Today we’re going to talk about How to Build Fraction Sense in the Elementary Grades.  Operating with fractions is what most textbooks focus on.  But in order to understand the operations, we first need to develop fraction sense.

I’m Christina Tondevold, The Recovering Traditionalist, and in this video we’re going to take a look at How to Build Fraction Sense in our quest to Build our own Math Minds so we can Build the Math Minds of our kiddos.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog.

Build Math Minds Facebook Group

Extending Children’s Mathematics: Fractions & Decimals: Innovations In Cognitively Guided Instruction by Susan Empson and Linda Levi

Graham Fletcher’s Progression of Fractions video

Number Talks: Fractions, Decimals, and Percentages by Sherry Parrish and Ann Dominick

Beyond Pizza & Pies by Julie McNamara & Meghan M. Shaughnessy

Beyond Invert & Multiply by Julie McNamara

Schools Don’t Spend Enough Time Helping Kids Build Fraction Sense

Aimee posted in the Build Math Minds Facebook group saying that she wants to improve her own personal understanding of fractions in order to improve her teaching of fractions and she asked for suggestions for professional development.

Lots of great resources were mentioned in that post, like the CGI book on Fractions & Decimals, Graham Fletcher’s Progression video,  Number Talks book, Beyond Pizza & Pies and Beyond Invert & Multiply.

A lot of the suggestions were focused on learning the operations (add, subtract, multiply, divide) but Aimee said she’s wanting to grow her personal understanding of fractions and that is way different than just learning how to operate with fractions.

We need to build our number sense around fractions and we need to build our students’ fraction sense as well.

Schools don’t spend enough time helping kids build fraction sense…the textbooks just jump straight into ‘here’s what a fraction is and here’s the steps to operate with them.”

Kids don’t get time to develop their fraction sense.

So that’s what I’d like to dig into with you.  How to Build Fraction Sense in the Elementary Grades.

Even if you teach the younger grades, I’m going to encourage you to watch this because my guess is that most of you didn’t get to develop your own fraction sense either.  I know it was a game-changer for me when I learned this as an adult and I finally made sense of fractions.

I have a full online course called The Flexibility Formula where we dive deep into each of these ideas I’m going to talk about.  There’s one for K-2 and one for 3-5 grade teachers.  In the 3rd-5th course, we look at how number sense not only impacts fractions but how it impacts multiplication & division and multi-digit addition & subtraction.

If you want more help with building your students’ number sense, aka their math flexibility, I’ll post a link to enroll in the courses below this video.

Let’s dive into the 8 Number Sense Concepts that help build Fraction Sense:

4 Early Numeracy Concepts

1. Subitizing

If you have taught the early grades, you know about subitizing. We spend a lot of time talking about subitizing with whole numbers and small amounts. Subitizing is the ability to instantly recognize. For instance, if I were to flash this image…

Could you tell how much it was without counting the dots? If you were, that means you were able to subitize that amount. We do a lot of work around this in the early grades but the more I work with kids of all ages, I really love focusing on subitizing even with fractional amounts.

Here’s what it looks like with fractional amounts…

Can you tell what fraction is shaded without counting? What about what was not shaded? Again, subitizing is the ability to see something quickly without needing to count. Your eyes recognized how much was there and you put it together in a fractional amount that ¼ was shaded and ¾ was not.

I’m not saying we need to do this for all kinds of fractions. This is a document that people inside the 3rd-5th grade Flexibility Formula course get where they can color or uncolor things and make their own subitizing cards for fractions.

We should be able to subitize the common fractional amounts. Things dealing with halves, fourths, thirds, maybe even fifths. I wouldn’t go higher than fifths because our ability to recognize and subitize even as adults ends around 5.

You can easily make these on your but I do want to caution you to only use 2 types of images for subitizing cards to begin with.

First is the Tape Diagram. All of the partitions on this one are vertical.

Next is the Area Model where partitions are vertical and horizontal.

I don’t like to start out with circles because I think its harder for kids to see. I also would not do parts of a set where you have 4 separate things and one of them is shaded in. We want to keep it contained and part of a whole.

2. Verbal Counting

The next concept is Verbal Counting. Clements and Sarama (where these concepts come from) have a much more academic sounding term for this, but Verbal Counting helps me remember exactly what the kids are able to do. 

