Number Sense for Upper Elementary

Fractions, Multiplication-Division, Number Sense | 0 comments

I’ve been getting lots of requests to talk about what number sense looks like for upper grades kiddos, especially upper elementary and even into middle school. I have been talking about these four number relationships that I first discovered through John Van de Walle and his group, through their book, Teacher Student-Centered Mathematics. It’s really geared towards kids that need to develop these four things around the numbers zero through 20.

However, the more I worked with them and the more I worked with kids of various grade levels, I began to realize that kids need this for all numbers. Let’s take a look at these four concepts and how they relate to multiplication and fractions. So if your kiddos seem to have no sense around multiplication and whether or not answers are reasonable, or if they can’t operate through sense making with fractions, then read on.

Number Relationship #1 Spatial Relationships (Multiplication)

Let’s get going with spatial relationships. We talk a lot about spatial relationships as being tied to subitizing. Now, we want kids to be able to instantly recognize a group of five so that when they move into multiplication they can use that understanding to help them better visualize multiplication. It’s not the only way to view multiplication but it starts out as seeing groups “of”. We want kids to be able to have a visual so that when they think of five groups of five, they have something to tie that to. So it starts out with things like that.

Number Relationship #2 One/Two More & Less (Multiplication)

I also really love using the rekrenrek for this, and this can really help bring out the idea of one and two more or less because when it comes to multiplication, it’s just not one more and two more, it becomes one group more or one group less. Or for example when you’re doing times seven, sevens are not friendly, but fives are. Instead of seeing groups of seven, if they can see groups of five and then two more in there, that helps them to better understand some of the relationships and get more fluent with their facts if they have these relationships built.

Number Relationship #3 Benchmark of 5 &10 (Multiplication)

The other big one is the benchmarks of five and ten. This plays a huge role when kids are doing multiplication. Let’s say you use a rekrenrek to show nine times six, sometimes it’s easier to not think of nine times six but instead ten times six. If all 10 beads are pushed over, that would be so much easier. If kids can think about how those numbers relate to the benchmarks, it plays such a huge role in understanding and improving their fluency.  So here’s a multiplication chart I have used before that shows the kind of the relationships we want kids to build when learning their multiplication facts.

The darker colors are the main facts that kids need to know and then the lighter colors are the ones that you can use the dark ones to help you out with. So, the dark red is the “times 10” and if you know your times’ tens, you can figure out your nines based on the relationship between one and two more and less. That’s the other key. You’ve got to have all of these number sense relationships working hand in hand. Same thing with the blue ones. If you know your “times fives,” that helps with your sixes and sevens. So the benchmarks of 5 and 10 still play a huge role as kids are moving into multiplication.

Number Relationship #4 Part-Part Whole (Multiplication)

The last number sense relationship is part-part-whole. This is where kids understand that parts can come together to make a whole or you can take a whole and break it into its parts. This is the distributive property in multiplication. Do kids understand that when they are doing five times seven, they can break it up and instead do five times five and then five times two? Or with a visual image of four times seven, do they see it as four groups of five and then another four groups of two, and that they can put those together to figure out what four groups of seven would be. Okay, that’s basically what part-part-whole turns into is the use of the distributive property

Number Relationship #5 Spatial Relationships (Fractions)

Now as we move into working with fractions, these four number sense relationships still apply. They play such a vital role. Let’s start with spatial relationships.

This is a common way that curriculums tend to show fractions, but I think they’re starting to move away because the research is showing this is not an appropriate way to begin a child’s understanding of fractions. This old method has kids to think of it as “out of”. Even I’m guilty of this sometimes and I have to watch myself so I don’t say “out of” when I’m talking about fractions. Instead of saying one out of three, it should be stated as one-third. Although the fraction should be one-third, we tend to say it as one out of three because of the visual, to us It looks like one out of three. It’s not that one out of three is wrong, it’s just that it should be a later version of fractions that we show to kids because it’s a ratio view of fractions. So then kids have a hard time wrapping their brains around it is because when they go to add one-third or one out of three and I add it to one out of two, then the number one things kids respond with is two-fifths. They just want to add the numerators, add the denominators and get their answer. If I have one-third and one-half as a visual why doesn’t it become two-fifths? I can tell you why, because the wholes aren’t the same size, but it doesn’t look like it to the kids. They’re seeing equal sized parts. If I just put those parts together, I get my answer, but that’s not true. So instead when we start working with fractions with kiddos, we want to use parts of a whole, not part of a set. Not to say you can’t later, but when they first start, the recommendations in the standards is to use part of a whole so that when they see one-third. They see it as one of the three pieces that make up the whole and when you go to add it to one-half you can’t add them because they aren’t the same sized pieces. We’ve got the same size whole and we don’t have same size pieces. So now I’ve got to think about how I can make those have the same size pieces to be able to add those together. So the visual picture that we are giving for kids around fractions is super huge and so try not to use parts of a set to begin with and instead show them part of a whole.

