Fractions are one of the most conceptually challenging topics for young learners, yet they form the foundation for future success in mathematics. Unlike whole numbers, which follow predictable patterns, fractions introduce new relationships and rules that can feel counterintuitive. A larger denominator represents a smaller piece, adding fractions isn’t as simple as counting up, and fractions don’t always behave in ways that align with students’ previous number sense.
Because fractions represent relationships rather than fixed quantities, students need conceptual understanding, not just memorized procedures. However, many traditional teaching approaches emphasize steps and tricks, like cross-multiplication or “keep, change, flip”, without deep reasoning. When students rely solely on rules, they may struggle to apply their knowledge to new situations. Instead, research supports the use of multiple representations; different ways of visualizing, modeling, and conceptualizing fractions, to help students develop a strong foundation.
By incorporating linear, area, and discrete models, students can explore fractions from different perspectives, make meaningful connections, and develop the flexibility needed to truly understand how fractions work.
The Three Essential Fraction Models: Linear, Area, and Discrete
Research has identified three dominant fraction models used in elementary classrooms: the linear model, the area model, and the discrete model. The linear model represents fractions as distances or points on a number line, helping students see fractions as part of the number system. The area model depicts fractions as shaded portions of a shape, reinforcing the concept of partitioning and equivalence. The discrete model represents fractions as sets of objects, making fractions tangible and intuitive for early learners. Each model highlights different aspects of fractions, and by integrating all three, teachers can support diverse learning styles, address misconceptions, and build a flexible understanding of fractions that applies across different mathematical contexts.
The Linear Model: Fractions as Length or Distance
The linear model represents fractions as points on a number line or as lengths of objects. Unlike models that focus on partitioning a whole, the linear model reinforces that fractions exist between whole numbers and are part of a continuous system.
Why It’s Important
Many students initially struggle to understand that fractions are numbers with precise locations on the number line. Without a strong linear representation, students often treat fractions as separate, disconnected values rather than part of a number system. Number lines help students develop fraction magnitude, equivalence, and ordering, which are essential concepts for reasoning about fractions in later grades.
Here are some examples of the linear model in action:
- Maintaining a growing number line in the classroom, where students progressively add fractions as they learn them.
- Exploring equivalent fractions and fractions greater than one by locating them on the same point of a number line.
- Using Cuisenaire rods or linking cubes to compare fractional lengths and develop a sense of relative size.
- Folding paper strips to visually explore and compare equivalent fractions.
Potential Challenges and Solutions
Research shows that students often struggle with number lines when first learning fractions (Larson, 1980). Unlike whole numbers, which are evenly spaced, fractions require students to think about relative size, partitioning, and equivalence, which are concepts that are not always intuitive. Some common errors include misplacing fractions or spacing fractions incorrectly. To support students, teachers can start with benchmark fractions like ½ and ¼ before introducing more complex values.It can also be helpful to introduce fraction families separately, starting with halves, fourths, and eighths before moving on to halves, thirds, and sixths. Once students are comfortable with these groupings, they can be combined on the same number line to deepen their understanding of relationships between fractions.
Hands-on tools such as fraction strips, Cuisenaire rods, and rulers can help reinforce these concepts. Encouraging discussions around fraction placement, equivalence, and comparisons also deepens understanding. When students gain confidence in using number lines, they develop a stronger grasp of fraction magnitude, ordering, and operations, setting them up for success in more advanced mathematical reasoning.
The Area Model: Fractions as Parts of a Whole
The area model represents fractions as shaded sections of a geometric shape, such as circles, rectangles, or grids. It’s one of the most commonly used fraction models, providing a visual and intuitive way to introduce partitioning, equivalence, and fraction operations.
Why It’s Important
The area model helps students see and physically divide a whole into equal parts. It makes equivalence, comparison, and fraction multiplication more concrete. For example, shading two out of four sections in a rectangle shows that ½ is equivalent to ²⁄₄, a concept far more intuitive than memorizing fraction rules.
This model is also useful for understanding comparisons. For example, if one rectangle is divided into thirds and another identical rectangle is divided into eighths, students can visually see that eighths are much smaller than thirds, even though the number 8 is greater than 3.
Here are some examples of the area model in action:
- Folding and cutting square origami paper to investigate unit fractions and understand the role of the denominator.
- Using fraction circles or pattern blocks to explore and compare equivalent fractions.
- Drawing and shading rectangles to visualize and compare different fractions.
Potential Challenges and Solutions
Students don’t always interpret area models correctly (Watanabe, 2002). Common mistakes include assuming that any given shape represents a whole rather than considering how it is partitioned, or focusing on counting pieces without recognizing that equal partitioning is necessary for accurate fraction representation.
To address these misconceptions, it’s crucial that teachers explicitly define the whole before asking students to determine fractions. Furthermore, using multiple shapes, such as circles, rectangles, and grids, helps prevent over-reliance on a single model. Providing hands-on practice with folding, shading, and cutting objects reinforces the concept of partitioning. By guiding students through these challenges, teachers help build a strong, transferable understanding of fractions using area models.
