# ZeroSum Ruler (home)

## Blogging on math education and other related things

### Adding Fractions With Pictures! (The Crisscross Method)December 3, 2012

Fraction Addition (And Subtraction): We’re not in kindergarten anymore

Addition and subtraction are only easy in elementary school.  Once middle school starts, continuing throughout any Math class taken that point forward, addition and subtraction are much harder than multiplication and division.  Why?  The Common Denominator.  To a kid who is not fluent in his multiplication facts, finding The Common Denominator is an exercise in torture.

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What is a common denominator?  A common denominator is a multiple of both denominators in a fraction addition (or subtraction) problem.  For example:

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In the above example, 6 is a common denominator of 2 and 3.  But is it the only one?  No.  How many common denominators are there between two fractions?  Infinite.  For example:

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Why would we want to use 7830 as a common denominator?  Why not?  The point is that any number that both denominators divide into evenly can act as a common denominator.  We are far less restricted than we thought.

So if we’re virtually unrestricted in choosing a common denominator, why not pick the one that is the product (multiply) of the two denominators?  For example:

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Just multiply the denominators to find a common denominator.  This is easy.

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At this point in the traditional method of adding fractions, we’d begin to ask our questions: “How many 8’s go into 16?”  Ok, 2.  “2 times 3 is …?”  Ok 6.  So 3/8  =  6/16 .  Though this process is easy to a person who is fluent in their multiplication and division, it will give reason for a non-fluent Math student to seize up.

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A great alternative way of adding fractions is the Crisscross Method of adding (and subtracting) fractions.  In this method, we use the common denominator just once (this method will not create two equivalent fractions to the original two) and multiply “crisscross” to find two new numerators.

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In  3/8  +  5/2, we’ll first multiply the denominators to find our new, common denominator:

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Next, we’ll multiply 3 • 2 (always starting our crisscross in the top left corner) to find the first missing numerator:

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And then 8 • 5 to find the second missing numerator:

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But why are we allowed to do this?  Let’s back up to see what really happened.-

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First, we found the common denominator 16 by multiplying the denominators (8 and 2) of both fractions.  We’re guaranteed that our denominator is common if we created it by multiplying the two original denominators to get it.  To get the first numerator 6, we multiplied the numerator of the first fraction (3) by the denominator of the second fraction (2).

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In the process, we multiplied both numerator and denominator by 2.  In other words, we multiplied  3/8 by  2/2 Any number divided by itself is just a fancy 1, and multiplying any number by 1 does not change the number’s value.  As a check to see if this process worked,  3/8  =  6/16 .  The old and new fractions are equivalent.

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The same is true to get the second numerator 40:

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Both numerator and denominator were multiplied by 8.  In other words, we multiplied  5/2  by 8/8, which is just a fancy 1.  Multiplying by 1 does not change a number’s value.  As a check,  5/2   =  40/16.  The old and new fractions are equivalent.

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Now we simply add the numerators:

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The Crisscross method also works for fraction subtraction – we’d have a subtraction in the numerator.  Why was this method not taught in school?

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Hurray for Fraction Addition (and Subtraction)!

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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!

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### Dirty Word of the Day: MemorizationOctober 21, 2012

As a high school Math teacher, I hear all the time, “I suck at Math!”, especially, considering that everything else in the world is found at the push of a button, when my students are faced with problems they can’t immediately solve.  I hear “I hate Math” when we’re solving equations, when we’re factoring, when we’re plugging x values back in to find angle sizes.   I hear it all the time.  but it’s when I hear it that got me thinking.

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Since 2003, I’ve been keeping a mental log of all the times I have heard “I hate Math” or “I suck at Math”, mainly because each one has left its own little crater on my Math soul.  I want – need, really - to figure out why kids feel this way.

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It took nearly 9 years, but yesterday I finally figured out why some kids hate Math with all of their being.  It’s because they can’t multiply.  This had been my suspicion for a few years, but yesterday it became clear that multiplication makes the difference.

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“But multiplying is easy, it’s not that.  Math is WAY harder than just multiplying!” you may say.  And I agree with you.  However, Math is 90% confidence, and when a kid loses this confidence because “multiplying is easy” and he can’t multiply, then he feels like a loser and closes off to the rest of his years of problem solving.

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The Conclusive Evidence

I had never seen a little kid do Math until yesterday when a former student of my husband came over with her Mother for lunch.  She’s in 4th grade now and has been having trouble with Math, so we sat down with her current homework: multi-digit multiplication problems.  The algorithm “multiply then carry, then multiply again and add what you carried” is a little weird, but she got that part.  Then all of a sudden out of nowhere, with fists slamming on homework…

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“I stink at Math!”

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“I hate Maaaaath!”

