# ZeroSum Ruler (home)

## Blogging on math education and other related things

### Manny Ramirez adds his fractions…January 7, 2011

Usually when adding fractions, we never ever ever ever ever add the denominators together.  That is, except for in baseball.  In a season of baseball, a “whole” is the entire season of at bats, not any one individual game. We won’t know what that whole is until the end of the season, so we keep adding the at bats (denominator), and tallying the numerator (hits), to find how many hits per at bats Manny has at any point in time during the season.  Weird, right?  But true!

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Scenario 1: Manny Ramirez’s batting average is 5/7 (ie .714, “Batting a 714!”, WOW!  Go Manny!) after two games: one game of 3/4 (three hits out of 4 at bats), and another game of 2/3 (two hits out of 3 at bats).  In other words, Manny has hit 5 times in 7 at bats, which was realized by adding the numerators and adding the denominators.

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But we’re told to never ever ever add denominators, so what happened?  What happened is simple: the “whole”, which is the basis of fractions, is defined here as the entire Baseball season at any point in time.  At this point in time, Manny’s whole season has consisted of 7 at bats.  The “whole” in baseball grows as each game progresses.  In fact, if we were to use the adding fractions algorithm to get a common denominator, we’d get 3/4 + 2/3 = 9/12 + 8/12 = 17/12!  Manny can’t possibly get 17 hits after 12 at bats!  That’s just nonsense!

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Wait, I don’t get it.  I hardly do, either.  But let’s try…

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Let’s think a bit more about Manny’s at bats.  Maybe if we thought of every at bat as its own whole, that is, each at bat is like a coin flip – he’ll either hit or not – we’d begin to understand what is happening.  Ah, we do!  BUT, we also have to keep in mind when we’re looking at his batting average: after 7 hits.  There is a common denominator here, it’s 7!  7 is the, albeit temporary until the next game, sample space.  When we look at 3/4 + 2/3 = 5/7, what we’re really looking at is (1/7 + 1/7 + 1/7 + 0/7) + (1/7 + 1/7 + 0/7) = 5/7!

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This can be seen with eggs, too…

Scenario 2: Here, we have 8/12 + 2/4.  If we add (8+2) and (12+4) we will get the fraction 10/16, and there are, in fact, 10 out of 16 spaces filled with eggs.  However, we completely disregarded the fact that the two containers are different sizes.  Let’s see what happens if we really ignore the discrepancy in container size:  If we first reduce 8/12 to “2/3” by chopping the numerators and denominators both by 4 (allowed!), and reduce the 2/4 to “1/2” by the old halfsies method (also allowed!), and then try adding the numerators and denominators together, we’ll end up with 3/5.  3/5 is definitely not the same as 5/8 (reduced from 10/16 by halfsies).  But why?

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We have to remember what we were doing, just like Manny had to remember that outfield is just as important as batting.  When we added the original numerators together (8+2) and the original denominators together (12+4) we were working with raw data, just like in the case of Manny Ramirez’s batting average.  What we really did was add (1/16 + 1/16 + 0/16, … you get the idea.  We defined the sample space as 16 because there are 16 total spaces for eggs, and we disregarded the different sizes of the containers.  If we first take the time to reduce the fractions, we change the fractions from ones that represent real information (actual egg numbers) to one that represents the proportion of eggs in each container.  Herein lies the problem.  How big is our whole?  We need to clearly define it.  If it’s 16, that’s fine if we consider 2 containers to be one whole.  But if we consider each container its own whole, we need to do things differently…

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If we are merely looking at how many eggs we have versus how many egg spaces, disregarding the discrepancy in egg carton size, we find that we have 10 eggs out of 16 total spaces.  16 is the whole.  This is useful information to have when baking a cake.  Or a few cakes and some French toast.  But if we first allow ourselves to reduce the egg carton fractions individually to 2/3 and 1/2, we change the problem from looking at one whole of 16 to two separate, differently-sized wholes of 3 and 2.  Once we do this, we enter into the realm of WHOLES.  And this is OK!  This is what fractions are all about!  There is a way to add wholes of different sizes; you just have to define how large you want your whole to be.

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But first, we have to remember an old mantra I heard somewhere, or didn’t hear anywhere, that Math is a Language.  Math is a language just as Portuguese is a language.  In Portuguese, you can’t talk in straight verbs, people would think you cracked your egg!  There are rules to follow when speaking Portuguese, and the same is true in math.  If we want to add 2/3 + 1/2, we absolutely can, but we first have to remember that each of these two fractions has already been given a clearly defined whole: one is the denominator 3 and the other is the denominator 2.  To add these portions of wholes, we have to first decide how large we want our end whole to be, and it can be any number.  It can be 1, 2, 1.17, 2.14, anything.  But what number makes sense, and more importantly, what number is easy to work with?  How about 6?

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Both 3 and 2 go into 6, so we can make the common baseball season, er, we can make the common denominator 6.  We do this by un-reducing the fractions:

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2/3 = 4/6 by multiplying the top and bottom both by 2.

1/2 = 3/6 by multiplying the top and bottom both by 3.

