Lesson 3

The purpose of this Math Talk is to elicit strategies and understandings students have for squaring values with negatives in different places. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to interpret quadratic 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?”

Ensure that students understand the difference between \(\text{-}x^2\) which is negative (since order of operations determines that the value for \(x\) is squared before multiplying by -1) and \((\text{-}x)^2\) which is positive (since order of operations determines that the values of \(x\) 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 square results in a positive value and no real numbers square to result in a negative value.

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 \(x\) based on the side length. In the associated Algebra 1 lesson, students reason to solve basic quadratic equations of the form \((x-a)^2 = b\). This activity presents a similar concept in a more concrete way.

Display the image for all to see. Ask students, “What is the area of the square? Explain your reasoning.” (9 since the area of a square is the side length squared.)

The purposes of the discussion are to reason about negatives that are being squared and to find values for basic quadratic equations using a concrete example. Ask students,

• “Is it possible for \(q\) to be a negative value for some values of \(p\)? Explain your reasoning.” (No, because any value multiplied by itself will result in a positive value. If \(p\) is positive, then a positive multiplied by a positive is still positive. If \(p\) is negative, then a negative multiplied by a negative is positive.)
• “Is it possible for \(p\) to be a negative value for some values of \(q\)? Explain your reasoning.” (Yes. For example \(p = \text{-}2, q = 4\) works since \(\text{-}2 \boldcdot \text{-}2 = 4\).)
• “What are the two solutions to the equation \(x^2 = 9\)?” (\(x = 3\) and \(x = \text{-}3\))
• “For the second square, what is an equation that represents the area based on the side lengths and given area?” (\((x+1)(x+1) = 35\) or equivalent)
• “After you found the side length of the third square, how did you find the value for \(x\)?” (I knew the side length had to be 10, so \(x - 3 = 10\). Solving that equation tells me that \(x = 13\).)

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. 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:

1. substitute values from the list in the equation until they find one that works.
2. substitute all of the values from the list in the equation to find all of the solutions.
3. reason about the value of the squared term, then find the values on the list.

The purpose of the discussion is to note that quadratic equations often have 2 solutions. Select previously identified students to share their solutions and methods. Ask students,

• “The first several equations have 2 solutions of the form \(a\) and \(\text{-}a\). What is different about the last equation?” (Since there is a value added inside the squared term, it does not follow the same pattern as the others.)
• “How does knowing that \(10^2 = 100\) and \((\text{-}10)^2 = 100\) help in finding the solutions to the last equation?” (It helps me see that \(n+3 = 10\) which gives one solution as \(n = 7\) and that \(n+3 = \text{-}10\) which gives the other solution as \(n = \text{-}13\).)
• “Does your answer to the first equation change if \(c\) represents the side length of a square?” (Yes, because lengths cannot be negative, so the only solution is 11.)