Lesson 11

In this warm-up, students reason about equations with quadratic expressions on both sides of the equal sign. They look for and use structure to solve the equations (MP7). Though each equation appears to be more complicated or to have more pieces than the preceding one, the underlying structure of all the equations is unchanged: \(\text {something}^2 = a^2\). The last equation is written in factored form, but because the factors are identical, students can see that it can also be written as \(\text {something}^2=a^2\). 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.

Invite students to share their responses and how they viewed the equations. If not mentioned by students, point out that the question can be thought of as: “When \(a\) is squared, it is equal to something squared,” so \(a\) must be equal to that something.

Explain to students that expressions that represent something squared are known as perfect squares. This term is 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 \(2x-9\).

The goal of this activity is to illustrate that a perfect-square expression can take different forms, some of which may not look like \(\text {something} \boldcdot \text {something}\) or \(\text {something}^2\). The work here prompts students to recognize structure in perfect-square expressions, particularly when written in standard form, preparing them to complete the square in an upcoming lesson.

Students are given a series of expressions that are clearly perfect squares, for example \((3x)^2\) or \((x+4)(x+4)\), and asked to rewrite them in standard form. As they repeatedly apply the distributive property to expand these expressions into standard form, students begin to recognize a pattern in how the two forms of expressions are related (MP8). They see that squaring an expression such as \((x+n)\) produces an expression in standard form in which the constant term is \(n^2\) and the linear term is \(2n\). Then, they use that insight to articulate why certain expressions in standard form (such as \(x^2-16x+64\)) can be described as perfect squares.

If students have trouble generalizing \((x+n)^2\) as \(x^2 + 2nx + n^2\) from working only with expressions, ask them to draw a rectangular diagram showing \(x\) and \(n\) along both sides of the rectangle and see if they can show on the diagram where the \(2n\) and \(n^2\) comes from. (If some scaffolding is needed, consider starting with numbers, for example, \((10+4)^2\), then \((x + 4)^2\) and then \((x+n)^2\).)

Invite students to share their responses and reasoning for the last question. Make sure students see the structure that relates the expression in standard form and its equivalent expression in factored form. Highlight that:

• In each given example, the constant term is a number squared (\(n^2\)) and the coefficient of the linear term is twice that number (\(2n\)).
• This is also true for the expression that contains fractions: \(\frac{1}{36}\) is \(\left( \frac16 \right)^2\) and \(\frac13\) is \(2 \boldcdot \frac16\).
• We call these expressions “perfect squares” because they can be written as something squared: \((x+3)^2\), \((x-8)^2\), and \((x+\frac16)^2\).
• In general, when \((x+n)\) is squared and expanded, we have: \(x^2 + 2nx + n^2\).

Tell students that knowing about quadratic expressions that are perfect squares can help us solve all kinds of equations in upcoming lessons.

Earlier in the unit, students solved equations with a squared variable on one side and a square number on the other, for example, \(x^2 =49\) or \(4x^2 = 100\). This activity allows students to see that more complicated quadratic equations can also be solved relatively easily when both sides of the equal sign are perfect squares. The work here motivates upcoming lessons on completing the square.

Arrange students in groups of 2. Display the two solutions from the task statement for all to see. Give students a moment to study and make sense of the two methods. Then, ask them to talk to their partner about what Han and Jada did at each step in their solution.

Next, ask students which method they prefer and why. They are likely to prefer Jada’s method since it takes far fewer steps, but some might prefer Han’s method because it is more familiar. Tell partners that they will take turns solving equations with each method and get a better feel for each method.

Select students to display their solutions for all to see. Ask students to reflect on the merits of the solution methods. Make sure they recognize that when equations are in perfect squares they are easier to solve because we can find their square roots.

Point out that all the equations in this activity already have perfect squares on both sides, but most equations that we need to solve do not. Tell students that in upcoming lessons, they will learn how to transform equations so that they have a perfect square and can be more easily solved.