Before students work on solving complex equations on their own, in this activity they examine the work (both good and bad) of others. The purpose of this activity is to build student fluency solving equations by examining the solutions of others for both appropriate and inappropriate strategies (MP3).

Encourage students to use precise language when discussing the different steps made by the four students in the problem (MP6). For example, if a student says Clare used the distributive property to move from to , ask them to be more specific about how Clare used the distributive property. Those specifics will help the whole class follow along. (Clare used the distributive property to re-write as .)

The purpose of this discussion is recognize some common move errors that can result in equations that are not equivalent.

Begin the discussion by asking, “How do you know when a solution to an equation is correct?” (One way to know it is correct is by substituting the value of into the original equation and seeing if it makes the equation true.)

Display Clare’s and Lin’s solutions for all to see.

Survey to see which solution they prefer. It is important to draw out that neither solution is better than the other. They are two ways of accomplishing the same task: solving for . Invite groups to share ways in which the steps that Clare and Lin took are alike and different, and annotate the two solutions with students’ observations. If none of the groups say it, point out that while the final steps may look different for Clare and Lin, their later steps worked to reduce the total number of terms until only an -term and a number remained—one on each side of the equation.

Display Mai’s and Noah’s incorrect solutions for all to see.

Invite groups to share an incorrect step that they found and what advice they would give to Mai and Noah for checking their work in the future.

The purpose of this lesson is to increase fluency in solving equations. Students will solve equations individually and then compare differing, though accurate, solution paths in order to compare their work with others. This will help students recognize that while the final solution will be the same, there is more than one path to the correct answer that uses principles of balancing equations learned in previous lessons.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their strategy for solving an equation. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. 

Students should take away from this activity the importance of using valid steps to solve an equation rather than following a specific solution path. Consider using some of the following prompts:

• “How many different ways did your group members solve each problem?”
• “When you compared solution paths, did you still come up with the same solution?” (Yes, even though we took different paths, we ended up with the same solutions.)
• “How can you make sure that the path you choose to solve an equation is a valid path?” (I can use the steps that we discovered earlier when we were balancing: adding the same value to each side, multiplying (or dividing) by the same value on each side, using the distribute property, and collecting like terms, whenever those steps are needed.)
• “What are some examples of steps that will not result in a valid solution?” (Performing an action to only one side of an equation and distributing incorrectly will give an incorrect solution.)