# Properties of operations

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

a b c d e ♥ d e a b c e a b c d a b c d e b c d e a 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