In this Warm-up, students reason about quadratic expressions that are perfect squares. They look for and use structure to rewrite expressions (MP7). Though each expression appears to be more complicated or to have more pieces than the preceding one, the underlying structure of all the expressions is unchanged. The last expression is written in factored form, but because the factors are identical, students can see that it can also be written as something2. Viewing a complicated expression as a perfect square prepares students to consider completing the square later.
Give students a moment to observe the list of equations, and ask them what they notice and wonder about the equations. Solicit a few observations and questions. Tell students that their job is to think about what a should be so that each equation is always true, regardless of what x is.
Each of these expressions is a perfect square, which means that each can be written as something multiplied by itself.
Rewrite each expression as something multiplied by itself in the form ()2. For example, 16x2 can be rewritten as (4x)2.
Invite students to share their responses and how they viewed the expressions. If not mentioned by students, point out that the question can be thought of as “When something is multiplied by itself, it is equal to something squared,” so the inside of the parentheses must be equal to that something.
Explain to students that perfect squares are related to what students learned in earlier grades about the area of a square. Multiplying the side length of a square by itself gives the area of the square, so we can say, for instance, that 25 is a perfect square because it is the area (in square units) of a square whose side length is 5 units. (2x−9)(2x−9) can be thought of as the area of a square with side length 2x−9.
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In this Warm-up, students reason about quadratic expressions that are perfect squares. They look for and use structure to rewrite expressions (MP7). Though each expression appears to be more complicated or to have more pieces than the preceding one, the underlying structure of all the expressions is unchanged. The last expression is written in factored form, but because the factors are identical, students can see that it can also be written as something2. Viewing a complicated expression as a perfect square prepares students to consider completing the square later.
Give students a moment to observe the list of equations, and ask them what they notice and wonder about the equations. Solicit a few observations and questions. Tell students that their job is to think about what a should be so that each equation is always true, regardless of what x is.
Each of these expressions is a perfect square, which means that each can be written as something multiplied by itself.
Rewrite each expression as something multiplied by itself in the form ()2. For example, 16x2 can be rewritten as (4x)2.
Invite students to share their responses and how they viewed the expressions. If not mentioned by students, point out that the question can be thought of as “When something is multiplied by itself, it is equal to something squared,” so the inside of the parentheses must be equal to that something.
Explain to students that perfect squares are related to what students learned in earlier grades about the area of a square. Multiplying the side length of a square by itself gives the area of the square, so we can say, for instance, that 25 is a perfect square because it is the area (in square units) of a square whose side length is 5 units. (2x−9)(2x−9) can be thought of as the area of a square with side length 2x−9.