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ArithmeticUnderstanding multiplication

Rounding numbers Factors and multiplesHere you will learn about the commutative property, including what it is, and how to use it to solve problems.

Students will first learn about the commutative property as part of operations and algebraic thinking in 3rd grade.

The **commutative property** states that when you add or multiply numbers, you can change the order of the numbers and the answer will still be the same.

For example,

When adding, you can change the Notice that even with a different order, |

This is also true when multiplying numbers.

For example,

When multiplying, you can change the Notice that even with a different order, |

The commutative property can be used to create friendly numbers when solving.

Friendly numbers are numbers that are easy to add or multiply mentally – like multiples of 10.

For example,

\begin{aligned} & 3+25+7 \\\\ & =3+7+25 \hspace{0.3cm} \text{ **Change the order of 25 and 7} \\\\ & =10+25 \hspace{0.65cm} \text{ **Adding 3 and 7 first, gives us 10 - a friendly number} \end{aligned}The commutative property lets us change the order and create friendlier numbers.

10 + 25 is easier to solve mentally than 3 + 25 + 7 = 28 + 7.

For example,

\begin{aligned} & 2 \times 8 \times 5 \\\\ & =2 \times 5 \times 8 \hspace{0.3cm} \text{ **Change the order of the 8 and 5}\\\\ & =10 \times 8 \hspace{0.65cm} \text{ **Multiplying 2 and 5 first, gives us 10 - a friendly number} \end{aligned}The commutative property lets us regroup and create friendlier numbers.

10 \times 8 is easier to solve mentally than 2 \times 8 \times 5=16 \times 5.

The commutative property can also be referred to as the commutative property of addition and the commutative property of multiplication, or more generally as the commutative law.

How does this relate to 3rd grade math?

**Grade 3 – Operations and Algebraic Thinking (3.OA.B.5)**Apply properties of operations as strategies to multiply and divide.

Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.)

3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.)

Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property.)

Use this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!

DOWNLOAD FREEIn order to use the commutative property:

**Check to see that the operation is addition or multiplication.****Change the order of the numbers and solve.**

Give an example of the commutative property using 4 + 9.

**Check to see that the operation is addition or multiplication.**

All the numbers are being added, so the commutative property can be used.

2**Change the order of the numbers and solve.**

Changing the order in the equation does not change the sum.

Give an example of the commutative property using 10 \times 6.

**Check to see that the operation is addition or multiplication.**

All the numbers are being multiplied, so the commutative property can be used.

**Change the order of the numbers and solve.**

10 \times 6 = 60 \; \longrightarrow \; 6 \times 10 = 60

Changing the order in the equation does not change the product.

Use the commutative property to create a friendly number and solve 6 + 32 + 14.

**Check to see that the operation is addition or multiplication.**

All the numbers are being added, so the commutative property can be used.

**Change the order of the numbers and solve.**

\begin{aligned} & 6+32+14 \\\\ & =6+14+32 \hspace{0.3cm} \text{ *Change the order of 32 and 14} \\\\ & =20+32 \hspace{0.8cm} \text{ *Adding 6 and 14 first gives us 20 - a friendly number} \\\\ & =52 \end{aligned}

Use the commutative property to create a friendly number and solve 3 \times 8 \times 3.

**Check to see that the operation is addition or multiplication.**

All the numbers are being multiplied, so the commutative property can be used.

**Change the order of the numbers and solve.**

\begin{aligned} & 3 \times 8 \times 3 \\\\ & =3 \times 3 \times 8 \hspace{0.3cm} \text{ *Change the order of the 8 and 3} \\\\ & =9 \times 8 \hspace{0.8cm} \text{ *Multiplying 3 and 3 first gives a 9 - a single digit number} \\\\ & =72 \end{aligned}

Notice that when multiplying, friendly numbers can also be single digit numbers. If you know your basic facts, it is easier to solve 9 \times 8 than solving 3 \times 8 \times 3=24 \times 3.

Use the commutative property to create a friendly number and solve 41 + 17 + 9.

**Check to see that the operation is addition or multiplication.**

All the numbers are being added, so the commutative property can be used.

