Lesson 1

A Towering Sequence

1.1: What’s Next? (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Here is a rule for making a list of numbers: Each number is 1 less than twice the previous number.

Pick a number to start with, then follow the rule to build a list of 5 numbers.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \(5 \boldcdot 2 - 1.\)

Activity Synthesis

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

1.2: The Tower of Hanoi (30 minutes)

Activity

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.

Launch

Students can use the embedded applet, which is also found at ggbm.at/sjnqSmeN.

Display a table for all to see that looks like this throughout the discussion:

number of discs 1 2 3 4 5 6 7
number of moves              

Ask students to read the rules to the puzzle, and give them a few minutes to solve the puzzle with 2 discs. Ask a student to demonstrate why it takes 3 moves to solve the puzzle with 2 discs. Before students start working, remind them they need to keep track of the number of moves needed for each new number of discs, and that they need to try to find the smallest number of moves. Encourage them to check in with those around them to see if anyone found a solution with fewer moves.

Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their explanations for how Jada used the solution for 3 discs to help her solve the puzzle for 4 discs. As students share their responses with their partner, listeners should press for details and clarity as appropriate based on what each speaker produces. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “What pattern did Jada notice?” or “How do you know your answer is the smallest number of moves?” Students can borrow ideas and language from each partner to strengthen their final product.
Design Principle(s): Optimize output (for explanation)
Representation: Develop Language and Symbols. If the applet is not available, provide physical manipulatives represent the discs and each tower. 
Supports accessibility for: Conceptual processing

Student Facing

In the Tower of Hanoi puzzle, a set of discs sits on a peg, while there are 2 other empty pegs.

A move in the Tower of Hanoi puzzle involves taking a disc and moving it to another peg. To move a disc, click on it, and then click on the peg where you want it to go. You cannot drag a disc. There are two rules:

  • Only move 1 disc at a time.
  • Never put a larger disc on top of a smaller one.

You complete the puzzle by building the complete tower on a different peg than the starting peg.

  1. Using 3 discs, complete the puzzle. What is the smallest number of moves you can find?
  2. Using 4 discs, complete the puzzle. What is the smallest number of moves you can find?
  3. Jada says she used the solution for 3 discs to help her solve the puzzle for 4 discs. Describe how this might happen.
  4. How many moves do you think it will take to complete a puzzle with 5 discs?
  5. How many moves do you think it will take to complete a puzzle with 7 discs?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

A legend says that a Tower of Hanoi puzzle with 64 discs is being solved, one move per second. How long will it take to solve this puzzle?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Set up either a physical puzzle with 2 discs or the digital version with 2 discs and display a table for all to see that looks like this throughout the discussion:

number of discs 1 2 3 4 5 6 7
number of moves              

Ask students to read the two rules of the puzzle. Next, invite students to name what moves are possible (move the top small disc to the middle peg) and which are not allowed (move both discs at once to the middle peg). Complete the puzzle for two discs, asking students to suggest moves to complete the puzzle in the fewest number of moves (3). Fill in the table for the number of moves needed for 2 discs. Tell students that now it is their turn to figure out the number of moves needed for different numbers of discs. Distribute objects with which to experiment with the puzzle. Alternatively, help students access the digital applet.

Ask students to read the rules to the puzzle and then give them time to solve the puzzle with 2 discs. Before students start working, remind them they need to keep track of the number of moves needed for each new number of discs, and that they need to try to find the smallest number of moves. Encourage groups to check in with those around them to see if anyone found a solution with fewer moves. If necessary, before students begin work on the rest of the questions, select a student to demonstrate why it takes 3 moves to solve the puzzle with 2 discs as a check that everyone understands the rules of the game.

Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their explanations for how Jada used the solution for 3 discs to help her solve the puzzle for 4 discs. As students share their responses with their partner, listeners should press for details and clarity as appropriate based on what each speaker produces. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “What pattern did Jada notice?” or “How do you know your answer is the smallest number of moves?” Students can borrow ideas and language from each partner to strengthen their final product.
Design Principle(s): Optimize output (for explanation)
Representation: Develop Language and Symbols. If the applet is not available, provide physical manipulatives represent the discs and each tower. 
Supports accessibility for: Conceptual processing

Student Facing

In the Tower of Hanoi puzzle, a set of discs sits on a peg, while there are 2 other empty pegs.

Three vertical pegs. On the first peg, 4 stacked discs, increasing in size from top to bottom.

