Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 58767
Rewriting Equations
Students are given a literal equation involving four variables and are asked to solve for the variable in the quadratic term.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, formula, variable, solving, rearranging
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_RewritingEquations_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 Rewriting Equations worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to apply strategies used in solving equations to rewrite formulas.
Examples of Student Work at this Level
The student:
Solves for the wrong variable.
Manipulates symbols and variables without any mathematical justification.
page 1 of 4 Subtracts b from every term in the equation.
Subtracts b twice from the same side of the equation.
Correctly subtracts b from both sides of the equation but is unable to solve further for v.
Questions Eliciting Thinking
What is the problem asking you to do? If all of the other variables were numbers, what would you do first? Second?
Is it possible to rewrite the formula to get v by itself? What would be your first step?
When solving equations, whatever is done to one side of the equation, must be done to the other side to keep the equation “balanced.” Does your work adhere to that
rule?
Instructional Implications
Review the four basic operations (i.e., add, subtract, multiply, and divide) and give the student the opportunity to determine the inverse of each. Provide feedback as
needed.
Use manipulatives (e.g., integer chips) to model the equation 10 = 6 + 4. Demonstrate that subtracting 2 from every term does not keep the equation equal, or
“balanced.” Help the student apply this understanding to the equation 10 = 6 + v, and then to the equation T = b + v. Finally, ask the student to rework the original
equation on a separate sheet of paper.
Review the reasoning that is used in solving equations and assist the student in applying it to formulas. Begin with simple three-variable formulas that require only one step
to solve. Then introduce the student to two-step and finally, multistep problems. Remind the student to use inverse operations when solving for the variable and correct
notation when showing work. Provide feedback as needed.
Consider implementing MFAS tasks Literal Equations (A-CED.1.4), Solving Literal Equations (A-CED.1.4), Solving Formulas for a Variable (A-CED.1.4) or Surface Area of a Cube
(A-CED.1.4), if not done previously.
Moving Forward
Misconception/Error
The student makes errors in using the Distributive Property when solving.
Examples of Student Work at this Level
The student:
Does not distribute the term 2d to both terms T and b, writing
.
Does not distribute the term 2d to every term in the equation, writing
.
Questions Eliciting Thinking
How do you multiply a binomial by a monomial?
Can you begin by first subtracting b from both sides of the equation? Why or why not?
What operation does the fraction bar represent? What is the inverse operation?
Instructional Implications
Provide feedback to the student regarding his or her error. Review the Distributive Property and how to correctly apply it when multiplying a part of an equation that
contains more than one term. Explain the difference between 2d·T – b and 2d( T – b).
Have the student solve the equation
for v.
Then have the student solve the equation
for v. Finally, ask the student to rework the original
equation on a separate sheet of paper.
Give the student additional multistep literal equations involving the use of the Distributive Property.
Making Progress
Misconception/Error
The student does not apply the inverse operation correctly when considering the squared term.
Examples of Student Work at this Level
The student:
page 2 of 4 Does not know that taking the square root is the inverse of squaring.
Attempts to take the square root of each side of the equation but is unable to do so correctly.
Does not simplify the equation completely, leaving the right side of the equation as
.
Does not extend the radical to the entire expression on the left side of the equal sign.
Questions Eliciting Thinking
What do you get when you square the number three? What operation can you use to “undo” squaring? If you divide nine by two, will you get three?
What are you solving for in this equation? Is v the same thing as
Is
the same thing as
? Would you consider
simplified?
? Do you see anything that you need to change in your answer? What does your answer seem to indicate?
Instructional Implications
Explain that the inverse of squaring is taking the square root. Begin with equations of the form
where c is a whole number. Then introduce the student to
equations in which the coefficient of x is different from one and c is a positive rational number. Be sure to emphasize that roots of quadratic equations occur in conjugate
pairs and require the student to explicitly show both roots. Provide the student with additional opportunities to solve formulas for a variable in the quadratic term.
Express to the student the importance of accurately writing his or her answer. Provide feedback to the student regarding any of the following: Clearly writing the decimal
point or comma in a numeral, extending the fraction bar completely across the numerator and/or denominator, and writing the exponent clearly as a superscript. Explain
how not doing these could result in answers that are misinterpreted.
Almost There
Misconception/Error
The student is unaware that roots of quadratic equations occur in conjugate pairs.
Examples of Student Work at this Level
The student rewrites the formula as
. When asked if there is another root of this equation, the student confidently asserts there is
not.
Questions Eliciting Thinking
Did you realize there is another root of this equation?
Can you find a solution of the equation
? Are there any other solutions? What is the square of -3?
Instructional Implications
Provide direct instruction on solving quadratic equations by taking square roots. Require the student to explicitly show both roots. Provide the student with opportunities to
solve quadratic equations in contexts in which only one root is a reasonable solution and in which both roots are reasonable solutions. Then provide the student with
additional opportunities to solve formulas with a variable in the quadratic term.
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 rewrites the equation as
.
page 3 of 4 Questions Eliciting Thinking
Why do roots of quadratic equations occur in conjugate pairs?
Can you show me an alternative, but still mathematically correct, procedure for solving this formula for v?
Instructional Implications
Ask the student to solve more complex formulas for specified variables.
Give the student formulas for which multiple methods of solution are possible. Ask the student to solve the formula in more than one way. Then ask the student to
compare the solutions and decide which, if either, is more efficient.
Challenge the student to find another way to solve for v. Provide assistance as needed. Ask the student to also try solving the equation for b and d.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Rewriting Equations 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-CED.1.4:
Description
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example,
rearrange Ohm’s law V = IR to highlight resistance R. ★
page 4 of 4