Fractions: Comparing Unlike Denominators

December 12, 2015 at 9:00 am (MATH) (, , , , , , , , , , , , , , , )

Crickets.  That’s the sound you would’ve heard if blogs could make noise.  Been a while since any real substance has been posted I know.  

Well, after spending Fridays bulking up on fraction knowledge, the unit came at last (last week actually).  We are now full tilt in the fraction realm: concentrating on their sizes, what it means to be a fraction, and how we can use mathematical tools to compare them?

Let’s start with what we know about fractions (for 4th grade concerns).  For one all fractions can, should, & must be divided equally or what us high fashion fractitioners know as equivalent groups.  Fractions should also be congruent (same shape, same size).  It’s all about that mythical creature known as fair.

It wouldn’t be fair to compare apples to oranges, this years class to last years class, Superman to the Justice League, or Larry Bird’s jumper to Steph Curry’s (Bird wins).

Who likes short shorts? #80's

Who likes short shorts? #80’s

If I told you that when I was a kid I’d come home and complete half of my homework, while at the same time my sister completed half of her homework, and yet I’d claim I had more work than she did, would you believe me?  What if I told you she was four years younger than I was and that she only had a spelling assignment each night, while I had spelling, math, and reading assignments to complete.  What then?  Is half still equal to half?  You want to talk about fair?  When one kid has ten spelling words and finishes five then gets to roam the neighborhood, while the other has a list of 24 words for spelling (not to mention the other subject matter to decipher), where is the justice?  I implore you ladies and gentlemen.  Where is the fair?  Can we compare these two halves to be equal?  Absolutely not, but an argument I did not win against my mother.  In order to compare these fractions we must think of how they can be rearranged prior to analyzing if one fraction is larger than another.  Therefore we must break, split, divide, group, etc into equivalent parts.    And here is where we get to the tools.

Carpenters need a nail gun to frame a stick built house.  Pete Rose needed a bat to beat Cobb’s all time hitting streak.  Number lines are one of the tools we need in order to compare and explain how greater one fraction is than another.

After we create our line, I instruct the kids to then find halfway – no matter if we are working with odd or even fractions.  You can always find half or the center mark.  For now, we’re concentrating on whole numbers 0-1 (but will get beyond one whole in time) and fractions halves-16th’s (sometimes beyond this as well).  So after locating the center and marking it as one half, we treat the number line as a teeter totter, going back and forth writing fractions until reaching the middle.  

For example, if working with 6ths we begin at each end with 0/6 (zero) as our beginning mark and then 6/6 (one whole) as the end.  So far we have one fraction on each side of the one half (halfway) mark, making it “fair”.  Making it equal.  Since we have an equal number we can ask ourselves, “What is half of 6?”  You betcha = 3.  Therefore 3/6 is the same as one half.  Now we can continue the method going back and forth among the center mark with 1/6, then to the other side of half to mark 5/6.  Now back to the left side of half to make 2/6, and finally skipping back to the right side of half to mark  4/6.  Voila!  We have number line divided into sixths.

Confused?  How do you think an eight year old feels after being exposed to this the first time?  How do you think I feel after explaining this concept?  But wait shoppers there’s more.

We have only made our tool, not yet comparing fractions.  Let’s stick with sixths.  If I want to know if 4/6 is <, >, or = to 1/2, then I can mark 4/6 on my number line I so heavily invested time in and see that it is past the halfway mark.  Now with practice some kids will be able to explain that 4/6 is > than 1/2 since half of six = three (3/6), therefore 4/6 is more because 4 is more than 3.  Yet not all kids can compute 1/2 multiplied by three on the numerator and denominator equates to 3/6.  Some of us need visuals to conceptualize this.  I can see from our model that 4/6 is indeed > than 1/2 (3/6) by  1/6 of a jump.

Note:  The model below was found via Google search and not my fav.  I would have marked zero as 0/6 and 0, as well as 1/2 and 3/6.  Until I get a tablet with a functioning camera we’re stuck with the interweb’s pics.   I digress.

 

Last lesson, you’re doing great focusing on the task at hand.  Gold CUB Paw.  Let’s throw in an odd number, thirds.  The kids would find 0/3 and one whole (3/3) to mark on their number line.  To locate 1/3 we’d stop and do something most public education has abolished and deemed as unlawful.  Thinking.  Think and ask yourself, “How does 3/3 and 6/6 relate?”  

Double the top, double the bottom.  Repeat: 1/3 doubled on the top =2, doubled the bottom = 6.  Same for 2/3.  Double 2 = 4, 3 again = 6.  Place these fractions in the same spot because we’ve found equivalent fractions folks.  Now if you’d start making a number line with thirds, it’s just as easy.  Find your center mark and teeter totter on each side, keeping an equal amount of fractions.  0/3 to left, 3/3 to the right.  1/3 to the left of half, 2/3 to the right of half.  I can look and explain to you 2/3 is > than 1/2 because it is past half way on the number line.

I can compare 2/6 and 2/3  as well.  Now we’re looking at unlike denominators, fancy speak for “the bottom number of a fraction isn’t the same”.  Highfaultin talk I tell you what.  I can see that 2/6 is < than 2/3 because 2/6 is before the number line and 2/3 is past it.  I could also explain that 2/3 is 2 jumps or more specific 2/6 jumps past 2/6 (aka 1/3 simplified).  The explaining why and by how much is the main focus when comparing the fractions.  Yes you can cross multiply/butterfly method but it will not give the student the ability to compare by how much one fraction is greater than another.  That’s not to say we do not use it, but we use it to compare as a back up.  They third method is changing fractions into common denominators which is also covered with creating the number line.  But for now, I think I’ve thrown enough heaters, and it’s time to bring in the lefty = class dismissed.

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2 Comments

  1. regina said,

    thanks for this!!

  2. Peggy said,

    In your homework analogy you left out a factor, time. So in fact, over time you and your sister may have had the same amount of homework and indeed it may have been fair.
    I resorted to pies/circles to help my kids with their homework last week and used a circular number line? It seemed to work but you would know more than I.

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