Understanding Fractions In Mathematics
Fractions are mathematical expressions that represent parts of a whole or a group. They are written in the form ⁿ⁄ₘ , where n, the numerator, tells us how many parts we have, and m, the denominator, indicates the total number of equal parts that make up the whole. Fractions are a cornerstone of mathematics because they allow us to represent numbers that fall between whole numbers, describe ratios and proportions, and perform operations that are not possible with whole numbers alone. Understanding fractions, therefore, is an important part of developing number sense and mathematical reasoning in young learners.
Numerators, Denominators, And Types Of Fractions
The denominator of a fraction determines the size of each part: the larger the denominator, the smaller each part becomes. For example, dividing a pizza into 4 equal parts, each slice is represented by the fraction ¼. These slices are larger than if the same pizza were divided into 8 equal parts, where each slice is represented by the fraction ⅛. The numerator of a fraction tells us how many of those parts we are talking about. For example, ¾ means we are talking about three out of the four pizza slices.
The relationship between the numerator and the denominator determines the type of fraction:
- Unit Fractions: A fraction with a numerator of 1, such as ¼ or ⅛. These fractions are the building blocks of other fractions and represent one part of a whole.
- Proper Fractions: A fraction where the numerator is smaller than the denominator, such as ¾. Proper fractions represent values less than 1.
- Improper Fractions: A fraction where the numerator is larger than or equal to the denominator, such as ⁵⁄₄ or ⁸⁄₈. Improper fractions represent values equal to or greater than 1.
- Mixed Numbers: A combination of a whole number and a proper fraction, such as 1¾. Mixed numbers provide an alternative way to express improper fractions and are often used in everyday contexts.
- Equivalent Fractions: Different fractions that represent the same value, such as ½ and ²⁄₄. Equivalent fractions can be calculated when both the numerator and denominator are scaled by the same factor.
Meanings Of Fractions
Fractions can have different meanings depending on the context in which they are used. Recognizing these meanings helps students build a flexible understanding of fractions and apply them to various mathematical situations.
Part-Whole Relationship With Fractions
One of the most familiar meanings of fractions is the part-whole relationship. In this interpretation, fractions represent a part of a whole that has been divided into equal pieces. For example, ¼ represents one part of a whole divided into four equal pieces. This understanding forms the foundation for many fraction concepts, including visual models and operations.
Fractions As Quotients
Fractions also represent division, showing how a quantity can be evenly split. For example, suppose you have 6 crayons to share among 3 friends. This sharing can be represented mathematically by the expression ⁶⁄₃ which means six divided by three. In this case, each friend receives 2 crayons, and the result is a whole number.
In contrast, fractions can also represent quotients that result in values less than 1. For example, ⅖ can be interpreted as the quotient 2 divided by 5, which equals 0.4. Here, the result is a decimal, reflecting that 2 cannot be evenly split into 5 whole parts, so the division produces a smaller, non-whole result.
It is important for students to explore both types of quotients so they can see how fractions bridge whole numbers and decimals. Understanding fractions as division in this way also lays the foundation for concepts like remainders, where some divisions don’t result in whole numbers, and for interpreting fractions as parts of a whole.
Fractions As Ratios
Fractions are also used to describe ratios, which compare two quantities. For example, ⅗ can represent a ratio of 3 parts to 5 parts, such as 3 apples for every 5 oranges. This interpretation emphasizes the relationship between the numerator and denominator as a comparison, rather than as parts of a whole.
Models of Part-Whole Fractions
Various models can be used to support understanding fractions by emphasizing how they describe equal parts of a whole.
Area Models For Fractions
Area models divide a two-dimensional shape, such as a circle, square, or rectangle, into parts with equal area. The shape itself represents the whole, and each part represents a fraction of the whole. For example, dividing a rectangle into four equal sections and shading one illustrates ¼.
Area models are especially useful for showing how fractions represent parts of a single object. They also help students understand relationships between fractions, including how to compare their sizes and identify equivalent fractions. For example, dividing a rectangle into eight equal parts and shading two of them, then comparing it to an identical rectangle divided into four parts with one shaded, demonstrates that ²⁄₈ is equivalent to ¼.
