Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 62555
Complete the Square - 2
Students are asked to solve a quadratic equation by completing the square.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, quadratic, equation, solve, complete the square
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_CompleteTheSquare2_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Complete the Square - 2 worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not demonstrate an understanding of completing the square.
Examples of Student Work at this Level
The student:
Attempts to solve the equation using a different method.
Is unable to identify a constant term to create a perfect square trinomial.
page 1 of 4 Questions Eliciting Thinking
What does it mean to complete the square?
What is a perfect square trinomial? Can you give me an example?
Could you complete the square if the leading coefficient is one instead of four?
What constant could you add to the expression
+ 6x to make it a perfect square trinomial?
Instructional Implications
Review the concept of a perfect square trinomial and show the student that a perfect square trinomial results from squaring a binomial. Have the student square several
binomials (including ones in which the coefficient of x is different from one) and assist the student in understanding the relationship between the constant in the binomial
and the constant and coefficients in the resulting trinomial. Guide the student to observe the features of the terms of a trinomial that indicate it is a perfect square. Provide
the student with a number of examples of the quadratic and linear terms of a trinomial (e.g., 3 + 18x) and ask the student to identify a constant so that the trinomial
becomes a perfect square. Then have the student rewrite each trinomial as the square of a binomial.
Review with the student the process of solving a quadratic equation of the form
= c by taking the square root of each side. Guide the student through several
examples emphasizing the reasons for each step in the process. Give the student additional equations to solve by first completing the square.
Consider using MFAS task Complete the Square - 1 (A-REI.2.4) if not previously used.
Moving Forward
Misconception/Error
The student completes the square but makes errors when rewriting the equation.
Examples of Student Work at this Level
The student identifies nine as the constant that will complete the square but:
Does not use the Addition Property of Equality and only adds nine (or 36) to one side of the equation.
Adds nine instead of 36 to the constant (13) on the right side of the equation.
Does not factor the perfect square trinomial correctly.
Questions Eliciting Thinking
How did you factor the trinomial? Did you check your factoring?
After you found that nine was the value needed to complete the square, what did you do?
I see that you added nine to this side of the equation to complete the square. What did you add to the other side of the equation? How did you determine this value?
Instructional Implications
Review with the student the process of completing the square to solve a quadratic equation. Encourage the student to be mindful of any value factored from the terms of
the trinomial and to take this factor into account when determining the appropriate value to add to the other side of the equation. Guide the student through several
examples emphasizing the reasons for each step in the process. Give the student additional equations to solve by completing the square.
If the student struggles to factor the perfect square trinomial, remind him or her that the constant in the binomial can be found in two different ways: by taking half of the
coefficient of the linear term or by taking the square root of the constant. Ask the student to find the constant term in each way and check that they are the same.
If the student factored out four on the left side of the equation, show the student that he or she could also divide all terms by four as an alternative. The student may
prefer this approach.
Almost There
Misconception/Error
The student makes an error when solving the equation.
Examples of Student Work at this Level
The student correctly completes the square, factors the perfect square trinomial, and rewrites the equation as
. However, the student makes an error in
solving this equation. For example, the student:
Only writes the positive root when taking the square root of
(and makes a computational error when solving the resulting equation).
page 2 of 4 Makes a computational error, for example, when adding -3 to
.
Includes extra solutions by writing each solution using the plus/minus symbol.
Questions Eliciting Thinking
What type of equation is this? How many solutions can a quadratic equation have? How many solutions should this equation have?
Did you take the square root of both sides of the equation?
Can you simplify your answer any further?
There is an error in the last step. Can you find it?
Instructional Implications
Provide feedback on any errors made and allow the student to revise the work. If the student found only one of the solutions, use a simplified example such as
= 4, to
remind the student there are two values that result in four when squared. Clearly identify the values and ask the student to revise his or her work by applying this result.
Remind the student that a fraction can be a perfect square. Provide additional examples of perfect square fractions and ask the student to find the square root of each.
Provide the student with a completed problem that contains errors. Have the student identify and correct the errors.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student adds 13 to both sides of the equation and divides all terms by four resulting in the equation
adding nine to both sides of the equation and rewrites the equation as
+ 6x + 9 =
takes the square root of each side of the equation rewriting it as x + 3 = ±
,
+ 6x =
. The student correctly completes the square by
. The student then factors the left side of the equation as
. The student
. The student then solves the two resulting equations and determines the solution set is {-
}.
Questions Eliciting Thinking
How can you check your solutions?
Do you think it would have been easier to solve this equation using another method? Why?
How can you use the process of completing the square to write a quadratic equation in two variables in vertex form?
How can you use graphing technology to check your solutions?
Instructional Implications
Challenge the student to solve, by completing the square, quadratic equations for which the coefficient of the linear term is an odd number.
Consider implementing MFAS task Completing the Square - 3 (A-REI.2.4).
Also consider using NCTM activities Two Squares Are Equal http://www.illustrativemathematics.org/illustrations/618 or Complete the Square
http://www.illustrativemathematics.org/illustrations/1690.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
page 3 of 4 Complete the Square - 2 worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.A-REI.2.4:
Description
Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x
– p)² = q that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the
quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic
formula gives complex solutions and write them as a ± bi for real numbers a and b.
page 4 of 4