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!
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You can teach fractions this way. But I would not make a fetish out of it. Talk first of equivalent fractions. Later let the kids choose. Addition of fractions is not about the formula a/b + c/d = (ad + bc)/bd or its LCD variant. It’s about finding a common unit of measurement. If kids understand that, i.e. is not much difference about the unit selection. Choosing for them just one possibility is hardly the best practice.
Thanks for the comment, Alexander.
The problem with teaching math to kids and using phrases like “it’s about finding a common unit of measurement” is that they don’t care about that. They care about fun. And cupcakes.
Make fractions about fun or cupcakes and you’re in. Later they can figure out how it all works together.
Interesting idea. I have taught fractions this way before and told the kids this is called the ‘macarena method’. See my blog post for explanation of this fab way of remembering the method. http://www.greatmathsteachingideas.com/2011/02/18/adding-fractions-the-macarena-method/
However, these days I teach adding fractions using this method: http://www.mathsmaster.org/MathsMaster.Org/Adding_and_subtracting_fractions_with_different_denominators.html
I find that the later method is understood by the kids and still retains the main concept that equivalent fractions are the same size. By ‘coming at it backwards’, by starting by adding fractions with like denominators then using equivalent fractions to find fraction sums with different denominators I find that the kids understand better when we start going the other way….
I LOVE the Macarena Method! I’m not so sure about that star thing. Looks like it takes a lot of effort. But I’m all about many methods for one task – everyone connects to different things.
The key is to get them to connect to anything first. After which they will more easily connect to all the other things.
Thanks for the comment and links, Will.
Alternate ways of doing things are wonderful, esp when they help kids get more to the root of what the tricks actually DO!
(Yeah, you and I are going to have fun talking about algorithms tomorrow night! 😉
Yep, Siggi. If they can’t get in through the front door, offer them the back door. If that still doesn’t work, try ever window in the house. And if they still can’t get in – by golly drill a hole in the roof!
I agree that the experience is painful at best of times, but providing students with another “one-size-fits-all” memory cue takes the meaning out of it. This looks very attractive only when compared to the algorithms that are usually taught. This in no way builds undersatnding. Yes it works, and the simple generalization in your recent video shows that, but it is does not enlighten adding fractions. I think teachers should strive to show students why we need common demominators, rather than try to show them tricks to side step them.
Thanks for your comment, Nat.
Teaching this method alongside finding common denominators will enhance them both. If a student then chooses to use this method instead of finding a common denominator, then it works. Indeed it shouldn’t be a “one-size-fits-all” method and I mightn’t have labeled it this way.
Instead of teachers trying to show why we need things, they can show a multitude of different ways to arrive at the same thing. English classes do this all the time when they say, “Compare and contrast the …” A teacher can’t know what’s going to click with a student. Even homeschool parents can’t always know what’s going to cause understanding. Instead of explaining, they (we) can offer various ways of experimentation or methods to help them explain to themselves.
bd is the common denominator. It may not be the LEAST common denominator, but it is A common denominator.
Good point, Maria. Although, you didn’t have to find it before coming up with the answer. So perhaps the title of the article should be “how to never find a common denominator before getting to the answer, again.”
Although one could argue that we never “found” a common denominator. We actually just arrived at the final answer and the denominator of that final answer happens to be one of the many possible common denominators.
Although this does find a common denominator, it always leads to non-reduced answers if the two denominators are not relatively prime.
I use a different method, one that finds the LCD as the LAST step in a very easy and algorithmic way.