In the previous section, you used the commutative property of addition and multiplication to determine whether or not two expressions are equivalent. In this section, you will investigate an additional property that you can use to determine if two expressions are equivalent.

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.

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Check Your Answer

Expression 1	Expression 2	Equivalent? (Yes/No)
3 + (x + 2)	(3 + x) + 2	Yes
x + (5 + 2x)	(x + 5) + 2x	Yes
(2 + 2x) − 3	2 + (2x − 3)	Yes
x + (2x + 3)	(x + 3x) + 2	No
(2 • x)(3)	2(x • 3)	Yes
2 − (3 − 4)	(2 − 3) − 4	No

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

• If you regroup two of three addends, are the expressions equivalent? How can you tell?

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  Check Your Answer

  Yes, because the scale is balanced and the scale contains the same number of blocks on both sides.

• If you regroup two of three factors, are the expressions equivalent? How can you tell?

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  Check Your Answer

  Yes, because the scale is balanced and the scale contains the same number of blocks on both sides.

• What do you notice about the pairs of expressions that are not equivalent?

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  Check Your Answer

  One pair has the same numbers, but coefficients are exchanged with constants. The other pair has regrouped subtraction.

• Is (24 ÷ 8) ÷ 2 equivalent to 24 ÷ (8 ÷ 2)? How can you tell?

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  Check Your Answer

  No. If you follow the order of operations, you perform the division inside parentheses first. Both expressions simplify to different numbers.
  (24 ÷ 8) ÷ 2 = (3) ÷ 2 = 1.5
  24 ÷ (8 ÷ 2) = 24 ÷ (4) = 6

Pause and Reflect

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

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Check Your Answer

The associative property is true for addition and multiplication, but not for subtraction or division.

Generalize the associative property using the numbers a, b, and c.

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Check Your Answer

Associative Property for Addition: (a + b) + c = a + (b + c)
Associative Property for Multiplication: (a × b) × c = a × (b × c)

Practice

1. Which of the following pairs of expressions are equivalent?
   1. 3 + (n − 5) and (3 + n) − 5
   2. 5x + (2x + 7.5) and (5x + 2x) + 7.5
   3. 6.5 − (3x − 10) and (6.5 − 3x) − 10
   4. (1 over 2 1 2 p + 1.1 + 7p and 1 over 2 1 2 p + (1.1 + 7p)
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   Need a hint?

   The associative property of equality is true only for addition and multiplication.
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   Check Your Answer

   I, II, and IV only
   For expression III, the terms are being subtracted, and subtraction is not associative. Expression I contains subtraction which can be rewritten as 3 + (n + 5) and (3 + n) + -5.
2. Three students wrote an expression to describe the perimeter of the triangle below.
   

   Which students wrote equivalent expressions?

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   Need a hint?

   The associative property states that you can regroup numbers being added together without changing the final sum.
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   Check Your Answer

   Amalia and Benjamin have equivalent expressions.
   Amalia wrote the expression 2m + (3m + 9) + 4 as the sum of the lengths of the three sides of the triangle.
   Benjamin wrote the expression (2m + 3m) + 9 + 4, which regroups the first two addends instead of the middle two addends.
   Cristobal wrote the expression 2m + (3m + 94), which incorrectly combines the digits 9 and 4 to make 94 instead of adding 9 and 4 to get 13.