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Properties of Multiplication in Decimal Operations

Associative and Commutative rules true for decimals

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Practice Properties of Multiplication in Decimal Operations
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Properties of Multiplication in Decimal Operations

Remember the students at the Science Museum from the Use Estimation to Check Answers when Multiplying Decimals Concept? Well, they have a new dilemma to work out and it involves math. Take a look.

Sarah and Kelly are having lunch at the Science Museum. They have each ordered a slice of pizza and a drink. Each slice of pizza is $3.25 and each drink is$2.00.

"We can just multiply to figure out how much we owe. I'll go first," Sarah said.

"It doesn't matter who goes first, we both have the same things," Kelly explained.

"No we have to put the money first and then the amount," Sarah argued.

Kelly wrote this on a piece of paper.

2(5.25)\begin{align*}2(5.25)\end{align*}

Sarah thinks the money amount must go first when you multiply. Kelly doesn't.

Who is correct?

Guidance

Previously we worked on using the properties of multiplication in numerical and variable expressions. Now we are going to apply these properties to our work with multiplying decimals and whole numbers.

What is a property?

A property is a rule that makes a statement about the way that numbers interact with each other during certain operations. The key thing to remember about a property is that the statement is true for any numbers.

The Commutative Property of Multiplication

The Commutative Property of Multiplication states that it does not matter which order you multiply numbers in, that you will get the same product.

a(b)=b(a)

What does this have to do with our work with decimals and whole numbers?

When we apply the Commutative Property of Multiplication to our work with decimals and whole numbers, we can be sure that the product will be the same regardless of whether we multiply the decimal first or the whole number first.

4.5(7) is the same as 7(4.5)

This means that we can multiply them in whichever order we choose. Our product will remain the same.

45×7315

We can also apply the Commutative Property of Multiplication when we have a problem with a variable in it.

Remember that a variable is a letter used to represent an unknown.

5.6a=a5.6

Here we haven’t been given a value for a, but that doesn’t matter.

The important thing is for you to see that it doesn’t matter which order we multiply, the product will be the same.

If we were given 3 as the value for a, what would our product be?

5.6(3)\begin{align*}5.6(3)\end{align*}

56×  3168

The Associative Property of Multiplication

We can also apply the Associative Property of Multiplication to our work with decimals and whole numbers.

The Associative Property of Multiplication states that it doesn’t matter how you group numbers, that the product will be the same. Remember that grouping refers to the use of parentheses or brackets.

6(3.4 ×\begin{align*}\times\end{align*} 2) =\begin{align*}=\end{align*} (6 ×\begin{align*}\times\end{align*} 3.4)2

We can change the grouping of the numbers and the product will remain the same.

This is also true when we have variable expressions.

5(6a)=(5×6)a

Once again, we can change the grouping of the numbers and variables, but the product will remain the same.

Look at these examples and determine which property is being illustrated.

Example A

4.5(5a) =\begin{align*}=\end{align*} (4.5 ×\begin{align*}\times\end{align*} 5)a

Solution: Associative Property of Multiplication

Example B

6.7(4) = 4(6.7)

Solution: The Commutative Property of Multiplication

Example C

5.4a = a5.4

Solution: The Commutative Property of Multiplication

Now back to the problem at the Science Museum.

Sarah and Kelly are having lunch at the Science Museum. They have each ordered a slice of pizza and a drink. Each slice of pizza is $3.25 and each drink is$2.00.

"We can just multiply to figure out how much we owe. I'll go first," Sarah said.

"It doesn't matter who goes first, we both have the same things," Kelly explained.

"No we have to put the money first and then the amount," Sarah argued.

Kelly wrote this on a piece of paper.

2(5.25)\begin{align*}2(5.25)\end{align*}

Sarah thinks the money amount must go first when you multiply. Kelly doesn't.

Who is correct?

Given the Commutative Property of Multiplication, Kelly is correct. It does not matter in which order you multiply. The products will be the same.

2(5.25)=\$10.50\begin{align*}2(5.25) = 10.50\end{align*}

This is the product. The Commutative Property of Multiplication is the answer.

Vocabulary

Multiplication
a shortcut for addition, means working with groups of numbers
Product
the answer from a multiplication problem
Estimate
an approximate answer-often found through rounding
Properties
rules that are true for all numbers
The Commutative Property of Multiplication
it doesn’t matter which order you multiply numbers, the product will be the same.
The Associative Property of Multiplication
it doesn’t matter how you group numbers in a multiplication problem, the product will be the same.

Guided Practice

Here is one for you to try on your own.

Which property is illustrated in this problem?

4(3.67)=(3.67)4\begin{align*}4(3.67) = (3.67)4\end{align*}

The only thing that changed in this problem is the order of the values being multiplied.

The Commutative Property of Multiplication is our answer.

Practice

Directions: Identify the property illustrated in each problem.

1. 4.6a = a4.6

2. (4a)(b) = 4(ab)

3. (5.5a)(c) = 5.5(ac)

4. ab = ba

5. 6ab = ab(6)

6. 6×4=4×6\begin{align*} 6 \times 4 = 4 \times 6\end{align*}

7. 5(ab) = (5a) x b

8. 7(8x) = (7 x 8)x

9. 2xy = 2yx

10. 3(4a) = (3 x 4)a

11. 6 x 7 x 4 = 4 x 7 x 6

12. abc = cab

13. xy(az) = x(yaz)

14. abcd = dcab

15. 2a(bc) = (2a)bc

Vocabulary Language: English

Estimate

Estimate

To estimate is to find an approximate answer that is reasonable or makes sense given the problem.
multiplication

multiplication

Multiplication is a simplified form of repeated addition. Multiplication is used to determine the result of adding a term to itself a specified number of times.
Product

Product

The product is the result after two amounts have been multiplied.
Properties

Properties

Properties are rules that work for given sets of numbers.
The Associative Property of Multiplication

The Associative Property of Multiplication

The associative property of multiplication states that regardless of how you group numbers in a multiplication problem, the product will be the same.
The Commutative Property of Multiplication

The Commutative Property of Multiplication

The commutative property of multiplication states that regardless of the order in which you multiply numbers in a multiplication problem, the product will be the same.