Tag: common denominator

  • Why It’s Okay to Add Denominators when Adding Fractions

    Why It’s Okay to Add Denominators when Adding Fractions

    I did some videos for avoiding finding a common denominator and why this trick works. Ever since then I’ve pondered what it would look like if you added without a common denominator.

    A mom on the Living Math forum was asking for advice on how to teach adding numerators and not adding denominators in fractions. So this topic is again in the front of my mind.

    First note, that you can add both numerators and denominators – this is called piecewise addition – but then you are getting into a whole new world of stuff. Actually, you’ll be experimenting with some goodies from Abstract Algebra (my field of study). We’ll talk about how and why this is good in a few paragraphs.

    What is adding fractions, anyway?

    For us in the “real world,” adding fractions is adding values that can’t be counted with fingers. It’s adding bits of things together – maybe whole things with bits, even.

    We have agreed to use things that look like

    some number on topsome number on bottom that isn’t zero\frac{\text{some number on top}}{\text{some number on bottom that isn’t zero}}

    to represent fractional pieces of a whole.

    When we add 12+14=34\frac{1}{2} + \frac{1}{4} = \frac{3}{4} we have said that our total is

    3 pieces of a whole pizzawhen the pizza is cut into 4 equal pieces\frac{\text{3 pieces of a whole pizza}}{\text{when the pizza is cut into 4 equal pieces}}
    Total: 3/4 of a “pizza” is blue.

    Quantity over value is important to young children.

    I remember taking money from Little Brother when we were kids. You might think I was stealing, but it was more of conning the innocent. He would have 3 dollar-bills. I would have 20 pennies.

    He would quite willingly trade his 3 monies for my 20 monies. For him, this was a matter of quantity. I knew the value, though. (Bad girl – sorry, LB I owe you. xo)

    This measuring of quantity, rather than value was important in LB’s development. Instead of someone teaching him how to measure value, he learned it through experience.

    This is why we should allow kids to add the denominators.

    Let’s use the example above and add the numerators and the denominators to see what that means.

    Adding 14+12=26\frac{1}{4} + \frac{1}{2} = \frac{2}{6} means that our total is

    2 pieces that we have6 pieces that were possible to have\frac{\text{2 pieces that we have}}{\text{6 pieces that were possible to have}}

    Notice here you can’t “reduce” the fraction – because this isn’t the value  as we know it.

    If you teach adding fractions using manipulatives or toys and allow adding denominators, soon your children will see that your   (on the top in the picture below) is very different than having their  , on the bottom.

    We have six total pieces.

    This will inspire them to work toward equality – and invent fractional addition (correctly) on their own!

    Tell us how it goes in the comments!

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  • LCM and GCF – And Something Really Cool

    LCM and GCF – And Something Really Cool

    In a previous post, we learned that you don’t have to use a LCM (least common multiple) to find a common denominator to add fractions. In another post, we learned that you can use primes instead of the GCF (greatest common factor) to reduce fractions.

    And now I’m going to demonstrate that finding the LCM and GCF of two numbers is still pretty cool.

    Apologies for the glare on the video. I tried a different method and clearly it is inferior to the other method.

  • Adding Fractions: Why Avoiding Common Denominators Works

    Adding Fractions: Why Avoiding Common Denominators Works

    I wrote this post showing that adding fractions can be done using the algebraic definition of addition of rationals:

    ab+cd=ad+bcbd\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}

    MathHeadInc (via twitter) has requested a video showing why this works.

    I aim to please.

    Here is the video showing why adding fractions using a common denominator is the same as the definition of addition of rationals:

    What do you think? Will this help convince your kids that “the trick” is okay to use? Share your experiences in the comments.

  • How to Never Find a Common Denominator Again

    How to Never Find a Common Denominator Again

    Do you keep struggling to teach common denominators? Do your kids just not “get it”?

    Well, it’s time to quit.

    Yep! You can teach (and do) fractions without ever finding a common denominator.

    The key is in the definition of addition for rational numbers. If you have two numbers  and  the sum of them is .

    Granted there is the technical issue of reducing, but using a little prime factorization will get you there without an issue. (And you don’t really HAVE to reduce, which I will discuss in a later post.)

    Here is an example:

    Here’s one with an obviously easy common denominator that works just fine with this method. In it I explain a little about why you would want to do it this way:

    You might argue that finding a common denominator is an important learning experience. And you’re right. In the same way that learning about death through the loss of a pet is an important learning experience. But if you can avoid all that pain, why wouldn’t you?

    If you teach this method first, kids will get annoyed with having to do so much reducing and discover the common denominator method for themselves. And that’s really what learning math is all about.

    What do you think? Can you teach fractions this way? Share your thoughts and experiences in the comments.

    Thanks to @padgets for our conversation about teaching fraction on #mathchat a few weeks ago. You inspired this post!