# ZeroSum Ruler (home)

## Blogging on math education and other related things

### The Coolest Math Class Ever. Seriously!June 6, 2011

I was lucky to get into Oliver Knill’s class Teaching Math With a Historical Perspective, one of the choices within the Harvard Extension School’s ALM in Mathematics for Teaching program.  It changed the way I think and the way I teach.  He explained complex topics, such as code breaking and non-verbal proofs, with such ease.  He was inspiring and made me look at math in a way I had never before – from a historical perspective!  His site is worth checking out, especially if you can not get into his class!  (Click on the red circle to go to Oliver’s site)

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### The long division algorithm, explained at lastMarch 29, 2011

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.

### US vs A+ Countries: Breadth vs Depth in Math. Which is better?December 6, 2010

(Click chart to enlarge)

Schmidt, William H., Wang, Hsing Chi., McKnight, Curtis C., J Curriculum Studies, 2005, volume 37, number 5, pages 525–559

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### When does -22 + 5 = -27?

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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### Graduate Thesis on… Negative Numbers?November 22, 2010

My 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.  These results indicate that the ZeroSum ruler works to improve student comprehension long-term even without the ruler.

Pretty exciting stuff.