# ZeroSum Ruler (home)

## Blogging on math education and other related things

### The Pain (really, actual PAIN) of Math AnxietyFebruary 11, 2013

Filed under: math anxiety — ZeroSum Ruler @ 8:09 pm
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Before you get excited about a way out of your Math homework tonight, it’s the anticipation of Math – not the actual act of solving problems – that causes some people actual physical pain.

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“We show that, when anticipating an upcoming math-task, the higher one’s math anxiety, the more one increases activity in regions associated with visceral threat detection, and often the experience of pain itself (bilateral dorso-posterior insula). Interestingly, this relation was not seen during math performance, suggesting that it is not that math itself hurts; rather, the anticipation of math is painful. Our data suggest that pain network activation underlies the intuition that simply anticipating a dreaded event can feel painful.”

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You can read the article When Math Hurts: Math Anxiety Predicts Pain Network Activation in Anticipation of Doing Math by Ian M. Lyons and Sian L. Beilock  here.

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We already knew that Math anxiety causes decreased brain function, explained here in Huffington Post’s article Math Anxiety Linked With Differences In Brain Functioning, Study Finds.  Even without an article stating so, this fact is obvious to any teacher, parent or even student who can solve for x during a warm up (with one eye closed while catching up with a friend) but then chokes on the subsequent quiz.  Now there’s proof that just thinking about our Math classes may be causing our kids physical pain.  I sort of feel bad.

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### Dirty Word of the Day: MemorizationOctober 21, 2012

As a high school Math teacher, I hear all the time, “I suck at Math!”, especially, considering that everything else in the world is found at the push of a button, when my students are faced with problems they can’t immediately solve.  I hear “I hate Math” when we’re solving equations, when we’re factoring, when we’re plugging x values back in to find angle sizes.   I hear it all the time.  but it’s when I hear it that got me thinking.

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Since 2003, I’ve been keeping a mental log of all the times I have heard “I hate Math” or “I suck at Math”, mainly because each one has left its own little crater on my Math soul.  I want – need, really - to figure out why kids feel this way.

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It took nearly 9 years, but yesterday I finally figured out why some kids hate Math with all of their being.  It’s because they can’t multiply.  This had been my suspicion for a few years, but yesterday it became clear that multiplication makes the difference.

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“But multiplying is easy, it’s not that.  Math is WAY harder than just multiplying!” you may say.  And I agree with you.  However, Math is 90% confidence, and when a kid loses this confidence because “multiplying is easy” and he can’t multiply, then he feels like a loser and closes off to the rest of his years of problem solving.

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The Conclusive Evidence

I had never seen a little kid do Math until yesterday when a former student of my husband came over with her Mother for lunch.  She’s in 4th grade now and has been having trouble with Math, so we sat down with her current homework: multi-digit multiplication problems.  The algorithm “multiply then carry, then multiply again and add what you carried” is a little weird, but she got that part.  Then all of a sudden out of nowhere, with fists slamming on homework…

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“I stink at Math!”

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“I hate Maaaaath!”

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Within the giant problem my husband gave her to do, she needed to multiply 8 x 5.  When it didn’t come immediately, she exploded.  And up went the walls.  Single-digit “multiplication is easy”, right?  Not if you don’t know it.  If you don’t know 8×5, then Math is the shittiest subject there ever was, ever is, or ever will be.  It totally blows.

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So what would help her?  Memorizing her multiplication tables.  Sounds simple and ridiculous, right?  Hold on a second.  Below are a few excerpts from an article, “Chess Experts Use Brains Differently Than Amateurs”:

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Experts use different parts of their brains than amateurs, maximizing intuition, goal-seeking and pattern-recognition, says a new study that examined players of shogi, or Japanese chess.

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Researchers believe that experts who train for years in shogi are actually perfecting a circuit between the two regions that helps them quickly recognize the state of the game and choose the next step.

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“Being ‘intuitive’ indicates that the idea for a move is generated quickly and automatically without conscious search, and the process is mostly implicit,” said the study.

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Math is a lot like chess: strategy, visualization of next moves, attack!  When a kid is a multiplication amateur, strategy can never develop, patterns will never be recognized, Math will always be counterintuitive.  Multiplication facts take a lot less time to master than chess.

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Memorization: Mathematicians’ Dirty Word

Memorizing sight words doesn’t make reading Harry Potter easy, but it does make it easier.  This is why we do it.  So why not memorize multiplication facts to make Math easier?  At some point between being a Math student and being a Math teacher, ”memorization” became a dirty word.  I agree that we shouldn’t force kids to memorize every Mathematical formula or the digits of pi, but I remember a deep sense of pride in having my multiplication facts memorized.  Maybe the way it was done – calling us up one by one to recite the facts to our 3rd grade teacher – was not the best method and probably contributed to my high-strung demeanor.  But when I got to pre-Algebra, I could cross-multiply; in Algebra I could quickly find factors; and in Geometry I could “plug it in” without a calculator.  All of these seemingly-unrelated abilities contributed to my feeling that Math wasn’t impossible.  I had confidence because I could multiply quickly.  I was fluent, solving came easy.  I could do more advanced problems because I had confidence.  I had confidence because I didn’t need to stop and think through every instance of multiplication.

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A recent brain study done by Dr. Carolyn McGettigan in the UK yielded unexpected results.  Contrary to hypothesis, the expert beatboxer uses less of his brain than the novice:

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The novice used many more brain areas, suggesting a need to plan each sound and a lack of automatic processing.

