Third-grade and fourth-grade teachers, you know the struggle is real when it comes to teaching fractions greater than one whole! Fractions can get a bit tricky for young mathematicians when we ask students to shift from thinking of them as parts of one whole to realizing they can be *greater* than one whole.

When beginning to learn about these fractions, we’re asking students to think more deeply about fractions and understand that fractions can look differently: numerators smaller than denominators, numerators larger than denominators, and even fractions standing next to whole numbers.

So how do we help students grasp this concept and build a rock-solid foundation? Let’s break it down!

## Talking about Fractions Greater than One

I know I talk a lot about representations, but that’s because of the huge role they play in helping students learn. Representing fractions in different ways and then making connections *between* those representations is critical in building students’ conceptual understanding. Ready to see what this looks like? Let’s start with verbal representations.

Language and vocabulary are an important part of teaching fractions. Numerator, denominator, greater than, less than, improper fraction, mixed number, and on and on! There are so many fraction-specific vocabulary terms.

While the words themselves allow us to speak precisely about the math, when building a conceptual understanding of fractions greater than one, or fractions in general, the language is not the most important part. Gasp! I know, I know. If a student cannot name a fraction like 12/9 as an “improper fraction,” I don’t get super stressed *as long as *they can identify it as a fraction that is greater than one, demonstrating to me that they understand the concept of improper fraction.

Connecting improper fractions to mixed numbers also helps solidify the idea that an improper fraction is a fraction greater than one. With mixed numbers, students can clearly see that the fraction is greater than one. Oh, and yes, being able to name a number as a “mixed number” isn’t a huge priority to me either when students are beginning this work. Instead, I prioritize students being able to verbalize their understanding of a mixed number being another way to represent an improper fraction and that they are both greater the one.

Instead of hyperfocusing on students using the precise words, to start let’s build that conceptual understanding while connecting that with the specific vocabulary as part of students’ verbal representations! Not allowing an immature vocabulary to hinder conceptual understanding is key. They will grow together as students investigate the concepts deeply!

## Getting Hands-On with Fractions Greater than One

Concrete representations give students something to physically manipulate helping them understand what the parts of a number mean. Some of my favorite manipulatives to use to teach fractions are pattern blocks, fraction strips, and fraction circles. When I can, I also like to use everyday materials like play dough and paper. Students are often familiar with sharing these items in their own lives which can make the leap to fraction work easier.

#### Pattern Blocks

Pattern blocks are great to teach fractions because of their flexibility. When beginning to learn about fractions, we might use the yellow hexagon to represent one whole and then use the other pieces to represent parts of that whole with the triangle representing ⅙, the rhombus representing ⅓, and the trapezoid representing ½.

With a hexagon representing one whole, students will likely already be familiar with why a rhombus represents ⅓ of the hexagon. From there, they can build ⅔ and 3/3. Now, when starting to work with fractions greater than 1, I often challenge students to show me a fraction like 5/3 with no prior instruction on improper fractions and watch the magic unfold. Students will sometimes build ⅗ instead of 5/3. Others will grapple with the idea that 3/3 is the whole hexagon and not know how there can be more than that. The discussion that ensues is rich!

However, when moving into fractions greater than one, we don’t have to stick with the hexagon as one. We can use any piece to represent the whole! For example, if we use the trapezoid to represent one whole, we can model 1⅓ or 4/3 using one trapezoid and one triangle or using three triangles. As you can see, pattern blocks help to connect to mixed numbers as well as improper fractions.

#### Fraction Strips & Fraction Circles

Fraction strips and fraction circles are math manipulatives that are made specifically to model fractions. When working with fractions greater than one, students will often need more than one set of strips or circles. Because of this and the added challenge of fractions greater than one, I like to have students partner up when they begin investigating the concept.

Working with fraction circles may feel familiar to students who have investigated fractions greater than one using food like cookies or pizzas. Similarly, fraction strips can be visually related to candy bars.

