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# Composition, Decomposition, Relationships, Representation, Context (page 2)

By Pearson Allyn Bacon Prentice Hall
Updated on Jul 20, 2010

### Decomposition

Once we have composed a countable unit we can go in an entirely different direction. Aside from simply counting the number of objects of that unit, we could also decide to partition, or decompose, one of those units. There is an entire world of mathematics found in decomposing units. A unit can be a geometric shape, and decomposing it into parts can begin a traditional study of fractions. A unit can be a group of objects, and a consideration of a portion of the group involves a different use of fractions. If our unit is readily divided into ten equal pieces, then we can introduce the idea of decimal numbers. If we decompose a unit into two different parts and then we study the relative sizes of those parts (Are the parts equal in size? Is one of the parts twice as large as the other?), then we introduce a study of ratio. If we can readily decompose our unit into one hundred equal parts, then we can work with percents. All kinds of measurement activities and understandings rely on decomposing as well. Rarely can we accomplish the precision that we need using whole numbers of measurement units exclusively. Precision in measurement, then, relies on the idea of decomposing units into meaningful parts.

A slightly different consideration of parts and wholes brings us to the heart of addition and subtraction. When we add two quantities together we are taking two parts and composing them into a single whole. When we take a single whole (group) and break it apart into two or more smaller groups we are doing a certain kind of subtraction. When we consider addition and subtraction in this light we can begin to see them as two sides of a coin rather than as two distinctly different operations and concepts. Addition and subtraction are not the only basic operations that have their roots in decomposing. Division as well can be best understood in this light. What is division if it is not taking a whole and dividing it into smaller parts? If the whole is a dozen objects, then we can use decomposing to answer questions about smaller groups into which our dozen objects can be broken.

Decomposing, then, opens up a wide range of mathematical topics for study. Decomposing is one of our powerful ideas. On one hand we can use the idea of composing as the basis for counting, and on the other hand we can use the idea of decomposing as the basis for analysis. If composing is the key idea behind counting, decomposing is the key idea behind fractions and all of the things we can do with fractions.

### Relationships

Another powerful idea emerges when we begin to consider the concept of relationships in mathematics. We can study relationships between numbers or between sets of numbers. We can also study relationships between shapes and between parts of shapes. We mentioned ratios in our introduction to decomposing; the examination and analysis of ratios leads us to the powerful idea of mathematical relationships. While it is true that the concept of a relationship between numbers is completely abstract, it is, nonetheless, one of the more important topics in mathematics. An ability to see relationships as something tangible, in spite of their abstract nature, will make it possible for a student of mathematics to progress into higher levels of mathematical study. As with composing and decomposing, the concept of relationship is found in numeric as well as geometric areas of mathematics. This is part of why we consider it to be a powerful idea.

Consider the set of numbers that would be used to represent pounds of bananas and the cost of the bananas. Tthe table below presents a possible set of such numbers. An examination of the relationship between pounds of bananas and cost of the bananas reveals a consistent pattern. Multiplying the number of pounds by \$0.15 yields the total cost of the bananas. Conversely, dividing the cost by \$0.15 gives us the number of pounds of bananas.  We say that the bananas are \$0.15 per pound. The term "per" indicates a relationship. Knowing this relationship we can compute the cost for any quantity of bananas, and we can compute the weight of any given purchase. Knowing a relationship between quantities gives us a lot of power. Studying relationships is a crucial part of a study of meaningful and useful mathematics.