Lesson 11

Finding Perfect Squares

These materials, when encountered before Algebra 1, Unit 7, Lesson 11 support success in that lesson.

11.1: Finding Perfect Squares (5 minutes)

Warm-up

In this warm-up, students practice noticing perfect squares including fractions, whole numbers, and expressions containing variables. 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.

Student Facing

Is each number or expression a perfect square? Explain how you know.

  1. \(\frac{1}{16}\)
  2. 9
  3. 39
  4. 121
  5. 324
  6. \(x^2\)
  7. \(100t\)
  8. \(49a^2\)
  9. \(3c^2\)
  10. \((x-1)^2\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is for students to recognize traits of perfect squares. Select students to share their solutions and reasoning. Ask students,

  • “When an expression is a product of a number and a variable, like \(49a^2\) or \(8p^2\), 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 \(a^2\) is also a perfect square, but 8 is not a perfect square even though \(p^2\) is, so \(8p^2\) is not a perfect square.)
  • “The expression \(3c^2\) is not a perfect square, but \((3c)^2\) 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 \(c\) are being squared, so the entire thing is a perfect square.)

11.2: Solving Perfect Square Equations (15 minutes)

Activity

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.

Student Facing

Solve these equations. Be prepared to explain your reasoning.

  1. \(x^2 = 16\)
  2. \(x^2 - 25 = 0\)
  3. \(x^2 +13 = 113\)
  4. \(3x^2 = 75\)
  5. \(121-x^2 = 0\)
  6. \(98 - 2x^2 = 0\)
  7. \((x-2)^2 = 100\)
  8. \((x+1)(x+1) = 9\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to remind students of methods for solving quadratic equations that can be written as a perfect square equal to a constant. Select students to share their solutions and reasoning. Ask students,

  • “Compare the equations in this activity to an equation like \((x-2)(x+3) = 12\). What is the same? What is different?” (They are all quadratic and can be written as an expression equal to a constant. The ones in the activity, though, can be written as something squared equal to a constant.)
  • “Why are there two solutions to each of these equations?” (Since they involve \(x^2\), there are usually 2 values that can square to the same number which provides a second solution.)

11.3: Row Game: Making Expressions Simpler (20 minutes)

Activity

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.

Launch

Arrange students in groups of 2. In each group, ask students to decide who will work on column A and who will work on column B.

Student Facing

Work independently on your column. Partner A completes the questions in column A only and partner B completes the questions in column B only. Your answers in each row should match. Work on one row at a time and check if your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find any mistakes.Write each expression using the fewest number of terms possible. 

Partner A

  1. \((4a)^2\)
  2. \(9b^2 + 39b^2 + b^2\)
  3. \(6c \boldcdot 6c\)
  4. \(4d \boldcdot 16d\)
  5. \((\frac{1}{4} k)^2\)
  6. \((\frac{1}{2} n) (\frac{1}{18} n)\)
  7. \((x+3)^2\)
  8. \((4y-1)(4y-1)\)

Partner B

  1. \(5a^2+11a^2\)
  2. \((7b)^2\)
  3. \(4c \boldcdot 9c\)
  4. \(8d \boldcdot 8d\)
  5. \(k^2 - \frac{15}{16} k^2\)
  6. \((\frac{1}{6} n)^2\)
  7. \((x+3)(x+3)\)
  8. \((4y-1)^2\)

 

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is for students to recognize some different forms of expressions that can be written as perfect squares. Select groups to share their solutions and methods for rewriting the expressions. Record the simplified expressions for all to see. Ask students,

  • “What do you notice about all of the solutions?” (They are all written as something squared.)
  • “Expand the last two expressions into standard form. Do you notice anything that might help you recognize them as quadratic expressions that could be written as something squared?” (\((x+3)^2 = x^2 + 6x + 9\) and \((4y-1)^2 = 16y^2 - 8y + 1\). It is ok at this stage if students do not notice any patterns or notice patterns that are specific to these examples. In the associated Algebra 1 lesson students will start to recognize these patterns.)