# 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 |

- Is there an identity element with respect to ♥?
- Are there inverse elements? If so, identify the inverse pairs. If not, provide a counterexample.
- Is ♥ commutative?
**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?

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

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

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

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