9 Early Fraction Concepts
What are Fractions?
According to the Kansas Mathematics Standards (2017), the formal definition of fraction is a number expressible in the form [latex]\frac{a}{b}[/latex] where a is the number of equal parts being referenced and b is the number of equal parts in the whole. But what does this mean?
The word “fraction” comes from the Latin word fractus or broken. A fraction describes how many parts of a certain size there are, for example, one-half, three-fourths, etc. Additionally, the top number (numerator) says how many parts you have, and the bottom number (denominator) says how many equal parts in the whole amount. Most importantly, a fraction is a number.
An understanding of fractions begins in first grade when students partition circles and rectangles into two and four equal parts, and describe the parts with the words halves, fourths, and quarters. In second grade, students partition circles and rectangles in two, three, and four equal parts, and describe those parts with the words halves, thirds, and fourths. Fractions become the major emphasis in third grade as students look at fraction symbols, and explore unit fractions.
“Students need significant time and experiences to develop a deep conceptual understanding” of fractions (Van de Walle, Karp, Bay-Williams, 2019, p. 338). Teachers typically begin fraction instruction with objects, paper folding, or even plastic fraction circles. But often we move too quickly past these models to abstract computation. Some experts even argue that drawings of fractions fail to be concrete enough for some students. The act of cutting and folding and manipulating is critical for all students.
Watch this video from Graham Fletcher, “The Progression of Fractions.”
Fraction Constructs
Fractions include many meanings such as part-whole, measurement, division, operator, and ratio. Understanding fractions includes the need to understand all of these different meanings. As you and your students begin to make sense of fractions, you must also understand all the possible concepts that fractions represent. The following constructs are based on the research from Van de Walle, Karp, and Bay-Williams (2019):
Fractions as Part-whole Comparisons. Part-whole comparisons is a good place to start when building a conceptual understanding of fractions. Part-whole comparisons can be shown by shading a region, part of a group, and measurement.
- Shading a region
For example, a circle can be divided into four equal parts. If three of those parts are shaded pink, then the fraction shaded is [latex]\frac{3}{4}[/latex].
- Part of a group
For example, I have five pets; three cats and two dogs. The fractional part of cats to all of my pets is [latex]\frac{3}{5}[/latex].
- Measurement
For example, in the fraction [latex]\frac{3}{8}[/latex], students use the unit fraction[latex]\frac{1}{8}[/latex]to count or measure that it takes 3 of those units to reach [latex]\frac{3}{8}[/latex].
Fractions as Division. Just