For the table below, answer the following questions considering the operations +, -, ×, ÷, ^:

- Which operations are commutative? Give a counter example for each non-commutative operation.

commutative: \(+, \times \);

\(1 - 2 \neq 2 - 1\)

\(1 \div 2 \neq 2 \div 1 \)

\( 1^2 \neq 2^1 \) - Which operations have an identity element? Please state them.

identity element for addition: 0

identity element for multiplication: 1

subtraction, division, and exponents do not have identity elements because the operations are not commutative - Which operations have inverse elements? Please state them.

inverse element for addition: \(-a\)

inverse element for multiplication: \( \dfrac{1}{a} \) - Which operations are associative? Give a counter example for each non-associative operation.

associative: \(+, \times\);

\( 10 - (9 - 8) \neq (10 - 9) - 8 \)

\( 10 \div (5 \div 1) \neq (10 \div 5) \div 1 \)

\( 1^{(2^3)} \neq (1^2)^3 \) - What is the full name of the distributive property? Please give an example for how the property works.

distributive property of multiplication and addition

\( a (b + c) = a \cdot b + a \cdot c \)

be careful as students will often "distribute" any time there is a coefficient in front of parentheses, e.g. \( a(b \cdot c) \rightarrow ab \cdot bc \)

What properties were used in the following equations?

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

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

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

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

inverse element of addition - \( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)

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

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

multiplicative identity - \( 29 \times \frac{7}{9} \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)

associative property of multiplication, multiplication inverse, multiplicative identity

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

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

\( (40 + 7) + 3 \), commutative property of addition

\( 40 + (7 + 3) \), associative property of addition

\( 40 + (10) \)

\( 50 \) - \( 25 \times (4 \times 27.2 ) \)

\( (25 \times 4) \times 27.2 \), associative property of multiplication

\( (100) \times 27.2 \)

\( 2720 \) - \( 24 \times 38 + 24 \times 12 \)

\( 24 (38 + 12) \), distributive property of multiplication and addition

\( 24 (50) \)

\( 1,200 \) (note: I did this mentally by 24 * 100 / 2 = 2400 / 2 = 1200) - \( 57^2 + (43 \times 57) \)

\( 57 \times 57 + (43 \times 57) \), rewrite the exponent in terms of multiplication

\( (57 + 43) 57 \), distributive property of multiplication and addition

(100) 57

5700

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