Now, you may think that this is a trick question but really, it’s a trick question because our students don’t have a solid foundation of fractions. And one of the ways that we build that solid foundation of fractions is by the visuals and representations that we provide kids around fractions.

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Blank Fraction Tiles from Hand2Mind

Well, I shouldn’t say deeply because it’s way deeper than what we can do in a short video but there are two things called partitioning and iteration around fractions that are super huge to develop for our students, yet, unfortunately, we tend to focus a whole lot only on partitioning.

And that leaves kids to when they see visuals like this, just counting how many pieces and then looking at how many are shaded. And that’s why the majority of kids say 1/3, but that’s not what we want kids to have an understanding of. It goes way deeper.

So, I’m going to turn this camera and point it down so we can get started with our learning today.

[To begin watching start at 1:56 in the video]

Most of our instruction around fractions is based on partitioning, which is when you give kids a whole and then you ask them, okay, if this is the whole, show me 1/4, and this might be done with a drawing, but it’s here they have to try to use these tiles to figure out, they’ve basically partitioned it into fourths, and then they can show you that this is a fourth.

Now, iteration is a little different because, in iteration, you start with the part, you start with this, and you say, if this is 1/4, what is the whole? So then they have to iterate that and maybe if you only gave them one, they could draw that and kind of repeat it over and over.
But if they have all the tiles, they would take four of them to create one whole, and this helps them understand that four 1/4 pieces make a whole. So, there’s a slight difference but it is a huge difference because it allows kids to really understand the value of an amount isn’t based upon just how many pieces you have.

If we want to look and tell how much this piece is worth, it’s worth whatever it takes to create the whole. So many kids have that misconception of we just count how many pieces, that’s our denominator, and then we count how many pieces are shaded and that’s our numerator. But really, the value of that shaded piece is dependent upon how it relates to the whole, so we need to iterate it to see how many it fills up of the whole.

Alright, so basically, we really want kids to be able to have experiences with both of those. We’re going to talk about how you can help kids model this idea of partitioning and iteration using the fraction tiles.

Using Blank Fraction Tiles to Model Partitioning

If this isn’t the whole, let’s say maybe this one is the whole.

And so when you change the whole, it makes them have to rethink the value of the amount because it’s not just written on there. So, if this is the whole, they have to think about what is the 1/3, and they have to kind of judge. They’re thinking in their mind. They’re actually partitioning that in their mind to figure out the size of what a third would be instead of just going over and being able to grab a fraction tile that says 1/3. This is from a different set, but this is the typical kind of fraction tiles that we see where if you say, show me 1/3, all they have to do is just find the one that says 1/3.

It’s not really building any understanding of fractions. So, one of the things you’ll get is that as kids are trying to figure out what the 1/3 is, you might get kids who do something like this, where they’re way off:

But they’re also doing iteration at the same time as partitioning, so it’s not like these are two totally separate ideas. They work hand in hand to really build a foundation for our students around fractions.

So, they’ll judge, they’ll come back, and they’ll say, okay, those weren’t quite big enough. And so maybe they’ll start with something like this.

They’ll try something a little bigger and then they realize, oh, gee, and think, if that’s too big, I can’t fit the third one in there. So they’re developing all of this spatial reasoning, as well as fraction reasoning as they are going through this idea of trying to find 1/3 of that whole. And that changes, like I said, dependent upon what the whole is.

So if I decide now that this [Orange] is a whole, how is that gonna impact my 1/3?

You can have these up here and have that discussion with students of, when this was the whole, these were 1/3. Now, if we want this to be the whole, what happens to our 1/3? So that’s one of the weird pieces when it comes to fractions is that 1/3 isn’t always equal to 1/3, right? It’s always dependent upon the size of the whole. The idea of 1/3 is that it is a third because it takes three of them to fill the whole. It takes three 1/3 to create the whole, whatever that whole is.

And so, again, they might start with something that they aren’t too sure is going to fit or not. And they’re like, oh, that didn’t quite do it for me, that one’s not gonna be it. And then this one. Oh, man, that’s not gonna fit either, that doesn’t work.

So they’re building all of these cool understanding as they are trying to think about this concept of a third, they are creating this foundational piece of fractions, which is that unit fraction, being able to understand that unit fraction is such a big deal, and the blank fraction tiles do that beautifully. And so this one is actually a nice prime example of where it doesn’t fit so nicely. There isn’t a 1/3 that goes nicely using these fraction tiles.

So, as kids are discovering this, and they’re like, what? Why don’t I have a third piece? Well, these fraction tiles aren’t cut in a way that every single one of them is gonna have a 1/3, so the fractions may not be able to do it nicely. And so that leads kids into having a discussion about, man, if the tiles don’t do it for us, how could we, what would it maybe look like to show 1/3 of that tile?

So you can blend in the concrete and the representational as we’re going through this, and you can also attach the abstract, the CRA, altogether. It’s wonderful when that all work together hand in hand, doesn’t it?

Using Blank Fraction Tiles to Model Iteration

Okay, so here’s a quick, little example of what it looks like to do an activity with these blank fraction tiles that are getting kids to truly model iteration. Like seeing those other ones where we’re doing the partitioning, you’re kind of doing iteration with it, but iteration, a classic example, is you just give the kids the part and you tell them how much it’s worth.

Let’s say, I might make this 1/5. And then you ask them to make a whole, or maybe you don’t do a whole, maybe you want them to make something bigger than a whole, but the idea is to help them start to understand that it takes five of these to make the whole. And that’s what a whole could look like. It doesn’t necessarily have to all be in a nice, little shape.

A kid could do something like this as well. The cool part is that you can build any kind of shape you want as long as they understand that they need five 1/5 pieces to create that whole. Let me get it so that there are no gaps, bringing all that fun stuff here, trying to make it look a little nicer there for you. That is a visual of a whole and the only thing that makes it that is because this is 1/5 and we need five 1/5 to create a whole. That’s the general idea of iteration, is using the part and iterating it over and over to create a certain amount. The part usually that you wanna start out with is a unit fraction.

If I change this and now this part is 1/4, what’s a whole look like? Now, it’s going to be different because we only need four of these to create a whole. If that’s 1/4, that equals one or four over four. Four 1/4 pieces create the whole. I have that off to the side and you couldn’t see it, it wasn’t in the screen there. That’s the idea we want kids to understand is that we start with our part, our unit fraction, and we iterate that over and over to build the whole. That’s iteration.

Partitioning is when we start with the whole and we ask them to partition or cut it to show a piece, whether it’s 1/4, 3/4, whatever it might be, you’re starting with the whole and then you’re going down to a part. Those are the two big pieces around fractions that we need kids to have experiences with and they help kids be able to answer questions like this and so that it’s not a trick question.

We see this as a trick question because kids don’t have experiences doing activities like this.

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