Math 25 Activity 2: Properties of Real Numbers – Commutative and Associative
Last activity, we discussed the different classifications for real numbers and reviewed applications
for operations using positive and negative numbers. Now we want to look at two properties of real
numbers that we use to make computations easier and to solve algebraic equations.
When we discuss properties of real numbers, the word “commutative” indicates when we can
switch the order of the numbers without affecting the outcome. One example is in addition: 5+3 is
the same as 3+5.
Take each example and first decide if the left and right sides of the equal signs are equivalent. That
would mean the equals sign makes the statement true. Then, using the explanation above, decide if
the commutative property was used in the example.
Example
2+3=3+2
2–4=4–2
2 5 = 5 2
26 = 62
2+3+4=4+2+3
2 (3 + 6) = 2  3 + 2  6
(2 + 3) + 6 = 2 + (3 + 6)
2+3=1+4
Are The Sides Equivalent?
Does it use the Commutative Property?
1. In your groups, answer the following question: For which operations from this list does the
commutative property hold true: addition, subtraction, multiplication, and division? Why or Why
not?
We use the following phrasing – “the commutative property of addition” and “the commutative
property of multiplication.”
2. Did you say that subtraction or division would work? Give an example of why the commutative
property does not necessarily work for all real numbers when the operation is subtract or divide.
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In math, operations are defined so that we can only perform an operation with two numbers at a time.
We usually write one number, the operation sign, and then the second number. When we have more
than two numbers, then we have to decide which two numbers and one operation we will do first.
Normally, when we have a situation with a choice, we use parentheses to indicate which pair of
numbers and operation we plan to work with first. The word “associative” means that in certain
circumstances we can regroup numbers without affecting the outcome, meaning, we could choose to do
a different operation other than the indicated first one.
One example is addition: (4+3) + 6 means do the (4+3) operation first, then add 6 second. That would be
7+6 which is finally 13. The associative property would mean that we could rewrite the original
expression (4+3) + 6 as 4+(3+ 6) which means add the 3+6 first, then add 4 second. That would be 4+9,
which is 13. See how it seems that the numbers remain in the same order, but the parentheses moved?
That is what we mean by “regroup”.
Take each example and first decide if the left and right sides of the equal signs are equivalent. That
would mean the equals sign makes the statement true. Then, using the explanation above, decide if the
associative property was used in the example.
Example
(2 + 6) + 11 = 2 + (6 + 11)
(2 + 3) – 7 = 2 + (3 – 7)
3(2  5) = (3  2)  5
(24  6)  2 = 24  (6  2)
6 – (7 – 2) = (6 – 7) – 2
9 (4 + 7) = 9  4 + 9  7
(2 + 3) + 6 = 2 + (3 + 6)
10 + [4 + (2 + 5)] = [10 + (4 + 2)] + 5
2[4(5  3)] = [2(4  5)]  3
6[7(4  2)] = [6(7  2)]  4
Are The Sides Equivalent?
Does it use the Associative Property?
Come up with new examples for when the associative property works and when it does not.
Examples of when Associative Property works
Examples of when Associative Property
does NOT work
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3. In your groups, answer the following question: For which operations from this list does the
associative property hold true: addition, subtraction, multiplication, and division? Why or Why
not?
We use the following phrasing – “the associative property of addition” and “the associative property of
multiplication.”
4. Did you say that subtraction or division would work? Give an example of why the associative
property does not necessarily work for all real numbers when the operation is subtract or divide.
5. What happens if the expression we are looking at mixes the operations? Will this property only
work if the operations are both addition or both multiplication? Use examples to support your
answer.
6. What happens if you have three or more operations? Would these properties still hold true? Use
examples to support your answer.
Your instructor will pick two or three groups to write their answer to each discussion question on the
board. Make sure your group agrees on your response to your assigned question then write your
answer on the board. The instructor will take a few minutes to review the answers with the class and
correct any incorrect answers.
Instructor!
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We use the fact that subtraction can be turned into addition of an opposite to use these two properties
when an expression has subtraction in it. 2 + 12 – 8 – 2 can be written as 2 + 12 + (-8) + (-2) so that we
can rearrange the numbers and group them however we want to begin operations. Going farther, we
say that these are properties of real numbers.
7. Would these properties be true if we are working with decimals or fractions? Why or why not?
Now that you have thought about the commutative and associative properties, you should be able to
apply them the expressions to help you with computations, and eventually, when solving equations.
Please take the example and demonstrate both the commutative and associative properties if you are
able. Then reduce the expression. Come up with two more examples with your group.
Example
5 + (3 + 12)
-6 + (6 + 7)
-5  (-3  2)
[-2  (-9)]  (-5)
-3 + 13 + 3
Commutative
Associative
Reduce
8. Which examples from the chart above demonstrate that commutative property can be important
because it can make computations easier?
Which examples demonstrate that associative property can be important?
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