In this activity, students extend their understanding of the order of operations to include expressions with exponents. They do so in the context of surface area, which provides a reason to find the value of an exponential expression before performing the multiplication.

The goal of this discussion is to make explicit the order of operations when finding the value of an expression involving exponents. 

Invite students to share their responses and reasoning. Point out that in finding the surface area, we need to find the area of one face of the cube, which is , before multiplying that number by 6. 

Tell students that sometimes it is not so clear in which order to perform the operations in an expression. However, there is an order that we all generally agree on, and when we want something done in a different order, we use parentheses, brackets, or other grouping symbols to communicate what to do first. In general:

• When an exponent occurs in the same expression as another operation, we evaluate the exponent first. For example, to find the value of , we calculate first, which is 16, and then multiply it by 3, which gives 48. 
• When an expression involves grouping symbols, we perform the operation inside them first. For example, for , we first multiply 3 and 4, which gives 12, and then square the 12 to get 144.

If students bring up PEMDAS or another mnemonic for remembering the order of operations, point out that PEMDAS can be misleading in indicating multiplication before division, and addition before subtraction. Discuss the convention:

• First, find the value of any expression in brackets or parentheses.
• Next, find the value of any expression with an exponent.
• Then, perform multiplication or division, from left to right.
• Lastly, perform addition or subtraction, from left to right.

In this activity, students use the order of operations to find the value of expressions with exponents. As they exchange explanations for their responses and work to reach an agreement, students practice constructing logical arguments, listening, and critiquing the reasoning of others (MP3).

The purpose of the discussion is to ensure that students understand and can apply the conventional rules for order of operations when working with expressions with exponents. Consider asking questions, such as:

• “Were there any expressions whose values were difficult to find? Why were they difficult?”
• “Did you disagree with your partner about the value in any of the rows? How did you settle the disagreement?”
• “What is something new that you learned about working with expressions with exponents?”