Fraction concepts are an important part of mathematics education. Developing a deep understanding of fractions lays the foundation for more advanced math topics. Using models is a useful strategy for building fraction sense. There are several common fraction models used in elementary and middle school math classes.
Area Model
The area model is a visual representation using length and width to show the relationship between the numerator and denominator of a fraction. This model connects fractions to geometry and parts of a whole. With the area model, a shape like a rectangle or circle is divided into equal parts. Some parts are shaded to represent the numerator, and the total number of parts represents the denominator.
For example, to model the fraction 3/4 with an area model:
- Draw a rectangle and divide it into 4 equal parts
- Shade 3 of the parts
- The shaded area represents 3/4
The area model helps students see that the denominator represents the total number of equal parts, and the numerator is how many of those parts are being considered. This model can be used with different shapes and with fractions greater than 1.
Benefits of the Area Model
There are several key benefits to using the area fraction model:
- Connects fractions to geometry and parts of a whole
- Visualizes the meaning of the numerator and denominator
- Allows fractions to be represented with concrete models
- Can model proper and improper fractions and mixed numbers
- Generalizable to different shapes like circles, squares, and triangles
Limitations of the Area Model
Some drawbacks to the area model include:
- Does not work for all fraction concepts like ratios or operators
- Difficult to draw precise models by hand
- Does not show benchmark fractions intuitively
- Static representation does not show the dynamic process of division
Number Line Model
The number line model represents fractions as points on a line segment divided into equal parts. It builds on students’ understanding of whole numbers on a number line. With the number line fraction model, the line segment is partitioned into the number of parts indicated by the denominator. Each part represents 1/n where n is the denominator.
To model a fraction on a number line:
- Draw a line segment and divide it into equal parts according to the denominator
- Starting from 0, count the number of parts indicated by the numerator and mark a point
- That point represents the fraction
For 3/4, divide the line segment into 4 equal parts between 0 and 1. Mark the point that is 3 parts away from 0. This illustrates the amount 3/4 between the benchmarks 0 and 1.
Benefits of the Number Line Model
Key advantages of the number line fraction model:
- Builds on students’ understanding of whole numbers on a number line
- Can model proper and improper fractions
- Allows comparing and ordering of fractions
- Shows fractions between two benchmark numbers like 0 and 1
- Can be horizontal or vertical
Limitations of the Number Line Model
Some limitations of this fraction model:
- Does not have a strong visual for parts of a whole
- Difficult to partition precisely by hand
- Number lines only show one representation at a time
- Challenging for modeling addition and subtraction
Set Model
With the set model, a fraction is represented by dividing a set of objects into equal groups. Circles or counters are often used to model fractions with sets.
To model a fraction with sets:
- Draw a set of objects like circles
- Divide the set into equal groups according to the denominator
- Circle or select the number of items indicated by the numerator
For example, to model 3/4 with sets:
- Draw 4 equal circles to represent 4 equal parts
- Circle or select 3 of the parts
The fraction represents 3 out of 4 equal parts.
Benefits of the Set Model
Key strengths of the set fraction model:
- Intuitive representation using concrete objects
- Allows fractions to be modeled physically and manipulated
- Can illustrate equivalence by rearranging parts
- Models part-whole relationships
- Easy to modify and expand
Limitations of the Set Model
Some drawbacks to the set model include:
- Difficult to draw precisely equal groups by hand
- Does not show fractions as one number intrinsically
- Tedious for modeling large denominators
- Static representation without inherent order
Comparing the Models
Each fraction model has strengths and limitations. Using a combination of models can help build students’ conceptual understanding. Here is a comparison of features for the three common fraction models:
| Model | Area | Number Line | Set |
|---|---|---|---|
| Represents part-whole relationship | Yes | Somewhat | Yes |
| Models proper and improper fractions | Yes | Yes | Yes |
| Compares and orders fractions | Limited | Yes | Limited |
| Shows benchmark fractions | No | Yes | No |
| Conceptual understanding | Moderate | Moderate | High |
| Ease of use by students | Moderate | Difficult | Easy |
The set model provides a strong conceptual foundation with physical representations students can manipulate. The number line builds understanding of order and benchmarks. The area model connects to geometry. Using all three models together can give students a deeper understanding of fractions from different perspectives.
Instructional Strategies with Fraction Models
Fraction models should be used as part of a comprehensive fractions curriculum. Here are some effective instructional strategies utilizing these models:
- Introduce models using familiar shapes and manipulate concrete objects
- Connect models to visual representations and mathematical symbols
- Ask students to explain and justify their reasoning using models
- Have students create their own fraction models with drawings or manipulatives
- Use models to solve word problems involving fractions
- Incorporate technology like virtual manipulatives and digital models
- Use multiple models to explore one fraction and make comparisons
- Progress from concrete to pictorial to abstract representations
Models should be used frequently to develop relational understanding. With guidance, students should make connections between representations. Instruction should focus on making sense of fractions with models before introducing procedures.
Real-World Connections
Relating fractions to real-life situations is an important part of concept development. Teachers can use models to represent fractions in contexts like:
- Cooking – fractions of recipes, portions, and measurements
- Time – fractions of hours, minutes, days, and weeks
- Money – fractions of dollar bills and coins
- Measurement – fractions of standard and non-standard units
- Art – fractions in geometric shapes and design
- Music – fractions of beats and notes
- Sports – fractions of game times and scores
Students should work with fraction models to represent these everyday scenarios. Making authentic connections helps reinforce the purpose of fraction skills.
Assessment Strategies
Fraction models can also be incorporated into assessments. Some examples of model-based fraction assessment tasks include:
- Represent fractions with two different models and compare them
- Create models to represent word problems and explain the models
- Identify and correct errors in fraction models
- Compare and order fraction models
- Solve multi-step problems using models
- Match fraction models to symbolic notations
Assessment should have students engage actively with fraction models, not just recognize them. This provides evidence of deeper conceptual understanding that transfers outside the classroom.
Conclusion
Fraction models are a valuable instructional tool for developing fraction understandings. The area, number line, and set models provide visual representations with different benefits. Using models can build conceptual knowledge of part-whole relationships, ordering, and equivalence. Models should be integrated with other strategies like real-world connections and discourse to support effective fraction learning.