Each time, ask whether anyone in the class solved it with fewer moves. Select students to share the smallest number of moves they found for 2, 3, and 4 checkers on each side. 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 \(\text 1.\) 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. Ask students how they would describe the rule for the next term in this sequence. A specific number in the list is called a term of the sequence. The list 1, 3, 7, 15, 31 is an example of a sequence. The bottom row of the table should now have the numbers 1, 3, 7, 15, and 31 filled in.Ĭonclude the discussion by telling students that in mathematics, we often call a list of numbers a sequence. 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. 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. 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.īegin 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. The goal of this discussion is to define sequence and term (of a sequence). 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. Encourage groups to check in with those around them to see if anyone found a solution with fewer moves. 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. Alternatively, help students access the digital applet.Īsk students to read the rules to the puzzle and then give them time to solve the puzzle with 2 discs. Distribute objects with which to experiment with the puzzle. Tell students that now it is their turn to figure out the number of moves needed for different numbers of discs. Fill in the table for the number of moves needed for 2 discs. Complete the puzzle for two discs, asking students to suggest moves to complete the puzzle in the fewest number of moves (3). 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). 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 discsĪsk students to read the two rules of the puzzle. 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.Īrrange students in groups of 2. If students don’t have individual access, projecting the applet during the launch would be helpful. 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. 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. 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). In this activity, students experiment with solving the Tower of Hanoi puzzle for different numbers of starting discs. The purpose of this activity is to define the word sequence.
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