Lesson 18

In the previous lesson, students completed the square to decide whether quadratic equations had real or non-real solutions. In this warm-up, students continue to make use of the structure of quadratic expressions to decide whether the solutions are real or non-real (MP7).

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’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 \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

Over several lessons, students have developed a concept of complex numbers. In this activity, students use that understanding to consider a situation where the quadratic formula leads to a negative number inside the \(\sqrt{}\) symbol. The focus of the activity is on how complex numbers come into play rather than performing the computation, which is the focus of the next activity.

Arrange students in groups of 2. Give a few minutes of quiet think time before asking students to share their ideas with their partner. If there is disagreement, encourage students to work to reach agreement. Follow with whole-class discussion.

Select students to share their responses and explain their reasoning. The main takeaway for discussion is that whenever the quadratic formula leads to a negative number inside the \(\sqrt{}\) symbol, it means the original equation has two non-real solutions.

In this row game activity, students solve equivalent quadratic equations given in different forms and then compare their results. The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3).

Look for groups where one partner uses quadratic formula while the other partner uses a different method for a specific row to share during the discussion. Also look for groups where partners disagree because their solutions look superficially different, for example if one partner gets \(x=2\pm \frac{\sqrt{32}}{2}\) while the other partner gets \(x=2 \pm \sqrt{8}\) for the first question.

Arrange students in groups of 2, assigning one student as partner A and the other as partner B. Explain to students that there will be two columns of problems and that they only do the problems in their column. Encourage students to use a different method than their partner on each question and verify that the solutions are equivalent.

If students still struggle using imaginary numbers to describe square roots of negative numbers, for example \(\sqrt{\text-8}\), consider asking,

• “What numbers square to make -1?” (\(\pm i\))
• “What numbers square to make 8?” (\(\pm \sqrt{8}\))
• “Then what numbers square to make -8?” (\(\pm i\sqrt{8}\)) Then remind students that, by convention, \(\sqrt{\text-8}\) refers to the number on the positive imaginary axis that squares to make -8, so \(\sqrt{\text-8}=i\sqrt{8}\).

Select previously identified students to share how they solved their equations and display their work for all to see. Make sure students explain why solutions that look different are actually equal. Ask students “What is the same with the two equations? What is different?” (One of the equations already involves a perfect square, while the other uses the quadratic formula or completing the square to find the solutions. Using the quadratic formula gives the same solution, but will often result in the solutions looking different superficially.)