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# Properties in Decimal Operations

## Associative and Commutative rules true for decimals

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Properties in Decimal Operations

Have you ever tried to reverse the order of something and end up with the same answer? Look at what Casey is trying to do.

Casey loves to ride his bike. Two days ago, he rode 9.5 miles. Then he rode 13.2 miles, and today he rode 11.5 miles. Casey is sure that he can ride in the reverse order for the next few days and still end up with the same mileage.

\begin{align*}9.5 + 13.2 + 11.5 = 11.5 + 13.2 + 9.5\end{align*}

This may seem very logical, but why is it true?

This is where properties come in. What Casey has done is illustrate one of the properties of addition. Do you know which one?

This Concept is all about properties. By the end of it, you will know which property Casey is using in his bike riding.

### Guidance

Previously we worked on how to add and subtract decimals both by using mental math and by completing the arithmetic on a piece of paper by lining up the decimal points.

We can also apply two properties to our work with decimals.

A property is a rule that remains true when applied to certain situations in mathematics.

We are going to work with two properties in this section, the associative property and the commutative property.

Let’s begin by learning about the commutative property.

The commutative property means that you can switch the order of any of the numbers in an addition or multiplication problem around and you will still receive the same answer.

4 + 5 + 9 = 18 is the same as 5 + 4 + 9 = 18

The order of the numbers being added does not change the sum of these numbers. This is an example of the commutative property.

How can we apply the commutative property to our work with decimals?

We apply it in the same way. If we switch around the order of the decimals in an addition problem, the sum does not change.

4.5 + 3.2 = 7.7 is the same as 3.2 + 4.5 = 7.7

Now we can look at the associative property.

The associative property means that we can change the groupings of numbers being added (or multiplied) and it does not change the sum. This applies to problems with and without decimals.

(1.3 + 2.8) + 2.7 = 6.8 is the same as 1.3 + (2.8 + 2.7) = 6.8

Notice that we use parentheses to help us with the groupings. When we regroup numbers in a different way the sum does not change.

Sometimes, we will have a problem with a variable and a decimal in it. We can apply the commutative property and associative property here too.

\begin{align*}x + 4.5\end{align*} is the same as \begin{align*}4.5 + x\end{align*}

\begin{align*}(x + 3.4) + 5.6\end{align*} is the same as \begin{align*}x + (3.4 + 5.6)\end{align*}

The most important thing is that the order of the numbers and the groupings can change but the sum will remain the same.

Look at the following examples and name the property illustrated in each example.

#### Example A

\begin{align*}3.4 + 7.8 + 1.2 = 7.8 + 1.2 + 3.4\end{align*}

#### Example B

\begin{align*}(1.2 + 5.4) + 3.2 = 1.2 + (5.4 + 3.2)\end{align*}

#### Example C

\begin{align*}x + 5.6 + 3.1 = 3.1 + x + 5.6\end{align*}

Now back to Casey and the bike riding. Here is the original problem once again.

Casey loves to ride his bike. Two days ago, he rode 9.5 miles. Then he rode 13.2 miles, and today he rode 11.5 miles. Casey is sure that he can ride in the reverse order for the next few days and still end up with the same mileage.

\begin{align*}9.5 + 13.2 + 11.5 = 11.5 + 13.2 + 9.5\end{align*}

This may seem very logical, but why is it true?

It is true because this is an example of the Commutative Property of Addition. Casey simply reversed the order of his mileage. The sum of his total miles will be exactly the same.

### Vocabulary

Properties
the features of specific mathematical situations.
Associative Property
a property that states that changing the grouping in an addition problem does not change the sum.
Commutative Property
a property that states that changing the order of the numbers in an addition problem does not change the sum.

### Guided Practice

Here is one for you to try on your own.

Name the property illustrated below.

\begin{align*}3.2 + (x + y) + 5.6 = (3.2 + x) + y + 5.6\end{align*}

This is an example of the Associative Property of Addition.

Why?

The order of the numbers did not change. The location of the parentheses did change. When the grouping of values changes in an expression, it is an example of the Associative Property of Addition.

### Practice

Directions: Identify the property illustrated in each number sentence.

1. \begin{align*}4.5 + (x + y) + 2.6 = (4.5 + x) + y + 2.6\end{align*}

2. \begin{align*}3.2 + x + y + 5.6 = x + 3.2 + y + 5.6\end{align*}

3. \begin{align*}1.5 + (2.3 + y) + 5.6 = (1.5 + 2.3) + y + 5.6\end{align*}

4. \begin{align*}3.2 + 5.6 + 1.3 + 2.6 = 3.2 + 2.6 + 5.6 + 1.3\end{align*}

5. \begin{align*}4.5 + 15.6 = 15.6 + 4.5\end{align*}

6. \begin{align*}(x + y) + 5.6 = x + (y + 5.6)\end{align*}

7. \begin{align*}17.5 + 18.9 + 2 = 2 + 17.5 + 18.9\end{align*}

8. \begin{align*}(x + y) + z = x + (y + z)\end{align*}

9. \begin{align*}5.4 + 5.6 = 5.6 + 5.4\end{align*}

10. \begin{align*}1.2 + 3.2 + 5.6 = 1.2 + 5.6 + 3.2\end{align*}

11. \begin{align*}3.2 + (x + y) + 5.6 = 3.2 + x + (y + 5.6)\end{align*}

12. \begin{align*}3.4 + x + y + .6 = .6 + y + x + 3.4\end{align*}

13. \begin{align*}2.2 + 4.3 + 1.1 = 1.1 + 2.2 + 4.3\end{align*}

14. \begin{align*}(1.2 + 3.4) + 7.6 = 1.2 + (3.4 + 7.6)\end{align*}

15. \begin{align*}8.9 + 9.3 + 3.1 = 9.3 + 8.9 + 3.1\end{align*}

### Vocabulary Language: English

Associative Property

Associative Property

The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).
Commutative Property

Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.
Properties

Properties

Properties are rules that work for given sets of numbers.