# 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!

-

Search Terms Sunday June 19, 2011

“show he pictures of fractions”

- -

“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”!

-

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

-

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

-

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!

-

“class”

Thanks!

-

“zero sum ruler”

You’re in the right place!

-

“show the fraction 16/5 by figures”

I’ve seen this one before…

-

“the zero sum ruler”

No really, look around!

-

“math students around the world 2010”

there are!

-

“blogs on math education failure”

One More to Graduate. Make that 50.000001%

Math manipulatives lead to student failure

-

“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.

-

“kathleen fick math”

Who the fick is she??

-

“math around the world, 1st grade”

They exist too!

-

“how to use the zero sum rule[r]”

-

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!

-

### 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!

-

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.

-

Whenever we divide, we’re asking “How many groups of this will fit into that?”  With, for example,

-

10 ÷ 2, we’re asking “How many groups of 2 will fit into 10?”  This is easy:

-

Here’s 10:

-

Here’s a group of 2:

-

We can easily see that 5 groups of 2 will fit into 10:

-

Unlike multiplication, division is not commutative.  We cannot divide backward and forwards and expect to get the same result.  For example:

-

10 ÷ 2 ≠ 2 ÷ 10

-

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.

-

We’ll start with a simple example where the result is a nice, easy whole number:

-

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.

-

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.

-

First, 3/4 :

-

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:

-

and then coloring in 5 of the rows:

-

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.

-

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:

-

18 out of the group of 20 orange boxes will fit.  And in fact,  3/4  ÷ 5/6    =    3/4   •  6/5    =    18/20

-

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  :

-

we’ll get ready to take  1/3  by dividing the entire area into 3 equal rows:

-

and coloring 1 of the 3 rows:

-

We see a group of 2 blue boxes.  As we’ve done before, let’s move the one on the outside into the inside:

-

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

-

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.

-

-

-

-

-

-

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

-

-

### 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.

-

contact blog author Shana Donohue: shanadonohue@gmail.com

### My Harvard Math for Teaching Thesis: Complete! And ready to share…March 20, 2011

After many many years of jumping through many many hoops, I am finally graduating with my MA in Mathematics for Teaching in May.  My thesis, Negative Number Misconceptions in High School: An Intervention Using the ZeroSum Ruler is right now at the printers being printed and bound.  I don’t know about you, but that instantaneous feeling of relief after taking a final exam or passing in a final paper stopped hitting me sometime in college.  So now, I’m just feeling a bit burnt out.  OK, completely burnt out.  But I’m sure it will hit me soon since it kind of needs to; I need to now get in a post-Bach program to get my Initial teaching license.  I like to do things backwards.

-

-

So here it is for download!  For all to read!  Or maybe to just glance.  In my study, the ZeroSum ruler proved effective in reducing eleventh grade error on integer addition and subtraction problems (especially with negative integers).  If I wasn’t so burnt out, I’d want to test it with younger kids.  Imagine how our world would be if my eleventh graders actually mastered integers when they learned them in, and only in, 7th grade.  But that’s in my thesis.]

### 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!

-

HaPpY Calculating!

-

### My Old Schoolhouse review debut!January 27, 2011

The ZeroSum Ruler was recently reviewed by The Old Schoolhouse Magazine!  You can read their full review – and get a glimpse at my old address where I fought a slumlord to the death of my career and almost me – at: The ZeroSum Ruler’s Old Schoolhouse Review!

-

“Math is a subject that students can sometimes fake their way through. They might not understand how a problem works, but given the formula, they can follow rules and get things to come out all right in the end. Faking can only get them so far, though. Eventually, they will either forget the formula or not be able to recognize it when arranged in an unfamiliar manner. What a math teacher wants to see is the light bulb moment–when a student doesn’t just use a formula but understands why it works.

Working with negative numbers is an abstract concept that many students have a hard time visualizing. How does one visualize what isn’t there? According to the website, the ZeroSum Ruler naturally brings this abstract “knowing” into concrete “showing”! This simple little device helps students see not only the negative numbers but also their relationship with other numbers.

For instance, a student might not see how subtracting 10 from 5 is actually the same as saying 5 + -10. Visualizing the process with the ZeroSum Ruler helps students see that when they are figuring out a real-life scenario, such as how much someone owes them, they are really counting forward in positive numbers.

-

The ruler itself is of laminated cardstock and is hinged at zero so that it can be folded, making the positive numbers line up with the negative numbers. This allows students to count forward the number they are subtracting or adding.

The true gem of the ZeroSum Ruler is the creator herself. Shana is passionate about math and making it reachable for students. Her website contains math videos and commentaries that help students see that math is fun, interesting, and relevant. She breaks things down in an easy-to-understand method, and she is also happy to help with math questions from students and teachers.

-

The ZeroSum Ruler is a great asset for students struggling with the concept of negative numbers. And its creator is a great help to parents struggling to teach those students.

-

Math is a subject that students can sometimes fake their way through. They might not understand how a problem works, but given the formula, they can follow rules and get things to come out all right in the end. Faking can only get them so far, though. Eventually, they will either forget the formula or not be able to recognize it when arranged in an unfamiliar manner. What a math teacher wants to see is the light bulb moment–when a student doesn’t just use a formula but understands why it works.

-

Working with negative numbers is an abstract concept that many students have a hard time visualizing. How does one visualize what isn’t there? According to the website, the ZeroSum Ruler naturally brings this abstract “knowing” into concrete “showing”! This simple little device helps students see not only the negative numbers but also their relationship with other numbers.

-

For instance, a student might not see how subtracting 10 from 5 is actually the same as saying 5 + -10. Visualizing the process with the ZeroSum Ruler helps students see that when they are figuring out a real-life scenario, such as how much someone owes them, they are really counting forward in positive numbers.

-

The ruler itself is of laminated cardstock and is hinged at zero so that it can be folded, making the positive numbers line up with the negative numbers. This allows students to count forward the number they are subtracting or adding.

-

The true gem of the ZeroSum Ruler is the creator herself. Shana is passionate about math and making it reachable for students. Her website contains math videos and commentaries that help students see that math is fun, interesting, and relevant. She breaks things down in an easy-to-understand method, and she is also happy to help with math questions from students and teachers.

-

The ZeroSum Ruler is a great asset for students struggling with the concept of negative numbers. And its creator is a great help to parents struggling to teach those students.”

-

Thank you, Old Schoolhouse Magazine!

-

You can purchase a ZeroSum Ruler eBook here: The ZeroSum Ruler on CurrClick or on my blog over there —>

-