In this activity, students encounter typical errors with signed numbers, operations, and properties. They are tasked with identifying which strategies are correct. For those that are not, students must describe the error that was made.

Since students are contrasting and critiquing different approaches, they are critiquing the reasoning of others (MP3).

The purpose of this discussion is to help students recognize some common missteps when rewriting expressions.

Invite students to share the errors they identified in the activity. Point out that since each given work in this activity contains at least two steps, students can use the steps to identify where the error occurs—where the expressions are no longer equal for a value of the variable.

Consider displaying these examples and asking student to explain the error each illustrates:

• (We can’t add or subtract terms that have variables from terms that are numbers.)
• (A negative times a negative is positive.)
• (When there are no parentheses, multiply before adding.)

Review the useful approach of rewriting subtraction operations as adding the opposite. If desired, demonstrate such an approach for the expression along with a box to organize the work of multiplying:

In this partner activity, students take turns making sense of expressions written in different ways. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

For the second and third rows, some students may not understand that the subtraction sign in front of the parentheses applies to both terms inside that set of parentheses. Some students may get the second row correct, but not realize how the third row relates to the fact that the product of two negative numbers is a positive number. For the last three rows, some students may not recognize the importance of the subtraction sign in front of . Prompt them to rewrite the expressions replacing subtraction with adding the inverse. 

Students might write an expression with fewer terms but not recognize an equivalent form because the distributive property has been used to write a sum as a product. For example,  can be written as or , which is equivalent to the expression  in column B. Encourage students to think about writing the column B expressions in a different form and to recall that the distributive property can be applied to either factor or expand an expression. 

Much discussion takes place between partners. Invite students to share how they used properties to generate equivalent expressions and find matches.

• “Which term(s) does the subtraction sign apply to in each expression? How do you know?”
• “Were there any expressions from column A that you wrote with fewer terms but were unable to find a match for in column B? If yes, why do you think this happened?”
• “What were some ways you handled subtraction with parentheses? Without parentheses?”
• “Describe any difficulties you experienced and how you resolved them.”

In this activity, students continue the work of generating equivalent expressions as they decide where to place a set of parentheses and explore how that placement affects the expressions. As students consider the differences in applying the order of operations based on the placement of parenthesis, they notice and make use of structure (MP7).

Students may not realize that they can break up a term like and place a parentheses, such as . Clarify that they may place the parentheses anywhere in the expression.

Invite a few students to share their strategy for placing parentheses to create an equivalent expression, and then an expression that is not equivalent. Because of the subtracted terms, there are limited options for creating an equivalent expression, but lots of possibilities for creating an expression that is not equivalent.