Lesson 13

The purpose of this Math Talk is to elicit strategies and understandings students have for operations on fractions and solving equations. These understandings help students develop fluency and will be helpful later in this lesson when students will need to use similar computations to solve equations.

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?”

In this activity, students encounter equations that contain rational numbers. They learn that, even though the numbers are not as straightforward as in equations they have previously seen, they can still solve the equations by completing the square. The less-friendly numbers and the more elaborate equations make the computations more challenging, so students need to attend to precision (MP6) and mind the steps for rearranging parts of the equations, as well as the signs of the numbers, especially negative numbers.

As students work, they may need to be reminded to rewrite the equation in a different form, isolate a term to make it easier to complete the square, apply the zero product property only when a product is equal to zero, and so on. Make note of common errors or hurdles and address them during whole-class discussion.

Give students a moment to look at the list of equations and notice how they are like or unlike other equations they have seen and solved before. Invite students to share some observations. They may notice, for instance:

• \((x-3)(x+1)=5\) does not have 0 on one side, so we cannot use the zero product property right away.
• \(x^2 + \frac12 x = \frac{3}{16}\) and \(x^2+3x+\frac84=0\) involve fractions.
• \(x^2+3x+\frac84=0\) has an odd number for the coefficient of the linear term.
• \((7-x)(3-x)+3=0\) has 0 on one side but the other side is a sum, not a product.
• \(x^2+1.6x+0.63=0\) has decimals.

Explain to students that each equation can be solved by completing the square and using the same reasoning as when we solved simpler-looking equations earlier.

Consider arranging students in groups of 2. If there is not enough time for students to answer all the questions, consider asking students to look at the list of equations and to choose one they think looks easier (or more familiar) and one that looks harder (or less familiar).

Some students may have a difficult time getting started given the increased complexity of these problems. Encourage these students to use completing the square to solve a more familiar problem first, such as \(x^2-10x+20=4\), and then to use the structure of the process they followed (MP7) as a guide to help them solve these more challenging problems.

Select students to share their solutions and display their work for all to see. If students found solutions that differ from each other, as time permits, invite these students to describe their thinking to the class, and encourage them to try to identify any errors and determine which solutions are correct. Discuss any common challenges or errors.

Make sure students understand that the process of solving by completing the square is the same whether the numbers in the equations are integers or other rational numbers. Elicit from students a generalization of the process. For example:

• Start with a quadratic expression in standard form on one side of the equal sign and a number (which could be 0) on the other side.
• Take half of the coefficient of the linear term and square it. The result is the constant term that would make the quadratic expression a perfect square.
• Perform the same operations to both sides of the equal sign so the constant term that is on one side is a perfect square that makes the expression a perfect square.
• Rewrite the expression (in standard form) as a squared factor.
• Find both values of the factor that, when squared, gives the number on the other side of the equal sign.

In this activity, students analyze worked examples of equations solved by completing the square. The work allows them to further develop their understanding of the method when used with rational numbers and to notice common errors. Identifying errors in worked examples is an opportunity to attend to precision (MP6) and to analyze and critique the reasoning of others (MP3).

Display the four equations in the task statement for all to see:

1. \(x^2 + 14x= \text-24\)
2. \(x^2 - 10x + 16= 0\)
3. \(x^2 + 2.4x = \text-0.8\)
4. \(x^2 - \frac65 x + \frac15 = 0\)

Arrange students in groups of 3–4. Assign one equation (or more, depending on time constraints) for each group to solve by completing the square. Once group members agree on the solutions, ask them to look for errors in the worked solution for the same equation (as the one they solved).

Display the four solutions in the task statement for all to see. Select students to share the errors they spotted and their proposed corrections.

To involve more students in the discussion, after each student presents, consider asking students to classify each error by type (not limited to one type per error) and explain their classification. Here are some examples of types of errors:

• Careless errors—for example, writing the wrong number or symbol, repeating the same step twice, leaving out a negative sign, or not following directions.
• Computational errors—for example, adding or subtracting incorrectly or putting a decimal point in the wrong place.
• Gaps in understanding—for example, misunderstanding of the problem or the rules of algebra, applying an ineffective strategy, or choosing the wrong operation or step.
• Lack of precision or incomplete communication—for example, missing steps or explanations, incorrect notation, or forgetting labels or parentheses.