Understanding Thirds In Mathematics
Thirds are created when a whole is divided into three equal parts. Each part is called one-third, written as ⅓, and is a type of unit fraction. In fractions involving thirds, the denominator (3) shows the number of equal parts in the whole, while the numerator tells how many of those parts are being considered. For example, ⅔ represents two out of the three equal parts.
One unique feature of thirds is their relationship with decimals: each third is equal to the repeating decimal 0.3̅3, which means ⅔ = 0.6̅6.
Thirds As A Benchmark Fraction
Thirds are considered benchmark fractions because they divide a whole into three equal parts, making them straightforward to visualize and recall. For young learners, this simplicity creates an accessible entry point into understanding fractions. Benchmark fractions like thirds serve as reference points for comparing, estimating, and reasoning about other fractions.
⅓ and ⅔ can be used as comparisons to halves, quarters, eighths, and other fractions.
Visualizing ⅓ on a number line makes it easier to estimate the size of related fractions like ⁴⁄₉ or ⅚.
Connecting Thirds To Sixths, Ninths And Twelfths
Understanding the relationship between thirds, sixths, ninths, and twelfths helps students see how fractions with denominators that are multiples of 3 are interconnected. This connection builds fluency in working with fractions, especially when comparing, finding equivalences, or solving problems involving fractions.
Thirds and Sixths
Since 6 is a multiple of 3, each third can be divided into two smaller, equal parts, creating sixths. This means that every fraction with a denominator of 3 can be rewritten with a denominator of 6 by doubling both the numerator and the denominator. For example, ⅓ = ²⁄₆ and ⅔ = ⁴⁄₆. This equivalence allows students to compare and combine fractions with denominators of 3 and 6 more easily. Visual models, such as fraction strips can help illustrate this relationship.
For example, dividing a strip into thirds and then further dividing each third into two equal parts demonstrates how ⅓ = ²⁄₆.
Thirds And Ninths
Since 9 is a multiple of 3, each third can be divided into three smaller parts, creating ninths. For example, ⅓ = ³⁄₉ and ⅔ = ⁶⁄₉ . Visual models, such as fraction strips, show how dividing each section of a strip representing thirds into three smaller sections produces equivalent fractions in ninths.
On a number line, marking thirds and ninths side-by-side helps students see how fractions like ⁴⁄₉ compare to ⅓.
Thirds And Twelfths
Since 12 is a multiple of 3, each third can be divided into four smaller parts to create twelfths. For example, ⅓ = ⁴⁄₁₂ and ⅔ = ⁸⁄₁₂. Using partitioned rectangles, students can divide each section representing thirds into four smaller parts to visualize this equivalence.
Comparing thirds and twelfths on a number line allows students to observe that ⁵⁄₁₂ is slightly smaller than ½ (equivalent to ⁶⁄₁₂), while ⁴⁄₁₂, which is equivalent to ⅓, is farther from ½ .
Teaching Strategies For Thirds
When teaching thirds, it’s helpful to provide students with concrete, visual, and real-world examples to build their understanding.
Use Visual Models When Working With Thirds
Visual models are an important way to help students develop a conceptual understanding of thirds as equal parts of a whole. Fraction strips or circles divided into three equal sections provide a clear representation of thirds, where each section represents one third.
These models also highlight the relative sizes of fractions, showing, for instance, that ⅔ is twice as large as ⅓. Partitioned rectangles serve a similar purpose, offering a visual way to compare thirds directly. For example, placing a rectangle divided into three parts next to one divided into six parts helps students recognize that ⅔ is equivalent to ⁴⁄₆.
Use Number Lines When Working With Thirds
Number lines offer a powerful tool for understanding thirds as specific numerical values and for connecting fractions to their positions within a continuum. When students mark ⅓, ⅔, and ³⁄₃ on a number line, they can see that thirds are evenly spaced and represent precise points between 0 and 1. This spacing emphasizes the idea that fractions are numbers, not just parts of a whole, and that thirds divide the distance from 0 to 1 into three equal intervals.
Comparing thirds to benchmark fractions like ½ further deepens understanding. For example, students can observe that ⅓ is smaller than ½, but ⅔ is larger. These comparisons help students develop number sense, as they begin to reason about fractions in terms of their relative sizes and distances from whole numbers.
