11/01/07 draft Page 1
Introduction to Fractions -Section 1
What is a fraction? Why is this type of number needed? The answers to these questions depend on who answers them. From a mathematician’s point of view, the defining of fractions (or more formally rational numbers) provides a type of number that by their very nature have properties that whole numbers and integers do not possess. From a fifth grader’s perspective, fractions allow us to address problems related to measurement and fair share division. This section will focus on the various definitions and conceptualizations of fractions, various ways to concretely model fractions, different representational forms of fractions, ordering fractions, and problem solving using fractions in a concrete, contextual sense.
Definition and conceptualizations of fractions
There is no standard definition of a fraction. Some sources will use the term fraction and rational number interchangeably. In some cases, the term fraction is used in reference to rational expressions, which can be composed of variables and polynomial expressions, or to make comparisons of real numbers, such as . For the purposes of this textbook, fraction and rational number will be defined as related, yet distinct, constructs. The definition used here relates to how fractions and rational numbers are usually taught in school mathematics. Before formal definitions of the terms are given, a review some of the different ways to conceptualize these types of numbers is presented.
The following are ways the representation can be interpreted from these different perspectives.
Parts of a whole: From this perspective means 3 equal sized parts out of 4 equal parts of a unit whole. This is the conceptualization with which most elementary students are introduced to fractions.
Quotient: From this perspective we view the fraction bar as another symbol representing division, hence means 3 divided by 4 and the quotient of this division is what is being represented. This is the perspective we use when we teach students to convert fractions to their decimal representation.
Ratio and Rates: Ratios are a broader concept of which fractions are a special case. Ratios and rates will be discussed in more detail in Chapter XX but for the purposes of the discussion here, means 3 units compared to 4 units of the same or different measure.
The ‘Parts of a Whole’ conceptualization is the one most easily supported by children’s understanding of division and fair-sharing, so this is the conceptualization that will be utilized for defining the terms ‘fraction’ and ‘rational number’.
Definition. A fraction is a number represented by two whole numbers, a and b, b ≠ 0 using the notation or a/b indicating that some unit whole has been divided or partitioned into b equal sized parts and a of them are under consideration. The a part of the fraction is called the numerator and the b part is called the denominator. The meaning of is relative to what unit whole it is referencing. A rational number is a number represented by two integers, a and b,
b ≠ 0 using the notation or a/b. We will expand this definition more in a later section.
Modeling fractions
As children are taught the definition of a fraction, one important consideration is what type of concrete and pictorial models will be used to represent this concept. Three types of common models used to represent fractions are set models, area models, and number line models.
A set model is appropriate when examples involving discrete objects are being used. If examples involve things that cannot be subdivided, but can be grouped in various ways, then this type of model is appropriate. For instance, when discussing a fraction in the context of a group of people or other living things, it is not appropriate to split a person or a dog into smaller pieces. For instance, having a problem with 2 ½ ponies just doesn’t make sense.
An example of a set model for ¾ might look like this:
Three-fourths of the happy faces are light.
An area model is another way that fractions are commonly represented for students. An area model is appropriate when giving examples involving objects that can be subdivided into any number of pieces. The model consists of some type of two-dimensional figure, such as a circle, square, rectangle, etc. While any type of figure can be used as an area models, some shapes are better than others. Circles are often used in elementary school textbooks, but have some limitations when used to represent operations with fractions. Squares or rectangles are easier to subdivide into odd numbers of equal sized pieces and are excellent models when used to teach fraction operations.
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An example of an area model might look like this:
Three-fifths of the large rectangle is shaded.
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A number line model is appropriate when using examples involving linear measurement. An advantage of number lines is that most children are already familiar with this instructional aid. A disadvantage is that number lines, for some children, are very abstract and if not presented carefully can actually lead to confusion. Consider this example of a common question that might be found in a homework set or standardized test over fractions.
Name the fraction represented by the diagram.
0 1 2 3 4
Some students will interpret this diagram as 11/25 because they will count all of the represented subdivisions, making that the denominator and then counting those represented by the arrow and making that the numerator. What they will fail to understand is that with a number line model, the distance from 0 to 1 is always the unit whole and any representation using a number line must be interpreted with that understanding.
When choosing any model, it is important that students have been given some guidance in how to interpret it. One of the key concepts in understanding fractions is relating the representation to its unit whole.
Consider this example:
What fraction is being represented by the shaded part of this diagram? The answer to that question depends upon how the unit whole for the diagram is interpreted. If the large rectangle is considered to be the unit whole, then the diagram represents 3/12. If the bold line in the middle of the diagram is seen as making two rectangles – then the unit whole consists of 6 small rectangles and the diagram is representing 2/6 in the left rectangle and 1/6 in the right rectangle or 3/6 altogether. This example not only shows the importance of identifying the unit whole for the situation, it is an example of how pictorial representations used to teach fractions need to be well defined or students may draw conclusions that the teacher hasn’t intended.
What if the above diagram is redrawn to look like this?
If the large rectangle only has 4 equal parts instead of 12 and the shaded sections were shifted to line up, what fractional part of the large rectangle is shaded? What has changed? Is the relationship ¼ the same as 3/12 for this rectangle? These are all questions related to understanding the definition of a fraction and lead to the next concept.
