Lesson 11

What are Perfect Squares?

11.1: The Thing We Are Squaring (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

In each equation, what expression could be substituted for \(a\) so the equation is true for all values of \(x\)?

  1. \(x^2 = a^2\)
  2. \((3x)^2=a^2\)
  3. \(a^2=7x \boldcdot 7x\)
  4. \(25x^2=a^2\)
  5. \(a^2=\frac14 x^2\)
  6. \(a^2=(x+1)^2\)
  7. \((2x-9)(2x-9)=a^2\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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 \(2x-9\).

11.2: Perfect Squares in Different Forms (15 minutes)

Activity

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.

Launch

Representation: Internalize Comprehension. Activate or supply background knowledge. Represent the same information through different modalities by using a diagram to write each expression in expanded form. As students progress through the questions, invite them to notice how each side of their diagrams will be same since the expressions are perfect squares. If students struggle on the last question, ask them to observe patterns of the entries in their diagrams (such as the lower left and upper right cells are the same) to help them diagram “in reverse,” starting from the standard form.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

  1. Each expression is written as the product of factors. Write an equivalent expression in standard form.

    1. \((3x)^2\)
    2. \(7x \boldcdot 7x\)
    3. \((x+4)(x+4)\)
    4. \((x+1)^2\)
    5. \((x-7)^2\)
    6. \((x+n)^2\)
  2. Why do you think the following expressions can be described as perfect squares?

    \(x^2+6x+9 \qquad x^2-16x+64 \qquad x^2+\frac13 x + \frac{1}{36}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Write each expression in factored form.

  1. \(x^4-30x^2+225\)
  2. \(x+14\sqrt{x}+49\)
  3. \(5^{2x}+6 \boldcdot 5^x + 9\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \((x+n)^2\).)

  \(x\) \(n\)
\(x\) \(x^2\) \(nx\)
\(n\) \(nx\) \( n^2\)

Activity Synthesis

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.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their explanations for the last question, present an incorrect answer and explanation. For example, “Some people may say they’re perfect squares because they have a term that is a square.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. For example, the author probably meant to say that each expression can be written as a squared linear expression. This will help students understand the language of “perfect squares” when referring to quadratic expressions and the relationship between the coefficients in standard and factored form.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

11.3: Two Methods (15 minutes)

Activity

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.

Launch

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.

Conversing: MLR8 Discussion Supports. In their groups of two, display the following sentence frames for students to use when describing their steps to solve each quadratic equation and questioning each others’ reasoning: “First, I _____ because . . .”, “I know _____ because . . .”, “Why did you . . .?”, and “How do you know . . . ?” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning when solving quadratic equations using multiple methods.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

Han and Jada solved the same equation with different methods. Here they are:

Han’s method:

\(\displaystyle \begin {align} (x-6)^2&=25\\(x-6)(x-6)&=25 \\x^2-12x+36&=25\\ x^2-12x+11&=0\\(x-11)(x-1)&=0\\ \\x=11 \quad \text{or} \quad x&=1 \end{align}\)

Jada’s method:

\(\displaystyle \begin {align} (x-6)^2&=25\\ \\x-6=5 \quad &\text{or} \quad x-6=\text-5\\ x=11 \quad &\text{or} \quad x=1 \end{align}\)

Work with a partner to solve these equations. For each equation, one partner solves with Han’s method, and the other partner solves with Jada’s method. Make sure both partners get the same solutions to the same equation. If not, work together to find your mistakes.

\((y-5)^2=49\)

\((x+4)^2=9\)

\((z+\frac13)^2=\frac49\)

\((v - 0.1)^2=0.36\)

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to call students' attention to the common elements of each method. 
Supports accessibility for: Visual-spatial processing; Conceptual processing

Lesson Synthesis

Lesson Synthesis

Reinforce the key points of this lesson by displaying some quadratic expressions such as:

  • \(x^2 +4x +8\)
  • \(x^2 +24x + 144\)
  • \(x^2+6x+16\)
  • \(x^2 - 40x +400\)

Discuss questions such as:

  • “Which of these quadratic expressions are perfect squares? Which are not?” (The second and fourth expressions are perfect squares. The other two are not.)
  • “How can you tell?” (When in standard form, an expression with perfect squares has a constant term that is a number squared and a linear term that is twice that number. In \(x^2 +24x + 144\), we can see 144 is 12 squared, and 24 is twice 12. In \(x^2 - 40x +400\), the term 400 is -20 squared and -40 is twice -20.)
  • “Why isn’t the third equation a perfect square? Isn’t 16 a perfect square?” (16 is 4 squared or -4 squared, but for the expression to be a perfect square, the coefficient of the linear term should be 8 (twice 4) or -8 (twice 4).)
  • “Suppose we have two equations: \(x^2 - 10x+ 16=9\) and \(x^2 - 40x + 400=9\). Which is easier to solve? Why?” (The second one, because the equation has a perfect square on each side. It can be rewritten as \((x-20)^2=9\). There are two numbers that can be squared to make 9, which are 3 and -3, so \(x-20=3\) or \(x-20=\text-3\).)
  • “Can’t we simply rewrite the left side of \(x^2 - 10x+ 16=9\) in factored form and solve?” (Rewriting the left side gives \((x-8)(x-2)=9\), but now we are stuck. We can’t use the zero product property because the expression does not equal 0.)

11.4: Cool-down - A Perfect Square (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

These are some examples of perfect squares:

  • 49, because 49 is \(7 \boldcdot 7\) or \(7^2\).
  • \(81a^2\), because it is equivalent to \((9a)\boldcdot(9a)\) or \((9a)^2\).
  • \((x+5)^2\), because it is equivalent to \((x+5)(x+5)\).
  • \(x^2-12x+36\), because it is equivalent to \((x-6)^2\) or \((x-6)(x-6)\).

A perfect square is an expression that is something times itself. Usually we are interested in situations in which the something is a rational number or an expression with rational coefficients.

When expressions that are perfect squares are written in factored form and standard form, there is a predictable pattern.

  • \((x+5)(x+5)\) is equivalent to \(x^2+10x+25\).
  • \((x-6)^2\) is equivalent to \(x^2-12x+36\).
  • \((x-9)^2\) is equivalent to \(x^2-18x+81\).

In general, \((x+n)^2\) is equivalent to \(x^2+(2n)x+n^2\).

Quadratic equations that are in the form \(\text {a perfect square} = \text {a perfect square}\) can be solved in a straightforward manner. Here is an example:

\(\displaystyle \begin {align} x^2 - 18x + 81 &= 25 \\(x-9)(x-9) &=25\\ (x - 9)^2 &= 25 \end {align}\)

The equation now says: squaring \((x-9)\) gives 25 as a result. This means \((x-9)\) must be 5 or -5.

\(\displaystyle \begin {align} x-9=5 \quad & \text{or} \quad x-9=\text-5\\ x=14 \quad & \text{or} \quad x=4 \end {align}\)