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Now we have our new common sample space, our new common whole, our new Common Denominator!  It’s 6!  Now we can add the numerators and come to 4/6 + 3/6 = 7/6.  The new common egg carton has 6 spaces for eggs and 7 eggs, or 1 carton and 1/6 of a carton.  We could make our sample space 12 and add 8/12 + 6/12 = 14/12, or one full carton of 12 with 2 eggs left over.

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But wait!  Why do we have one egg left over in the first addition and 2 eggs left over in the second addition?  Remember, we’re no longer talking real eggs here; we left real eggs behind when we decided to look at each carton individually and throw sample space 16 [rightfully] out the window.  We are talking “proportion of the whole”, and with fractions, we can decide however big we want our whole to be.  How many at bats will Manny have?

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For a picture tour on how to deal with fractions: Multiplying Fractions With Pictures!

### 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!

### 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…

### Patterns in i?November 29, 2010

You can imagine my surprise at the end of last school year when, on my tutoree’s final online examination, the imaginary number i was everywhere.  “WHAT?” I thought, “There was just one small section of one small chapter on i in the textbook and here it is, on my students’ final exam, EVERYWHERE.”  At best, it was frustrating.  Sure, math is math, but different publishers tend to focus on different topics, and i was not on of those topics Glencoe included much of.

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For five years, I had taught Algebra 1 and loved it.  The kids loved me and I loved all of their “ooooh, I get it!”s.  But this year had been different because I was moved up to Algebra 2.  So I set my mind to teach this slightly more advanced Algebra (at least with Glencoe it’s only slight), brushed up during the summer, got my curriculum down pat, taught a rough year right up until the final exam and….

BOOM!  i!

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Patterns are everywhere, especially in math.  The imaginary number i is no exception.  The number’s value follows an interesting and very distinct pattern, repeating itself every fourth iteration.  The pattern it DOES NOT fit is into a regular one in Glencoe’s Algebra 2 textbook.  I was mad that my students and I had worked so hard only to be sidelined by a final exam not connected to Glencoe at all.

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So this year I changed.  I taught i first!  We wouldn’t be stopped!  If the “patterns_of_i.xls” sheet over there in the margin for you to download and use in your classes is not enough, I’d be more than happy to email you more.  You can reach me at sdonohue@post.harvard.edu

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I quit this year, jumped a sinking ship, really.  It was horrible leaving the kids- like I was going on maternity leave and never coming back.  But it was the decision I had to make so that I could focus on my thesis, my health and on finding a job where I would be respected.  What they say about finding happiness first before you can pass it on is true.  What they also say about not doing school part-time unless your job is also part-time is also true.

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Now I’m finishing my thesis and looking for a new job, oh, and emailing you files to use in your classes.  I have thousands that I’ve made over the years that I’d love to share with you.

Go i!

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### Easy Factoring Trinomials…by grouping!November 27, 2010

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When I was a kid in high school, I remember trying seemingly endless combinations of numbers that would factor each assigned trinomial.  Well, you can kiss those hours of work good-bye!  Factoring by grouping is not only faster, it’s SIMPLE!  Just a few steps lie between you and complete trinomial factoring success…

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### Awesome Euclid and his AlgorithmNovember 19, 2010

One of my favorite things that I learned while in graduate school for math education was the Euclidean Algorithm for finding the Greatest Common Factor of two numbers.  If you click on the  picture to the left, you’ll get to a very informative YouTube video on the Algorithm.  It’s a bit boring, but very educational, and it shows exactly how to go about using Euclid’s method to find the biggest number that divides into two numbers.

(The screenshot to the left will bring you to the YouTube video on the Euclidean Algorithm)

The alternative, but mainstream, way using factor trees and circling primes always confused my students.  “Do I count the 3 twice since I circled it as a factor in both 81 and 57?”

If Euclid’s method was the mainstreamed one, math would be a lot more interesting and one more confusing topic could be checked off the list.  Euclid, you rock!

(The screenshot here of the kids is a funny video about Euclid and his algorithm)

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### Let’s grow some grass!November 7, 2010

Who knew?  Students love growing grass.  Let me elaborate…. Students love taking care of their grass as they watch it grow- enough to get them to do some pretty complicated algebra.

What started out as a simple week-long project that incorporated a bit of environmental sciences (my undergrad background) into algebra became a 10-week long project spanning the curriculum from ecology to linear extrapolation.  It’s been a long time in the making, but beyond a shadow of a doubt this lesson is one of the most engaging that I have created.   My students love the life aspect of the project and hardly complain about doing some pretty complex algebra.

Now, with the magic of a WordPress widget called ”My Shared Files    BOX” (on the sidebar), I was able to upload the Growing Grass Project files onto my blog for all to use!

All three files are important, but the excel workbook includes everything the student needs to create a final portfolio piece, including a formatted final excel sheet that the student can type into and cell directions.

I’m very excited about this and hope that if you do use the project, that you will add a comment to this post on how it went.  I also have other files that go along with the starter ones I posted.

I guarantee that your students will be engaged in their learning and that you will find ways to link most of algebra 1  to what comes up along the way.

WARNING!  This lesson takes on a life of its own!  Proceed with caution!

Now Let’s grow some grass!

p.s. For supplementary files, you can email me at ZeroSumRuler@gmail.com.  The files include ones on scatter plots and lines of fit as well as a PowerPoint and activity on linear inter- and extrapolation.