Associative Law of Addition and Multiplication

A closing note on Average Joe's Non-associative Averaging: Now that you have practiced using the associative property with addition and multiplication, let's revisit Average Joe and his averaging tendencies to see exactly why it fails to be associative. Just like we use the symbols $+$ and $\times$ for addition and multiplication, let's introduce our own symbol for averaging two numbers. Let us write $a \oplus b$ to mean ""the average of a and b"" so that:

$a \oplus b =\frac{a+b}{2}$

It should seem straightforward to check that averaging is, in fact, commutative so that $a \oplus b = b\oplus a$, but without parentheses, an expression like $a \oplus b \oplus c$ does not make sense. For example, you can check

$(2 \oplus 10) \oplus 14 = 6 \oplus 14 =10$
$2 \oplus (10 \oplus 14)=2\oplus 12=7$

Which terms are grouped together changes the final result. Poor Joe cannot make sense of $2 \oplus 10 \oplus 14$, and neither can anyone else because averaging two numbers is not associative.