Proportional Relationships and Equations

5 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time, followed by a whole-class discussion.

Narrative

This is the first Which Three Go Together? routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together? Why do they go together?” 

Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepens their awareness of connections across representations.

This Warm-up prompts students to compare four expressions. In making comparisons, students have a reason to use language precisely. The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of algebraic expressions.

Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group, and then together find as many sets of three as they can.

Student Task

Which three go together? Why do they go together?

A

​​​​525 \boldcdot 2

B

4?=204 \boldcdot {?}=20

C

x+5x+5

D

5x5x

Sample Response

Sample responses:

A, B, and C go together because:

  • They have a symbol that represents the operation.

A, B, and D go together because: 

  • They all represent multiplication.
  • They all involve multiplying by 5.

A, C, and D go together because: 

  • They are expressions, but not equations.
  • They include the numeral 5.

B, C, and D go together because: 

  • They have a symbol representing an unknown number.
Activity Synthesis (Teacher Notes)

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology they use, such as “sum,” “product,” “factor,” “term,” “expression,” “equation,” “variable,” or “unknown,” and to clarify their reasoning as needed. Consider asking:

  • “How do you know . . . ?”
  • “What do you mean by . . . ?”
  • “Can you say that in another way?”
Standards
Building On
  • 6.EE.2·Write, read, and evaluate expressions in which letters stand for numbers.
  • 6.EE.A.2·Write, read, and evaluate expressions in which letters stand for numbers.

10 min

10 min