Properties of operations

Consider the operation ♥ defined by the table below. For example, b ♥ d = c).

a b c d e
a d e a b c
b e a b c d
c a b c d e
d b c d e a
e c d e a b
  1. Is there an identity element with respect to ♥?
  2. Are there inverse elements? If so, identify the inverse pairs. If not, provide a counterexample.
  3. Is ♥ commutative?
  4. Challenge: Is ♥ associative?

Source: Big ideas in mathematics for future middle grades teachers and elementary grade specialists: Big ideas in algebra, by Koker and Szydlik


Identify the properties

What properties were used in the following equations?

  1. \( a + (n + 49) = (a + n) + 49 \)
  2. \( 6\frac{4}{5} \times \frac{1}{2} = \frac{1}{2} \times 6\frac{4}{5} \)
  3. \( (2b)c = 2(bc) = 2bc \)
  4. \( z + -z = 0 \)
  5. \( 29 \times \frac{7}{9} \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)
  6. \( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)
  7. \( xy^2 + 0 = xy^2 \)
  8. \( 1 \cdot x = x \)

Use properties to make the following computations easier to do mentally.

  1. \( (7 + 40) + 3 \)
  2. \( 25 \times (4 \times 27.2 ) \)
  3. \( 24 \times 38 + 24 \times 12 \)
  4. \( 57^2 + (43 \times 57) \)

Source: Reconceptualizing mathematics, Chapter 12.1, by Sowder, Sowder and Nicholson