Equivalent or equal fractions
Fractions that represent the same relative amount are called equivalent or equal fractions. Every fraction (actually every real number) has an infinite number of representations. Let’s consider the fraction . This fraction can be represented with the following area model:
One third of the large rectangle is shaded. By dividing each of the one third sections into two equal sections, the model can be changed to represent the fraction .
This process of dividing each of the sections into equal sized subsections can continue to represent an infinite number of different fractions that all relate to the same relative amount.
This model could represent . But all of the diagrams have the same sized rectangle (the unit whole), with the same amount of the area of that rectangle shaded blue and the same amount of the area of the rectangle not shaded. The only difference between the three diagrams is how many divisions of the area are shown. This is the concept behind equivalent or equal fractions.
Symbolically this concept can be defined as follows: Given any fraction , for which
b ≠ 0, we can represent the same relative amount in an infinite number of ways by multiplying both the numerator and denominator by the same natural number n, therefore
. When is such that a and b have no common factors greater than 1, then we say that the fraction is in simplest form, reduced form, or in lowest terms.
There are several methods for determining if two fractions are equivalent.
Method 1 – Use cross products
For example, determine if and are equal? Using cross products it can be shown that , so the two fractions are representing the same relative amount.
Method 2 – Simplify both fractions
Another way to determine if and are equal fractions is to simplify both fractions and show they are equal to the same fraction.
and . Since both and equal , they must equal each other.
Method 3 – Convert both fractions to their decimal representations (Note: this method is not appropriate for students who haven’t studied decimals)
The conceptualization that =, leads to the use of the decimal representation of this quotient to determine if the two fractions are equal. Since both and are equal to , then by the transitive property the two fractions must equal the same relative amount. (Decimal representations of fractions will be discussed more in the next chapter.)
Ordering fractions
Research on how children learn about fractions has revealed that given opportunities to interact with various types of fraction manipulatives and interesting and challenging problem situations to work through, children will develop their own techniques for ordering fractions. These ‘intuitive ordering techniques’ are grounded in a developing fraction number sense. Traditional textbooks use more computationally oriented techniques of ordering fractions. Both are presented here.
Intuitive ordering techniques
• Same denominator method. If the two fractions have the same denominator, that means the ‘parts’ are all the same size, so comparing the fractions consists of determining which fraction is describing more parts, so compare the numerators. The fraction with the larger numerator is larger.
Rule – given
Example: Given and , since 2 < 5, we determine that .
• 1/n type fractions. If the two fractions both have a numerator of 1, then compare the denominators. The larger the denominator, the smaller the size of 1 part. The fraction with the smaller denominator is the larger fraction.
Rule – given
Example: Given and , since 3 < 8 then .
• Same numerator method. This is a generalization of the 1/n method.
Rule - given
Example: Given and , since 5 < 13 then
• (n-1)/n type fractions. This method examines how close to being a unit whole each fraction is. The fraction with the smaller missing piece is the larger fraction.
Rule – given
Example: Given and , since the missing fifth sized piece is larger than the missing eighth sized piece, then .
• Comparison to ½ method. This is a benchmarking method where the two fractions are compared to the value ½. Most children can easily divide by two, even if the dividend is an odd number. One way to use this method is to divide each denominator by 2, then compare the numerator to this result. If the numerator is less than the denominator divided by 2, then the fraction is less than ½. If the numerator is more than the denominator divided by 2, then the fraction is more than ½. Another way to use this method follows.
Rule -
Example: Given and , we determine that because 4 < 9÷2 and because 7> 12÷2. So using the transitive property, .
Traditional textbook methods
• Find a common denominator. This reflects the first of the intuitive techniques listed above. A representation that gives both fractions a common denominator is found and the fraction with the larger numerator is the larger fraction. For example, which is smaller 11/20 or 4/7?
Therefore, 11/20 < 4/7.
· Convert to decimal. As discussed before, this method is not appropriate for students who haven’t studied decimals, but this is a common method of comparing fractions once students understand the concept of fractions as representing the division of the numerator by the denominator.
AN IMPORTANT PROPERTY
Fractions have an interesting property that natural numbers, whole numbers, and integers do not possess. Consider this question. Name a whole number between 0 and 1? The answer, of course, is there are no whole numbers between 0 and 1. But if the questions is changed to “Can you name a fraction between 0 and 1?”, the answer is “yes”. What is amazing is that there are an infinite number of fractions between 0 and 1 and between any two fractions in that interval, there are an infinite number of fractions. This concept is called the density property.
Given any two numbers a and b in a set, if there exists another number c such that a < c < b, then the set of numbers exhibits the density property.
Improper fractions and mixed numerals
One misconception that students often develop is that all fractions are less than 1. Obviously, this is not the case. Fractions that are between 0 and 1 are called proper fractions. Another way to define this is to say that given the fraction if a < b, then the fraction is considered a proper fraction. Fractions for which the numerator and denominator are equal can be simplified and are also whole numbers or non-negative integers. Fractions in which the numerator is greater than, or equal to, the denominator are called improper fractions. More formally, given the fraction , then the fraction is an improper fraction.