Unit 8, Lesson 12
Completing the Square (Part 1)
Algebra 1
12.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-SSE.A.2
Use the structure of an expression to identify ways to rewrite it. For example, see as , thus recognizing it as a difference of squares that can be factored as .
Instructional Routines
NoneMaterials
NoneActivity Narrative
This activity reinforces the meaning of perfect squares and the fact that a perfect square can appear in different forms. To recognize a perfect square, students need to look for and make use of structure (MP7).
Launch
NoneActivity
NoneSelect all expressions that are perfect squares. Explain how you know.
Student Response
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Building on Student Thinking
Activity Synthesis
Display the expressions for all to see. Invite students to share their responses, and record them for all to see. For each expression that they consider a perfect square, ask them to explain how they know. For expressions that students believe aren’t perfect squares, ask them to explain why not.
For the last expression, , students may reason that it is not a perfect square because:
- The constant term 20 is not a perfect square. (Most students are likely to reason this way.)
- If 20 is seen as , then the coefficient of the linear term would have to be , not 10.
Though students have been dealing mostly with rational numbers, the second line of reasoning is also valid and acceptable.
12.2
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
In this activity, students begin to complete the square. They start by transforming given perfect squares from standard form to factored form, and vice versa. Then, they are given an incomplete expression in standard form that contains only the squared term and linear term. Students need to decide what constant term makes the expression a perfect square, and then write the equivalent expression in factored form. To accomplish these tasks, students must rely on the structure they noticed in an earlier lesson about the relationship between the standard and factored forms of perfect squares (MP7).
Launch
Arrange students in groups of 2. Give students a few minutes of quiet think time, and then ask them to discuss their responses with their partner. Follow with a whole-class discussion.
MLR2 Collect and Display. Circulate to listen for and collect the language that students use as they explain how to rewrite perfect squares from standard form to factored form. On a visible display, record words and phrases, such as “perfect square,” “half,” “linear coefficient,” and “constant term.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Advances: Conversing, Reading
12.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
Earlier, students identified that an expression in standard form can be written as a perfect square and learned to write it that way. Here, they learn to use that skill to solve quadratic equations.
Launch
Consider keeping students in groups of 2.
Display the two solution methods in the Task Statement for all to see. Ask students if the quadratic expression in the original equation is a perfect square. Then, ask them to study the methods and make sense of the steps. Afterward, discuss with students:
- “How are the two solution methods alike?” (They both involve making the expression on the left a perfect square. They are the steps for solving the same equation, so the solutions are the same.)
- “How are they different?” (In the first solution, Diego first subtracted 9 from each side and then added 25 to complete the square. In the second, Mai added 16 to 9 because that is what is needed to make 25.)
Tell students that either method works, but some people prefer the first approach because adding the constant term to both sides of the equation so that it is on the other side of the equal sign (the right side, in this case) allows them to see what constant term is needed to make a perfect square on the left side. They also find it to be less prone to errors.
MLR8 Discussion Supports. Display sentence frames to support students in producing statements about how features of the strategies are alike and how they are different. Examples: “_____ and _____ are alike because . . . .” “_____ and _____ are different because . . . .” “One thing that is the same is . . . .” “One thing that is different is . . . .”
Advances: Speaking, Conversing
Advances: Speaking, Conversing
Representation: Internalize Comprehension. Provide a blank two-column graphic organizer for students to record their ideas as they compare and contrast the two solution methods. Label the left column “alike” and the right column “different.” Encourage students to use the organizer to take notes and then prepare their ideas to share with the whole class. If students are unsure how to start, tell them to number the steps in the examples and compare each step individually, placing it in the correct spot on the diagram.
Supports accessibility for: Organization, Attention
Supports accessibility for: Organization, Attention
Lesson Synthesis
To close the lesson, help students connect the new method to earlier skills. Discuss questions such as:
- “Earlier we solved by completing the square. Could we have solved by rewriting it in factored form and using the zero product property? Why or why not?” (Yes, the equation can be rewritten as and solved easily.)
- “Is solving that equation by completing the square a quicker way?” (In this case, no.)
- “Can be solved by rewriting it in factored form?” (It can be written in factored form as , but that’s as far as we can go.)
- “Can it be solved by completing the square?” (Yes. To make it a perfect square we can add 16 to each side, which makes the equation , or , which can be solved by finding square roots of 36.)
- “When might we prefer to complete the square than to rewrite an equation in factored form?” (Not all equations can be written in factored form, so it is not always possible to solve that way. In those cases, we can solve by completing the square.)
Tell students that in upcoming lessons, we will look at other examples of equations that cannot be easily rewritten in factored form but can be solved by completing the square.
Student Lesson Summary
Turning an expression into a perfect square can be a good way to solve a quadratic equation. Suppose we wanted to solve .
The expression on the left, , is not a perfect square, but is a perfect square. Let’s transform that side of the equation into a perfect square (while keeping the equality of the two sides).
- One helpful way to start is by first moving the constant that is not a perfect square out of the way. Let’s subtract 10 from each side:
- And then add 49 to each side:
- The left side is now a perfect square because it’s equivalent to or . Let’s rewrite it:
- If a number squared is 9, the number has to be 3 or -3. Solve to finish up:
This method of solving quadratic equations is called completing the square. In general, perfect squares in standard form look like , so to complete the square, take half of the coefficient of the linear term and square it.
In the example, half of -14 is -7, and is 49. We wanted to make the left side To keep the equation true and maintain equality of the two sides of the equation, we added 49 to each side.