Verbal Counting is just the ability to count out loud and in order. We spend a lot of time with kids in the early grades doing this with whole numbers. But how often do we do this with fractions? How many times have you asked your students, “start at 8/3 and count by 2/3rds”? 

Counting is foundational to building a sense of numbers. If we can’t count, it’s really hard to operate with them. Addition and subtraction is based on counting. That’s how kids first learn to add and subtract. So it should be how they first learn to add and subtract with fractions as well. 

Starting at 8/3rds and counting by 2/3rds can be complicated. So start at 0 and count by 2/3rds instead. Are we able to do that? A lot of us struggle with this to begin with but it’s just like counting. It’s 0/3rds, 2/3rds, 4/3rds, 6/3rds, 8/3rds, and so on. We want them to see how counting with fractions is really similar to counting with whole numbers. 

3 & 4. Object Counting and Cardinality

Just being able to count is not the full picture. We don’t want them to just list the number names in order, we want them to attach those counts to something. Just because they can count to 10 fast, it doesn’t mean they actually understand counting. They need to attach those counts to something.

Cardinality is when they’re done counting,  they understand the last word that they say says how many. So if they count five things, and you ask them “how many are there?” Can they say five, or do they just go right back and count again?

A word of caution when we’re getting kids to count objects.

Kids want to count the stuff.

We teach them this also. We say “how many are shaded?” so we get the 1.

Then we say “how many total parts are there” and they count to  4. Then we tell them to put a slash between them.

That’s a fine place to start, but it leads to kids seeing an image like this…

And they say “well there’s 2 shaded, and 3 total parts, so it’s ⅔”. This understanding of seeing how many parts are shaded and how many total parts there are is a foundational piece of fractions but its limiting kids. It’s known as partitioning. Where we have the whole and we cut it into the parts and we look at the parts individually. How many parts are shaded, and how many parts are total.

Another view of fractions is what’s known as iteration. Iteration is the understanding that what I call the shaded part depends on what the whole is and is called that based on how many iterations it takes to make the whole. So I have to take that piece and iterate it as many times as it takes to fill the whole.

That’s a different view of fractions but it is a big piece we need to get kids to understand. This is something that also helps with object counting and seeing the repetitions of that piece of the unit over and over is basically the object counting of fractions.

So if I want to determine the size of this yellow piece…

It is dependent on what the whole is worth. So if I say the green piece is 1, it’s the whole. So what’s yellow? To be able to figure that out, I have to iterate it until I can see how many iterations it takes to fill the whole.

Doing that, we can see that the original yellow piece is ⅓. But, it’s only ⅓ if the green is the whole. The value is dependent upon the size of the whole. If the whole changes, then the value changes.

So if I use this example…

And I now say the gray one is worth 1, now the yellow is no longer worth ⅓ and I have to iterate it over and over to figure out its value.

That object counting with fractions and cardinality is knowing the amount once the counting is done.

4 Number Relationships

1. Spatial relationships

Spatial relationships is having a visual of amounts, but being able to look at how it relates to other amounts. So if I just have this image and ask the kids how much is shaded…

It’s building an understanding of ½ and what ½ looks like but we aren’t building relationships to any other fractional amounts. So building this spatial relationship is about attaching another quantity to it and talking about the relationship.

What do you notice is the same? What do you notice is different? How does ½ relate to ⅓? You have to have the visuals for kids because if I just put up the fraction ½ and the fraction ⅓ and I ask kids to talk about how they relate to each other and how they compare it doesn’t build sense around those fractional amounts. We need the visuals. That’s the spatial part that helps build relationships around quantities. 

1. One/Two More Less

We spend a lot of time doing one/two more less with whole numbers. What’s one more? What’s one less? What’s two more? What’s two less? What’s ten more? What’s ten less? Kids need this around fractions as well.

If we have this image of 2/4, we can ask questions like, “what’s ¼ more? What’s ¼ less? What’s 2/4 more? 2/4 less?” You can even ask “what’s 1 whole more or less?” Ultimately, we want kids to be able to instantly tell us what’s one more of whatever unit fraction we’re working in. If the picture shows thirds, I would ask what’s ⅓ more or less. It makes a big difference to attach visuals or manipulatives instead just the digits of the fractions on paper. 

