# ZeroSum Ruler (home)

## Blogging on math education and other related things

### Adding Fractions With Pictures! (The Crisscross Method)December 3, 2012

Fraction Addition (And Subtraction): We’re not in kindergarten anymore

Addition and subtraction are only easy in elementary school.  Once middle school starts, continuing throughout any Math class taken that point forward, addition and subtraction are much harder than multiplication and division.  Why?  The Common Denominator.  To a kid who is not fluent in his multiplication facts, finding The Common Denominator is an exercise in torture.

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What is a common denominator?  A common denominator is a multiple of both denominators in a fraction addition (or subtraction) problem.  For example:

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In the above example, 6 is a common denominator of 2 and 3.  But is it the only one?  No.  How many common denominators are there between two fractions?  Infinite.  For example:

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Why would we want to use 7830 as a common denominator?  Why not?  The point is that any number that both denominators divide into evenly can act as a common denominator.  We are far less restricted than we thought.

So if we’re virtually unrestricted in choosing a common denominator, why not pick the one that is the product (multiply) of the two denominators?  For example:

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Just multiply the denominators to find a common denominator.  This is easy.

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At this point in the traditional method of adding fractions, we’d begin to ask our questions: “How many 8’s go into 16?”  Ok, 2.  “2 times 3 is …?”  Ok 6.  So 3/8  =  6/16 .  Though this process is easy to a person who is fluent in their multiplication and division, it will give reason for a non-fluent Math student to seize up.

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A great alternative way of adding fractions is the Crisscross Method of adding (and subtracting) fractions.  In this method, we use the common denominator just once (this method will not create two equivalent fractions to the original two) and multiply “crisscross” to find two new numerators.

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In  3/8  +  5/2, we’ll first multiply the denominators to find our new, common denominator:

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Next, we’ll multiply 3 • 2 (always starting our crisscross in the top left corner) to find the first missing numerator:

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And then 8 • 5 to find the second missing numerator:

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But why are we allowed to do this?  Let’s back up to see what really happened.-

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First, we found the common denominator 16 by multiplying the denominators (8 and 2) of both fractions.  We’re guaranteed that our denominator is common if we created it by multiplying the two original denominators to get it.  To get the first numerator 6, we multiplied the numerator of the first fraction (3) by the denominator of the second fraction (2).

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In the process, we multiplied both numerator and denominator by 2.  In other words, we multiplied  3/8 by  2/2 Any number divided by itself is just a fancy 1, and multiplying any number by 1 does not change the number’s value.  As a check to see if this process worked,  3/8  =  6/16 .  The old and new fractions are equivalent.

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The same is true to get the second numerator 40:

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Both numerator and denominator were multiplied by 8.  In other words, we multiplied  5/2  by 8/8, which is just a fancy 1.  Multiplying by 1 does not change a number’s value.  As a check,  5/2   =  40/16.  The old and new fractions are equivalent.

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Now we simply add the numerators:

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The Crisscross method also works for fraction subtraction – we’d have a subtraction in the numerator.  Why was this method not taught in school?

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Hurray for Fraction Addition (and Subtraction)!

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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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### The Language of Math PosterAugust 19, 2011

Below is a poster I hang in my classroom every fall.  Each year it grows longer as more and more terms come up for the different operations of math.  When I was a kid, no one told me to look out for these words, or that math was even a language at all, which made word problems pretty tough.  By clicking on the poster you will be sent to the original Excel file on Google Docs.  Do you have any words to add?

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### Income and Debt with ZeroSum RulerJuly 10, 2011

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One of the ZeroSum ruler’s main purposes is to calculate debt/income problems.   In the problem “I owe you \$12 and pay you back just \$7. How much do I still owe you?” how do you come to your answer?   Do you count backwards from \$12 to \$7?   Or do you count forwards from \$7 to \$12?   No really, how much do I owe you?   How did you figure this out?    The ZeroSum ruler allows the student to count forwards instead of backwards just like we do in real life!    So why do we make our kids count backwards in school?

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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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### 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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### Math manipulatives lead to student failureMay 19, 2011

During a 4th grade substituting assignment, the teacher left a set of word problems for the kids to do.  A bunch of these word problems involved division, and the students were directed to use their counting blocks.  As I walked around the room, I saw kids doing just about everything a kid will do with giant leggo-type blocks.  There were guns, there were swords, there were towers.  Some kids were using the blocks to work the word problems, but many of the students who wanted to use them for good were having trouble.  My role morphed from teaching math to teaching the kids how to use the counting blocks.  One word problem called for dividing 125 by a variety of numbers.  There is a large margin of error while counting 125 of anything, and with a string of problems that all rely on a 100% accurate count, it felt to me that the kids’ time could have been better spent.  When do manipulatives cross the line from helpful to hurtful?

A great article titled Teacher Learning and Mathematical Manipulatives: A Collective Case Study About Teacher Use in Elementary and Middle School Mathematics Lessons  by Laurel Puchner, Ann Taylor, Barbara O’Donnell and Kathleen Fick, outlines one of the many problems that can arise while using manipulatives in math.  This article is a worthwhile read, especially for those teachers wondering why manipulatives don’t seem to work as well as advertised.

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contact blog author Shana Donohue: shanadonohue@gmail.com