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
to represent fractional pieces of a whole.
When we add 
we have said that our total is
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 
means that our total is
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.
This will inspire them to work toward equality – and invent fractional addition (correctly) on their own!
Tell us how it goes in the comments!
Manipulatives in the pictures are from the Playful Patterns set from Discovery Toys. Buy them here (scroll to the bottom of that page).
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Hi Bonnie,
I’m not sure I’m with you on this method. Having children explore mathematics to discover how it all works is a great idea, of course. But wouldn’t they become confused if they try adding the way you describe here, and possibly never straighten it out?
Love to hear what you think!
I guess you’d have to judge the kid before doing something like this. I would say, though, that if the child was confident in exploring and looking at a variety of things in different ways, then it wouldn’t be a problem.
This is an exercise in seeing things differently as well as noticing how you don’t HAVE to do things in math like you’re told. If you want to build a bridge, you MUST do things like we’ve agreed on, but when you’re experimenting and playing, then the world of math is your own.
The biggest frustration having people say, “There’s only one way to do math” or “there’s only one answer in math.” If teachers continue to count problems wrong if they’re merely done differently (or the result is in a different form) than the teacher prefers, then we will continue to have only algorithmically minded kids. If we help children see that they really can do anything they want, then we’ll encourage them down the path of creative and mathematical thinking.
The “mediant,” which is what you get by adding numerators and adding denominators is actually a very useful thing in some contexts, but I have not tried it in teaching kids to add fractions.
I usually teach kids to add fractions by finding the LCD as the LAST step, because it is so much faster and simpler than the traditional method. If we insist that kids must find the LCD first, they cannot get started on problems like 11/42 + 5/48. If we allow the kids to find the LCD *after* they find the answer’s numerator, life is so much simpler.
hmmm!
Your curiosity, Donna, makes me curious!