Have you ever trained for a running event?

While training, Travis had a daily routine. On Monday, began his day with a 3.2 mile run followed by 1.2 miles of sprints on the track at school. On Tuesday, he began with the sprints and then ran 3.2 miles.

Except for the order, was his training any different?

Which mathematical property does this represent?

**This Concept is about identifying the commutative and associative properties of addition in decimal operations. By the end of it, you will know which property is illustrated in this situation.**

### Guidance

By now you know how to add decimals, round and add decimals, and estimate and add decimals. Now we need to examine decimal addition through the lens of the two mathematical properties which pertain to addition: the ** Commutative Property of Addition** and the

**.**

*Associative Property of Addition*In the examples that we worked with in the last section, we added three decimal addends in the order in which they appeared. What would happen if we added them in a different order? Would the sum change?

**The Commutative Property of Addition states that** *the order of the addends does not change the sum***.**

Let’s test the property using simple whole numbers.

\begin{align*}4 + 5 + 9 = 18 && 5 + 4 + 9 = 18 && 9 + 5 + 4 = 18\\ 4 + 9 + 5 = 18 && 5 + 9 + 4 = 18 && 9 + 4 + 5 = 18\end{align*}

As you can see, we can add the three addends (4, 5, and 9) in many different orders. The Commutative Property of Addition works also works for four, five, six addends, and it works for decimal addends, too.

**What about the Associative Property of Addition?**

**The Associative Property of Addition states that** *the way in which addends are grouped does not change the sum***.** The Associative Property uses parentheses to help us with the grouping. When parentheses are inserted into a problem, according to the order of operations, you perform that operation first. What happens if you move the parentheses? This is where the Associative Property comes in.

Once again, let’s test the property using simple whole numbers.

\begin{align*}(4 + 5) + 9 = 18 && (5 + 4) + 9 = 18 && (9 + 5) + 4 = 18\end{align*}

Clearly, the different way the addends are grouped has no effect on the sum. The Associate Property of Addition works for multiple addends as well as decimal addends.

**These two properties are extremely useful for solving equations.** An ** equation** is a mathematical statement that two expressions are equal. For example, we can say that \begin{align*}15 + 7 = 24 - 2\end{align*} since both sides of the equation equal 22.

A ** variable equation** is an equation such as \begin{align*}12 + x = 14\end{align*}. It is called a variable equation because it includes an algebraic unknown, or a variable.

Find the value for \begin{align*}x\end{align*} in the following equation, \begin{align*}71.321 + 42.29 = x + 71.321\end{align*}

Here is where we can use the Commutative Property of Addition. We know that one side of the equation is equal to the other side of the equation therefore we just substitute the given value in for \begin{align*}x\end{align*} since the second addend did not change

**Our answer is \begin{align*}x\end{align*} is equal to 42.29.**

**We can use the Associative Property for both expressions and equations. Let’s look at an expression.**

Find the value when \begin{align*}x\end{align*} *is* 2 *and* \begin{align*}y\end{align*} *is* 3. \begin{align*}(x+4)- y\end{align*}

This is a ** variable expression**. Notice that there are two given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, but that those values could be any numbers. The key here is that we could move the parentheses and the two variable expressions would have the same value. We can find the value of this expression by substituting 2 and 3 into the expression.

\begin{align*}(2+4)- 3\end{align*}

**The value of this variable expression is 3.**

**What happens if we move the parentheses?**

\begin{align*}x+(4-y)\end{align*}

We substitute the given values in for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}2+(4-3)\end{align*}

**The value of this expression is 3.**

**Because of the Associative Property, both variable expressions have the same value regardless of the groupings.**

Now it's time for you to identify which property is being used in the following examples.

#### Example A

\begin{align*}3.4 + 3.2 + 5.6 = 5.6 + 3.4 + 3.2\end{align*}

**Solution: Commutative Property of Addition**

#### Example B

\begin{align*}(3.4 + 3.2) + 5.6 =3.4 + (3.2+5.6)\end{align*}

**Solution: Associative Property of Addition**

#### Example C

True or false. When using the Associative property the order of the numbers always changes.

**Solution:False, only the position of the grouping symbols changes.**

Here is the original problem once again.

While training, Travis had a daily routine. On Monday, began his day with a 3.2 mile run followed by 1.2 miles of sprints on the track at school. On Tuesday, he began with the sprints and then ran 3.2 miles.

Except for the order, was his training any different?

Which mathematical property does this represent?

When you look at Travis' training schedule, you will see that the components are exactly the same. He ran the same distance on and off the track. The order of the elements did change.

**This is an illustration of the Commutative Property of Addition.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Decimal
- a part of a whole represented by digits to the right of a decimal point.

- Estimate
- to find an approximate solution to a problem.

- Rounding
- a method of estimating where you rewrite a decimal or whole number according to the place value that it is closest to.

- Commutative Property of Addition
- states that the order in which you add numbers does not impact the sum of those numbers.

- Associative Property of Addition
- states that the grouping of addends does not impact the sum of the addends. The groupings are completed using parentheses.

### Guided Practice

Here is one for you to try on your own.

Hassan’s recipe for potato soup calls for 14.25 kilograms of potatoes. He already has 6.12 kilograms. How many more kilograms of potatoes does he need to buy?

**Answer**

In this problem, we are trying to find the number of kilograms of potatoes Hassan needs to buy.

Let’s call that value \begin{align*}x\end{align*}.

We know how many kilograms he already has, and we know the total kilograms he needs, so we can write the equation

\begin{align*}6.12 + x = 14.25\end{align*}.

Now we can solve for the value of \begin{align*}x\end{align*} using mental math. \begin{align*}x\end{align*} needs to be a value that, when added to 6.15, equals 14.25.

Let’s begin with the whole numbers. \begin{align*}6 + 6 = 12; \ 6 + 8 = 14\end{align*}; the whole number portion of \begin{align*}x\end{align*} is 8.

Now let’s look at the decimals. \begin{align*}.12 + ? = .25\end{align*}. The decimal parts differ in both tenths and hundredths, but if we think of them as whole numbers it is easier to see their relationship \begin{align*}12 + 13 = 25\end{align*}, so \begin{align*}.12 + .13 = .25\end{align*}.

Combining the whole number portion of \begin{align*}x\end{align*} and the decimal portion of \begin{align*}x (8 + .13)\end{align*}, we get \begin{align*}8.13.x = 8.13\end{align*}. Remember to put the units of measurement in your answer!

**The answer is 8.13 kilograms.**

### Video Review

Here are videos for review.

Khan Academy Video on Associative Law of Addition

Khan Academy Video on Commutative Law 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*}

Directions: Find the value of \begin{align*}x\end{align*} in the following equations.

9. \begin{align*}.8603 + .292 = x + .8603\end{align*}

10. \begin{align*}(2.65 + x) + 19.35 = 22 + 2.115\end{align*}

11. \begin{align*}.306 + 1.076 = (.782 + x) + .294\end{align*}

12. \begin{align*}6.174 + 76.41 = 76.41 + x\end{align*}

Directions: Solve each problem.

13. Lamont measures the amount of water he drinks. His results for four consecutive days were as follows: 0.6 liters, 0.72 liters, 0.84 liters, 0.96 liters. If the pattern continues, how much water will Lamont drink on the fifth day?

14. Barbara is starting a jewelry making enterprise. At the supply store, she spent $19.19 on beads and $6.81 on wire and clasps. If she left the store with $24.50, how much did she start with?

15. A playground has a length of 18.36 yards and a width of 12.24 yards. What is the perimeter of the playground? (Remember: \begin{align*}P = l + l + w + w\end{align*}.)