Properties of operations

Which operations (+, -, ×, ÷, ^) have the properties given?

Properties + - × ÷ ^
Commutative
Associative
Identity element

Note: An identity element for an operation gives the same value as was started with. For example, the identity element for addition is zero (0) because \( a + 0 = a = 0 + a \).

Identify the properties

What property (or 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. \( (2b)c = 2(bc) \)
  4. \( (2b)c = c(2b) \)
  5. \( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)
  6. \( (29 \times \frac{7}{9}) \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)

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

  1. \( (7 + 40) + 3 \)
  2. \( 5 \times (2 \times 27.2 ) \)
  3. \( 24 \times 8 + 24 \times 12 \)
  4. \( 57^2 + (43 \times 57) \)

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