Today, we learned the Associative and Commutative properties of addition and multiplication. We
defined those properties as follows:
Four Properties of Arithmetic:
The Commutative Property of Addition: If a and b are real numbers, then a + b = b + a.
The Associative Property of Addition: If a, b, and c are real numbers, then (a + b) + c = a + (b + c).
The Commutative Property of Multiplication: If a and b are real numbers, then a × b = b × a.
The Associative Property of Multiplication: : If a, b, and c are real numbers, then (ab)c = a(bc).
We then used these properties to prove that (x + y) + z = (z + y) + x.
Our proof looked like this:
(x + y) + z
= x + (y + z)
= x + (z + y)
=(z + y) + x
Associative Property of Addition
Commutative Property of Addition
Commutative Property of Addition
We also looked at flow charts of more difficult proofs and filled in the property that was used to move
between parts of the flow chart using these abbreviations:
C+ for the commutative property of addition
C× for the commutative property of multiplication
A+ for the associative property of addition
A× for the associative property of multiplication
For example, the following flow chart:
We filled in like this:
Finally, we tried one practical application for the associative and commutative properties whole
numbers. We asked, what would be an easy way of finding the sum 53 + 18 + 47 + 82? Using the
associative and commutative properties of addition, we can first add 53 + 47, which equals 100, then
add 18 + 82, which also equal 100, and find that the sum of all four numbers is 200.