The Discrete Model: Fractions as Sets of Objects
The discrete model represents fractions using groups of individual objects such as counters, connecting cubes, or even pieces of candy or students in the classroom. Unlike the area or linear models, which show fractions as part of a continuous whole, the discrete model emphasizes part-whole relationships within a set.
Why It’s Important
The discrete model is especially useful for young learners who are transitioning from whole-number thinking to fractional reasoning. Because young children are already comfortable with counting, using sets of objects makes fractions less abstract.
For example, if a teacher presents 12 counters and asks, “What fraction are red?”, students can see that 3 out of 12 are red, representing ³⁄₁₂. This approach helps students grasp fractions intuitively before moving to more complex models.
Here are some examples of the discrete model in action:
- Using a collection of counters, cubes, or other small objects and asking students to sort them into groups that represent different fractions. For example, if 3 out of 6 counters are red, the fraction would be ³⁄₆.
- Grouping students based on shared characteristics and representing the group as a fraction. For example, if 6 out of 18 students packed their lunch, the fraction would be ⁶⁄₁₈.
- Using fun-size packs of candies to name fractions of a set. For example, students can open a bag of candied-covered chocolates and name the fraction of the candies that are blue, green, and yellow.
Potential Challenges and Solutions
Students who are used to thinking about fractions as parts of a single object may find it difficult to recognize one whole as a collection of multiple items. Using visual and hands-on strategies, such as drawing a boundary around a group or physically organizing objects, can help students adjust to this new perspective. Regular exposure to real-world examples and visual models further reinforces their understanding and builds confidence.
By engaging with sets through hands-on activities and meaningful contexts, students can develop a flexible grasp of fractions as they relate to groups of objects. This foundational understanding will support them as they progress to more advanced concepts like ratios and proportions.
Why Multiple Representations Matter
No single model can fully capture the complexity of fractions because fractions are more than just parts of a whole. Fractions represent ratios, distances, division, and relationships between quantities, and they appear in various mathematical contexts. Relying on only one type of model can lead to restricted thinking and misconceptions, making it difficult for students to apply their understanding in new situations.
When students are exposed to only one way of thinking about fractions they often struggle to transfer their knowledge. For example, a student who learns fractions exclusively through circular area models may have difficulty placing fractions on a number line because they have never thought about fractions as numbers with precise locations. Similarly, a student who works only with discrete sets may not recognize that fractions can also represent continuous quantities.
Using multiple representations allows students to develop a more flexible and connected understanding of fractions. Seeing fractions in different ways; on a number line, within an area model, and as a set of objects, strengthens their ability to compare fractions across different contexts, apply fraction concepts to real-world problems, and develop flexibility in mathematical thinking. For instance, students begin to see that ½ of a pizza and ½ of a distance on a number line represent the same proportion, even though they look different. They also learn to apply fractions in various situations, such as measuring ¾ of a cup of flour, dividing ¾ of a group of marbles, or walking ¾ of a mile, understanding that fractions take different forms but serve the same purpose.
Research supports the idea that exposure to multiple fraction models leads to deeper understanding and better long-term retention (Carpenter et al., 1999; Watanabe, 2002). When students experience fractions in diverse ways, they make stronger connections, solve problems more effectively, and approach fractions with confidence. By intentionally incorporating linear, area, and discrete models into instruction, teachers can provide a solid foundation that moves students beyond memorized procedures and toward true mathematical reasoning.
Conclusion
Teaching fractions through multiple representations is key to helping students build a deep and flexible understanding of this essential mathematical concept. Each model; the linear, area, and discrete model, offers a unique perspective that helps students move beyond rote memorization and isolated rules.
The linear model reinforces that fractions are numbers with precise locations on a number line, strengthening their understanding of fraction magnitude and ordering. The area model highlights partitioning and equivalence, allowing students to visually compare and operate with fractions. The discrete model bridges whole-number thinking to fractions by representing them as sets of objects, supporting the development of proportional reasoning.
By incorporating all three models into instruction, teachers can ensure that students see fractions in multiple ways and make meaningful connections across different contexts. This approach enhances problem-solving skills, deepens conceptual understanding, and increases confidence in mathematical thinking. Instead of viewing fractions as abstract or disconnected from real life, students begin to see their relevance in measurement, sharing, data analysis, and proportional reasoning. These skills are not only essential for advanced math but also for everyday experiences.
When students grasp fractions as numbers, parts of wholes, and relationships between quantities, they are better prepared for more complex topics like fraction operations, algebra, and proportional reasoning. More importantly, they develop the ability to approach mathematical problems with confidence and flexibility rather than relying on memorized procedures. By embracing multiple representations, we shift the focus from simply working with fractions to truly understanding them, ensuring that students gain a lasting, transferable foundation that will support their mathematical growth for years to come.