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Dr Carolyn McGettigan, a neuroscientist at University College London, compared magnetic resonance imaging (MRI) scans during two tasks – counting and beatboxing.

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Dr McGettigan says: “When you think about an expert you might think they activate extra bits of the brain – not just the bits you use to make sounds, but something exciting and different that you might not expect.”

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“What we have at the moment is a demonstration that being an expert doesn’t mean you activate more of your brain. The phrase ‘less is more’ is sort of appropriate here.”

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Thinking is Overrated

Thinking is important, but not that important.  When it comes to the building blocks of any language, a level of fluency is essential.  Without it, reading is exhausting, and things that are exhausting are avoided.  I doubt that my husband – a true bookworm - would have read the entire Harry Potter series [more than once] if he had to individually sound out each word.  It just wouldn’t have happened.  J.K. Rowling would have never earned the necessary funds to  go on to write The Casual Vacancy (is it any good?) if everyone struggled through her Harry Potter books.  Is it any mystery that kids hate solving equations or finding missing side lengths if they have to “sound out” each instance of multiplication?

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So what to do?

We have to stop making our kids think so much!  It’s exhausting them.  When I was a kid, my parents gave me this Math toy that tricked me into learning my multiplication facts.  Flashcards for facts up to 12×12 are also great.  It’s got to be fun, not forceful, of course, but it’s got to happen.  It will make all the difference later and is really that simple.

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### Is Common Core meeting its Goal?May 21, 2012

Is the original goal of Common Core being lost in the upper grades?

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One major difference between the U.S. and so-called ”A+ Countries” is, while we focus on breadth, they focus on depth.  While there is a natural progression throughout a student’s school years from one math topic to another in these high-achieving countries, in the U.S. we seem to have a “throw at the wall and see what sticks” mentality.  For example, in grade 8 we cover 32 unique mathematical topics.  In high-achieving countries this number is just 18.

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The new Common Core curriculum aims to bring our focus back to depth in the lower grades but seems to miss this mark once the abstract maths are reached.  While it is true that more topics have been cut out than added in most grade levels, topics traditionally covered in Algebra 2 (and some may say pre-Calculus and above) – piecewise functions, limits, logarithms, areas under curves, Algebraic proofs, and rational function graphing to name a few – are now part of Algebra 1.  Does adding so many advanced topics to the Gateway of Higher Math (ok, I’m biased) do what Common Core initially set out to do?

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Below is a comparison of the math topics taught each year in A+ countries (first chart) and those covered in the U.S. (second chart) each year (compiled by Professor W.H. Schmidt).  These comparison charts were created before, and as a support for, Math reform in the U.S.  Still, to meet the new upper-grade Common Core Standards, school districts are turning to hybrid-type courses: “Algebra/Geometry/Stats Year 1″, etc. (Yes, that’s ONE year’s course) to meet all of the new high school requirements.  While the Common Core sets out for mastery at the Elementary level, did it really mean to hybridize high school math?  If depth is more important that breadth, what are we doing?

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### If Pigeons can do it, so can youDecember 24, 2011

Here is a fascinating article on the mathematical intelligence of…. pigeons

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http://news.discovery.com/animals/pigeons-math-animals-111222.html

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### Are these searches for real? (Search Terms 6/19/2011)June 20, 2011

My blog keeps track of the search terms that have led people to me.  Some of them make sense.  Other ones?  I always wish that I knew the people who did these searches because I would like to help them with their math questions.  Below is a partial list of search terms from June 19, 2011 and explanations for the people who may have done the searching.  The 7th one down might be worth reading.  Enjoy!

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Search Terms Sunday June 19, 2011

“show he pictures of fractions”

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“show the fraction 16/5 on figures”

Ok, this is a search term I live for.  Somewhere out there is a confused little kid trying to finish his homework, or a mom trying to help her kid finish his homework, and I want to help.  It makes me sad that this kid did not get his answers in class.  So I will attempt to explain “16/5 with figures”!

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We have 1/5 because 1 of the 5 slices is green.  Now we need to take 16 of them…

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This is extremely messy.  We have so many empty slices (4 of every 5 slices are empty!).  So, let’s condense….-

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And if we remember that fractions are all about making wholes, we count that we have “three wholes and 1/5 left over” or 3 and 1/5.  Please email me if you need more background or help of any kind!

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“class”

Thanks!

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“zero sum ruler”

You’re in the right place!

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“show the fraction 16/5 by figures”

I’ve seen this one before…

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“the zero sum ruler”

No really, look around!

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“math students around the world 2010”

there are!

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“blogs on math education failure”

One More to Graduate. Make that 50.000001%

Math manipulatives lead to student failure

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“pi the whole number”

Pi is not a whole number.  It’s not even rational.  If your teacher sent you on a quest for information about the whole number pi, tell your teacher that’s her request is an irrational one.

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“kathleen fick math”

Who the fick is she??

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“math around the world, 1st grade”

They exist too!

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“how to use the zero sum rule[r]”

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Most of this post is completely unnecessary, but the one about fractions is completely necessary.  I gain [serious] blog-posting inspiration from your search terms, so am looking forward to seeing more tomorrow and being forced to write more about important math why’s!   Thank you!

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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.

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

contact blog author Shana Donohue: shanadonohue@gmail.com

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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:

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

contact blog author Shana Donohue: shanadonohue@gmail.com

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