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Within the giant problem my husband gave her to do, she needed to multiply 8 x 5.  When it didn’t come immediately, she exploded.  And up went the walls.  Single-digit “multiplication is easy”, right?  Not if you don’t know it.  If you don’t know 8×5, then Math is the shittiest subject there ever was, ever is, or ever will be.  It totally blows.

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So what would help her?  Memorizing her multiplication tables.  Sounds simple and ridiculous, right?  Hold on a second.  Below are a few excerpts from an article, “Chess Experts Use Brains Differently Than Amateurs”:

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Experts use different parts of their brains than amateurs, maximizing intuition, goal-seeking and pattern-recognition, says a new study that examined players of shogi, or Japanese chess.

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Researchers believe that experts who train for years in shogi are actually perfecting a circuit between the two regions that helps them quickly recognize the state of the game and choose the next step.

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“Being ‘intuitive’ indicates that the idea for a move is generated quickly and automatically without conscious search, and the process is mostly implicit,” said the study.

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Math is a lot like chess: strategy, visualization of next moves, attack!  When a kid is a multiplication amateur, strategy can never develop, patterns will never be recognized, Math will always be counterintuitive.  Multiplication facts take a lot less time to master than chess.

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Memorization: Mathematicians’ Dirty Word

Memorizing sight words doesn’t make reading Harry Potter easy, but it does make it easier.  This is why we do it.  So why not memorize multiplication facts to make Math easier?  At some point between being a Math student and being a Math teacher, ”memorization” became a dirty word.  I agree that we shouldn’t force kids to memorize every Mathematical formula or the digits of pi, but I remember a deep sense of pride in having my multiplication facts memorized.  Maybe the way it was done – calling us up one by one to recite the facts to our 3rd grade teacher – was not the best method and probably contributed to my high-strung demeanor.  But when I got to pre-Algebra, I could cross-multiply; in Algebra I could quickly find factors; and in Geometry I could “plug it in” without a calculator.  All of these seemingly-unrelated abilities contributed to my feeling that Math wasn’t impossible.  I had confidence because I could multiply quickly.  I was fluent, solving came easy.  I could do more advanced problems because I had confidence.  I had confidence because I didn’t need to stop and think through every instance of multiplication.

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A recent brain study done by Dr. Carolyn McGettigan in the UK yielded unexpected results.  Contrary to hypothesis, the expert beatboxer uses less of his brain than the novice:

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The novice used many more brain areas, suggesting a need to plan each sound and a lack of automatic processing.

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Dr Carolyn McGettigan, a neuroscientist at University College London, compared magnetic resonance imaging (MRI) scans during two tasks – counting and beatboxing.

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Dr McGettigan says: “When you think about an expert you might think they activate extra bits of the brain – not just the bits you use to make sounds, but something exciting and different that you might not expect.”

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“What we have at the moment is a demonstration that being an expert doesn’t mean you activate more of your brain. The phrase ‘less is more’ is sort of appropriate here.”

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Thinking is Overrated

Thinking is important, but not that important.  When it comes to the building blocks of any language, a level of fluency is essential.  Without it, reading is exhausting, and things that are exhausting are avoided.  I doubt that my husband – a true bookworm - would have read the entire Harry Potter series [more than once] if he had to individually sound out each word.  It just wouldn’t have happened.  J.K. Rowling would have never earned the necessary funds to  go on to write The Casual Vacancy (is it any good?) if everyone struggled through her Harry Potter books.  Is it any mystery that kids hate solving equations or finding missing side lengths if they have to “sound out” each instance of multiplication?

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So what to do?

We have to stop making our kids think so much!  It’s exhausting them.  When I was a kid, my parents gave me this Math toy that tricked me into learning my multiplication facts.  Flashcards for facts up to 12×12 are also great.  It’s got to be fun, not forceful, of course, but it’s got to happen.  It will make all the difference later and is really that simple.

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### Wanna be a Math Hero? Answer these questions!September 17, 2012

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These Math students need YOUR help.

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If you’ve checked all recent posts on Facebook, refreshed your Twitter page until it can be refreshed no more, all of your Pinterest friends seem to be on vacation and your email is all read, why not answer some Math questions?

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I’ve been addicted to this site since last night, which in 2012 terms is an eternity.  All I can imagine are kids all over the US toiling away at their Math homework, one hand on head, one wrapped around a pencil, foregoing food, sleep, showering, just to get tomorrow’s math work complete in time for their teachers to put a small check in the corner.   Hey, maybe a few teachers are stickerers, I don’t know.  Personally, I’m a grape-flavored stamper.  So here I come to the rescue!  The THANK YOU! emails are cool to get; I do feel a bit like a hero today.

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Questions range from “Plz help me graph y = 45x + 40” to “What is the square root of 1 – i?  So try it out!  It’s a great way to put that advanced degree to good use!

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### The Math Book by Clifford Pickover is awesome!July 11, 2012

“Have you read The Math Book?”

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“Which math book?”