**Change the order of the numbers and solve.**

\begin{aligned} & 41+17+9 \\\\ & =41+9+17 \hspace{0.3cm} \text{ *Change the order of 17 and 9} \\\\ & =50+17 \hspace{0.8cm} \text{ *Adding 41 and 9 first gives us 50 - a friendly number} \\\\ & =67 \end{aligned}

Use the commutative property to create a friendly number and solve 3 \times 5 \times 4.

**Check to see that the operation is addition or multiplication.**

All the numbers are being multiplied, so the commutative property can be used.

**Change the order of the numbers and solve.**

\begin{aligned} & 3 \times 5 \times 4 \\\\ & =3 \times 4 \times 5 \hspace{0.3cm} \text{ *Change the order of the 4 and 5} \\\\ & =12 \times 5 \hspace{0.7cm} \text{ *Multiplying 3 and 4 first gives a 12 - a basic fact} \\\\ & =60 \end{aligned}

Notice that when multiplying, friendly numbers can also be numbers that are basic facts. If you have memorized the basic multiplication facts from 1-12, it is easier to solve 12 \times 5 than solving 3 \times 5 \times 4=15 \times 4.

- Be intentional about choosing problems where the commutative property makes solving easier, since it is not always useful or necessary in all solving situations.

- Instead of just telling students the commutative property definition, draw attention to examples of the commutative property when they naturally occur in daily math activities. Record the different examples you see in the classroom on an anchor chart. Over time, students will start recognizing and using the property on their own. Then, after there are sufficient examples, you can introduce students to the property name and definition by using their own examples.

- Include plenty of student discourse around this property so that students understand changing the order of numbers when adding or multiplying does not change the final result. This could include students sharing their thinking or critiquing the thinking of others.

**Using the commutative property for subtraction or division**

The commutative property only works when changing the order of the numbers doesn’t change the answer. This is not true for subtraction or division and they are considered non-commutative arithmetic operations.

For example,

11-5 = 6 \; AND \; 5-11 = -6

Changing the order of the numbers, changes the answer.

**Thinking there is only one way to use the commutative property to solve with friendly numbers**

Sometimes there is more than one way to use the commutative property when solving.

For example,

\begin{aligned} & 6 \times 4 \times 5 \hspace{2.1cm} 6 \times 4 \times 5 \\ & =5 \times 4 \times 6 \hspace{1.7cm} =6 \times 5 \times 4 \\ & =20 \times 6 \hspace{2.05cm} =30 \times 4 \\ & =120 \hspace{2.4cm} =120 \end{aligned}

**Confusing the order of operations**

Equations are always solved moving from left to right. It is not necessary to formally introduce students to the order of operations, but they need to understand and read equations in this way. Otherwise the commutative property may not mean anything to them.

This commutative property topic guide is part of our series on properties of equality. You may find it helpful to start with the main properties of equality topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other related topic guides in this series include:

1. Which of the following equations shows the commutative property?

11 \times 6=(10+1) \times 6

11 \times 6=11 \times 6

11 \times 6=6 \times 11

11 \times 6=11+11+11+11+11+11

The commutative property says that changing the order in the equation does not change the product.

11 \times 6=66 \; \longrightarrow \; 6 \times 11=66

2. Which of the following equations shows the commutative property?

11 + 5 + 9 = 16 + 9

5 + (11 + 9) = 5 + 20

5 + 11 + 9 = 9 + 11 + 5

5 + 9 + 11 = 5 + 20

The commutative property says that changing the order in the equation does not change the sum.

\begin{aligned} & 5+11+9 \hspace{0.7cm} \longrightarrow \hspace{0.7cm} 9+11+5 \\\\ & =16+9 \hspace{2.1cm} =20+5 \\\\ & =25 \hspace{2.65cm} =25 \end{aligned}

3. Which of the following equations shows how to solve 2 \times 9 \times 5 using the commutative property?

\begin{aligned}
& 2 \times 9 \times 5 \\
& =2 \times 5 \times 9 \\
& =10 \times 9
\end{aligned}

\begin{aligned}
& 2 \times 9 \times 5 \\
& = 2 \times(9 \times 5) \\
& = 2 \times 45
\end{aligned}

\begin{aligned}
& 2 \times 9 \times 5 \\
& =18 \times 5
\end{aligned}

\begin{aligned}
& 2 \times 9 \times 5 \\
& =(2 \times 9) \times 5 \\
& =18 \times 5
\end{aligned}

