In this activity, students use repeated reasoning to find patterns in the powers of . Students find that powers of correspond to a repeating sequence . This activity goes beyond the depth of understanding required to address the standard.

Monitor for groups who use these strategies to jump ahead in the pattern to get or . The strategies are listed in order of efficiency.

• Continue counting through the pattern until reaching 38 or 100 matching one of the values they have found to each number.
• Use that exponents that are multiples of 4 correspond to the 1 in the pattern, then adjust the position in the pattern based on a multiple of 4 near the target power.
• Make a connection between the position in the pattern and the remainder when dividing the exponent by 4. Then, rewrite the expression in the form .

The purpose of this discussion is to share the pattern for powers of and how to quickly find the value for any whole-number power of .

Invite previously selected groups to share their strategy for finding large powers of . Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How long does it take for the pattern to repeat? How does that show up in each strategy?”
• “Which strategy would you use to find the value of ?”

If time allows, students may benefit from visualizing that multiplying a complex number by can be seen as a counterclockwise rotation by 90 degrees around the number 0 on the complex plane. Ask a student to provide a complex number, then sketch the position of the number on the complex plane. Then, ask students to multiply the number by increasing powers of and sketch the location of each value on the complex plane. Ask students what they notice about numbers in the complex plane.

The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3). In a row game, pairs of students do different problems that have the same answer. If there are discrepancies in their answers, students must communicate with each other to resolve those discrepancies.

Look for groups who disagree and then work to reach agreement, to share during the Activity Synthesis. Also look for any groups who initially agree, but upon deeper inspection, were both incorrect.

If students stop with the answer in the last problem, consider saying:

• “What do you know about rewriting powers of ?”

• “How could writing out the repeated factors of help you to rewrite the equation in the form ?”

Select previously identified groups to share how they resolved disagreement, or how they figured out they were both incorrect. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”