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• 13 Rules That Expire article
• Graham Fletcher’s guest vlog post

So the first one we’re gonna take a look at is, all of these, actually, are around fractions, but this first one is about the idea that when we have fractions that are greater than one, we should always write them as a mixed number.

Fractions Greater Than 1 Should Always Be Written As A Mixed Number

Yes, they can be written as a mixed number, but there are often times where it is beneficial to leave that fraction as a fraction greater than one. Let’s say they’re doing a problem and they come up with an answer to part of the problem that is 17/8.

Well, if they transfer that into the mixed number of 2 1/8, but then the next part of the problem has them subtracting 3/8, that makes it even more difficult to solve the next part of the problem. Instead if they had just left it as 17/8 and then went to subtract 3/8, it’s super quick to do that problem. And then, if they need to, at the very end, yes, you can put it as a mixed number to figure out what that value is.

But there are often times where you do not need to be turning those fractions that are greater than one into mixed numbers. And as kids get into the upper grades, a lot of the work that they do that involve fractions greater than one, like dealing with slope, it is always left as a fraction or as a fraction greater than one, instead of transferring it into a mixed number.

Now the rest of these are all language that we use when we talk about fractions that might not necessarily expire, but they need to expire from our vocabulary because it’s doing a disservice to our students in building their understanding around fractions.

Stop Saying “Out Of”

When we say “out of,” like for the example, ⅔. If I say that as “two out of three,” it’s a ratio view of fractions, which is a more complex version of fractions, and it also makes kids think that it’s two separate things. Two out of three, instead of seeing it as 2/3. It is one number; a fraction is one number.

We also need to be saying that one number with the attribute that it attaches to. I don’t know if that’s the right word there, but basically, we need to put it in context, people, because fractions only make sense if they are in a contextual situation. 2/3, I need to know what it’s of. Is it 2/3 of cup, is it 2/3 of a gallon, is it 2/3 of $1 million? Those are very different things, and so when we say a fractional amount, it’s not just two out of three. It’s 2/3 of something, and we need to attach that “of something” to the fractional amount for those fractions to truly make sense.

Now one of my favorite variations of how we should be saying fractions is attaching the unit that we’re working in. So with that example 2/3, we can say just “two-thirds”, but if we kind of break it apart a little bit and say the number and the unit fraction as kind of two pieces that come together, it builds such powerful understand for kids.

So instead of just “two-thirds”, often it’s helpful to say, “2 one-thirds.” A slight variation in how we say it, but man, it makes a big difference for kids to understand that 2/3 is 2 one-thirds. I could have 2 one-thirds of a cup, or 2 one-thirds of $1 million, whatever it is. But it helps them see that to know what 2/3 is, you’ve got to know what that unit fraction is of 1/3, and then we get two of those. It’s such a powerful way to talk about fractions, and it’s way better than saying “out of.”

Stop Saying “Reducing”

We do the best we can with the knowledge we have at the time, and once we’ve learned better, we do better.

One of the things I want to encourage you to do is to do better when we’re talking about equivalent fractions. We’re not making them bigger, we’re not making them smaller, we’re not reducing, because that’s what that phrase reducing makes us think about, is that it’s getting smaller, and that’s not the case. We need kids to understand that all we’re doing is simplifying.

I like that phrase so much better, because that’s really what we’re doing. When you see 5/10, we aren’t reducing it to 1/2. It’s still the same. But we’re simplifying 5/10 and writing it as 1/2 because 1/2 is a simpler fraction to operate with, to think about, and be able to visualize than 5/10. So the idea of simplifying fractions creates this better imagery for students than reducing does.

Now on a side note of this one, all too often, every single problem we give students that involve fractions, once they get their answer, the textbook, the worksheet, whatever, says “Solve, then simplify.” Why? Why do they need to simplify every single answer? The only reason in real life that we ever simplify a fraction is if we’re then going to go on and use it to operate a little bit deeper with the problem. I want kids to understand when it’s appropriate to simplify a fraction, and when it’s okay to just leave it. If all I’m doing is operating and I added and all I need is my answer, that’s fine. That’s all they need. But oftentimes we need to simplify if we’re going to go on and use that answer to help solve another part of a problem.

So my advice is, don’t make kids simplify just for the sake of simplifying. Teach them to understand that you simplify maybe because you want to connect it to a simpler fraction, or we simplify it because we need to then use that fraction and then do some more operations with it, and it’s easier if we simplify that fraction, to begin with before we move on to do more operations.

Stop Saying “Top Number And Bottom Number”

Okay, our last one is to stop saying top number and bottom number. I was so guilty of this. It’s still hard. If you watched the video a few weeks ago that Graham Fletcher did, he even said it. In his video, he was talking about top and bottom number. It is so easy for us to just say that instead of saying, “Numerator, denominator,” right? Top and bottom is easier to say. And it tells the kids which number is which.

Numerator, denominator, we have to explain all of that stuff, right? It’s so much easier and it works super quick to just tell top and bottom, but it does a disservice to our students because it facilitates that understanding, or misunderstanding, I should say, that a fraction is two different numbers.

When we say the top number and bottom number, they see it as two separate numbers that we’re just putting together and separating with a line.

We want them to understand that a fraction is one number. 2/3 is one number on a number line. It’s not a two and a three, right? So it’s a simple thing to change, but it really is very powerful in what it builds for our students’ understandings around fractions.

Alright, I hope that this one helped you build your math mind, so that you can build the math minds of your students.

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