Properties of operations
Which operations (+, -, ×, ÷, ^) have the properties given?
Properties
+
-
×
÷
^
Commutative
🙂
❌
🙂
❌
❌
Associative
🙂
❌
🙂
❌
❌
Identity element
0
❌
1
❌
❌
Note: An identity element for an operation gives the same value as was started with. For example, the identity element for addition is zero (0) because \( a + 0 = a = 0 + a \).
Identify the properties
What property (or 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
\( (2b)c = 2(bc) \)
associative property of multiplication
\( (2b)c = c(2b) \)
commutative property of multiplication
\( \frac{1}{2} \cdot (6 + \frac{4}{5}) = \frac{1}{2} \cdot 6 + \frac{1}{2} \cdot \frac{4}{5} \)
distributive property of multiplication and addition
\( (29 \times \frac{7}{9}) \times \frac{9}{7} = 29 \times (\frac{7}{9} \times \frac{9}{7}) = 29 \times 1 = 29 \)
associative property of multiplication, then 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
\( 40 + (10) \)
\( 50 \)
\( 5 \times (2 \times 27.2 ) \)
\( (25 \times 4) \times 27.2 \), associative property of multiplication
\( (100) \times 27.2 \)
\( 2720 \)
\( 24 \times 8 + 24 \times 12 \)
\( 24 (8 + 12) \), distributive property of multiplication and addition
\( 24 (20) \)
\( 480 \) (note: I did this mentally by 24 * 2 * 10 = 48 * 10 = 480)
\( 57^2 + (43 \times 57) \)
\( 57 \times 57 + (43 \times 57) \), 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
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