Like pattern blocks, students can also use fraction strips and fraction circles to investigate the connection between improper fractions and mixed numbers. They make the connection even more explicit because the pieces labeled 1 can take the place of the equivalent number of smaller fraction pieces.

#### Number Lines

Number lines are also a wonderful way to model a fraction using visual representations. Once students are beginning to understand fractions greater than one, it’s critical that they understand fractions as a number like any other. Often, students only see fractions as part of a whole and don’t understand them as a number on the number line. Representing a fraction greater than one on a number line also solidifies students’ conceptual understanding of fractions in general.

To reinforce students’ general understanding of fractions, you might ask them to model 5/3 on a blank number line. To complete this task, students would need to partition the number line into equal-sized pieces and then count the tick marks to label 5/3. When discussing this model it would be beneficial to relate this model to representing 5/3 with fraction strips.

A more complex task would be to ask students to place a smaller fraction, such as ⅓ or 3/3, on a number line with 5/3 already labeled. This task requires students to consider how many pieces the number line should be partitioned into and then where the given fraction would be located.

Finally, remember to make connections between all representations that students are using. It’s important for them to see *all* of the representations as connected, like understanding the 5/3 represented with five blue rhombuses has the same value as 5/3 represented on a number line. Seeing all of these representations of fractions greater than one builds conceptual understanding, anchoring them when things get even more complex.

## Symbolic Representations of Fractions Greater than One

It’s so important to not rush into *solely* representing fractions abstractly which is why I saved this representation for last. We want students to be able to have something to hang their hat on when interacting with only abstract representations.

When working with abstract representations of fractions greater than one whole, I want to ensure that students are able to first identify fractions equivalent to one whole. I typically do this by having students build fractions that are equivalent to 1 and having them notice the pattern in the fractions they create (for example, that the numerator and denominators are the same number).

From there, one of my favorite activities is a card sort. I simply write fractions that are less than 1, equal to 1, and greater than 1 on index cards. They can be any fractions and the cards don’t need to be anything fancy. Then working in pairs or small groups, I ask students to sort the cards any way that makes sense to them. Students will sort them in all kinds of ways – groups of round numbers and pointy numbers, groups of fractions with the same digits, a group with fractions that have their favorite numbers and a group that doesn’t. For this first pass, anything goes!

After a discussion on how students sorted and their reasoning, I have them sort the cards again, this time one group being fractions less than 1, one group being fractions equivalent to 1, and the third group being fractions greater than 1 (improper fractions and mixed numbers). The most important part of this activity is the discussion when students justify their placement of the fractions. The discussion allows students to learn from each other and allows me to assess students’ understanding of comparing fractions to one whole.

## Games and Activities for Fractions Greater than One

This post wouldn’t be complete without mentioning some of my favorite games and activities to teach improper fractions! My fraction Post-It note templates can help make any lesson extra special. Who doesn’t like those bright little sticky squares?! Students can use them when solving problems to model fractions greater than one whole in their math notebooks or even to prove their thinking when sharing with the whole class.

## Free Fraction Post-It Notes

Looking for a concrete way to help your students visualize fractions? Just stick, print, and let your students color in their fractions to create helpful visuals in your classroom!

I also love connecting abstract representations of improper fractions and mixed numbers with visual models by playing Fraction Bingo! This can be such a simple, but awesome class game for practicing and reviewing fractions greater than one. You can find Fraction bingo (and tons of other games and activities) in my Fraction Mega Pack.

I also love using dominoes to select improper fractions to model and convert into mixed numbers. I love the game below in particular because it asks students to create the visual representation from the abstract. All of these fraction activities and games make learning about fractions greater than one just a little more fun!

In a nutshell, conceptual understanding is key to students truly mastering fractions greater than one. I hope some of these ideas have resonated with you. Do you have other tricks up your sleeve? Comment below or shoot me an email to share the wisdom. Until next time!

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