Untitled Document

You will use a balance scale model to determine whether or not two expressions are equivalent. In this section, you will focus on how the commutative property helps you to determine whether or not two expressions are equivalent.

Interactive exercise. Assistance may be required. Use the interactive below to determine whether or not the given expressions are equivalent. Click and drag the blocks representing the first expression onto the left pan of the balance scale. Use the blocks to represent the second expression on the right pan. If the balance scale balances, then the expressions are equivalent. Record your results in a table like the one shown. Click the Next button to see the next pair of expressions.

Copy the following table into a word processing or spreadsheet app or program. Use the results from the interactive to complete the table.

Expression 1	Expression 2	Equivalent? (Yes/No)
2x + 4	4 + 2x
3(2x)	2(3x)
3x + 2	2x + 3
x − 3	-3 + x
2x − 4	4 − 2x
4x + 1	1 + 4x

Interactive popup. Assistance may be required.

Check Your Answer

Expression 1	Expression 2	Equivalent? (Yes/No)
2x + 4	4 + 2x	Yes
3(2x)	2(3x)	Yes
3x + 2	2x + 3	No
x − 3	-3 + x	Yes
2x − 4	4 − 2x	No
4x + 1	1 + 4x	Yes

Use your completed table and the interactive to answer the questions that follow.

If you change the order of the terms when you add them, are the expressions equivalent? How can you tell?
Interactive popup. Assistance may be required.
Check Your Answer
Yes, because the scale is balanced and the scale contains the same number of blocks on both sides.
If you change the order of the factors being multiplied, are the expressions equivalent? How can you tell?
Interactive popup. Assistance may be required.
Check Your Answer
Yes, because the scale is balanced and the scale contains the same number of blocks on both sides.
If you exchange the coefficient of x and the constant term, are the expressions equivalent? How can you tell?
Interactive popup. Assistance may be required.
Check Your Answer
No, because the scale is unbalanced and the scale does not contain the same number of blocks on both sides.
If you change the order of the terms when you subtract them, are the expressions equivalent? How can you tell?
Interactive popup. Assistance may be required.
Check Your Answer
No, because the scale is unbalanced and the scale does not contain the same number of blocks on both sides.
The figure below shows how 4 cookies can be divided among 2 students (4 ÷ 2) and how 2 cookies can be divided among 4 students (2 ÷ 4).

Does the order in which numbers are divided affect the answer? Explain.

Interactive popup. Assistance may be required.
Check Your Answer
Yes. If you divide two numbers in a different order, you get two different answers. In this example, 4 ÷ 2 = 2, so when you divide 4 cookies among 2 students, each student gets 2 cookies. But 2 ÷ 4 = 1 over 2 1 2 , so when you divide 2 cookies among 4 students, each student gets 1 over 2 1 2 of a cookie.

Pause and Reflect

The commutative property of equality states that when the numbers are rearranged, the result of the operation is the same. For which operations does the commutative property hold true?

Interactive popup. Assistance may be required.
Check Your Answer
The commutative property is true for addition and multiplication, but not for subtraction or division.

Generalize the commutative property using the numbers a and b.

Interactive popup. Assistance may be required.
Check Your Answer
Commutative Property for Addition: a + b = b + a
Commutative Property for Multiplication: ab = ba

Practice
1. Which of the following pairs of expressions are equivalent?
  1. 3x − 5 and 5 − 3x
  2. 2x + 0.5 and 0.5 + 2x
  3. 4(6.4)m and 4(m)(6.4)
  4. 81 over 3 1 3 p − 1.1 and −1.1 + 81 over 3 1 3 p
  Interactive popup. Assistance may be required.
  Need a hint?
  The commutative property of equality is true only for addition and multiplication.
  
  Interactive popup. Assistance may be required.
  Check Your Answer
  II, III, and IV only
  For expression I, the terms are being subtracted, and subtraction is not commutative.
2. Luke modeled an expression using algebra tiles. The expression he modeled is shown below.
  
  Luke claims he modeled the expression 4x + 9, but his friend, Yesenia, claims he modeled the expression 9 + 4x. Who is correct? Explain your answer.
  
  Interactive popup. Assistance may be required.
  Need a hint?
  The commutative property is true for addition and multiplication, meaning that you can add numbers in any order or multiply numbers in any order.
  
  Interactive popup. Assistance may be required.
  Check Your Answer
  Both students are correct. According to the commutative property of addition, 4x + 9 and 9 + 4x are equivalent expressions.
3. One of Shelley’s homework problems asks her to add the expressions 2x + 4 and 3x + 2.5. She writes the expression (2x + 3x) + (4 + 2.5) from the sum of the two original expressions. Did Shelley write an equivalent expression? How do you know?
  Interactive popup. Assistance may be required.
  Need a hint?
  The commutative property is true for addition and multiplication, meaning that you can add numbers in any order or multiply numbers in any order.
  
  Interactive popup. Assistance may be required.
  Check Your Answer
  Yes. Shelley used the commutative property to rearrange the order of the terms being added together.
  (2x + 4) + (3x + 2.5) = (2x + 3x) + (4 + 2.5)

Pause and Reflect

Practice