Lesson 6

This warm-up prompts students to compare four equations. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the equations in comparison to one another. To allow all students to access the activity, each equation has one obvious reason it does not belong. Encourage students to find reasons based on the structure of the equation (MP7). During the discussion, listen for important ideas and terminology.

Arrange students in groups of 2–4. Display the equations for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their group. In their groups, tell each student to share their reasoning about why a particular question does not belong, and together, find at least one reason each question doesn’t belong.

Ask each group to share one reason why a particular equation does not belong. Record and display the responses for all to see. After each response, ask the class whether they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as coefficient or solution. Also, press students on unsubstantiated claims.

The goal of this activity is for students to notice the structure of equations. Any way of sorting is fine, but the discussion should land on explaining how equations involving an expression like \(p(x+q)\) are different from ones that have an expression like \(px+q\). Monitor for different ways groups choose to categorize the equations, but especially for categories that distinguish between these two types of expressions. As students work, encourage them to refine their descriptions of equations using more precise language and mathematical terms (MP6).

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the equations, they should work with their partner to come up with categories, and then take turns sorting each equation into one of their categories, explaining why they are doing so. If necessary, demonstrate this protocol before students start working.

Distribute one set of cards to each group of students. Give students 5 minutes to work with their partner, followed by a whole-class discussion.

Select groups of students to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between equations that involved expressions of the form \(px+q\) vs \(p(x+q)\). Attend to the language that students use to describe their categories and equations, giving them opportunities to describe their equations more precisely. Highlight the use of terms like coefficient, sum, product, variable, and solution. 

This activity is an opportunity to put together the learning of the past several lessons: correspondences between tape diagrams, equations, and stories, and using representations to reason about a solution. The focus of this activity is still contrasting the two main types of equations that students encounter in this unit.

Keep students in the same groups. 5 minutes of quiet work time followed by sharing with a partner and a whole-class discussion.

For each story, select 1 or more groups to present the matching diagram, their equation, and their solution method. Possible questions to ask:

• “How were the diagrams alike? How were they different?” (They have the same numbers and a letter. One has 3 equal groups and an extra bit, the other just has 3 equal groups, but each group is a sum.)
• “How were the stories alike? How were they different?” (They were both about distributing 90 flyers. In one story, Lin makes a series of moves to each volunteer. In the other story, she gives each volunteer the same amount, but then there are some left over.)
• “What parts of the story made you think that one diagram represented it?”
• “Explain how you reasoned about the story, diagram, or equation to find the value of the variable.”