Length Models For Fractions
Length models use linear representations, such as strips of paper or number lines, to show fractions as distances or intervals. These models help students understand that fractions are specific numerical values and positions along a continuum.
A natural way to introduce length models is by having students fold strips of paper into equal parts. Starting with benchmark fractions, such as halves and fourths, students can build a foundation for understanding how wholes can be divided into equal parts. Folding and labeling the sections helps students visualize fractions as equal divisions along a length.
Once students are comfortable with physical models, they can transition to representing fractions as points on a number line, deepening their understanding fractions as precise numerical values. Supporting this transition with prior experience using number lines to locate whole numbers and intervals ensures students have the foundation to grasp fractions in this format.
Set Models For Fractions
Just like in place value, where a key transition occurs when children begin to see groups of ones, tens, etc. as single units, this model requires students to view a set of objects as a single “whole.” In the set model of fractions, the entire collection of objects represents the whole, and each subset corresponds to a fractional part. Developing this understanding is essential for students to grasp how fractions describe equal parts of a group. This conceptual shift is necessary for understanding fractions beyond simple part-whole relationships with physical objects.
In this example, the whole is represented by the 8 counters. 3 shaded counters out of the 8 counters represents the fraction ⅜.
Benchmark Fractions
Certain fractions, such as ¼, ½, and ¾, are considered benchmark fractions because they serve as reference points for understanding fractions due to their simplicity, frequency, and intuitive connections in both mathematics and everyday life. These fractions divide the whole into familiar and manageable parts, making them useful reference points for comparing, estimating, and reasoning about other fractions.
Operations With Fractions
While some concepts from whole number operations apply to fractions, many do not transfer directly. Understanding fractions requires building conceptual knowledge of each operation (addition, subtraction, multiplication, and division) rather than relying on procedures from whole numbers.
Addition And Subtraction Of Fractions
Addition and subtraction of fractions require students to work with fractions that have the same-sized parts, or a common denominator. This ensures that the parts being considered are comparable.
When fractions have the same denominator, addition and subtraction are straightforward because the denominators already represent equal-sized parts of the whole. Since the size of each part doesn’t change during the operation, the denominators remain unchanged. Only the numerators (i.e., the number of parts being considered) are added or subtracted.
An example of this is shown with the equation ⅜ + ²⁄₈ = ⅝ . In the sum, ⅝, the total number of parts considered is 5, and the size of each part (the denominator) stays the same (eighths).
When denominators differ, finding a common denominator is necessary. It is necessary to rewrite fractions as equivalent fractions with the same denominator, allowing the numerators to be added or subtracted.
For example, to add ¼ and ⅙, we first find a common denominator (12). We rewrite ¼ as its equivalent fraction ³⁄₁₂, and ⅙ as ²⁄₁₂. Now the fractions can be added together giving us ³⁄₁₂ + ²⁄₁₂ = ⁵⁄₁₂.
Common Misconceptions About Fractions
It is common as students begin working with fractions for them to bring misconceptions from their understanding of whole numbers. Addressing these early helps prevent confusion later on.
Misconception: Adding or Subtracting Both Numerators and Denominators
A common mistake is thinking that ⅕ + ²⁄₇ = ³⁄₁₂, by simply adding across the top and bottom. However, this ignores the fact that the parts represented by the fractions are not the same size. Visual models can help students see that combining fractions requires a shared denominator so that the parts are equal and comparable.
Misconception: Thinking a Larger Denominator Means a Larger Fraction
Students often assume that a larger denominator means the fraction is larger (e.g., thinking ¹⁄₁₂ is greater than ¼). This confusion comes from their experience with whole numbers, where bigger numbers usually mean more. However, in fractions, the denominator tells us how many equal parts the whole is divided into. The larger the denominator, the more pieces there are, and the smaller each piece becomes. Use area models, or have students fold fraction strips to help clarify this idea.
