Lesson 16

In this warm-up, students must decide if each quadratic expression is a perfect square. This review of perfect squares will be helpful later in the lesson when students complete the square to solve various quadratic equations.

Select a few students to explain which expressions are perfect squares and how they know. The important idea to review is that the binomial \((x+n)\) squares to become \(x^2+2nx+n^2\), so in order for such a quadratic expression to be a perfect square of a binomial, its constant must be the square of half the coefficient of \(x\).

This activity is optional because it revisits content from an earlier course.

The purpose of this activity is for students to analyze two valid methods of solving a quadratic equation (completing the square and the quadratic formula) that result in equivalent answers that look different. This gives students the opportunity to review the two methods. Students make viable arguments and critique the reasoning of others as they inspect the logic of each method and decide whether the different-looking answers are equivalent (MP3).

Arrange students in groups of 2. Encourage students to explain their reasoning to their partner as they work, and if there is disagreement, they should work to reach agreement. Remind students of the meaning of \(\pm\) notation. For example, a result of \(x=2 \pm \sqrt{5}\) after using the quadratic formula means the original equation has two solutions: \(x=2+\sqrt{5}\) and \(x=2-\sqrt{5}\).

When analyzing Elena’s work, some students may not remember that \((x-3)^2=2\) implies \(x-3=\pm \sqrt{2}\). Remind these students that for any expression \(A\), if \((A)^2=2\) then the value of \(A\) could be positive or negative.

The purpose of the discussion is to establish that Han’s solutions are equivalent to Elena’s, Han correctly used the quadratic formula, and Elena correctly completed the square. Begin the discussion with a few questions to check students' understanding of the two methods, such as:

• “Why does Elena add 2 to each side as her first step?” (Elena adds 2 to each side so that the left hand side will be a perfect square. The expression \(x^2-6x+7\) is not a perfect square, but \(x^2-6x+9\) is. Half of the coefficient of \(x\) is -3, and \((x-3)^2=x^2-6x+9\).)
• “When Han used the quadratic formula, why does positive 6 show up even though \(b=\text-6\)?” (Han had to substitute -6 for \(b\) in the quadratic formula, but the formula itself uses \(\text-b\). So \(\text-b=\text-(\text-6)=6\).)

Then, ask students whether they agreed with either Han or Elena and to explain their reasoning. Press students on unsubstantiated claims and include others in the discussion by asking whether they agree. If not mentioned by students, point out that Elena’s solutions look different than Han’s. Ask students how it might be possible for their solutions to look different if they both solved the problem correctly.

This activity is optional because it revisits content from an earlier course.

This activity gives students the opportunity to review solving quadratic equations. Look for students who use different strategies to compare during discussion.

Arrange students in groups of 2. Encourage students to share their responses with their partner, and if there is disagreement, work to reach agreement. Encourage students to use a different method than their partner on each question and verify that the solutions are equivalent.

Select 2–3 previously identified pairs to share their strategies and reasons for choosing a particular solution method. If two pairs solved the same problem but got answers that look different, ask the class to verify that their solutions are equivalent.