Definitions:

An operation, \(\star\), has an **identity**, \(e\), if \(a \star e = a\) and \(e \star a = a\). For example, the identity for addition is 0 because \( a + 0 = a\) and \(0 + a = a\).

- Does multiplication have an identity element? If so, what is it?
- Does subtraction have an identity element? If so, what is it?
- Does division have an identity element? If so, what is it?

An operation, \(\star\), has an **inverse**, \(x\), if \(a \star x = e\) and \(x \star a = e\). For example, in addition \(-a\) is the inverse of \(a\) because \( a + (-a) = 0\) and \((-a) + a = 0\).

- Does multiplication have an inverse? Why or why not?
- Does subtraction have an inverse? Why or why not?
- Does division have an inverse? Why or why not?

An operation, \(\star\), is **commutative** if \(a \star b = b \star a\). For example, addition is commutative because \(2 + 3 = 3 + 2\), which works for any two numbers.

- Is multiplication commutative? If so, give an example. If not, give a counter-example.
- Is subtraction commutative? If so, give an example. If not, give a counter-example.
- Is division commutative? If so, give an example. If not, give a counter-example.

An operation, \(\star\), is **associative** if \(a \star (b \star c) = (a \star b) \star c\). For example, addition is associative because \(1+ (2 + 3) = (1 + 2) + 3\), which works for any three numbers.

- Is multiplication associative? If so, give an example. If not, give a counter-example.
- Is subtraction associative? If so, give an example. If not, give a counter-example.
- Is division associative? If so, give an example. If not, give a counter-example.

What properties were used in the following equations? Use the vocabulary from CCSSM (page 90, table 3).

- \( a + (n + 49) = (a + n) + 49 \)

associative property of addition - \( 6\frac{4}{5} \times \frac{1}{2} = \frac{1}{2} \times 6\frac{4}{5} \)

commutative property of multiplication - \( (2b)c = 2(bc) \)

associative property of multiplication - \( z + -z = 0 \)

additive inverse - \( 29 \times \frac{7}{9} \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)

associative property of multiplication, multiplicative inverse, multiplicative identity - \( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)

distributive property of addition and multiplication - \( xy^2 + 0 = xy^2 \)

additive identity - \( 1 \cdot x = x \)

multiplicative identity

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

- \( (7 + 40) + 3 \)

\( (40 + 7) + 3 \) by the commutative property of addition

\( 40 + (7 + 3) \) by the associative property of addition

answer is \( 40 + 10 = 50 \) - \( 25 \times (4 \times 27.2 ) \)

\( (25 \times 4) \times 27.2 \) by the associative property of multiplication

answer is \( 100 \times 27.2 = 2720 \) - \( 24 \times 38 + 24 \times 12 \)

\(24 \times (38 + 12) \) by the distributive property of addition and multiplication

answer is \( 24 \times 50 = 120\) - \( 57^2 + (43 \times 57) \)

\(57 \times (57 + 43) \) by the distributive property of addition and multiplication

answer is \( 57 \times 100 = 5700 \)

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

Suppose Δ is a new operation that only works for numbers 1 through 5. The table below gives the results of the operation. For example, 2 Δ 5 = 4.

Δ | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 4 | 5 | 1 | 2 | 3 |

2 | 5 | 1 | 2 | 3 | 4 |

3 | 1 | 2 | 3 | 4 | 5 |

4 | 2 | 3 | 4 | 5 | 1 |

5 | 3 | 4 | 5 | 1 | 2 |

- Does Δ have an identity? If so, what? yes, 3
- Does Δ have inverses? If so, what are they? yes, 1 and 5, 2 and 4
- Is Δ commutative? Why or why not? yes, values are symmetric along main diagonal
- Is Δ associative? If so, give an example. If not, give a counter-example.

example: 1 Δ (2 Δ 3) = 1 Δ 2 = 5 and (1 Δ 2) Δ 3 = 5 Δ 3 = 5

note: would need to test all 10 combinations to be sure