In this Warm-up, students practice noticing perfect squares including fractions, whole numbers, and expressions containing variables. Generalizing their understanding of perfect squares beyond whole numbers involves looking for and expressing regularity in repeated reasoning (MP8). In the associated Algebra 1 lesson, students recognize quadratic expressions in standard form as perfect squares. The practice here will help students prepare for that work.

The purpose of the discussion is for students to recognize traits of perfect squares. Select students to share their solutions and reasoning. Here are some questions for discussion:

• “When an expression is a product of a number and a variable, like or , how do we know if it is a perfect square?” (Both the number and variable must be perfect squares. In the examples, 49 is a perfect square and is also a perfect square, but 8 is not a perfect square even though is, so is not a perfect square.)
• “The expression is not a perfect square, but is. Explain the difference.” (In the first expression, 3 is not being squared and is not a perfect square itself, so the expression is not a perfect square. In the second expression, both the 3 and are being squared, so the entire thing is a perfect square.)

In this activity, students solve quadratic equations that can be written as a perfect square equal to a constant. In the associated Algebra 1 lesson, students recognize patterns of quadratic equations that can be written as perfect squares. The work of this activity supports students in solving those equations after they have been written as perfect squares. As students strategize how to approach each problem they make sense of problems and persevere in solving them (MP1).

In this activity, students work to write simpler versions of expressions. All of the expressions in this row game can be written as a perfect square. Students work in pairs, and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive at the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner’s work or explaining their reasoning. In the associated Algebra 1 lesson, students look for quadratic expressions that can be written as perfect squares. Recognizing a perfect square in different forms will support students for the Algebra 1 lesson.