Lesson 1

The purpose of this task is for students to generate a list of numbers based on a rule. Since this activity is also meant to help set the expectation that students are responsible for creating mathematical objects, like lists of numbers, and making sense of activities (MP1), there is no need to identify any specific vocabulary such as sequence, term, or arithmetic at this time. In the next activity, students will generate a list of numbers from a puzzle and then describe the pattern in the list.

As students are working, monitor for students who have generated a successful sequence and invite them to share during the discussion.

Arrange students in groups of 2. Tell students there are many possible answers. After brief quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.

Avoid demonstrating a possible sequence of 5 numbers for the whole class first in order to set up the expectation that students are responsible for making sense of the activities.

If a student has trouble getting started, ask them to first pick a number. Once they’ve picked a number, ask what the next number would be, according to the rule. Then invite them to continue.

Some students may interpret “1 less than twice a number” as \(1-2x\) (where \(x\) is the number). Ask these students questions like:

• “What’s 1 less than 5?” (4)
• “What’s 1 less than 5 times 2?” (9)
• “What’s 1 less than 100?” (99)
• “What’s 1 less than 100 times 2?” (199)

Write some of these statements on the board along with their translation into mathematical expressions. For example, write “1 less than 5 times 2” next to \(5 \boldcdot 2 - 1.\)

Invite previously identified students to present their lists. Ask the other students to verify their calculations.

The purpose of this activity is to define the word sequence. In this activity, students experiment with solving the Tower of Hanoi puzzle for different numbers of starting discs. They look for a pattern 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 (MP8). 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.

This activity works best when each group has access to manipulatives or devices that can run the GeoGebra applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet during the launch would be helpful.

Monitor for students writing clear explanations for Jada’s reasoning and for the number of moves needed for 5 discs to share during the discussion.

Students may incorrectly interpret the rules or presume 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.

The goal of this discussion is to define sequence and term (of a sequence). This is also an opportunity to establish classroom norms regarding the flow of an activity 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 challenge them to do it, or demonstrate the minimal solution.

Invite previously identified students to share their explanation for Jada’s strategy and the number of moves needed to complete a puzzle with 5 discs. The bottom row of the table should now have the numbers 1, 3, 7, 15, and 31 filled in.

Conclude the discussion by telling students that in mathematics, we often call a list of numbers a sequence. The list 1, 3, 7, 15, 31 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 \(\text{current}=2 \boldcdot \text{previous}+1.\) There is no need to introduce formal notation or discuss a specific rule for finding term \(n\) at this time, but if students suggest these (time permitting), welcome their explanations.

This optional 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 Tower of Hanoi activity for the discussion of this activity.

This activity works best when each student has access to manipulatives or devices that can run the GeoGebra applet, because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the applet during the launch is helpful.

Monitor for students writing clear explanations for Noah’s reasoning and for the number of moves needed for 7 checkers on each side to share during the discussion.

Students may incorrectly interpret the rules or presume 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.

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 challenge them to do it, or demonstrate the minimal solution. 

Invite previously identified students to share their explanation for Noah’s strategy and the number of moves needed to complete a puzzle with 3 checkers on each side. The bottom row of the table should now have the numbers 3, 8, 15, and 24 filled in.

If this activity was used instead of the Tower of Hanoi, make sure to 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 \(n\) at this time, but if students suggest these (time permitting), welcome their explanations.