Problem 88 - Introductory Algebra for College Students

Question

You can transpose the letters in the word "conversation" to form the phrase "voices rant on." From "total abstainers" we can form "sit not at ale bars." What two algebraic properties do each of these transpositions (called anagrams) remind you of? Explain your answer.

Step-by-Step Solution

Answer

The two main algebraic properties that these anagrams remind of are the commutative and associative properties. The commutative property is represented through the re-ordering of letters to create a new phrase, whereas the associative property is seen with the formation of new words (or groups of letters) out of the original phrase.

Step 1 - Establish understanding of relevant algebraic properties

First, the commutative property should be understood; it states that the order in which operations are performed does not change the result. This can be exemplified by the equation \(a + b = b + a\). The associative property, on the other hand, explains that how numbers are grouped in an operation does not affect the outcome, like in the equation \((a + b) + c = a + (b + c)\).

Step 2 - Link properties to anagrams

Anagrams are basically a re-ordering or re-grouping using the same letters to form different words or phrases. So the commutative property can be likened to the re-ordering of the letters, because the change in their positions does not change the fact that these are the same letters. The associative property can be associated with the formation of new groups of letters to form new words out of the same letters.

Step 3 - Communicate findings

Both the anagrams 'conversation' and 'voices rant on', and 'total abstainers' and 'sit not at ale bars' remind you of the commutative and associative properties of algebra. The letters in both pairs of words/phrases can be re-ordered (commutative property) and re-grouped (associative property) without distorting the original collection of letters.