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.

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. Here are some questions for discussion:

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

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.

The purpose of the discussion is to note that quadratic equations often have 2 solutions and that there are multiple ways of making sure both solutions are found.

Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “Did anyone solve the problem the same way but would explain it differently?”
• “Are there any benefits or drawbacks to one approach compared to another?”

If time allows, here are some additional questions for discussion:

• “The first several equations have 2 solutions of the form and . 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 and help in finding the solutions to the last equation?” (It helps me see that , which gives one solution as and that which gives the other solution as .)
• “Does your answer to the first equation change if represents the side length of a square?” (Yes, because lengths cannot be negative, so the only solution is 11.)