The commutative property says that changing the order in the equation does not change the product.

\begin{aligned} & 2 \times 9 \times 5 \\\\ & =2 \times 5 \times 9 \hspace{0.3cm} \text{ *Change the order of 9 and 5} \\\\ & =10 \times 9 \hspace{0.65cm} \text{ *Multiplying 2 and 5 first gives us 10 – a friendly number} \end{aligned}

4. Which of the following equations shows how to solve 37 + 28 + 23 using the commutative property?

\begin{aligned}
& 37+28+23 \\
& =37+(28+23) \\
& =37+51
\end{aligned}

\begin{aligned}
& 37+28+23 \\
& =37+23+28 \\
& =60+28
\end{aligned}

\begin{aligned}
& 37+28+23 \\
& =65+23
\end{aligned}

\begin{aligned}
& 37+28+23 \\
& =(37+28)+23 \\
& =65+23
\end{aligned}

The commutative property says that changing the order in the equation does not change the sum.

\begin{aligned} & 37+28+23 \\\\ & =37+23+28 \hspace{0.3cm} \text{ *Change the order of 23 and 28} \\\\ & =60+28 \hspace{1cm} \text{ *Adding 37 and 23 first gives us 60 – a friendly number} \end{aligned}

5. Which of the following equations shows how to solve 8 \times 4 \times 5 using the commutative property to create a friendly number?

\begin{aligned}
& 8 \times 4 \times 5 \\
& =8 \times(4 \times 5) \\
& =8 \times 20
\end{aligned}

\begin{aligned}
& 8 \times 4 \times 5 \\
& =32 \times 5
\end{aligned}

\begin{aligned}
& 8 \times 4 \times 5 \\
& =(8 \times 4) \times 5 \\
& =32 \times 5
\end{aligned}

\begin{aligned}
& 8 \times 4 \times 5 \\
& =8 \times 5 \times 4 \\
& =40 \times 4
\end{aligned}

The commutative property says that changing the order in the equation does not change the product. Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

\begin{aligned} & 8 \times 4 \times 5 \\\\ & =8 \times 5 \times 4 \hspace{0.3cm} \text{ *Change the order of 4 and 5} \\\\ & =40 \times 4 \hspace{0.6cm} \text{ *Multiplying 8 and 5 first gives us 40 – a friendly number} \end{aligned}

6. Which of the following equations shows how to solve 16+18+22 using the commutative property to create a friendly number?

\begin{aligned}
& 16+18+22 \\
& = (16+18)+22 \\
& = 34+22
\end{aligned}

\begin{aligned}
& 16+18+22 \\
& =34+22
\end{aligned}

\begin{aligned}
& 16+18+22 \\
& =18+22+16 \\
& =40+16
\end{aligned}

\begin{aligned}
& 16+18+22 \\
& = 16+22+18 \\
& = 38+18
\end{aligned}

The commutative property says that changing the order in the equation does not change the sum. Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

\begin{aligned} & 16+18+22 \\\\ & =18+22+16 \hspace{0.3cm} \text{ *Change the order of all the numbers} \\\\ & =40+16 \hspace{1cm} \text{ *Adding 18 and 22 first gives us 40 – a friendly number} \end{aligned}

No, you can solve the numbers as they appear in the equation, without changing the order. The commutative property just gives you flexibility to add or multiply in a different order.

Yes, the commutative property can be used with integers, rational numbers and any real number, as long as they are all being added or multiplied.

The associative property of addition states that you can change the grouping of numbers when adding (using parentheses) and the sum will still be the same. The order of operations changes, but not the written order of the numbers in the equation. The commutative property of addition says you can change the written order of the numbers when adding and the sum will still be the same.

It is one of the properties of numbers for mathematical operations. This property states that any number added to 0 will still result in the same number (0 + a = a) or any number multiplied by 1 will still result in the same number (1 \times a=a).

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