# ZeroSum Ruler (home)

## Blogging on math education and other related things

### New (free) ZeroSum ruler – for teaching addition with negative numbersSeptember 30, 2012

Below is a new version of the ZeroSum ruler.  This one needs no hardware to construct, just scissors and glue.  You can download, print and use this proven tool right now by clicking on the picture, which will bring you to the PDF file that contains 2 ZeroSum rulers.

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### Subtraction as “as compared to”December 10, 2010

I learned the best thing today at an interview, which I know sounds a bit weird.  Usually at interviews it’s all about what I’ve done and where I see myself in 5 years and whatnot.  This interview was far different, and better, and awesome.  Today at my interview, I learned something really great about… subtraction.

Let’s take a problem like “7 - 2“.  We can read the subtraction sign as “as compared to“.  When we find 7 and we find 2 on a number line, comparably they are 5 apart.  And indeed, 7 - 2 = 5.

But does it work for subtracting negatives?  let’s check…

“7 - -2″.  Seven as compared to negative 2.    7 and -2 are 9 apart, and in fact    7 -  -2   =   7 + 2 =   9!

This is a SUPER model that, as my boyfriend just said, opens a whole new world.  Who knew at 33 I’d still be learning subtraction!

Awesome!

### 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.