Using the Commutative Property of Addition

Oct 13, 2025 by wbrum.org 3 views

Understanding the Problem

In this article, we will explore how to find the total length of two pieces of string. We will use basic mathematical concepts and terms from the Word List to solve this problem. The two pieces of string are 12 inches long and 9 inches long, respectively.

Defining the Problem

To find the total length of the two pieces of string, we need to add their individual lengths together. This is a simple arithmetic operation that involves combining two or more numbers to get a total value.

Using the Word List

The Word List includes terms such as "addition," "arithmetic," "numbers," and "total." We will use these terms to explain the concept of adding lengths of strings.

Step 1: Identify the Lengths of the Strings

The first piece of string is 12 inches long, and the second piece is 9 inches long. We need to identify these lengths as the numbers we will add together.

Step 2: Add the Lengths Together

To find the total length of the two pieces of string, we need to add their individual lengths together. We can use the following equation:

12 inches + 9 inches = ?

Using the Commutative Property of Addition

The commutative property of addition states that the order of the numbers being added does not change the result. In other words, we can add the numbers in any order and still get the same result. For example:

12 inches + 9 inches = 9 inches + 12 inches

Using the Associative Property of Addition

The associative property of addition states that when we add three or more numbers together, the order in which we add them does not change the result. For example:

(12 inches + 9 inches) + 5 inches = 12 inches + (9 inches + 5 inches)

Solving the Equation

Now that we have identified the lengths of the strings and added them together, we can solve the equation:

12 inches + 9 inches = ?

To solve this equation, we need to add the two numbers together. We can use a number line or a hundreds chart to help us visualize the addition process.

Using a Number Line

A number line is a visual representation of the numbers on a line. We can use a number line to help us visualize the addition process.

We can start at 12 inches and count up 9 inches to find the total length.

Using a Hundreds Chart

A hundreds chart is a table that shows the numbers from 1 to 100. We can use a hundreds chart to help us visualize the addition process.

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

We can start at 12 inches and count up 9 inches to find the total length.

Solving the Equation

Now that we have identified the lengths of the strings and added them together, we can solve the equation:

12 inches + 9 inches = 21 inches

Conclusion

In this article, we have explained how to find the total length of two pieces of string. We used basic mathematical concepts and terms from the Word List to solve this problem. The two pieces of string are 12 inches long and 9 inches long, respectively. We added their individual lengths together to find the total length, which is 21 inches.

Word List

Addition
Arithmetic
Numbers
Total
Commutative property
Associative property
Number line
Hundreds chart

References

[1] "Mathematics for Dummies." By Mark Ryan. John Wiley & Sons, 2013.
[2] "The Joy of Mathematics." By Alfred S. Posamentier. Prometheus Books, 2004.
[3] "Mathematics: A Very Short Introduction." By Timothy Gowers. Oxford University Press, 2002.
Frequently Asked Questions: Adding Lengths of Strings =====================================================

Q: What is the total length of two pieces of string, one 12 inches long and the other 9 inches long?

A: The total length of the two pieces of string is 21 inches.

Q: How do I add the lengths of two strings together?

A: To add the lengths of two strings together, you can use the commutative property of addition, which states that the order of the numbers being added does not change the result. For example:

12 inches + 9 inches = 9 inches + 12 inches

Q: What is the commutative property of addition?

A: The commutative property of addition is a mathematical concept that states that the order of the numbers being added does not change the result. For example:

a + b = b + a

Q: What is the associative property of addition?

A: The associative property of addition is a mathematical concept that states that when we add three or more numbers together, the order in which we add them does not change the result. For example:

(a + b) + c = a + (b + c)

Q: How do I use a number line to add the lengths of two strings together?

A: To use a number line to add the lengths of two strings together, you can start at the length of one string and count up the length of the other string. For example:

Q: How do I use a hundreds chart to add the lengths of two strings together?

A: To use a hundreds chart to add the lengths of two strings together, you can start at the length of one string and count up the length of the other string. For example:

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

Q: What is the difference between the commutative and associative properties of addition?

A: The commutative property of addition states that the order of the numbers being added does not change the result. For example:

a + b = b + a

The associative property of addition states that when we add three or more numbers together, the order in which we add them does not change the result. For example:

(a + b) + c = a + (b + c)

Q: How do I use the commutative and associative properties of addition to add the lengths of two strings together?

A: To use the commutative and associative properties of addition to add the lengths of two strings together, you can rearrange the numbers being added to make the calculation easier. For example:

12 inches + 9 inches = 9 inches + 12 inches

Q: What is the total length of three pieces of string, one 12 inches long, one 9 inches long, and one 6 inches long?

A: To find the total length of three pieces of string, you can add the lengths of the individual strings together. For example:

12 inches + 9 inches + 6 inches = 27 inches

Q: How do I use a hundreds chart to add the lengths of three strings together?

A: To use a hundreds chart to add the lengths of three strings together, you can start at the length of one string and count up the lengths of the other two strings. For example:

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
4	5	6	7	8	9	10	11	12	13	14	15	16
5	6	7	8	9	10	11	12	13	14	15	16	17
6	7	8	9	10	11	12	13	14	15	16	17	18
7	8	9	10	11	12	13	14	15	16	17	18	19
8	9	10	11	12	13	14	15	16	17	18	19	20
9	10	11	12	13	14	15	16	17	18	19	20	21
10	11	12	13	14	15	16	17	18	19	20	21	22
11	12	13	14	15	16	17	18	19	20	21	22	23
12	13	14	15	16	17	18	19	20	21	22	23	24
13	14	15	16	17	18	19	20	21	22	23	24	25
14	15	16	17	18	19	20	21	22	23	24	25	26
15	16	17	18	19	20	21	22	23	24	25	26	27
16	17	18	19	20	21	22	23	24	25	26	27	28
17	18	19	20	21	22	23	24	25	26	27	28	29
18	19	20	21	22	23	24	25	26	27	28	29	30
19	20	21	22	23	24	25	26	27	28	29	30	31
20	21	22	23	24	25	26	27	28	29	30	31	32

	1	2	3	4	5	6	7	8	9	10	11	12
1	2	3	4	5	6	7	8	9	10	11	12	13
2	3	4	5	6	7	8	9	10	11	12	13	14
3	4	5	6	7	8	9	10	11	12	13	14	15
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