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 sequences. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the lists of numbers in comparison to one another. In a later activity, students will learn a precise definition for "sequence," so there is no expectation that students use the word at this time.

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.

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 increase, decrease, or rate, 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?”

Math Community
After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.

Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.

After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.

The purpose of this activity is for students to understand what a sequence is by creating one based on a puzzle. In this activity, students experiment with solving the Tower of Hanoi puzzle for different numbers of starting discs. Students look for a pattern (MP8) in the sequence generated by listing the number of moves needed to solve the puzzle for different numbers of discs and then describe the pattern informally.

The other purpose of this activity is to establish that students are expected to try things out, look for patterns, explain their thinking, justify their responses, and listen respectfully to their classmates. When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

In the digital version of the activity, students use an applet to explore the puzzle. The applet allows students to move the discs step by step and allows only valid moves. Use the digital version if students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, displaying the applet for all to see would be helpful during the Launch.

This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured partner meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions, such as, “What did you mean by _____?” and “Can you say that another way?” Finally, students write a second draft of their response, reflecting ideas from partners and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.

This activity is provided in case an additional activity is needed to reinforce the lesson goal. Alternatively, if students are familiar with the Tower of Hanoi puzzle, use this activity instead, and use the structure of the Launch and discussion for the activity “The Tower of Hanoi” for the discussion of this activity. Students experiment with solving a checker jumping puzzle, similar to the Tower of Hanoi puzzle, for different numbers of starting checkers. They look for a pattern in the sequence generated by listing the number of moves needed to solve the puzzle for different numbers of starting checkers (MP8) and then describe the pattern informally.

In the digital version of the activity, students use an applet to explore the puzzle. The applet allows students to move the checkers step by step and allows only valid moves. Use the digital version if students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, displaying the applet for all to see would be helpful during the Launch.

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.