Properties of operations

For the table below, answer the following questions considering the operations +, -, ×, ÷, ^:

  1. Which operations are commutative? Give a counter example for each non-commutative operation.
  2. Which operations have an identity element? Please state them.
  3. Which operations have inverse elements? Please state them.
  4. Which operations are associative? Give a counter example for each non-associative operation.
  5. What is the full name of the distributive property? Please give an example for how the property works.

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. \( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)
  6. \( xy^2 + 0 = xy^2 \)
  7. \( 1 \cdot x = x \)
  8. \( 29 \times \frac{7}{9} \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)

Use properties to make the following expressions easier to do mentally. Compute the simplified expression.

  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