 The idea of one/two more or less is the first concept where we are adding one or two more or less. With fractions, it can be really hard for kids to understand that  7 / 8 + 2 / 8 does not equal 9 / 16. You can tell them until you’re blue in the face that you only add the numerators and the denominators stay the same. But if they don’t have experience with counting with fractions, they will still think they need to add the denominators. 

We need to build their sense of fractions. When we do that, they will understand that the size of the pieces does not change, just the amount of those pieces when we add and subtract. All of these number sense concepts work together. 

1. Benchmarks

Benchmarks are super important to build as well. In the early years, it’s the benchmarks of like 5, 10, 50, and 100. But when it comes to fractions, we need to build their benchmarks to halves and wholes. So 1 / 2, 1, 1 ½, 2, 2 ½, etc. Also the equivalents of those so 1/2 , 2/2, 3/2, 4/2, etc. We want them to see that this can work with any denominator. So 3/6 and 6/6 or 10/20 and 20/20. We need them to understand those equivalences and how the numbers relate to those halves and wholes.

It’s not just to have them memorize and understand equivalents. It helps kids with comparing fractions as they move forward and also with addition and subtraction. When kids understand how fractions relate to those benchmark numbers, when they start adding and subtracting they’re going to use that faction sense to make problems friendlier.

We talk so much in the early grades about making problems friendly. When it comes to fractions, make it a whole number. If we took 2 3/4 + 3 2/4, could we take any of those numbers and make it a whole number to make the addition easier? Kids will naturally do that if they’ve built their fraction sense. Kids naturally see that 2 and 3/4 is darn close to 3. They will want to round it to 3 or add some. Whatever strategy they choose to add and subtract fractions, they will choose it if they’ve built a sense of fractions.

1. Part-Part-Whole

Part-Part-Whole is all those activities in the young grades where we take a whole and break it into all the parts that you can make. For instance if we ask them to break down 5 into all the ways you can make up 5. Kids would give you 4+1, 3+2, 5+0, and so on. So it’s similar to that but now we’re doing it with fractions.

Can you decompose 5/8 up into different parts? What kind of parts can you get if you have 5/8 ?

It’s the same as decomposing 5. We still have 5 things…5, 1/8th pieces. I can have 4, 1/8th pieces and 1, 1/8th piece. 4 pieces and 1 piece makes 5 pieces. You can have 3/8 + 2/8, or 5/8 + 0/8. Whatever way I decompose 5, I can decompose 5/8 in the same way.

So why is this important? We don’t want to do this just to make them do it. This builds a sense of what these fractions actually are and then they can use that understanding when they go to add, subtract, multiply, and divide. If they understand that 5/8 can be 4/8 + 1/8 , 3/8 + 2/8 , 5/8 + 0/8, could they use one of those things to help them make 2 and 3/4 + 3 and 5/8 friendlier?

Here is an example of what that might look like.

We’ve seen kids say “2/8 is the same thing as 1/4 and I need 1/4 to make 2 and 3/4 a whole.” So they will pull the 2/8 out and they know what’s left over. Then the problem becomes 3 + 3 and 3/8. That’s a whole lot easier than doing 2 and 3/4 + 3 and 5/8. But they have to have a sense of fractions in order to be able to do that.

These 8 number sense concepts are what builds number sense around fractions for our students.

Conclusion

This is just an overview of the 8 number sense concepts and inside the course we go deeper into these, how to assess students, how they help kids understand the operations, and experiences you can provide your students to help them build these concepts. 

I say it all the time, but you can’t directly teach these to kids.  They need to play around with the numbers and build their number sense.  Number Sense can’t be Taught, it’s Caught.  That holds true even with fractions number sense.

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

Want to dive deeper into what fraction sense is and how to help your students build it? 

Join The Flexibility Formula 3rd-5th.  We go into building a sense of numbers for fraction, multiplication & division, and multi-digit addition & subtraction.  If you work with younger grades, I also have The Flexibility Formula K-2 that shows you how to help your students build number sense for 0-20 and then extend that to numbers beyond 100.  Use the buttons below to find out more about the courses and enroll.

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