Background
Models of Fractions
Students need to have many opportunities to experience many different interpretations of fractions in order to develop a robust, mature understanding. A deep understanding will provide a solid foundation. There are several different ways to model fractions, and it is important to expose the students to all of them in a variety of ways. Variation is the key to creating a robust understanding and flexible thinking.
Students are typically taught about fractions with part-whole comparisons. The part-whole comparison is just one interpretation of fractions. If students are only exposed to one or two models they will develop a limited understanding and will incorrectly generalize that fractions can only be represented in those one or two ways. The different types of models are listed below.
Area Models
Area models can be depicted with squares, rectangles, circles, or liquid volume. Singapore Math recommends using squares and rectangles since they are easier for children to draw. Circles are a little more complicated especially when representing thirds and fifths. The rectangle below is an example of an area model depicting the fraction ¼ by the shaded region.
Linear or Measurement Models
The linear or measurement models include number lines, bars, rulers and scales. Often the fractional parts are not clearly identified on rulers and scales. When using number lines, it is important to emphasize that the unit is the linear distance of the unit and not the point indicating the interval. One example of a linear or measurement model is the bar below, also depicting the fraction 1/4 by the shaded region:
Set Models
Set models are a collection of objects. Fractions can be depicted using set models in many ways. For example, the fraction 1/3 can be shown by beginning with 15 soda bottles, five of which are root beer, so that one third of them are root beer. The collection of shapes below is another example of a set model.
In this collection, 3/8 of the shapes are squares, 1/4 are triangles, and so on.
Constructs of Fractions
Fractions are a difficult concept for children to fully grasp because fractions are multifaceted. In addition to the different types of models there are also many different uses for fractions. Five interrelated interpretations, or constructs have been identified. The five constructs are part-whole, measure, operator, quotient and ratio and rate. 2 Part-whole comparisons are the most dominant construct in education, and unfortunately this construct is limited and leads to misconceptions. Fractions as measures are important and are often illustrated using a number line. When fractions are used as operators, they enlarge or reduce the size of a number or an object. An example of operators would be two times or one half of a whole. Symbolically, fractions represent a quotient. When a fraction is written such as 1/3, the midline means divide. Representing fractions as ratios and rates is another type of construct. A ratio compares two quantities of the same type such as two out of three of the students. A rate compares two unlike quantities as in miles per hour. The many interpretations of fractions demonstrate the underlying complexity that is too often simply represented with a pie.
Division in Fractions
Division is the common thread that runs through all of the constructs. Understanding fractions as division is essential to teaching students to develop a robust understanding of fractions. The operation of division has two distinct models: equal share and partitioning. An example of an equal share is 15 toy cars that need to be put into three equal groups. The partitioning model could also be called the measurement model. A similar problem demonstrating the partitioning or measurement model would be to begin with 15 toy cars and ask students how many packages of five toy cars there are. Both models are equally important and need attention when teaching students how to work with fractions.
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