# 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