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.

The goal of this discussion is to define “sequence” and “term (of a sequence).” This is also an opportunity to establish classroom norms for transitioning from working time to a synthesis of the main ideas, listening to other students’ explanations, and formally naming important mathematical concepts or objects after students have had an opportunity to interact with them.

Begin the discussion by asking students, “How many moves does it take to complete the puzzle with 1 disc?” (Just 1, since the puzzle specifies the discs have to end up on a different peg.) Select students to share the smallest number of moves they found for 3 and 4 discs. Each time, ask whether anyone in the class solved it with fewer moves. If no one finds the minimal solution, ask students to keep looking if time allows, or share the minimum number of moves and demonstrate the minimal solution.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last question, in which they describe Jada’s thinking. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, the bottom row of the table should now have the numbers 1, 3, 7, and 15 filled in. Tell students that in mathematics, we often call a list of numbers a sequence. The list 1, 3, 7, 15 is an example of a sequence. A specific number in the list is called a term of the sequence. Ask students how they would describe the rule for the next term in this sequence. After a brief quiet think time, select 2–3 students to share their thinking, and write down any notation they come up with to describe the recursive rule, such as There is no need to introduce formal notation or discuss a specific rule for finding term at this time, but if students suggest these and time allows, welcome their explanations. Use the rule to fill in 31, 63, and 127 for 5, 6, and 7 discs, respectively.

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.

If students didn’t complete the activity “The Tower of Hanoi,” the goal of this discussion is to define “sequence” and “term (of a sequence).” This is also an opportunity to establish classroom norms for transitioning from working time to a synthesis of the main ideas, listening to other students’ explanations, and formally naming important mathematical concepts or objects after students have had an opportunity to interact with them. If students did complete the activity “The Tower of Hanoi,” encourage students to use “sequence” and “term” in their explanations.

Select students to share the smallest number of moves they found for 2, 3, and 4 checkers on each side. Each time, ask whether anyone in the class solved it with fewer moves. If no one finds the minimal solution, ask students to keep looking if time allows, or share the minimum number of moves and demonstrate the minimal solution.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last question where they describe Noah’s thinking. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

If this activity was used instead of “The Tower of Hanoi,” after Stronger and Clearer Each Time, define sequence and term for students before inviting students to describe a rule for the next term in the pattern. After a brief quiet think time, select 2–3 students to share their thinking and write down any notation they come up with to describe the recursive rule, such as “first add 5, and then keep adding the next odd number.” There is no need to introduce formal notation or discuss a specific rule for finding term at this time, but if students suggest these and time allows, welcome their explanations.

Students may think there is no better solution once they find one solution that works. Encourage groups to regularly switch who is in charge of moving the discs and to check with other groups around them once they think they have found a solution with the fewest number of moves.

Students may think that there is no better solution once they find one solution that works. Encourage groups to regularly switch who is in charge of moving the checkers and to check with other groups around them once they think they have found a solution with the fewest number of moves.