Number Relationship #6 One/Two More & Less (Fractions)

As we move into one and two more and less, it’s not the whole one and two, but it’s the one and two of the unit that we’re using. So, if I’m working with fourths and the kids can do two-fourths and three-fourths, do they know what one-fourth more or two-fourths more is? Do they know one-fourth less or two-fourths less? So this relates a lot to counting. You know when we are counting whole numbers, we can count up and back and that tells us what’s one more, one less. But do we do that with fractions? If you haven’t been, you should. Getting kids to count with fractions is super huge to help them understand one and two more and less.

Number Relationship #7 Benchmark of 5 &10 (Fractions)

The other big one is the benchmarks. When it comes to fractions, the benchmarks aren’t going to be five and ten anymore. They become the half and the wholes. So once you work past one, then it becomes one and a half, two, two and a half, and three. Those are your benchmark fractions as they move forward. Kids should be able to take that and say where would three-fourths go. Not that they have to slice it into fourths, but they should know it’s in between a half and one, and hopefully they know it’s directly in half of half and one. Then when it comes to things like 12 thirteenths, I don’t want kids marking off 13 tick marks. I want them to know it’s super close to a whole and then same thing as they move past the whole. Can they understand that 20 thirteenths is really 20 one-thirteenth pieces? And is that more than a whole? Is it two? Is it one and a half? Is it close to one and a half? More than one and a half? Less than one and a half? I don’t need to know exactly where 20 thirteenths goes, but I want them to understand what it’s close to.

That’s what the benchmarks are about. Benchmarks become super helpful when they start having to order fractions from least to greatest or greatest to least. Basically, if they can place those fractions on a number line, they put them in order. Later, try to place some fractions in order, but don’t use common denominators, don’t try all of the butterfly stuff that we sometimes teach kids when they’re comparing. Use the benchmarks and see how do the fractions relate to those benchmarks and see if you can place them on the number line using that.

Number Relationship #8 Part-Part Whole (Fractions)

Our last part is part-part-whole. When it comes to fractions, it’s not any different than what it was with whole numbers. We want kids to be able to decompose five all the ways possible to decompose five. Like one and four, two and three, and so on. We want them to see all the ways that they can break apart five. Same is true when we get to five-eighths or any version of a fraction. I want them to be able to break that apart. I want them to understand that five-eighths is made up of five one-eighth pieces.

Just like five is made up of five individual things, five-eighths is made up of five one-eighth pieces (⅛ + ⅛ + ⅛ + ⅛ + ⅛). Then they can break that into lots of different ways. I can have one-eighth and four-eighths (⅛ + 4/8). I can have two-eighths and three-eighths (2/8 + ⅜). It also helps with some of the equivalence; if I know that I have one-eighth and four-eighths, four-eighths is one-half. So, it helps me to know how five-eighths relates to the half. It all works together.

All of these number sense relationships work hand in hand to build a really huge understanding of fractions and about multiplication. Now the last thing I will talk about in part-part-whole is don’t forget, we don’t want kids to just be able to break apart the number into all the different ways, we want them to actually use it. How does being able to break apart five-eighths help you when you go to add? If I’m adding one-half and five-eighths, how does that help me? What if I had seven-eighths and five-eighths? What might I break the five-eighths into? What if I had three-fourths and five-eighths? What would it be helpful to break the five-eighths into to make that problem friendlier? Again, if they know their benchmarks and how fractions relate to benchmarks, that makes it so much easier. So I’m not going to tell you any answers there. If you want to, put it in the comments of this post and tell me how did you use a half plus five-eighths and how did you decompose five-eighths to help you out?

Conclusion

So, remember that number sense is developed gradually by exploring numbers, visualizing them, and relating them in lots of ways that are not limited by traditional algorithms. That still holds true when it comes to multiplication and fractions. Our kiddos need time to play with these amounts, play with their understanding of multiplication, play with their understanding of fractions.  I hope that this has helped you build your math mind about how older kids need to develop their number sense when it comes to multiplication and to fractions.

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I’ve been getting lots of requests to talk about what number sense looks like for upper grades kiddos, especially upper elementary and even into middle school. I have been talking about these four number relationships that I first discovered through John Van de Walle and his group, through their book, Teacher Student-Centered Mathematics.