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The Math Book?”

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“Umm, ??”

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“There’s a The Science Book, too!”

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#*%^&!!!!!

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No really!  There IS a book called The Math Book  and it’s by the fascinating and mind-boggling book-producing author Clifford A. Pickover, who you can even follow on Twitter

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Before I was introduced to this book, I would have been the red converser in the above dialogue.  I mean, surely the title The Math Book was taken back in the stone ages.  How could there possibly also be a The Science Book (and a The Physics Book and a The Medical Book, etc.) in modern times?

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I’m not sure how it happened, just that it happened and that it’s awesome.  The Math Book was published in 2009 and has a cooler format than just about any other math book I have ever cracked open (possibly tied with Glencoe’s high school math textbooks, but those are textbooks so are in a different category altogether) and certainly cooler than any other book about math I have ever read.  As you turn each page, you are surprised by yet another full-page picture relating to the previous page’s concise summary of some even stranger mathematical topic.   Here we can see the page on Zeno’s Paradox, a mathematical problem that has plagued mathematicians for centuries.  Did you know that Math states you’ll never make it out of your bedroom door?  If you get up to leave (dinner’s ready!), at some point you’ll be halfway between your computer chair and the door.  At some point later you will be halfway between the halfway point and the door.  You journey to the door is marked by an endless series of halfway points, and you can always take half of whatever distance you have left to travel (think the molecular level… and beyond).  Hope you stashed snacks!

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Ant stride lengths, cicada life cycles, spirals, knots and Murphy’s Law (doh!), fractals as used in animation, Cryptography, magic squares, slide rules, imaginary numbers, Möbius strips, Zero, monkeys typing on keyboards, monsters!, and dimensions far past our comfortable 3 are just some of the many topics, broken down by year, surveyed in this wonderful book.  If you’re a math teacher of students who often ask for extra credit, this book is a great jump-off.  The kids can look through the book for something that catches their fancy and do additional research until their heart’s – and page number requirement’s - content.   This is a truly awesome book for all ages.

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contact blog author Shana Donohue: shanadonohue@gmail.com

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### Is Common Core meeting its Goal?May 21, 2012

Is the original goal of Common Core being lost in the upper grades?

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One major difference between the U.S. and so-called ”A+ Countries” is, while we focus on breadth, they focus on depth.  While there is a natural progression throughout a student’s school years from one math topic to another in these high-achieving countries, in the U.S. we seem to have a “throw at the wall and see what sticks” mentality.  For example, in grade 8 we cover 32 unique mathematical topics.  In high-achieving countries this number is just 18.

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The new Common Core curriculum aims to bring our focus back to depth in the lower grades but seems to miss this mark once the abstract maths are reached.  While it is true that more topics have been cut out than added in most grade levels, topics traditionally covered in Algebra 2 (and some may say pre-Calculus and above) – piecewise functions, limits, logarithms, areas under curves, Algebraic proofs, and rational function graphing to name a few – are now part of Algebra 1.  Does adding so many advanced topics to the Gateway of Higher Math (ok, I’m biased) do what Common Core initially set out to do?

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Below is a comparison of the math topics taught each year in A+ countries (first chart) and those covered in the U.S. (second chart) each year (compiled by Professor W.H. Schmidt).  These comparison charts were created before, and as a support for, Math reform in the U.S.  Still, to meet the new upper-grade Common Core Standards, school districts are turning to hybrid-type courses: “Algebra/Geometry/Stats Year 1″, etc. (Yes, that’s ONE year’s course) to meet all of the new high school requirements.  While the Common Core sets out for mastery at the Elementary level, did it really mean to hybridize high school math?  If depth is more important that breadth, what are we doing?

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### Grade 8 Math MCAS Review – PreziMay 2, 2012

Math MCAS are quickly approaching.  Below is a multiple-choice 8th Grade Math MCAS review Prezi (phew!) covering 25 question types that students often get wrong.  This Prezi is viewable (and downloadable) at http://prezi.com/8me0m014oo_u/mcas-prep/ (or by clicking the picture).   Please email me if you’d like editing rights to this Prezi.

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Thank you to Paolo Tosolini for his AWESOME YouTube tutorial on how to create a cool Prezi.

### Archimedes says “cone + sphere = cylinder”?April 14, 2012

Filed under: algebra,math education,number theory,proof — ZeroSum Ruler @ 5:19 pm
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Archimedes of Syracuse (87 BC – 212 BC), before the eternal time sucks of Facebook and DrawSomething, sat and imagined a cone and a sphere balancing one side of a scale, and on the other side balancing against these two objects was a cylinder.  The sphere he imagined has the same diameter as height of the cylinder and the cone and he reasoned that the scale would remain perfectly balanced.

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But would it?  Using just their volume formulas, which Archimedes did not yet have, can we see that this works?

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Does sphere + cone = cylinder?

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