fraction division problems, let’s look at 1/2 ÷ 1/6 in a slightly different way. First, we’ll set the stage with 1/2:
Here we have a circle and we colored half of it. This next part is where things can get weird. Remember how, in 10 ÷ 2, the 10 set the stage before we began bundling groups of 2? If we instead thought about 10 ÷ 2 as 10/1 ÷ 2/1 , we can begin to see why this problem was so easy: the 10 and the 2 already shared a common denominator. Just as we did there, we’ll create a common denominator in this problem. The easiest way to do this is to superimpose the 1/6′s denominator atop the 1/2 and see what shakes out:
When we divide the entire region into 6 equal pieces, essentially turning 1/2 into 3/6 , it will become very easy to then take 1/6 :
Just like in 10 ÷ 2, we now ask “How many 1/2 fit in 1/6 ?” In other words, how many green pie pieces fit into the original blue 1/2 ?
3 do. And in fact, 1/2 ÷ 1/6 = 1/2 • 6/1 = 3.
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Fraction division hasn’t earned its own post based on easy problems like 1/2 ÷ 1/6 . This problem was easy for a couple reasons: the answer was a whole number and it was very easy to create a common denominator. Next, let’s look at a slightly harder fraction division problem in a still slightly different way:
This problem is more difficult for a few reasons. First, the result will not be a whole number. Second, the result will be a fraction less than one. Third, the denominators 4 and 6 don’t overlap very easily, so we’ll need to create a common denominator that is larger than both 4 and 6. We’ll deal with these first two reasons as we work through the problem. To mitigate the third reason this problem is more difficult, we’ll create a larger common denominator. Fortunately, this larger common denominator will appear naturally as we begin to draw the problem.
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First, 3/4 :
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To show 3/4 , we divided an area into 4 columns and pink-boxed 3 of them. Keeping the entire area in mind as we have done before, we will now get ready to take 5/6 by first creating 6 rows:
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and then coloring in 5 of the rows:
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By using columns to show the first fraction, and rows to show the second fraction, we naturally created a common denominator of 24. This will happen every time we use the column and row method.
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Now let’s ask our question: “How many 5/6′s (orange boxes) fit into 3/4 (pink outline) ?” In other words, how many of the orange boxes from the group of 20 will fit into the pink-boxed 18 area? So it’s a bit easier to visualize, let’s move as many orange boxes inside as will fit:
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18 out of the group of 20 orange boxes will fit. And in fact, 3/4 ÷ 5/6 = 3/4 • 6/5 = 18/20
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So far we have seen a whole number answer and a fractional answer less than 1. In this third example, we’ll look at the last type of fraction division problem – one that yields a mixed number. This next problem was asked of me twice during two different interviews for middle school Math teaching positions in the Boston Public Schools:
Because we’ll need to create a larger common denominator here, as 2 and 3 don’t easily overlap, we will use the column and row method. Starting with 1/2 :
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we’ll get ready to take 1/3 by dividing the entire area into 3 equal rows:
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and coloring 1 of the 3 rows:
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We see a group of 2 blue boxes. As we’ve done before, let’s move the one on the outside into the inside:
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2 of the 2 blue boxes (2/2) will fit into the pink-boxed area. Additionally, another 1 out of the 2 blue boxes (1/2) will also fit:
And in fact, 1/2 ÷ 1/3 = 1/2 • 3/1 = 3/2
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With the new Common Core Standards, kids are being asked to divide fractions beginning in 5th grade. As with anything, once we develop a pattern for fraction division, showing the process with pictures becomes easy. Once a kid can see and feel what is happening in these problems, the process of dividing fractions will begin to make more sense.
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Please check the comments below for some good additional information on why the division algorithm works.
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Also see Multiplying Fractions with Pictures!
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Also see Multiplying 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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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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Public school? Supersize my class! January 25, 2011
There are two upsides of a larger class: more diversity among students and the drive to succeed (and/or not act out in front of peers). I found that when I was teaching if too many students were out on a day, the kids who were in school felt it should be a “free day”. Kids weren’t there to bounce ideas off each other or to push each other.
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So does class size matter? It definitely does. Even when I had less students in class and the ones who were there were less motivated to do work, I was better able to connect to the students who were there. There were less kids to reach. And isn’t that what good teaching is half about?
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Read the article The Class Size Debate on Huffington Post.
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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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Let’s grow some grass! November 7, 2010
Who knew? Students love growing grass. Let me elaborate…. Students love taking care of their grass as they watch it grow- enough to get them to do some pretty complicated algebra.
What started out as a simple week-long project that incorporated a bit of environmental sciences (my undergrad background) into algebra became a 10-week long project spanning the curriculum from ecology to linear extrapolation. It’s been a long time in the making, but beyond a shadow of a doubt this lesson is one of the most engaging that I have created. My students love the life aspect of the project and hardly complain about doing some pretty complex algebra.
Now, with the magic of a WordPress widget called ”My Shared Files BOX” (on the sidebar), I was able to upload the Growing Grass Project files onto my blog for all to use!
All three files are important, but the excel workbook includes everything the student needs to create a final portfolio piece, including a formatted final excel sheet that the student can type into and cell directions.
I’m very excited about this and hope that if you do use the project, that you will add a comment to this post on how it went. I also have other files that go along with the starter ones I posted.
I guarantee that your students will be engaged in their learning and that you will find ways to link most of algebra 1 to what comes up along the way.
WARNING! This lesson takes on a life of its own! Proceed with caution! 
Now Let’s grow some grass!
p.s. For supplementary files, you can email me at ZeroSumRuler@gmail.com. The files include ones on scatter plots and lines of fit as well as a PowerPoint and activity on linear inter- and extrapolation.
