Variables & Constants

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Algebra developed as a language to express patterns &amp; relations between numbers &amp; shapes.

All number &amp; shape patterns have entities which change and those that do not change. Those that change are called Variables &amp; those that do not change are called Constants. One of the simplest ways to introduce these ideas to students is through the idea of &ldquo;changing&rdquo; and &ldquo;not changing&rdquo; in patterns. Let us consider the following number patterns.

Number Patterns

&ldquo;--&ldquo;indicates that this pattern can be continued as long as we want. But to continue the pattern, we need to &ldquo;understand&rdquo; the essence of the pattern.

We have given an explanation of the essence of the pattern in the last row. (In the classroom it is suggested that this be shown only after students try to express their own understanding of the pattern).

Two terms which keep on occulting in the explanation are &ldquo;changing&rdquo; and &ldquo;not changing&rdquo;. Their meanings are apparent from the context. A &ldquo;changing&rdquo; number can have many values. A &ldquo;non-changing&rdquo; number has a fixed value.

How do we summarize each of these patterns into a simple statement which conveys the meaning&rdquo; of the pattern? We do it by using the language of Patterns!

Language of Patterns

We use the following strategy to describe the patterns.

We use any letter from the alphabet to indicate a &ldquo;changing&rdquo; number. (We will explain the reason for this choice in the next chapter). The &ldquo;not-changing&rdquo; number is used as it is.

So Pattern 1 can be written as &ldquo;3 + a&rdquo;. Pattern 2 can be written as &ldquo;7 – a&rdquo;. Pattern 3 can be written as  &ldquo;b + b&rdquo;. Pattern 4 can be written as &ldquo;4 X a&rdquo;. Pattern 5 can be written as &ldquo;a + b&rdquo;. Pattern 6 can be written as &ldquo;a X a&rdquo;. These results are summarized below for convenience.

The bottom row has expressions in pattern language. Each of these expressions is an abstraction of a series of number patterns (shown in first table) with changing &amp; not changing entities.

Patterns &amp; Pattern Language

It would be a good practice if students practice Pattern language in both perspectives.


 * 1) Given a pattern of numbers, summarize the pattern in Pattern Language
 * 2) Given a patter in terms of the pattern language, write down a few examples of the pattern it could represent.

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