# ZeroSum Ruler (home)

## Blogging on math education and other related things

### 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”

What about “she” 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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### Why we add fractions the way we do… a visual tourJanuary 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!

### Awesome Euclid and his AlgorithmNovember 19, 2010

One of my favorite things that I learned while in graduate school for math education was the Euclidean Algorithm for finding the Greatest Common Factor of two numbers.  If you click on the  picture to the left, you’ll get to a very informative YouTube video on the Algorithm.  It’s a bit boring, but very educational, and it shows exactly how to go about using Euclid’s method to find the biggest number that divides into two numbers.

(The screenshot to the left will bring you to the YouTube video on the Euclidean Algorithm)

The alternative, but mainstream, way using factor trees and circling primes always confused my students.  “Do I count the 3 twice since I circled it as a factor in both 81 and 57?”

If Euclid’s method was the mainstreamed one, math would be a lot more interesting and one more confusing topic could be checked off the list.  Euclid, you rock!

(The screenshot here of the kids is a funny video about Euclid and his algorithm)

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### Is Quirky a scam?November 18, 2010

Filed under: invention — ZeroSum Ruler @ 6:13 pm
Tags: , , ,

I paid my \$10.

I wrote all about the ZeroSum Ruler.

I submitted a picture, wrote my bio, wrote all about the ruler, and…..

Nothing.

No really, what is “Quirky”??

Still, I did manage to see that five people, probably unemployed and bored, decided to click on my product number to see what it was.  Here are their comments and my replies:

Becky Blatchford:

“I like this. If it is inexpensive enough say under \$10 I’d bet moms would buy this up!”

Me:

“Thanks Becky.  Hopefully they will when more people find out about it.”

socrtwo:

“Interesting and simple.”

Me:

“Thanks scortwo”

Michael Kloeckner:

“I love it, I cannot believe this is not out there somewhere”

Me:

“Thank you Michael. Now only to find a venture capitalist to back production and marketing!”

Dave McKey:

“I agree with Aaron. no need to fold the ruler.  good luck!”

Me:

“Thanks Dave.  I’ll think about the fold.”

Aaron Dale:

“I watched the video on your website and the only question I have is, why fold the ruler? Just make the ruler so it has negative numbers along one edge and positive along the other edge. But if a kid can use a tool to combine negative and positive numbers, they should use a calculator. It would be best for the kid to understand why they get the answer they get instead of counting the difference on the ruler. It seems like cheating.”

Me:

“I see what you’re saying about the folding.  it would be a lot easier to make without the fold.  The reason I made it fold was so that it starts out looking like the normal numberline the kids use in school.  The folding then shows the relationship between negatives and positives.  When a kid tells me -22 + 5 = -27, there’s a problem.  When this kid is in 11th grade, there’s a bigger problem.  When this kid is not just one kid but many, we’ve got a huge problem.  My thesis results are in and the ruler caused students to make 1/2 the mistakes as before the ruler.  True, data isnt’ everything, but I never thought it woule make such a difference. ”

So what is Quirky?  Can I build a website, undercut Quirky by charging anyone with an idea \$8 to write all about it only to have it disappear into a “relist” button one week later?  I mean, everyone has an idea, most everyone has \$8, this is the best scam going, right?

Am I missing something?