The purpose of this Math Talk is to elicit strategies and understandings students have for squaring values with negatives in different places. It encourages students to think about how exponents affect expressions and to rely on what they know about negative numbers and fractions to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students will need to be able to interpret quadratic equations.

To evaluate the expressions, students need to look for and make use of structure (MP7).

The goal of this discussion is to review students’ strategies for evaluating expressions involving negative numbers, fractions, and exponents.

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?”
• “What connections to previous problems do you see?”

Ensure that students understand the difference between , which is negative (since the order of operations determines that the value for is squared before multiplying by -1), and , which is positive (since the order of operations determines that the value of is multiplied by -1 first, inside the parentheses, before the new value is squared).

In this activity, students first find some values that square to another value. The discussion of this item reminds students of the role negative numbers play in this relationship. In particular, that a negative value squared results in a positive value, and no real numbers square to result in a negative value. Students look for and make use of structure (MP7) when they treat binomials as one quantity, rather than as a pair of numbers.

In the final question, students reason about the area of squares to find a value for the side length and, in some cases, go a step further to find a value for based on the side length. In the associated Algebra 1 lesson, students reason to solve basic quadratic equations of the form . This activity presents a similar concept in a more concrete way.

In this activity, students match values to equations they solve. Students should recognize that quadratic equations usually have 2 solutions and should consider both positive and negative values in their solutions. When students strategize how to match values to equations, they make sense of problems and persevere in solving them (MP1). In the associated Algebra 1 lessons, students solve similar equations by inspection. This activity allows students to select from a list of values rather than discovering them on their own.

Monitor for students who use these different strategies:

• Substitute values from the list in the equation until they find one that works.
• Substitute all of the values from the list in the equation to find all of the solutions.
• Reason about the value of the squared term, then find the values on the list.