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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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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We’ll stop taking long division for granted in a minute, but first, let’s take a general look at division. What is division? Division is the shortcut for subtracting one value from another value repeatedly until we reach zero. We say that “90 divided by 10 is equal to 9″ because 90-10-10-10-10-10-10-10-10-10 = 0. Using the distributive property, we can rewrite this as 90 – 9(10) = 0.
If we then made steps to solve this equation, we’d get:
90 – 10(9) = 0
+10(9) + 10(9)
90 = 10(9) 90 is “ten, nine times”. Division is the inverse of multiplication.
But division also works in situations where we would not eventually subtract to end with zero. For example, we may end with the number 5. “95 divided by 10 is equal to 9.5″ because 90-10-10-10-10-10-10-10-10-10-5, and that last 5 is one-half (.5) of the 10 we had been subtracting from 90. This 5 is also known as a “remainder”. The result of “95 divided by 10” can then be written as “nine with a remainder of 5”. This remainder is a smaller number than the number being repeatedly subtracted, so does not quite fit in (“remains” outside).
95 – 10(9) = 5
+10(9) + 10(9)
95 = 5 + 10(9)
95 = 10(9) + 5 [commutative property]
95 is “ten nine times with five left over”
However, subtracting repeatedly can be tedious for large numbers, such as “950 divided by 10”, so the long division algorithm was developed. Formatting “950 divided by 10” to the algorithm, we would write:
The sideways “L” is referred to as the “division bar” in this paper.
Here, the decimal can be expanded to 950.00 because of the definition of our number system. This also makes room for any possible “remainders”, or numbers less than 10 that are left over once we subtract all the 10’s from 950 that we can.
To work this algorithm, we want to first ask ourselves, “how many 10’s can be subtracted from 9?” We want to remember both: that our definition of division is “repeated subtraction” and that “multiplication is division’s inverse”. 10×9 is 90, not “9”. We’d answer with “zero” and imagine a “0” above the 9 in 950. Here, we’ll put in a zero, but usually we wouldn’t.
Usually, when the number above the division bar is not zero, we would move to the second step of the algorithm. But since our “0” is imaginary, we won’t move to step two quite yet.
Starting the algorithm over, we’d then move in from the “9” and look at “95” and ask ourselves, “How many 10s cam be subtracted from 95?” We’d answer with “9” and place it above the tens spot in 950.
Our placement of the “9” actually represents a “90” as it is vertically aligned with the 950’s tens placement. Had we placed this “9” over the 9 in 950, it would have represented a 900. By placing our “9” over the 5 in 950, we’re saying that “ninety 10s can be subtracted from 950”, which we know to be true. If you notice the placement of this “90”, you will see that its 9 is vertically aligned with the 950’s hundreds placement. Does this make sense? (remember the definition of division as multiplication’s inverse!)
Continuing to step two of the algorithm, we write this “90” below the “95” to find the next remainder, remembering that this “90” actually represents a “900” because of its placement.
We are left with “5”, which is actually a “50” because of its placement below the tens spot of the 950. The “5” in the original 950 is in the tens spot, therefore our remainder of 5 is actually a remainder of 5×10, or 50.
The algorithm then tells us to “bring down” the next digit in the original 950 and repeat the algorithm. But what is the purpose of this “bringing down”? We always need to keep place value intact and our “5” is really a “50”. By bringing down the “0”, we accomplish this. Also, remembering that our “9” is a 90 because of place value and that division is repeated subtraction, we have:
950 – 10(90), which yields 50 (not 5)
We bring down the “0” from 950 and place it next to our 5 remainder to make a “50”, creating the 50 we already knew to be true.
If that remainder of “5” had been a “0”, we’d be moving into the ones place. Because it is a “5” and aligned in the tens spot of “950”, we’re still working within the tens. We then ask ourselves, “How many 10s can be subtracted from 50?”, and continue on with the algorithm.
We end here by saying that “950 divided by 10 is 95”. In the division algorithm, when we end with a “0”, we stop the algorithm because it means that the divisor (10) can be subtracted from the divisee “950” an even amount of times with no remainders. But we can use this same algorithm when dividing any number by any other number.
Comments welcome!
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.]
The ZeroSum ruler improved my student’s understanding
of integers
by 62%
in a very short 2 weeks.
Surpassed even my high expectations!
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HaPpY Calculating!
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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!
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“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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.”
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Thank you, Old Schoolhouse Magazine! 
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You can purchase a ZeroSum Ruler eBook here: The ZeroSum Ruler on CurrClick or on my blog over there —>
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The ZeroSum Ruler is on CurrClick! In the FREE informational eBook, you can learn all about the ZeroSum Ruler, how it works, and what makes it a different and more INTUITIVE way to learn. 