A move in the Tower of Hanoi puzzle involves taking a disc and moving it to another peg. There are two rules:

  • Only move 1 disc at a time.
  • Never put a larger disc on top of a smaller one.

You complete the puzzle by building the complete tower on a different peg than the starting peg.

  1. Using 3 discs, complete the puzzle. What is the smallest number of moves you can find?
  2. Using 4 discs, complete the puzzle. What is the smallest number of moves you can find?
  3. Jada says she used the solution for 3 discs to help her solve the puzzle for 4 discs. Describe how this might happen.
  4. How many moves do you think it will take to complete a puzzle with 5 discs? Explain or show your reasoning.
  5. How many moves do you think it will take to complete a puzzle with 7 discs?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

A legend says that a Tower of Hanoi puzzle with 64 discs is being solved, one move per second. How long will it take to solve this puzzle? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: term and sequence.
Supports accessibility for: Conceptual processing; Language

1.3: Checker Jumping Puzzle (30 minutes)

Optional activity

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.

Launch

Students can use the embedded applet, which is also found at ggbm.at/rzbzad2q.

Ask students to read the rules to the puzzle and then give them time to solve the puzzle when there are 3 spaces and one checker on each side. Remind them they need to keep track of the number of moves needed, since the goal is to find the smallest number of moves. Encourage groups to check in with those around them to see if anyone found a solution with fewer moves. Before groups begin work on the rest of the questions, select a student to demonstrate using the applet why it takes 3 moves to solve the puzzle when there are 3 spaces and one checker on each side as a check that everyone understands the rules of the game.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects. Provide blue and red checkers if the applet is not available. 
Supports accessibility for: Conceptual processing

Student Facing

Some checkers are lined up, with blue on one side, red on the other, and with one empty space between them. A move in this checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backwards is not allowed, and 2 checkers cannot occupy the same space.

You complete the puzzle by switching the colors completely: ending up with blue on the right, red on the left, and with 1 empty space between them.

Drag the checkers to move them.

  1. Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?
  2. Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?
  3. Make guesses about the number of moves for 2 and 4 checkers on each side, then test your guesses.
  4. Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe how this might happen.
  5. How many moves do you think it will take to complete a puzzle with 7 checkers on each side?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Set up either a physical puzzle with 2 discs or the digital version with 2 discs and display a table for all to see that looks like this throughout the discussion:

number of checkers
on each side
1 2 3 4 5 6 7
number of moves 3            

Ask students to read the rules of the puzzle. Next, invite students to name what moves are possible (move the rightmost blue checker one space to the right) and which are not allowed (move the leftmost blue checker to the middle open space). Complete the puzzle for 2 checkers on each side, asking students to suggest moves to complete the puzzle in the fewest number of moves (8). Fill in the table for the number of moves needed. Tell students that now it is their turn to figure out the number of moves needed for different numbers of checkers. Distribute objects with which to experiment with the puzzle. Alternatively, help students access the digital applet.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects. Provide blue and red checkers if the applet is not available. 
Supports accessibility for: Conceptual processing

Student Facing

Some checkers are lined up, with blue on one side, red on the other, with one empty space between them. A move in this checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backwards is not allowed, and 2 checkers cannot occupy the same space.

A puzzle.

You complete the puzzle by switching the colors completely: ending up with blue on the right, red on the left, with 1 empty space between them.

  1. Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?
  2. Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?
  3. Estimate the number of moves needed if there are 2 or 4 checkers on each side, then test your guesses.
  4. Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe how this might happen.
  5. How many moves do you think it will take to complete a puzzle with 7 checkers on each side?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

Arrange students in groups of 2. Ask students to come up with a sequence of 5 numbers that follows some rule and write it down on a piece of paper. After a brief quiet think time, tell students to exchange papers with their partners and then try to find the next term in their partner's sequence along with the rule they used.

Conclude the lesson by inviting students to share a sequence and rule their partner wrote that they found interesting. Display these for all to see while students share.

1.4: Cool-down - Next? (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

A list of numbers like 3, 5, 7, 9, 11, . . . or 1, 5, 13, 29, 61, . . . is called a sequence.

There are many ways to define a sequence, but one way is to describe how each term relates to the one before it. For example, the sequence 3, 5, 7, 9, 11, . . . can be described this way: the starting term is 3, then each following term is 2 more than the one before it. The sequence 1, 5, 13, 29, 61, . . . can be described as: the starting term is 1, then each following term is the sum of 3 and twice the previous term.

Throughout this unit, we will study several types of sequences along with ways to represent them.