Rationale
The rationale for teaching division of fractions in an Algebra II class is based on several critical mathematical concepts and real-world applications. Following are some key points:
Fraction Division using the partitioning model
The first insight students learn from the arithmetic operation of division comes from the partition model. For example, if I have six books and need to give them to three students, how many books do each of them get? In this case, we are partitioning a set of six books into three sets, and the answer is two, as each set will have this number of books. However, when we divide by a fraction, the partition model is not very helpful. For instance, how can we interpret the division of 6 apples by 1/2? We need instead to think of division in terms of measurement. In this case, we are given the size of the portions, and the question is, how many portions of ½ apple are there in 6 apples?
Fraction Division using the definition of copies of b in a
A general definition of division reads as follows (Adding it Up 2001). In the division of a/b, where a and b could be an integer or rational numbers, ask the question, how many copies of b are there in a? This general definition is used throughout the unit, reinforcing to the students that it is possible to have a general definition that extends their prior conceptions and is universally applicable to all the problems they may find.
Fraction Division as Inverse Multiplication
From the point of view of the Rules of Arithmetic, the division of fractions can be seen as the inverse operation of multiplication (as shown in the Appendix). Thus, 1/a, by definition, is the result of dividing a quantity (playing the role of a unit) into equal parts.
So, 1/a = a-1. Then b/a = b * 1/a,
And, (b/a)-1 = b-1 (1/a)-1 = (1/b) a = a/b. All these equations are a consequence of the Inverse Rule, plus the Associativity and Commutativity Rules (see Appendix).
Moreover, in general, just as multiplying two fractions together results in a fraction, dividing one fraction by another also produces a fraction. Since dividing is the same as multiplying by the multiplicative inverse, which is a fraction, this follows from the fact that the product of fractions is a fraction.
Extending Fraction Operations
Algebra II builds upon the foundation of arithmetic and elementary algebra. By introducing the division of fractions, students expand their understanding of fraction operations beyond addition and subtraction, which are typically taught at earlier grade levels. Students also reinforce and consolidate their knowledge of fractions by seeing their application in real-life practical problems which involve their division.
Understanding Fractional Quotients
The learning and practicing of the division of fractions allow students to interpret the result as a fractional quotient. Realizing this is very important since fractions and ratios are generally seen as different mathematical entities when they are equivalent. For example, dividing 2/3 by 1/4 can be interpreted as "How many 1/4-sized pieces can fit into a 2/3-sized piece?" That is, what is the ratio of 2/3-sized pieces to 1/4-size pieces? This idea is essential for working with rational numbers and understanding real-world applications involving ratios and proportions.
Rationalizing Complex Fractions
Algebra II often deals with more complex algebraic expressions, including rational expressions. Division of fractions provides a foundation for simplifying and rationalizing complex fractions, where the numerator and denominator can contain fractions or algebraic expressions. It is common for students to get discouraged when they see problems that involve fractions and even more when they are present in ratios. Therefore, students must learn and practice complex fractions.
Applications in Science and Engineering
The division of fractions has numerous applications in fields such as physics, engineering, and chemistry. For example, calculating rates, proportions, concentrations, and dilutions often involve dividing fractions. Teaching division of fractions equips students with the necessary mathematical tools to solve problems encountered in these areas. The students gain confidence in their mathematical abilities, as they can apply the abstract concepts of mathematics to STEM applications, which can further motivate them to pursue post-secondary education in these fields.
Problem-Solving and Critical Thinking
The division of fractions requires students to think critically and apply mathematical reasoning to solve problems. As noted in the introduction, math is always about understanding problems, especially word problems, where the students need to use their previous knowledge to decode the information, translate it into symbolic expressions, and then form an equation. Significantly, the Algebra problems involving the division of fractions encourage logical thinking, pattern recognition, and the ability to manipulate algebraic expressions. These problem-solving skills are crucial for success in higher-level mathematics and many other disciplines.
Preparation for Higher-Level Math
The division of fractions is a stepping stone to more advanced mathematical concepts. A clear understanding of the meaning of division of fractions, with the use of different visual and symbolic methods, generates an understanding that lays the groundwork for topics like operations with rational expressions, solving equations involving fractions, and working with complex numbers, all of which are covered in later advanced algebra, precalculus, and calculus courses. (Hiebert 1997, Heid 2005)
In summary, teaching the division of fractions in an Algebra II class provides students with a deeper understanding of fractions, promotes problem-solving skills and critical thinking, and prepares them for more advanced math topics. It also offers practical applications in various scientific and engineering fields, motivating the students to pursue these academic fields and contributing to developing a well-rounded mathematical foundation.
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