# ZeroSum Ruler (home)

## Blogging on math education and other related things

### 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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### 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.

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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.]

### Was math discovered or invented?March 14, 2011

On this Pi Day in 2011, I find myself wondering if math was discovered or invented.   The Pythagoreans believed that math was discovered and that whole numbers ruled the universe.  They also did not believe in irrational numbers, of which we now know there are an infinite amount.  So then was math invented?  I’m not sure….

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### Discover something new! The ZeroSum Ruler FREE eBookDecember 21, 2010

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I am excited to bring to you this free version of the ZeroSum Ruler’s eBook!  In this book, you will find pictures and explanations of how the ZeroSum Ruler is different yet more intuitive in its way of teaching integer addition and subtraction.  if you like what you read, there is a version of the eBook on CurrClick that includes a ZeroSum Ruler, practice problems, and some more fin pictures.  Enjoy!  (click the picture below to be directed to your free download!)

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### overkilling negatives?May 8, 2010

I know the ruler seems a bit overkill for a simple subject like adding positives and negatives, but I teach 11th grade in Boston and it’s the biggest stumbling block for even my students taking my advanced algebra class.

The problem is that kids are taught a “noun-verb” way of solving problems like “-12 + 7″. They are told to find -12 (noun, static number) and count up 7 spaces (verb, movement) to the right to see what number they land on. This is fine in a classroom with a number line taped to the desk, but it doesn’t teach the kids how to think about the numbers and a lot of kids will get this problem, and ones like it, wrong. It only gets worse with “x + 12 = 7 (solve for x)” or “y + 12x = 7x + 3 (solve for y)”. It’s the same problem over and over again, just disguised.

The problem with the number line and the “noun-verb” way of solving is that it’s not the way we think. It’s not even the way we are taught in school to solve these problems. In the Boston 7th grade curriculum is a book called “Accentuate the Negative” where the very first page of text has a caption over a kid’s head that reads something along the lines of “I owe my dad \$4. I have -\$4″. So this business of “owing” comes into play very early.

If I owed you \$12 (-12) and I only paid you back 7 (+7), how much would I still owe you? Asked like this, it’s a simple problem. You’d count up from 7 until you got to 12, knowing that the answer would be in “owe”, or negative. In school however, the kids are told to start at -12 and count up 7 spaces. This is completely backwards from how we think.

So to get to my ruler…. The ZeroSum ruler allows a kid to find -12, find 7, fold the ruler in half and count the space between the two numbers’ absolute values. This is what we do when we are finding out how much someone owes us, and this is really the way we think. In time, and to answer your question about what a kid would do with numbers beyond -25 and +25, a kid would start to see the relationship between positives and negatives and that if you “owe” more than you “pay” (if the negative is further away from break even (zero) than the positive) then the answer will take a negative sign. But it’s really the space between the absolute values we are counting.

### So, how much do I owe you?April 19, 2010

You friend borrows \$22 from you.  He pays you back \$15 the next day.  How much does he still owe you?  Asked this way, it’s obvious he owes you \$7.  But give a kid the problem -22 + 15, and the answer mysteriously becomes, well, mysterious.

WHY?

My students can certainly tell me how much I would still owe if I borrowed \$22 and paid just \$15 back. Like us, they’d probably count up from 15 to get to 22. But give a student the problem “-22 + 15″, and all bets are off.

For this number sentence, we are taught in school to find “-22″ on a number line and count to the right 15 spaces to find the number we land on. But this is not what we do in real life to find out how much someone still owes. There is a huge disconnect here.  In real life, we count up from 15 to 22, keeping a tally on our fingers of how many numbers we pass by.  We would never count up 15 from -22 to find how much someone owes us!  It’s no wonder students have difficulty with negative numbers with the way we are taught!

To plug my product, the ZeroSum ruler allows a student to count the spaces from 15 to -22 by folding the ruler in half at the pivot and counting from 15 to +22. When the positives are aligned with their negatives, they’re essentially finding the difference between the absolute values of -22 and 15.  This is the way we think and therefore a more natural way to learn.