This Warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items, in comparison to one another.

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any complex number terminology they use, such as “real part,” “complex plane,” or “complex numbers,” and to clarify their reasoning as needed. Consider asking:

• “How do you know_____?”
• “What do you mean by _____?”
• “Can you say that in another way?”

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 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.