# ZeroSum Ruler (home)

## Blogging on math education and other related things

### Solving equations with the ZeroSum RulerDecember 18, 2010

The ZeroSum Ruler came from a need to teach my algebra students how to balance equations such as “solve for x in: 22x + 17 = 3x + 5″, where the student has to either subtract 3x – 22x or 5 – 17 to get to the answer. My students were having a lot of difficulty with this, answering that “3x – 22x = -25x”, and so on. As it turned out, this was the BIGGEST mistake my students were making in algebra, which boils down to simple integer subtraction!

(Click the picture over there <— to go to the ruler’s video)

The ZeroSum Ruler has proven to increase understanding of integer addition and subtraction (click here to read the RESULTS of my thesis study) by 62%! One other method of teaching this idea is the “red chip/black chip” model where an amount of red (negative) chips cancel out a same amount of black chips. Sure this method works, but the chips get cumbersome. The ZeroSum Ruler works the same way as the chips except without the chips. No more buying multiple sets of checkers to perform integer addition and subtraction!

Another method that is in widespread use to teach problems such as “-22 + 17″ is the rigid number line. With this example, a student would be directed to find -25 and count 17 spaces to the right. A real-world example of “-25 + 17″ might be “Jim borrowed \$25 from you and has paid you back just \$17. How much does he still owe you?” In this problem, which is exactly “-25 + 17″, it is easier- and intuitive- to count up from positive 17 to 25. But how would that work on a number line? It doesn’t! But, the ZeroSum Ruler is foldable, allowing its positive numbers to align with their negative counterparts and therefore allowing students to solve integer addition and subtraction intuitively.

The ZeroSum Ruler eBook contains a ZeroSum Ruler cut-out to put together, simple instructions on how to construct and use the tool, and practice problems.  You can purchase a ZeroSum Ruler eBook through CurrClick or through SmashWords

HaPpY Calculating!

### When does -22 + 5 = -27?December 6, 2010

My graduate thesis is a study of the long-term effects the ZeroSum ruler has on eleventh grade student understanding of negative integers.  By eleventh grade, students should easily be able to answer “-22 + 5 =”, but on a diagnostic test given to 57 students, 40.35% of the students answered this problem incorrectly.  Why does this matter?  It matters because it shows that students did not learn the relationship between negative and positive numbers in elementary or middle school.  By the time they get to me in eleventh grade and need to be fluent in equation manipulation, answering “-22 + 5 = -27″ is a real problem.

My thesis was set up the following way:

1: Diagnostic test: eight simple sums and differences of integers  (ie: ’22 + 5=”) without a ZeroSum ruler or calculator

2: Introduction to the ZeroSum ruler with examples

3: Three activities, spaced out over 2 weeks,  using the ZeroSum ruler

4: A post test within days of the last activity (no ZeroSum ruler or calculator)

5: A delayed retention test one month after the last activity (no ZeroSum ruler or calculator)

Because the attendance rates of students in Boston Public Schools is not the best, especially by the 11th and 12th grades,  a subgroup of 31 students was identified who took the diagnostic test, participated in at least 2 of the 3 activities with the ZeroSum ruler, took the post test, and took the delayed retention test.  The data shows a 62% decrease in student error from the diagnostic test to the delayed retention test because of the ZeroSum Ruler!  These results indicate that the ZeroSum ruler works to improve student comprehension long-term even without the ruler.

Pretty exciting stuff.

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### ZeroSum Ruler eBook!December 5, 2010

An eBook, complete with a ZeroSum Ruler cut-out, may be coming soon to CurrClick.com!

In the meantime, you can download a ZeroSum Ruler eBook and cut-out for \$4.00 through the ”Buy Now” button below, hosted by PayLoadz.com and PayPal…

### 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.

### Thank you Allan Cohen!April 5, 2010

Allan Cohen writes a blog called “Classrooms Without Walls” and gave me a shout out at:

http://www.gather.com/viewArticle.action?articleId=281474978153616 .  How super!   Thank you Allan!